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Brock University Faculty of Business Department of Finance, Operations, and Information Systems FNCE 4P17 Derivatives, Part II Winter 2009 Instructor: Hatem Ben Ameur Office: TA 318 Email: [email protected] Phone: (905) 688–5550 x 5874 Class time: Wednesdays 15:30–17:00 and Fridays 14:00–15:30 (TA 307) Description: A derivative is a contract whose return depends on the price movements of some underlying assets. There are three main families of derivative contracts: options, futures, and swaps. They all have the ability to reduce risk; thus, are widely used for hedging purposes. This course covers advanced topics related to options, futures, and swaps. We focus on their structure, evaluation, and hedging properties. Complex products, such as exotic options, callable bonds, and swaptions are considered. Materials: The following textbook is required, and will be used frequently during the term. 1. Robert W. Kolb and James A. Overdahl, 2007, Futures, Options, and Swaps, 5th Edition, Blackwell. The following textbooks offer useful alternate explanations, and are listed in increasing order of difficulty. 2. John C. Hull, 2005, Fundamentals of Futures and Options, 6th Edition, Prentice Hall. 3. Don M. Chance and Robert Brooks, 2007, An Introduction to Derivatives and Risk Management, 7th Edition, Thomson. 4. John C. Hull, 2008, Options, Futures, and Other Derivative Securities, 7th Edition, Prentice Hall. Monitoring the news about financial derivatives is recommended as an important complement to classroom work. Selected articles from business newspapers such us The Financial Times, The Globe and Mail, The National Post, and The Wall Street Journal will be discussed in class. Grading: The grading policy is based on: • four assignments each worth 5%; • two midterm exams each worth 30%; • and a quiz worth 20%. The midterm exams are cumulative. The assignments will improve your skills, and help you prepare for the exams. For each assignment, you will be given selected problems to solve, articles to comment on, and empirical experiments to implement. A quiz is planned at the last week. Office hours will be fixed during the first class. In every aspect of the course, students must comply with the Brock University Honour Code. Outline: Week 1–2 3 4 5–6 7 8 9 10 11–12 Chapter Risk-Neutral Pricing Option Sensitivities Pricing American Options Options on Stock Indexes, Foreign Currencies, and Futures – First Midterm Exam Pricing Corporate Securities Exotic Options Interest-Rate Options – Second Midterm Exam Long-Term T-Bond Futures Contract Swap Contracts and Swaptions – Quiz Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Derivatives, Part II Hatem Ben Ameur Brock University – Faculty of Business Finance, Operations, and Information Systems FNCE 4P17 Winter 2010 1 Hatem Ben Ameur 1 Derivatives, Part II Brock University, FNCE 4P17 Risk-Neutral Pricing Topics Covered: 1 The Present-Value Principle 2 Pricing in the Binomial Model 3 Risk-Neutral Pricing of …nancial assets 4 Pricing in the Black and Scholes Model 2 Hatem Ben Ameur 1.1 Derivatives, Part II Brock University, FNCE 4P17 The Present-Value Principle Let t be the present time and T an investment horizon. The discount factor from time T to time t, indicated by t;T , is the present value at time t of one dollar to be received with certainty at time T . The compound factor from time t to time T is the future value to be received with certainty at time T of one dollar invested at time t. In a perfect market, the compound factor is the inverse of the discount factor: 1 ct;T = . t;T In a perfect market, an asset whose future cash ‡ows and their occurrence dates are known in advance with certainty can be evaluated using the present-value principle: PVt = X tn>t 3 t;tn CFtn , Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 where t;tn , for tn > t, is the discount factor from time tn to time t, CFtn the future cash ‡ow (in dollars) promised at time tn, and PVt the asset’s present value at time t (in dollars). For this equation to hold, the asset’s cash ‡ows and their occurrence dates have to be known in advance with certainty. Exercise: Can a Treasury bond be evaluated using the PV principle? Give the assumptions under which a stock can be evaluated using the PV principle. Does the PV principle apply for a corporate bond, a European call option, or a forward/futures contract? Exercise: The nominal interest rate is …xed at rnom = 4% (per year), and interest is compounded semi-annually. Give the semi-annual interest rate r:5 (in % per six months), which is relevant to compound interest, the compound factor c0;1, and the discount factor 0;1. Compute the equivalent annual interest rate r1 (in % per year), which 4 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 would apply if interest were compounded annually. This is the e¤ective interest rate. Compute again the compound factor c0;1 and the discount factor 0;1. Compute the equivalent monthly interest rate r1=12 (in % per month) and quarterly interest rate r0:25 (in % per quarter). Compute again the compound factor c0;1 and discount factor 0;1 . Recognize that r:5 , r1 , r1=12 and r:25 are all equivalent rates. The continuously compounded interest rate rc (in % per year) is a nominal rate, which would apply if interest were compounded at each second (or even a fraction of a second). Compute again the compound factor c0;1 and the discount factor 0;1. Recall that a n ! ea, when n ! 1, 1+ n where a 2 R and n 2 N. An arbitrage opportunity is an investment strategy that guarantees a riskless pro…t. Suppose the PV principle applies for a …nancial asset. If the asset is not traded at its (fair) present value, arbitrage 5 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 would be possible. Borrow at the risk-free rate, and buy the asset if it is undervalued. Short sell the asset, and save at the risk-free rate if the asset is overvalued. 1.2 Pricing in the Binomial Model We consider a market for a saving account (the riskless asset) and a stock (the risky asset). Trading activities take place only at the current time t0 = 0 and at horizon t1 = T . No trading is allowed in between. All positions are then closed at the horizon. In addition, the stock price is assumed to move from its current level S0 according to a one-period binomial tree: S0 up S1 = uS0 p % & 1 p . S1down = dS0 t0 = 0 t1 = T 6 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 The stock price can rise by a factor of u or drop by a factor of d, where u > d. The probabilities p and 1 p de…ne the physical probability measure P , under which investors evaluate likelihoods and make decisions in the real world. The parameters u and d can be seen as volatility parameters. The greater is u d, the higher the volatility of the stock return. Example: Assume that the stock price is currently quoted at $100, and can either increase by 25% or decrease by 20%. The factors u and d are u = 1:25, and d = 0:8. 7 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 In this case, the one-period binomial tree is up S0 % & t0 = 0 S1 = uS0 = 1:25 100 = 125 . S1down = dS0 = 0:8 100 = 80 t1 = T The price can move upward from $100 to $125, or downward from $100 to $80. The (periodic) risk-free rate is indicated by r, and is expressed in % over the time period [t0, t1]. The binomial tree is arbitrage free if, and only if, the following property holds: d < 1 + r < u. up Indeed, in the case of an upward movement S1 = uS0, the rate of return on the stock u 1 (in % per period) 8 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 must exceed the risk-free rate r (in % per period). Otherwise, the stock would be overpriced, and an arbitrage opportunity would arise. Similarly, the risk-free rate r must exceed the rate of return on the stock under a downward movement, that is, d 1. Example (continued): The risk-free rate is r = 7% (per period). Is the model arbitrage free? The binomial tree is simple, but viable. It is used here to go through the fundamentals of options pricing. The goal now is to characterize the (present) value of a European call option C0 in the one-period binomial tree, as a function of the stock price S0, the option strike price X , the volatility parameters u and d, and the risk-free rate r. Consider a European call option on a stock with a maturity date T = t1 and a strike price X . 9 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 The hedge portfolio consists of holding shares of the stock and a single signed call option on that stock with an initial value of H0 = S0 C0 , S1 C1 . and a terminal value of H1 = For a speci…c level of , the hedge portfolio is riskless, and, by the no-arbitrage principle, must earn the risk-free rate. A formula for the (present) value of the call option can therefore be derived. The value of the hedge portfolio moves along the binomial tree as follows: up H0 % & up H1 = S1 up C1 . H1down = 10 S down C1down Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 The hedge portfolio is riskless if, and only if, up H1 = H1down. Solving for gives up = C1 up S1 C1down S1down = C , S which depends on the known parameters S0, X , u, and d. Given the hedge parameter , the hedge portfolio, as a riskless investment, should earn the risk-free rate: H0 = S0 C0 up H1 H1down H1 = = = . 1+r 1+r 1+r Solving for C0 gives up p C1 + (1 p ) C1down C0 = , 1+r 11 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 where p = 1+r d 2 (0; 1) . u d The probabilities p and 1 p de…ne the so-called riskneutral probability measure P , which is not related in any way to the physical probability measure P . In sum, the value of the European call option can be expressed as a weighted average (an expectation) of its promised cash ‡ow, which is discounted at the risk-free rate: C1 j S0 , C0 = E 1+r where E [:] is the expectation sign under the risk-neutral probability measure P , C0 the call-option value at time t0, and C1 the call-option value at time t1. The pricing formula discounts the risky cash ‡ow of the call option by the risk-free rate as if investors were risk neutral, while they are not. This is done via a major 12 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 correction. We move from the real world, seen under the physical probability measure P , to a risk-neutral world, seen under the risk-neutral probability measure P . Example (continued): Consider a European call option on the previously mentioned stock with a strike price of X = $100. 1. Compute the risk-neutral probabilities for upward and downward movements. 2. Draw the one-period binomial tree for the call option. 3. Compute the hedge ratio. 4. Use the formula to compute the value of the call option. 13 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 5. Check that the hedge portfolio earns the risk-free rate. 6. Compute in two di¤erent ways the value of its associated European put option. The risk-neutral probabilities are: 1+r d u d 1 + 7% 0:8 = 1:25 0:8 = 0 :6 p = and 1 p = 0:4. The one-period binomial tree for the stock, the call op- 14 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 tion, and the hedge portfolio are: up S0 = 100 C0 = 14:02 = 0:556 H0 = 42:02 t0 = 0 % & S1 = 125 up C1 = 25 up H1 = 0:56 125 S1down = dS0 = 0:8 C1down = 0 H1down = 0:56 80 t1 = T where 0 :6 25 + 0:4 0 C0 = , 1 + 7% 25 0 , = 125 80 H0 = 0:56 100 14:02. The call-option value veri…es: C0 = S0 H0, C1 = S1 H1, and 15 25 = 45 100 = 80 , 0 = 45 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 where C1 is the call-option value at time t1, that is, the call-option payo¤ in this example. In other words, one can fully replicate the call option using a strategy that consists of holding shares of the underlying stock and borrowing at the risk-free rate. If the signer decides to replicate the option, he insures the promised payment to the option holder at maturity. A market in which one can fully replicate all derivative contracts is called a complete market. The binomial tree and the Black and Scholes model are two examples. Exercise: Compute the present value of the stock in the binomial tree. The stock price, discounted at the risk-free rate, is said to verify the martingale property. 16 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 The multi-period binomial tree is a simple extension of the one-period binomial tree. Both are arbitrage free under the same condition, that is, d < 1 + r < u. In a multi-period binomial tree, the stock price moves according to a Markov process, and veri…es the martingale property. The call-option value at time t can then be expressed as follows: Ct+1 Ct = E j Ft 1+r Ct+1 =E j St , 1+r where Ft represents the information available to investors up to time t. This formula holds for both European call and put options. Pricing European options in a multi-period binomial tree is done as follows: 17 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 1. Start at maturity, and evaluate the option; 2. Use the risk-neutral pricing formula, and evaluate the option backward in time; 3. Stop at the origin. Exercise (continued): Extend the one-period binomial tree to a three-period binomial tree. Evaluate the European call option, and its associated put option. Check that the call-put parity holds. 1.3 Risk-Neutral Pricing In a risky environment, it is possible to extend the presentvalue (PV) evaluation principle, as long as the market model is arbitrage free. The martingale property is a 18 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 su¢ cient condition for the market model to preclude arbitrage. The present value of an asset can be expressed as follows: 2 3 X 4 PVt = E t;tn CFtn j Ft5 , tn>t where E [ ] is the expectation sign under the risk-neutral probability measure, t;tn , for tn > t, the discount factor from time tn to time t, CFtn the cash ‡ow promised by the asset at time tn, and Ft the information available to investors up to time t. This formula supports market risk, credit risk, and random payment dates. This formula discounts the risky cash ‡ows of a …nancial asset using their associated risk-free rates, while higher rates should be used. An adjustment is therefore made: the expectation is computed under the risk-neutral probability measure, rather than the physical probability measure. This is the risk-neutral pricing formula, which holds as long as the market model is arbitrage free. 19 Hatem Ben Ameur 1.4 Derivatives, Part II Brock University, FNCE 4P17 Pricing in the Black and Scholes Model Black and Scholes considered a frictionless market for a stock (the risky asset) and a saving account (the riskless asset) in which trading takes place continuously. The stock price process fStg is assumed to follow a geometric Brownian motion characterized by ST = SteR , where R is the continuously compounded rate of return on the stock over [t, T ]. The annualized rate of return on the stock is R= (T t) (in % per year). The rates of return on the stock over successive and non-overlapping time intervals are independent. The Black and Scholes model turns out to be arbitrage free. Under the risk-neutral probability measure P , the rate of return on the stock R is random, and follows a normal distribution: p 2 R= r =2 (T t) + T tZ , 20 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 where r is the continuously compounded risk-free rate (in % per year), the volatility of stock returns (per year), T t the option’s remaining time to maturity (in years), and Z a random variable that follows the standard normal distribution N (0, 1). The stock price process is said to be lognormal, since the natural logarithm of a future stock price, given the current stock price, follows a normal distribution. Black and Scholes used a hedge portfolio, and derived a formula for a European call option on a stock paying no dividend: Ct = = h E e r(T t) max (0, ST X ) j St N (d1) St Xe r(T t)N (d2) , i where d1 d2 ln (St=X ) + r + 2=2 (T p = T t p = d1 T t. 21 t) , and Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Here, St is the current stock price, X the option’s strike price, the volatility of stock returns, T t the option’s remaining time to maturity, r the risk-free rate (in % per year), and N ( ) the cumulative function of the standard normal distribution. " Please establish the second term of the Black and Scholes formula. # The hedge portfolio used in the Black and Scholes model consists of a short position on the call option and a long position on shares of the underlying stock. This portfolio is similar to the one used in the binomial model with the di¤erence that it has to be continuously adjusted. The Black and Scholes model is not realistic, but remains very useful in practice. 1.5 Assignment 1 Exercise 1: This exercise shows how to replicate a European call option in the Black and Scholes model using 22 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 the hedging strategy, based on the underlying stock and the saving account. Consider 1; 000 European call options on a stock paying no dividend. The size of each option contract is of 100. The initial stock price is St = $49, the strike price X = $50, the risk-free rate r = 5% (per year), the volatility = 20% (per year), and the time to maturity T t = 20=52 = 0:3846 (in years). 1 Check that the call-option value is Ct = $240; 052:73. 2 Compute the coe¢ cient at the start. The stock price at maturity is ST = $57 14 ; thus, the call option expires in the money. 3 Compute the call-option payo¤ paid by the seller to the buyer at maturity. 23 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 The path of the stock price, observed once a week, was as follows: week0 49 (in dollars), week1 48 81 , week2 47 38 , week3 50 41 , week4 51 34 , week5 53 18 , week6 53, week7 51 78 , week8 51 38 , week9 53, week10 49 78 , week11 48 12 , week12 49 87 , week13 50 38 , week14 52 81 , week15 51 78 , week16 52 87 , week17 54 78 , week18 54 85 , week19 55 78 , week20 57 41 . 4 Use the hedging strategy, rebalanced weekly, to replicate the European call option. Use Excel, and …ll in the following hedging table. Explain. Week # 0 1 ... 19 20 Stock Price 49 48 18 Delta Shares Purch. Cost Cum. Cost Interest ($000) 55 78 57 14 5 Repeat this exercise for the following path of stock prices: week0 49 (in dollars), week1 49 34 , week2 52, 24 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 week3 50, week4 48 38 , week5 48 41 , week6 48 34 , week7 49 58 , week8 48 14 , week9 48 41 , week10 51 18 , week11 51 12 , week12 49 87 , week13 49 78 , week14 48 43 , week15 47 12 , week16 48, week17 46 41 , week18 48 81 , week19 46 85 , week20 48 18 . 6 Simulate a random path for the stock price and do again question 4 . De…ne your code parameters as follows: s for St, X for the option strike price, sigma for , Tau for the remaining time to maturity T t, r for the risk-free rate, a and b for the lower and upper bound of the interval [a, b], respectively, and delta for the hedge ratio . 7 Set the number of observation dates N as an additional parameter. Do again question 6 with increasing N 2 f20, 40, 80, 160g. De…ne the hedge error e (epsilon) and show that e ! 0 when N ! 1. This question is not mandatory, but very instructive. It is worth an additional 10 points in the …rst mid-term exam (the deadline for question 7 ). 25 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Exercise 2: In the Black and Scholes model, the stock price follows a lognormal process, which is a Markov process. The following transition parameter keeps track of the behaviour of the stock price. 1. Show that the following transition parameter can be expressed as P (ST 2 [a, b] j St) 0 ln (b=St) r @ p =N 0 [email protected] ln (a=St) r p T 2 =2 T t) (T t 2 =2 t (T t) 1 1 A A, where N (:) is the cumulative distrubution function of N (0; 1). 2. Comment on the case where a = X and b = 1. Give an example, and support your argument by selected numbers. 26 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 3. Set St = $100 and r = 4%. Fill in the following Tables for P (ST 2 [a; b] j St) . Set T t = 1 (year) when changes and = 20% (par year) when T t changes. Intrepret your results. T [a; b] [70; 80] [80; 90] [90; 100] [100; 110] [110; 120] 10% 20% 30% :5 t 1 2 . 27 Hatem Ben Ameur 2 Derivatives, Part II Sensitivity Analysis Topics Covered: 1. Single-Option Sensitivity Analysis 2. Portfolio Sensitivity Analysis 28 Brock University, FNCE 4P17 Hatem Ben Ameur 2.1 Derivatives, Part II Brock University, FNCE 4P17 Single-Option Sensitivity Coe¢ cients The value of an option contract, v , depends on several inputs, including the underlying asset price S , the volatility of asset returns , the option’s remaining time to maturity T t, and the risk-free rate r. The sensitivity of an option with respect to its underlying asset price is v , S where S is a small change in the underlying asset price, and v is the associated change in the option value. All other inputs are assumed to be …xed. A sensitivity coe¢ cient cannot directly be computed from real-life observations. Indeed, the inputs that a¤ect the option value change together over time. A sensitivity coe¢ cient can, however, be computed via a market model, especially when a closed-form solution is available for the 29 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 option value. The Black and Scholes model is an example. The delta coe¢ cient, indicated by , is the …rst partial derivative of v with respect to S , that is, = @v @S v , S!0 S when the right-hand quantity exists. = lim Exercise: The call-option parameters are S = $100, X = $100, = 0:3 (per year), T t = 180 (days), and r = 8% (per year). Use Table 14.1, and approximate at S = $100. Approximate the change in the call-option value if the stock price drops by $0:5. The gamma coe¢ cient, indicated by 30 , is the second Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 partial derivative of v with respect to S , that is, @ 2v = @S 2 @ @v = @S @S @ , = @S when the right-hand quantity exists. Table 14.4 and Table 14.5 provide and , as functions of some relevant parameters, for European call and put options in the Black and Scholes model. [Please explain Figure 14.1 on page 478.] Exercise: Comment on Figure 14.4 on page 484, and Figure 14.13 on page 491 in conjunction with Figure 14.1 on page 478. Please adjust the shape of the curve of . 31 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 The …rst-order Taylor expansion is S, v where S is a small change in the stock price. All other parameters are kept …xed. Exercise: Use a Taylor expansion of order one, and approximate the change in the call-option value if the stock price drops by $0:5. Compare with the true value. The second-order Taylor expansion, which is more accurate than the …rst-order one, is 1 ( S )2 . 2 Again, all the parameters that a¤ect the option value are kept …xed, except the stock price. v S+ Exercise: Use a second-prder Taylor expansion, and approximate the change in the call-option value if the stock price drops by $0:5. Compare with the true value. Use Table 14.6 on page 480. 32 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 The vega coe¢ cient, indicated by V , is the …rst partial derivative of v with respect to , that is, V= @v @ = lim !0 v , where is a small change in the volatility of stock returns. Here, all the parameters that impact the option value are kept …xed, except the volaltility of stock returns. Exercise: How does a call-option value behave as a function of the volatility of stock returns? Answer the same question for a put-option value. Use Table 14.4, Table 14.5, and Figure 14.8. Check if Figure 14.8 is correct. Exercise: Approximate the change in the put-option value if the volatility rises by 1%. Compare with the true value. Use Table 14.6 on page 480. 33 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 The theta coe¢ cient, indicated by , is the …rst partial derivative of v with respect to t, that is, @v @ (T t) v , = lim t!0 t where t is a small change in time, for example, one day from the present time. Here, all parameters that impact the option value are kept …xed, except the time index. = @v = @t Exercise: How does the call-option value behave as a function of the time index? Use Table 14.4. The call-option value decreases over time. This is the time-decay phenomenon for a call option. [Please explain the time-decay phonomenon from Figure 14.1.] 34 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Exercise: A call-option holder knows with certainty that he will lose value over time. Is it pro…table to hold a call option? Explain. All the parameters that impact the option value are kept constant, except the stock price and the time index. Then, the …rst-order Taylor expansion for a multivariate function can be written as t. S+ v Exercise: Approximate the change in the call-option value if the stock price rises by $0.5 over one trading day. There are 252 trading days per year. Compare with the true value. Use Table 14.6 on page 480. Finally, the rho coe¢ cient, indicated by , is the …rst partial derivative of v with respect to r, that is, @v = @r = lim r!0 35 v , r Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 where r is a small change in the risk-free rate. Here, all parameters that impact the option value are kept …xed, except the risk-free rate. Exercise: Approximate the change in the call-option value if the stock price rises by $0.5 over one trading day, and the risk-free rate drops by 1%. Compare with the true value. Use Table 14.6 on page 480. The single-option sensitivity coe¢ cients, as de…ned above, can be extended to all derivative contracts. 2.2 Portfolio Sensitivity Coe¢ cients Consider a potfolio of derivative contracts and their underlying assets. The value of this portfolio is a linear combination of its basic components. For each component, compute the sensitivity coe¢ cient. Then, apply 36 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 the same linear combination, and obtain the associated portfolio-sensitivity coe¢ cient. Consider the hedge portfolio considered in the previous chapter: P = S C, where 2 R is the delta coe¢ cient computed at S . The delta coe¢ cient of this portfolio is @P = @S = @S @C @S @S 1 = 0. The hedge portfolio is therefore insensitive to small changes in the stock price. This portfolio is called neutral. A neutral portfolio is not necessarily neutral, V neutral, neutral, or neutral; however, neutral portfolios and the like can be obtained. For example, a portfolio combining two call options and their common underlying asset can be made neutral. Indeed, the 37 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 value of the portfolio is P = nsS + n1C1 + n2C2, where nS , n1, and n2, are the number of shares, …rst call option, and second call option, respectively. Solve the following equation for nS , n1, and n2: ( ns + n1 1 + n2 2 = 0 , 0 + n1 1 + n2 2 = 0 and determine the associated neutral portfolio. Example: Set ns = 1, 1 = 0:6151, 2 = 0:4365, 1 = 0:0181, and 2 = 0:0187. Show that n1 = 5:1917 and n2 = 5:0251. Identify the associated neutral portfolio. Following the same rules, a sensitivity analysis can be done for simple and complex strategies that combine derivative contracts and their underlying assets. A long straddle is a portfolio of a long position on a call option and a long position on a put option, both with 38 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 the same parameters. The pro…t function of the straddle, under the assumption that it is held till maturity, is = CT C0 + PT = max (0, ST X) P0 C0 + max (0, X ST ) P0, where CT and C0 are the call-option values at maturity and at the start, respectively; and PT and P0 are the putoption values at maturity and at the start, respectively. Similarly, the pro…t function of the straddle, under the assumption that it is held up to time t, is = Ct C0 + Pt P0, where Ct and Pt are functions of the stock price at time t, among other parameters. [Please explain Figure 14.15 on page 496.] Exercise: Consider the straddle associated to the exercise price X = $100, described in Table 14.7 on page 495. Consider the straddle value at the start, that is, C0 + P0, as a function of the stock price, the volatility of stock 39 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 returns, the remaining time to maturity, and the risk-free rate. Compute its sensitivity coe¢ cients, and interpret your results. How will the straddle behave if the stock price drops by $1 and the volatility rises by 2% over the next three trading days? A strangle is similar to a straddle except that its associated call and put options have di¤erent strike prices. [Please explain Figure 14.16 on page 497.] Exercise: Consider the strangle de…ned in Figure 14.16 on page 497. Answer the same questions as for the straddle. A butter‡y spread is a portfolio that employs three call options associated to the same parameters except for their strike prices. Consider the butter‡y spread taken from Table 14.7 on page 495, which consists of a long position on the call 40 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 option with a strike price of X1 = $90, a long position on the call option with a strike price of X3 = $110, and a short position on two call options with a strike price of X2 = $100. [Please explain Figure 14.17 on page 499.] Exercise: Consider the butter‡y spread in Figure 14.17 on page 499. Answer the same questions as for the straddle. Consider two call options associated to the same parameters except for their strike prices. A bull spread is a portfolio of a long position on the call option with the lowest strike price and a short position on the call option with the highest strike price. Exercise: Consider the bull spread de…ned in Figure 14.18 on page 500. Answer the same questions asked for the straddle. 41 Hatem Ben Ameur 2.3 Derivatives, Part II Brock University, FNCE 4P17 Assignment 1 Part 2 Do exercises no 1, 2, 3, 4, 9, and 16 on pages 505–506. Exercise: Consider the call option in Table 14.6. 1 Use the formulas in Table 14.1–Table 14.4 and compute the call-option sensitivity coe¢ cients in Table 14.6. 2 Use Excel and draw Figure 14.1, Figure 14.9, and Figure 14.13. Comment your results. 42 Hatem Ben Ameur 3 Derivatives, Part II Brock University, FNCE 4P17 Pricing American Options Topics Covered: 1. Exercise Value and Holding Value 2. Pricing American Options in the Black and Scholes Model (a) The American Call Option (b) Transition Parameters for Pricing American Options 3. Pricing American Options in the Binomial Model 43 Hatem Ben Ameur 3.1 Derivatives, Part II Brock University, FNCE 4P17 Exercise Value and Holding Value The American version of a European option gives its holder the additional right to exercise the option early, before maturity. Thus, an American option is worth more than its associated European option. The di¤erence in value, called the early exercise premium, stems from the early-exerce feature. Let Vt (s) be the value of an American option, and vt (s) be the value of its associated European option at time t when St = s. Uppercase letters are used for American options, and lowercase letters are used for their European counterparts. Let Vte (s) and Vth (s) be the exercise value and the holding value of an American option at time t when St = s, respectively. The exercise value is also called the intrinsic value. 44 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Example: The exercise value of a vanilla call option is Cte (s) = max (0, s X ) , for t T, where St = s is the current underlying asset price, and X is the exercise price. Similarly, the exercise value of a vanilla put option is Pte (s) = max (0, X s) , for t T. The holding value of an American option is Vth (s) =E h t;u Vu j St i =s , where E [ ] is the expectation sign under the risk-neutral probability measure, u is the …rst decision date after t, u = t is the discount factor from time u to time t, and St = s is the stock price at time t. The holding value veri…es Vth (s) 0, for all s. with the convention that VTh (s) = 0, for all s. 45 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 The holding value represents the future potentialities of the contract, discounted back to the current date. Except for a few exceptions, the holding value cannot be computed in a closed form, and must be approximated in some way. The reason is that Vu itself depends on an expectation, which itself depends on an expectation, and so on. In sum, the holding value is a high-dimensional integral. Computing the holding value is the most challenging issue in pricing American options. The value of an American option is Vt (s) = max Vte (s) , Vth (s) , for all s, and the optimal exercise policy consists of exercising the option at time t when the underlying asset is at St = s if, and only if, Vte (s) > Vth (s) , and holding the contract for at least another moment if, and only if, Vte (s) Vth (s) . 46 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 This is to say that the option holder acts in an optimal way. A European option can be seen as a particular case of an American option, where the exercise values veri…es Vte (s) = 0, for t < T and all s, and VTe (s) = VT (s) , for all s. The following general results hold: Vt (s) Vth (s) vt (s) , for all s, and Vt (s) Vte (s) , for all s. The exercise premium at time t when St = s is de…ned as Vt (s) vt (s) 0. [Please explain Figure 15.1 on page 514.] 47 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 The time value of an American option at time t when St = s is de…ned as Vt (s) Vte (s) 0. The larger the time value of an American option, the longer is its early exercise. 3.2 Pricing American Options in the Black and Scholes Model 3.2.1 The American Call Option Call-option values admit the following bounds: ct St Xe r(T t), and Ct St 48 X. Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 An American call option on a stock that pays no dividend cannot be exercised optimally before maturity. Indeed, one has Cte = St X St Xe r(T t) ct Cth. Exercising the call option before maturity is clearly suboptimal. When the underlying stock pays dividends, however, it may be optimal to exercise the call option early, just before an ex-dividend date. " Please draw two paths for the stock price, with and without dividend paying. # Consider a call option on a dividend-paying stock. Let D1; : : : ; DN be the dividends, and t1; : : : ; tN their associated ex-dividend dates over the option’s life [t, T ]. Fisher Black approximated the American call-option value as follows. 49 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 1. Adjust downward the current stock price by the PV of all dividends: St = St N X Dne r(tn t). n=1 This is somewhat equivalent to adjust Stn downward by Dn at tn, for n = 1; : : : ; N . 2. Compute the European call-option value using the Black and Scholes formula: c = BS (St , X , , T t, r) . 3. For each ex-dividend date tm, adjust the call-option’s strike price downward by the PV of all remaining dividends, including the one that is about to go exdividend, that is, Xm = X X Dne r(tn tm). n m This is equivalent to adjust upward the stock price by the same amount. 50 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 4. Compute the American call-option value using the Black and Scholes formula, under the assumption of an early exercise just before tm, that is, cm = BS (St , Xm, , tm t, r ) . 5. The approximate call-option value and its optimal exercise policy are given by cb = max fc; c1; : : : ; cN g . The intuition behind this approximation is that the American call-option can be exercised only prior to an exdividend date or at maturity. Example: The parameters of the American call option are St = $60, X = $60, = 20% (per year), T t = 180 (days), r = 9% (per year), D1 = D2 = $2, t1 = 60 (days), t2 = 150 (days). Approximate its value and identify its optimal exercise policy. Check that the (approximate) optimal exercise policy, as given by Black’s 51 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 approximation, consists of exercising the call option early, at time t2. The optimal exercise policy, provided by Black’s approximation, is deterministic while it must be expressed as a function of time and the stock price as follows: exercise the call option early at the ex-dividend date tn if, and only if, the stock price is su¢ ciently high at that time. Merton extended the Black and Scholes model to a …nancial asset that pays dividends continuously at a …xed rate (in % per year). Think of the underlying asset as a stock index. The extension is easily done by adjusting downward the current stock price as follows: St = St e (T t) . Similarly, in the Black-Scholes-Merton model, an American call option can be exercised early, at any time before maturity if, and only if, the stock price is su¢ ciently high. 52 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Along the same lines, an American put option on a stock can be exercised early at any time before maturity, whether or not the stock pays dividends. The optimal exercise policy is as follows: exercise the put early if, and only if, the stock price is su¢ ciently low. [Please explain Figure 15.3 and Figure 15.4 on page 524.] Several approximations have been proposed for pricing American options. Some of them are available in the textbook. All of them su¤er from two major disadvantages. They are neither accurate nor general. 3.2.2 Transition Probabilities for Pricing American Options Transition probabilities under the risk-neutral probability measure completely characterize the dynamics of the stock price. The following numerical procedure can be 53 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 used for pricing American options as long as the transition probabilities are known in a closed form. n o Let G = a0 = 0; : : : ; ap; ap+1 = 1 be a grid points for the stock price. Any choice for the ai, for i = 1; : : : ; p, will work as long as they cover a large domain and verify a ! 0, when p ! 1. Consider the transition probabilities Stn+1 2 [ai, ai] j Stn = ak , for all i and k, which are known in a closed form in the Black-ScholesMerton model, among other models. P Consider an American option characterized by its exercise value Vte, which can be exercised early at the decision dates t0; : : : ; tN = T . Assume that the option-value function is known at a decision date tn+1 and the grid points G . This is not a strong assumption since the option value is known everywhere at maturity. The numerical procedure acts backward in time from tn+1 to time tn, as follows. 54 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 1. Interpolate the value function at time tn+1 by a piecewise constant approximation: Vbtn+1 Stn+1 = p X n+1 i i=0 I Stn+1 2 [ai, ai] , where I ( ) is the indicator function, and Vbtn+1 (ai), for all i. n+1 i = 2. Move backward in time from time tn+1 to time tn, and consider the holding value Vthn (ak ) =E h e r(tn+1 tn)Vtn+1 Stn+1 j Stn = ak , which is not known in closed form. 3. Approximate the holding value function as follows: Vethn (akh) i r(t t ) b n+1 n V =E e tn+1 Stn+1 j Stn = ak = P e r(tn+1 tn) X n+1 i i Stn+1 2 [ai, ai] j Stn = ak . 55 i Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 4. Approximate the option value at tn and at the grid points G using Vetn (ak ) = max Vten (ak ) , Vethn (ak ) , for all ak in G , and identify the optimal exercise policy. 5. Interpolate Vetn (ak ), for all ak in G , to Vbtn (s), for all s > 0. 6. Follow the same procedure, and go from time tn to time tn 1, backward in time to time t0. 7. Reach any level of desired precision by increasing the grid size p since Vbt0 St0 ! Vt0 St0 , when p ! 1. This numerical procedure has been proposed by Ben Ameur et al for American call and put options in European Journal of Operational research, and for American Asian options in Management Science. It is used by several …nancial institutions all over the world. 56 Hatem Ben Ameur 3.3 Derivatives, Part II Brock University, FNCE 4P17 Pricing American Options in the Binomial Model The binomial model can be adjusted to be consistent with the Black-Scholes-Merton model: VtBM ! VtBSM, when t ! 0. The convergence speed is believed to be 0:5, meaning that the error e =j VtBM VtBSM j , is cut by half if the time increment t is divided by 4. The …rst part of this result is hard to prove, and the second is an open question. For the binomial model to be consistent with the BlackScholes-Merton model, set u=e p t, 1 e(r ) t d = , and p = u u d 57 d . Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Example: Consider an American call option on a stock that pays dividends continuously at a …xed rate = 13:75% (per year). The option parameters are as follows: St0 = $60, X = $60, = 20% (per year), T t = 180 (days), r = 9% (per year), and t = 90 (days). Compute its early exercise premium, and identify its optimal exercise policy. Please check that u = 1:104412, d = 0:905460, p = 0:416663, and that the two-period binomial tree for the stock price is S0 = 60 % & S1u = 66:26 S1d = 54:32 58 % & % & S2uu = 73:18 S2ud = S2du = 60 . S2dd = 49:19 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 If the stock pays discrete dollar dividends over the option’s life, Black’s intuition applies to adjust the binomial tree, though with a minor modi…cation. 1. Adjust downward the stock price at the present date t0 by the PV of all dividends over the option’s life: St0 = St0 PV all dividends. This is somewhat equivalent to adjust St downward by Dt, for all ex-dividend dates. 2. Create the binomial tree as usual. 3. At each decision tn, adjust upward the stock price by the PV of all remaining dividends (including the one that is to be paid). 4. Go backward through the adjusted binomial tree, and ompute options values as usual. 59 Hatem Ben Ameur 3.4 Derivatives, Part II Brock University, FNCE 4P17 Assignment 2 Part 1 Read the chapter, do the examples, and exercises no 1–7 on page 537. This part is to be prepared for the midterm exam, but should not be handed in as part of Assignment 2. Do exercises no 8, 17, 19. Exercise (Bonus – worth 10 points in the second mid-term exam): This exercise is to be handed in after the reading week. Use VB-Excel. In all cases, your program must be fully automated. 1 Implement Black’s procedure for pricing an American call option on a stock that pays discrete dollar dividends. Let the number of ex-dividend dates ‡exible. Solve the example shown in the textbook. 2 Implement the analytical approximation of American option prices, and draw Figure 15.3 and Figure 15.4. Solve the examples shown in the textbook. 60 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 3 Implement the binomial tree on a stock that pays dollar dividends. Let the number of ex-dividend dates ‡exible. Solve the examples in the textbook. Report the binomial trees reported in the textbook. 61 Hatem Ben Ameur 4 Derivatives, Part II Exotic Options Topic Covered: 1. Forward-Start options 2. Instalment Options 3. Compound Options 4. Chooser Options 5. Barrier Options 6. Binary Options 7. Asset-or-Nothing Options 62 Brock University, FNCE 4P17 Hatem Ben Ameur Derivatives, Part II 8. Lookback Options 9. Asian Options 63 Brock University, FNCE 4P17 Hatem Ben Ameur 4.1 Derivatives, Part II Brock University, FNCE 4P17 Forward-Start Options Standard call and put options are known as vanilla options. Other option contracts, known as exotic options, are traded on exchanges and in over-the-counter markets. They are supposed to match hedgers’needs and requirements. Forward-start options are examples. Under a forward-start option, the holder pays the premium before the option starts at parity. There are three key dates: the date the contract is signed t, the date the option starts T1, and the date the option matures T2. Date T1 is called the grant date. Forward-start options are often used in compensation packages for executives. Proposal: The presentation on executive options will cover forward-start options, as well as other option contracts used in compensation packages. 64 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Forward-start options can be evaluated using the riskneutral evaluation principle. Consider a European forwardstart option in the Black and Scholes model. The value of the European forward-start option at the start is vt =E h e r(T1 t)vTBS ST1 ; X 1 = ST1 ; ; T2 T1; r j St , since it starts at parity. On the other hand, one has vTBS ST1 ; X = ST1 ; ; T2 1 = ST1 f ( ; T2 T1; r ) , T1; r where f is a function of the volatility of the stock logreturns , the option’s life T2 T1, and the risk-free rate, but not the stock price ST1 . Using the martingale property, the value of the European forward-start option is vt = E = St h i r(T t) 1 e ST1 j St f f BS ( ; T2 T1; r ) = vtBS (St; St; ; T2 T1; r ) , whether it is a call or a put option. 65 ( ; T2 i T1; r ) Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 If the stock underlying the option pays continuous dividends at a constant dividend rate (in % per year), the martingale property is adjusted as follows: E h e r(T1 t)ST1 j St i =e (T1 t) S , t and the value of the European forward-start option is consequently adjusted as follows: vt = = = h i r(T t) 1 E e ST1 j St f ( ; T2 T1; r; e (T1 t)St f BSM ( ; T2 T1; r; ) e (T1 t)vtBSM (St; St; ; T2 T1; r ) . ) All in all, compute the value of a European forward-start option at the grant date as if the stock price were equal to the current stock price, and then discount this value back from the grant date to the start date at the dividend rate. Valuing American forward-start options can be done backward in time through a stochastic dynamic program, as it can be done for American vanilla options. 66 Hatem Ben Ameur 4.2 Derivatives, Part II Brock University, FNCE 4P17 Instalment Options The holder of a European instalment option must pay instalments at certain decision dates to keep the option alive. Therefore, at each decision date, the holder must choose between the following: 1. to pay the instalment, which keeps the option alive till the next decision date; 2. not to pay the instalment, which puts an end to the contract. The main bene…ts for the holder of an instalment option are twofold. First, risk management with instalment options is ‡exible. Like American options, instalment options are well suited for hedging cash ‡ows whose timing is uncertain. Second, the hedging cost structure with instalment options is ‡exible too. Instead of paying lump 67 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 sums for hedging instruments, corporations may enter an instalment option at a low initial cost, and adjust the instalment schedule according to their cash forecasts and liquidity constraints. This feature is particularly attractive for corporate treasurers who massively hedge interest-rate and currency risks with forwards, futures, or swaps, because option contracts imply an entry cost that may be incompatible with a temporary cash shortage. Consider an instalment option with positive instalments 0 ; : : : ; N , to be paid at the decision dates t0 = 0; : : : ; tN = T . The initial instalment 0 plays the role of the option’s premium. By the risk-neutral evaluation principle, the net holding value of this instalment option at decision date tn is vtnh (s ) n = E [e r(tn+1 tn)vtn+1 (Stn+1 ) j Stn = s] n, and the overall value is vtn (s) = max 0, vtnh (s) , for n = 0; : : : ; N n 68 1. Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 The value of the instalment option at maturity is vtN (s) = ( max (0; s X ) , for an instalment call option . max (0; X s) , for an instalment put option The exit strategy at time tn is as follows: put an end to the option’s life if, and only if, ( Stn < an, for an instalment call option , Stn > bn, for an instalment put option where an and bn are two thresholds that are functions of the option parameters. The threshold is determined by solving for s = Stn the following equation: vtnh (s) = 0. n One way of pricing the instalment option is via backward iteration: from the known function vtN , compute vtN 1 , then from vtN 1 compute vtN 2 , and so on, down to vt0 . Neither the value function vtn , for n = 0; : : : ; N 1, nor the threshold vn (an or bn) is known in a closed form. They must be approximated in some way. Stochastic 69 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 dynamic programming can be used (Ben Ameur et al., 2006, European Journal of Operational Research). Example: Consider an instalment call option with only one intermediate instalment 1 to be paid at t1 < t2 = T . The net holding value at the decision date t1 is r(T t1 ) c (S ) j S = s] cnh t1 1 T T t1 (s) = E [e = E [e r(T t1) max (0; ST X ) j St1 = s] = cBSM St1 ; X; ; T t1; r; 1, t1 1 where X is the option’s strike price, and the overall value is ct1 (s) = max 0; cBSM St1 ; X; ; T t1 t1; r; 1 . This is the payo¤ from a European call option with maturity t1 and strike price 1 on a call option with maturity T and exercise price X , the underlying call option being written on a dividend-paying stock. Options on options are also known as compound options. There are mainly four types of compound options: calls on calls, calls on puts, puts on calls, and puts on puts. 70 Hatem Ben Ameur 4.3 Derivatives, Part II Brock University, FNCE 4P17 Chooser Options The holder of a chooser option has the right to determine whether the chooser will become a vanilla call or put option by a speci…ed choice date. Chooser options are also known as as-you-like-it options. For simplicity, assume that the call and put options have the same exercise price and maturity date. Chooser options may be useful for hedging market risk, which strongly depends on the occurrence of a future event. Example: In 1993, the North American Free Trade Agreement (NAFTA) was being discussed, but had not yet been agreed upon. NAFTA was known to be bene…cial to the Mexican peso. Before the NAFTA agreement, chooser options were e¤ective for hedging the Mexican peso risk. If NAFTA went through, the chooser option would then become a call option; otherwise, the chooser option would become a put option on the Mexican peso. 71 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 There are mainly three key dates to consider: the evaluation date t, the choice date T1, and the expiry date T2. If T1 = t, one has vt = max cBSM , pBSM , t t and if T1 = T2, one has vT = max (cT , pT ) = (ST X ) I (ST > X ) + (X X) , S T ) I (ST and, by the risk-neutral evaluation principle, vt = cBSM + pBSM , t t which is the value of the associated straddle. Now, if T1 2 (t, T2), one has vT1 BSM = max cBSM T1 , pT1 BSM r(T2 T1 ) = max cBSM T1 , cT1 + Xe ST1 e = cBSM T1 + e )(T2 T1 ) (T2 T1 ) max 72 0, Xe (r (T2 T1 ) ST1 , Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 where the right-hand side consists of a portfolio of a call option with a strike price of X and a maturity date of T2 and e (T2 T1) put options with a strike price of Xe (r )(T2 T1) and a maturity date of T1. The riskneutral evaluation principle gives vt = cBSM (St; X; ; T2 t; r; ) + t (r )(T2 T1 ) ; ; T e (T2 T1)pBSM S ; Xe t 1 t 4.4 t; r; Barrier Options A barrier option acts as a vanilla option under the assumption that the underlying stock price reaches a barrier along its path. There are four families of barrier options: down-and-in, down-and-out, up-and-in, and up-and-out barrier options. They are also known as knock-in and knock-out options. [Please discuss the payo¤ from a down-and-in call option.] 73 . Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 A barrier option is a path-dependent option since its …nal payo¤ does not only depend on the stock price at maturity, but also on its behaviour over the option’s life. Barrier options are not traded; however, they are useful at modeling knock-in and knock-out events. Example: Several corporate securities can be interpreted as derivatives on a …rm’s value. For example, Merton’s model of a corporate discount bond assumes that the …rm can only go bankrupt at maturity, which is a major limitation. One way to improve Merton’s model is to de…ne bankruptcy as Vt < bt, for t 2 [0, T ] , where Vt is the …rm’s value at time t, and bt is a barrier related to the debt’s amortization. Instalment and compound options are also useful at modeling corporate coupon bonds, since bondholders can put an end to the corporate bond if certain instalments are not paid. 74 Hatem Ben Ameur 4.5 Derivatives, Part II Brock University, FNCE 4P17 Binary Options A binary option pays a certain amount if an event happens, but nothing otherwise. This event is usually related to the performance of an underlying …nancial asset. A cash-or-nothing call option pays its holder a …xed amount if the underlying …nancial asset exceeds a given strike price X at maturity T . The value of a binary option is Binary ct = = = h E e r(T t)AI (ST > X ) e r(T t)AP (ST > X ) e r(T t)AN dBSM , 2 i where A is the amount to be paid if the binary option expires in the money, and N dBSM is the normal dis2 tribution function evaluated at dBSM . Along the same 2 lines, the value of a cash-or-nothing put option is Binary pt = Ae r(T t)N 75 dBSM . 2 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 An asset-or-nothing call option promises its holder the underlying asset under the condition that the underlying asset exceeds a given strike price at maturity. This is a plain vanilla call option with X = 0 whose value is Binary ct = St e (T t) N dBSM 1 . The value of an asset-or-nothing put option is Binary pt 4.6 = St e (T t) N dBSM . 1 Lookback Options The payo¤ function from a lookback option is based on extreme values. The payo¤ from a European option to buy an underlying asset at the lowest price along its path is ST min St, [0, T ] 76 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 and the payo¤ from a European option to sell at the highest price is max St [0, T ] ST . Under Black, Scholes, and Merton’s model, lookback options can be evaluated in a closed form, since the properties of lognormal processes and their extreme values are known. 4.7 Asian Options The payo¤ from an Asian option is based on average prices, rather than terminal prices. Asian options are useful in markets where there are major players that can manipulate market prices, at least locally in time. Examples include oil and oil-product markets. 77 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 For example, a European Asian call option is characterized by the following …nal payo¤: max 0, S T St0 + + StN X = max 0, N +1 ! X . Other exotic options do exist, such as exchange options, basket options, rainbow options, and gap options, among others. They may be helpful to hedge for speci…c risks, or better analyse fundamental …nancial securities. 4.8 Assignment 2 Part 2 Read the chapter, and do exercises no 1, 5, and 6 on page 616. 78 Hatem Ben Ameur 5 Derivatives, Part II Brock University, FNCE 4P17 Options on Stock Indices, Currencies, and Futures Topics Covered: 1. Adjusting Basic Models for Continuous Dividends 2. Options on Stock Indices 3. Options on Foreign Currencies 4. Options on Futures 79 Hatem Ben Ameur 5.1 Derivatives, Part II Brock University, FNCE 4P17 Options on Stock Indices Merton extends the Black and Scholes model for a stock that pays dividends continuously at a …xed rate (in % per year). From now on, the Black and Scholes model will be indicated by BS, and the Black-Scholes-Merton model by BSM. In BSM, e (T t) shares of stock at time t is equivalent to one share of stock at time T , the dividends being reinvested in additional shares over [t, T ]. Thus, BSM call- and put-option values are obtained by exchanging St in BS by St e (T t) . For example, the BSM formula for a European call option is cBSM = N (d1) Ste t (T t) 80 Xe r(T t)N (d2) , Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 where d1 = = ln Ste (T t) =X p ln (St=X ) + r d2 = d1 p p + r + 2 =2 (T T t + 2 =2 (T T t) t t) , and t. T The binomial model can also be adjusted for a stock that pays continuous dividends. The adjustment is done through risk-neutral probabilities: e(r ) d . p = u d Options in BSM admit the following bounds: cBSM t St e pBSM t Xe r(T t) (T t) Xe r(T t), and 81 St e (T t) . Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Exercise: Use the no-arbitrage property, and establish these lower bounds. The put-call parity for European options in the BSM is r(T t) = pBSM + S e cBSM + Xe t t t (T t) , and for American options is St e (T t) X CtBSM PtBSM St Xe r(T t). Exercise: Use the no-arbitrage property, and establish the put-call parity for European call and put options. 5.2 Options on Stock Indices Options on stock indices are traded in exchanges and over-the-counter markets. Some indices track the movement of the overall stock marker, and others track the movement of particular industries. 82 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Example: The S&P 100 and S&P 500 stock indices are based on selected 100 and 500 US stocks, respectively. Options on S&P 100 are of American style, while options on S&P 500 are of European style. Since the delivery of a stock index involves high transaction costs, stock-index options are settled in cash using the following payo¤ function: Cte = m max (0; It X) , max (0; X It) , for a call option, and Pte = m for a put option. Here, It is the stock-index level at time t, m is the multiplier, and X is the option’s strike price. The scalar m plays the role of the option’s size, and is usually set at 100. Other exchange-traded options related to stock indices are the so-called LEAPS, CAPs, and FLEX options. 83 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 A stock index is a …nancial asset that provides its holder with several discrete dividends throughout all over the year. The assumption that dividends are distributed continuously at a …xed rate is acceptable in this context. The dividend rate can easily be infered from futures prices, or estimated from historical observations. Thus, the BSM formula can be used for pricing. Exercise: A European call option on a stock index has the following parameters: It = 350, X = 340, = 20%, T t = 150 days, r = 8% (per year), = 4% (per year). Set m = 1. Check that cBSM = $25:92. t Compute pBSM . t A call option on a stock index can be used to hedge against upward movements of the stock market, and a put option on a stock index, to hedge against downward movements of the stock market. 84 Hatem Ben Ameur 5.3 Derivatives, Part II Brock University, FNCE 4P17 Foreign Currency Options Options on foreign currencies are traded both in exchanges and in over-the-counter markets. For example, options on the Canadian dollar are traded on the Philadelphia Stock Exchange. A foreign currency continuously pays its holder interest at the foreign risk-free rate. Thus, BMS applies with = rf , where rf is the foreign risk-free rate (in % per year). For example, the European call-option value is rf (T t) cBSM = N ( d e ) S t 1 t Xe r(T t)N (d2) , where d1 d2 ln Ste rf (T t)=X + r + 2=2 (T t) p = T t ln (St=X ) + r rf + 2=2 (T t) p = , and T t p = d1 T t. 85 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Example: Consider a European call on the British pound with the following parameters: St = 1:4 (US$ per British pound), X = 1:5 (US$ per British pound), = 50% (per year), T t = 200 days, r = 8% (per year), and rf = 12% (per year). Use a …ve-period binomial tree, and price the European call option. Compute the value of the associated put option. The parameters of the binomial tree are u=e t = 1:1180, d = 1 = 0:847452, u and e(r rf ) t p = u d d = 0:445579. Figure 16.1 gives the …ve-period binomial tree. Riskneutral evaluation is then used to value the call option. The result is cBSM = 0:1519 (US$ per British pound). t Given the call-option value, compute the put-option value. 86 Hatem Ben Ameur 5.4 Derivatives, Part II Brock University, FNCE 4P17 Options on Futures Contracts Unlike options on stocks, options on futures contracts are settled in cash. Let t, T1, and T2 T1 be the current date, the option’s maturity date, and the futures’delivery date, respectively. The futures’delivery date needs not be equal to the option’s maturity date. Suppose that the futures contract is active at the current date t. If a call option on a futures contract is exercised at t, the call-option holder acquires a long position in the underlying futures contract plus a cash amount equal to m ft X , where m is the futures’ size, t the current date, ft the futures price at t, ft the last settlement price, and X the option’s strike price. 87 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Example: Consider a futures call option on copper with a strike price X = $0:70 (per pound of copper). One futures contract is on 25,000 pounds of copper. The futures price for delivery in one month is ft = $0:81 (per pound of copper), and the last settlement price is ft = $0:80 (per pound of copper). If the call-option holder exercises his right, he is given a long position on the futures contract and a cash amount of 25; 000 (0:80 0:70) = $2; 500. If desired, the position in the futures contract can be closed out immediately, which would leave the investor with the following cash amount: m ft X +m ft = m (f t X ) = 25; 000 (0:81 0:70) = $2; 750. ft The futures price for delivery at T is related to its underlying asset price: ft = Ste(c y)(T t), 88 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 where St is the price of the futures’ underlying asset at t, c the cost of carry, and y the convenience yield. The cost of carry measures the storage cost plus the interest required to …nance the asset less the income earned on the asset. For example, for 1- a non-dividend paying stock, set c = r and y = 0, 2- a stock index, set c = r and y = 0, 3- a foreign currency, set c = r rf and y = 0, and 4- a commodity held for consumption, the general formula applies. For an investment asset underlying the futures contract, one has ft = Ste(r )(T t) , which is equivalent to St = fte (r )(T t) . Substituting this expression for St in the BSM formula gives Black’s formula, which turns out to be independent of . Indeed, the holding in‡ows and out‡ows are already included in the futures price. 89 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 The call-option value is r(T t) cB = N ( d f e ) t 1 t Xe r(T t)N (d2) , where d1 d2 ln fte r(T t)=X + r + 2=2 (T p = T t ln (ft=X ) + 2 (T t) =2 p = , and T t p = d1 T t. t) Black’s formula is equivalent to the BSM formula with St replaced by ft, and by r, as if a futures option were an option on an underlying asset that continuously pays dividends at the domestic risk-free rate. Example: Discuss the option on the stock index futures on page 546. European as well as American futures options can be evaluated through a binomial tree, adjusted as follows: p = 1 u 90 d . d Hatem Ben Ameur 5.5 Derivatives, Part II Brock University, FNCE 4P17 Assignment 2 Part 2 Read the chapter, do the examples, and exercises no 1–5 on page 564. This part is to be prepared for the midterm exam, but should not be handed in as part of Assignment 2. Do exercises no 6 and 8. The question dealing with Barone-Adesi and Waley’s formula is not mandatory. Exercise: The sensitivity parameters in the BS model are known in closed form (see Chapter 2). Give these parameters in closed form in the BSM model, and explain your approach. Give the associated formulas for options on stock indices, foreign currencies, and futures. In this context, single- as well as multiple-asset sensitivity analysis work, modulo some adjustments. 91 Hatem Ben Ameur 6 Derivatives, Part II Brock University, FNCE 4P17 Pricing Corporate Securities Topics Covered: 1. Common Stock as a Call Option on the Firm 2. Senior and Junior Debts 3. Callable Bonds 4. Convertible Bonds 5. Warrants 92 Hatem Ben Ameur 6.1 Derivatives, Part II Brock University, FNCE 4P17 Common Stock as a Call Option Consider a model for a company with a simple capital structure, consisting of one common stock and a purediscount bond. The …rm value Vt at the current time t veri…es Vt = St + Bt, where St and Bt are the stock price and bond value at time t, respectively. This equation holds under the condition that the company is active, that is, Bt . Vt Since the corporate bond bears some credit risk, the PV evaluation principle cannot be used. Risk-neutral pricing is a viable alternative. Here, the stock and the corporate bond are interpreted as option contracts on the …rm. The following result, established by Merton, is a clue to pricing corporate securities. The common stock can be 93 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 seen as a European call option on the …rm with an exercise price and expiry date equal to the bond’s face value and maturity, respectively. Indeed, if the …rm value at maturity exceeds the bond face value, that is, FV, VT then FV ST = VT 0. Conversely, if VT < FV, then the …rm will go bankrupt, the bondholder will take control of the …rm, and the stock will end up worthless, that is, ST = 0, and BT = VT . In sum, the stock price at maturity can be expressed as ST = max (0; VT 94 FV) . Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 This results in interpreting the stock as a call option on the …rm. [Please explain Figure 17.1 on page 567.] The current stock price is then St = cBS t; V ; r ) , t (Vt; FV; T where cBS t is the Black and Scholes formula for a European call option, and V is the volatility associated to the …rm, but not to the stock. Exercise: Single-asset sensitivity analysis applies in this context. Specify the impact on the stock price of a small change in the …rm value, time to maturity, …rm volatility, and risk-free rate. In all cases, the corporate-bond value at maturity is BT = VT = VT = FV ST max (0; VT max (0; FV 95 FV) VT ) . Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Two useful interpretations result from the above equations. Holding the corporate bond is equivalent to 1. holding the entire company and selling a European call option to the stockholder to buy the company for the bond’s face value, 2. holding a riskless bond and selling a put option to the stockholder to sell the company for the bond’s face value. [Please explain Figure 17.2 on page 568.] The put-call parity gives Bt = Vt = FV cBS t (Vt; FV; T e r(T t) t; V ; r) pBS t (Vt; FV; T t; V ; r) , showing that the corporate bond is worth less than its associated riskless bond. The di¤erence in values represents the credit worthiness of the issuing company. The higher 96 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 the put-option value, the lower is the credit worthiness of the issuing company. The common stock and the corporate bond are interpreted as derivative contracts whose values, St and Bt, depend on the …rm value Vt and its volatility V . There are several methods used to approximate Vt and V , some of them from St and S . Exercise: Portfolio-sensitivity analysis applies in this context. Specify the impact on the corporate-bond value of a small change in the …rm value, time to maturity, …rm volatility, and risk-free rate. Moody’s KMV o¤ers a default-prediction model that is based on option theory and its applications for corporate securities. 97 Hatem Ben Ameur 6.2 Derivatives, Part II Brock University, FNCE 4P17 Senior and Subordinated Debts Debt contracts di¤er by their securitization levels. For example, senior debts are reimbursed …rst (if possible), and subordinated debts are reimbursed next. Subordinated debts are also called junior debts. Consider a company with a simple capital structure that consists of one common stock, a pure-discount senior bond with a face value of FVs, and a pure-discount junior bond with a face value of FVj , both maturing at the same j time T . Let Bts and Bt be the senior- and junior-bond values at time t, respectively. Three scenarios can happen at maturity: 8 s! > V < FV > T > > > s = V , and B j = 0 > > S = 0, B T T > T T > > s j < FVs VT < FV + FV ! . s s = FVs , and B j = V > S = 0, B FV > T T T T > > s + FVj ! > > V FV > T > > j > : ST = VT FVs + FVj , BTs = FVs, and BT = FVj 98 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Whatever happens at maturity, these equations can be summerized as FVs + FVj ST = max 0, VT , for the stock, BTs = FVs max (0, FVs VT ) , for the senior debt, and j BT = max (0, VT FVs) max 0, VT FVs + FVj for the junior debt. In sum, the common stock can be interpreted as a European call option on the …rm with a strike price of FVs+ FVj ; the senior bond as portfolio of a riskless bond with a face value of FVs and a signed European put option with a strike price of FVs; and the junior bond as a portfolio of a European call option with a strike price of FVs, and signed European call option with a strike price of FVs+FVj . The riskless bond and all option contracts expire at the corporate bond’s maturity. [Please explain Figures 17.3 and 17.4 on page 571.] 99 , Hatem Ben Ameur 6.3 Derivatives, Part II Brock University, FNCE 4P17 Callable Bonds A callable bond can be redeemed by the issuer before maturity for a known call price. A callable can be seen as a portfolio of a straight bond that contains an embedded call option at the discretion of the issuer. Corporate bonds are typically callable, while T-bonds are seldom callable. Some callable bonds can be redeemed at any time, and others at speci…ed dates before maturity. However, the investor is often protected against the call feature over a certain protection period, for example, the …rst …ve years of the bond’s life. Unlike vanilla call options, call options embedded in bonds are not traded alone, but rather live within their associated bonds. 100 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 The value of a callable bond is always lower than the value of its associated straight bond, for it is less desirable. Thus, the value of the embedded call option is straight Ct = B t Bt , where Ct is the value at time t of the embedded call straight option, Bt and Bt are the values at time t of the straight bond and the callable bond, respectively. Here, capital letters are used since the bond with its embedded call option can be interpreted as an American derivative contract. Clearly, the bond issuer is better to call the bond early when interest rates are low, and …nance its activities at a lower cost. The optimal exercise policy for the bond issuer is as follows: redeem the bond before maturity if, and only if, the interest rate is su¢ ciently low. [Please explain Figure 17.5 on page 572.] 101 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Pricing callable bonds is complex, and can be achieved through a dynamic program as for American vanilla options. For simplicity, consider a T-bond that can be redeemed at each coupon date. Assume that the callable bond value is known at a given future date tn+1.This is not really a limitation since the bond value is known at maturity, that is, BT = 1 + c, where c is the last coupon. Risk-neutral evaluation gives Bnh(r) = E " n+1 n # Bn+1 rtn+1 j rtn = r , where frg is the interest-rate process, n+1= n is the discount factor over [tn, tn+1], Bnh the bond holding value at the current decision date tn, and Bn+1 the (overall) bond-value function at the next decision date tn+1. Though highly complex, this expectation can be e¢ ciently computed under several dynamics for the interest rate, which are consistent with the no-arbitrage principle. See the paper by Ben Ameur et al published in Journal of Economic, Dynamics, and Control. 102 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 The payo¤ at time tm from the issuer to the investor is ( cn + c, if the issuer calls , Bnh(r) + c otherwise where cn is the call price at time tn, which is known at the start. The issuer must redeem the bond at tn if, and only if, the holding value exceeds the exercise bene…t, that is, Bnh(r) > cn, which is equivalent to r < rn , where r = rtn is the current interest-rate level at time tn, and rn is a threshold at time tn that identi…es the optimal redemption policy. The overall value function of the callable bond at time tm is Bn(r ) = min cn, Bnh + c. 103 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Finally, the embedded call option can be computed using the equation straight Ct = B t Bt . Achieving these steps backward in time from maturity to the start provides the bond value function at each decision date, and identi…es its optimal redemption policy. This methodology can be adjusted a bit to accommodate corporate callable bonds. " 6.4 Please discuss the callable bond issued by the Swiss Confederation. # Convertible Bonds A convertible bond can be converted into shares of the issuing …rm at the discretion of the bond holder. 104 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 The conversion ratio is the number of shares received by the investor upon conversion, indicated here by N . The conversion ratio N is often …xed at the start, such that S0 < B0, N to preclude immediate conversion. For simplicity, consider a European convertible discount bond with maturity T , face value FV, which can be converted into shares at maturity. At maturity, the …rm will default, or the bond will pay its holder the larger between its face value and its conversion value. In case of default, that is, VT < FV, one has BT = VT , and ST = 0, 105 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 where VT is the …rm value, BT the bond value, and ST the stock price at maturity. Otherwise, the …rm value ver…es FV, VT and the bondholder will decide whether to convert the bond or not. The bond value is BT = max (FV, N ST ) = FV + max (0, N ST FV) . Please notice that the known amount FV does not represent a riskless straight bond, but a straight bond issued by the …rm. Indeed, this payment is uncertain, and will only be received under the condition that FV. VT All in all, the convertible bond can be interpreted as a portfolio of an otherwise identical corporate bond and a call option on N shares of stock with an exercise price equal to the bond’s face value. 106 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Proposal: A possible presentation could focus on coupon corporate bonds, which include both call and conversion features. This presentation would include callable and convertible bonds with ‡exible call prices and conversion factors. 6.5 Warrants A warrant is similar to a call option except that it is written by the underlying company, and typically has a long maturity. If a warrant is exercised, the underlying company issues a new share for delivery. A dilution e¤ect takes place, and stockholders lose some value. Exercising the warrant makes sense only when the resulting share price exceeds the exercise price. 107 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Consider a …rm with a simple capital structure, that is, M shares of stock and N European warrants. The …rm value and the stock price just before maturity are related as follows: VT ST . =M At maturity, they are related as follows: VT = M ST + N max (0, ST X) . If the warrant expires out of the money, the …rm value and the stock price move along continuous paths; otherwise, the …rm value remains continuous, while the stock price jumps downward at maturity. Thus, under exercise at maturity, one has VT =M ST = VT =M ST + N wT =M ST + N (ST 108 X) , Hatem Ben Ameur Derivatives, Part II which can be summarized as M ST + N ST = M +N Brock University, FNCE 4P17 X . The warrant value at maturity is wT = max (0, ST M = max 0, = max 0, = = M M +N 1 1+ N M X) ST + N M +N M ST M +N max 0, ST max 0, ST X X M X N +M X X . By the law of one price, the warrant value and its associated call-option value are related in the same way: wt = 1 1+ N M cBS t N St + w t , X , S , T M t, r , where S is the volatility of the stock log-returns, including the warrant. This is a formula for the warrant value 109 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 as a function of the warrant value, which can be solved numerically. Proposal: A possible presentation could focus on the relationship between option theory and investment projects, known as real options. This presentation would include options to explore, develop, extend, postpone, switch, and shut down a real project. 6.6 Assignment 3 Part 1 Read the chapter, and do exercises no 1, 2, 3, 5, 6, 7, and 8 on pages 576 and 577. 110 Hatem Ben Ameur 7 Derivatives, Part II Brock University, FNCE 4P17 The Long-Term T-Bond Futures Contract (Revisited) 7.1 Futures Contracts A forward contract commits two parties, one to buy and the other to sell an underlying asset at a known future delivery date (maturity) for an agreed-upon delivery price also known as the forward price. No payment is exchanged up front. Forward contracts are traded over the counter; therefore, forward-market participants are subject to both market and credit risk. If prices rise, the long party records a gain since he committed to buying the underlying asset at a lower price, and the short party records a loss. The opposite happens when prices fall. The cumulative loss recorded by 111 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 one party at maturity can be so great that he can fail to ful…ll his obligations. A futures contract acts as a forward contract except that it is traded on an exchange, and thus is subject to a margin system and a daily settlement of gains and losses. Unlike a forward contract, which is settled at maturity, a futures contract is settled at the end of each trading day, thereby keeping the loss and counterparty risk at a very low level. For each futures contract, the exchange de…nes a set of terms and conditions related to the contract’s size, quotation unit, minimum price ‡uctuation, grade, trading hours, delivery terms, daily price limits, and delivery procedures. First of all, when a futures contract is issued, each party is invited to open a margin account with his broker, and to deposit an initial margin. The broker is invited to do 112 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 the same with his clearing …rm, which in turn, does the same with the clearinghouse. Next, the trading process continues until the day’s end. The settlement price is then registered, based on the last futures price(s). A new settlement price is revealed at the end of each trading day. Finally, the futures contract is marked to the market at the end of each trading day. The contract is closed and reopened at the new settlement price, and the value of both parties is consequently reset to zero. The di¤erence between the new and the last settlement prices, if positive, is subtracted from the seller’s margin account, and added to the buyer’s margin account. If the di¤erence in prices is negative, then the opposite is done. This is the daily settlement system. When a margin account falls below a certain level, known as the maintenance margin, the investor receives a margin 113 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 call to bring the margin balance back up to the initial margin level. The margin system and daily settlement drastically reduce counterparty risk, since losses, which are monitored every day, remain limited. To avoid physical delivery, positions on futures contracts are usually closed before the delivery month, and thus settled in cash. Liquidity makes it possible to leave futures markets at any time before maturity. 7.2 The T-Bond Futures Contract: The Margin and the Daily Settlement Systems The T-bond futures contract, traded on the CBOT, calls for the delivery of $100,000 T-bonds with a minimum remaining life of 15 years at the …rst delivery date, within 114 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 a delivery month. Thus, at the …rst delivery date, a deliverable T-bond has at least 15 years to maturity or to its earliest call date. De…ne the reference T-bond underlying this futures contract as a hypothetical T-bond with a maturity of 20 years and a coupon rate of 6% (per year). Delivery months are March, June, September, and December. The initial margin is $2,500, and the maintenance margin is $2,000. This futures contract is one of the most widely traded in the world, mainly because its ability to hedge long-term interest-rate risk. In the following Table, we consider a short position on the CBOT T-bond futures contract, which was taken on 08/01 at the quoted futures price of 97 27. The futures price was thus at $97 + 27=32 per $100 of principal, which corresponded to $97; 843:75 per $100; 000 of principal amount. The seller deposited $2; 500 into his margin account. The futures price fell to $97; 406:25 at the end of the …rst trading day. This was the new 115 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 settlement price, the old one being $97; 843:75. The seller recorded a gain, and the buyer recorded a loss. The absolute di¤erence between the two settlement prices, that is, $437:50, was subtracted from the buyer’s margin account, and added to the seller’s margin account. The balance of the seller’s margin account then rose to $2; 937:50. The futures price rose to $97; 781:25 at the end of the second trading day. This was the new settlement price, the old one having been $97; 406:25. The seller recorded a loss, and the buyer a gain. The absolute di¤erence between the two settlement prices, that is, $375:50, was subtracted from the seller’s margin account, and added to the buyer’s margin account. The balance of the seller’s margin account was then at $2; 562:50. . . On 08/08, the balance of the seller’s margin account fell to $1250 just after the contract was marked to the market, which is lower than the maintenance margin of $2000. A margin call of $1250 was issued the same day, and honoured the next trading day, on 08/11. On 08/18, the last trading day, the settlement price fell from $100; 781:25 to $100; 500:00, and the seller then recorded a gain of $281:25. This trader then closed his position by inversion, left the market, and withdrew the remaining balance of $3; 031:25. 116 Hatem Ben Ameur 7.3 Derivatives, Part II Brock University, FNCE 4P17 Embedded Options The hypothetical T-bond is typically not traded on the market, and thereby cannot be delivered. Thus, the short trader is given the right to deliver alternative long-term T-bonds. This is the choosing option, which is also called the quality option. To make the delivery fair for both parties, the price received by the short trader is adjusted according to the quality of the T-bond delivered. This adjustment is made via a set of conversion factors, which are de…ned by the CBOT as the prices of the eligible T-bonds at the …rst delivery date under the assumption that interest rates for all maturities equal 6% per year, and that interest is compounded semi-annually. The T-bond, which is actually selected for delivery, is known as the cheapest to deliver. The short trader is given the right to deliver the underlying T-bond within a delivery month. This is the timing 117 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 option. The delivery process takes place according to special features, that is, the delivery sequence and the endof-month delivery rule. The delivery sequence consists of three consecutive business days: the position day, the notice day, and the delivery day. During the position day, the short trader can declare his intention to deliver until 8:00 p.m., while the CBOT closes at 2:00 p.m. (Central Standard Time). This six-hours option is known as the wild card play. On the notice day, the short trader has until 5:00 p.m. to state which T-bond will be actually delivered. The delivery then takes place before 10:00 a.m. on the delivery day, against a payment based on the settlement price on the position day. Finally, during the last seven business days before maturity, trading on the T-bond futures contract stop while delivery, based on the last settlement price, remains possible according to the delivery sequence. This is the end-of-month option. 118 Hatem Ben Ameur 7.4 Derivatives, Part II Brock University, FNCE 4P17 Evaluation So far, papers in the literature have considered essentially the quality option under a ‡at term-structure of interest rates. The futures contract is then equivalent to a forward contract. Let r be the level of interest rate over the contract’s life [0, T ], and let (c, M ) be the deliverable T-bond with maturity M and coupon rate c. The theoretical futures price g0 and the cheapest to deliver (c , M ) are obtained by solving the following equation: max f(c;M )g0 (c;M ) pT (c:M; r ) = 0, where f(c;M ) and pT (c:M; r ) are the conversion factor and the fair value of the T-bond (c, M ) at maturity, respectively. The conversion factor of the T-bond (c; M ), de…ned as f(c;M ) = pT (c; M; 6%) , 119 Hatem Ben Ameur veri…es Derivatives, Part II Brock University, FNCE 4P17 8 > < > 1, if c > 6% f(c;M ) : = 1, if c = 6% . > : < 1, if c < 6% Thus, the conversion factor f(c;M ) adjusts the futures price g0 upward if a high-quality T-bond (c, M ) is delivered, and downward if a low-quality T-bond (c, M ) is delivered. No adjustment is made if the reference T-bond (6%, 20) is delivered. The conversion factors make delivery somewhat fair for both counterparties, but they are not perfect. Exercise: Show that the cheapest to deliver bears an extreme coupon. For simplicity, assume that the contract’s expiry date is a coupon date. Identify the cheapest to deliver as a function of the interest rate r. The last question is a bonus question, and is not required for the second midterm exam. Ben-Abdallah, Ben-Ameur, and Breton (2009) (GERAD report, and Journal of Banking and Finance) provided a 120 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 full representation of this futures contract, with all its embedded options. The most important results are: 1. The quality option is the most valuable embedded option (around 2% of par for viable interest-rate levels); 2. The timing option is the second most valuable embedded option (around 0.2% of par for viable interestrate levels); 3. The end-of-month delivery rule and the wild card play are less important (around 2 and 0.2 basis points of par and less, respectively). 121 Hatem Ben Ameur 8 8.1 Derivatives, Part II Brock University, FNCE 4P17 Sample of Exams First Midterm Exam Exercise 1 (20%): Consider a three-period binomial tree for a non-dividend-paying stock with the following parameters: S0 = 100 (in dollars), u = 1:25, d = 0:8, r = 7% (per period), and p = 1+r d . u d 1. Show that this market model is arbitrage free. 2. Draw the three-period binomial tree for the stock. 3. Evaluate the European put option on this stock with a strike price X = 100 (in dollars) and a maturity T = t3. 122 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 4. Evaluate the associated American put option, and identify its optimal exercise policy. 5. Evaluate the early-exercise premium associated to this put option. 6. Consider an Asian put option characterized by the following exercise value: Pten = max 0, X S tn St0 + + Stn = max 0, X , n+1 for n = 0, 1, 2, 3 . Answer again questions no 3, 4, and 5. Exercise 2 (20%): Consider two European call options with the same parameters except for their strike prices. A bull spread with calls consists of a long position on the call option c1 with the lower strike price X1, and 123 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 a short position on the call option c2 with the higher exercise price X2. The current time is indicated by t, and the maturity by T . The bull spread’s pro…t function is de…ned as: = vT = c1T vt c1t (c2T c2t) , where vt and vT are the values of the bull spread at time t and T , c1t and c1T are the values of c1 at time t and T , and c2t and c2T are the values of c2 at time t and T , respectively. The following Table provides the sensitivity parameters of the two call options. c1 Strike price Value Delta Gamma Theta Vega Rho 90 16:33 0:7860 0:0138 11:2054 20:4619 30:7085 124 c2 100 10:30 0:6151 0:0181 12:2607 26:8416 25:2515 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 The current stock price St is at $100, the volatility at 30% (per year), the remaining time to maturity T t at 180 days, and the risk-free rate r at 8% (per year). 1. Plot the curve of the bull spread’s pro…t, as a function of the stock price at maturity ST . 2. Explain why this strategy is called a bull spread. 3. Suppose that the current stock price rises by $0.1. What is the impact on the bull spread’s value? What is the impact on the bull spread’s pro…t? Use Delta, then use Delta and Gamma. 4. Is this bull spread subject to time decay? Explain. 5. Suppose that the current price will rise by $0.1, and that the volatility will drop by 1% over the next two 125 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 business days. What is the impact on the current bull spread’s value? What is the impact on the bull spread’s pro…t? Exercise 3 (20%): Consider an arbitrage-free market model for a non-dividend-paying stock and a savings account that grows at the continuous risk-free rate r (in % per year). Consider a call and a put option on this stock with an exercise price X and the remaining time to maturity T t. Lower-case letters are used to indicate European-option values, and upper-case letters are used to indicate American-option values. The …rst …ve questions deal with the American call option, and the last …ve questions deal with the American put option. 1. Prove that ct St Xe r(T t), for all parameters. 126 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 2. Provide an intuitive explanation for the following property: Cth ct, for all parameters, and give an example where Vth > vt, for all parameters. 3. Provide an intuitive explanation for the following property: Ct max (0, St X ) , for all parameters. 4. Prove that an American call option on a non-dividendpaying stock cannot be exercised optimally before maturity. 5. Use a graph, and explain why an American call option can be exercised optimally before maturity when the underlying stock pays dividends before maturity. 127 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 6. Use the same graph, and plot the exercise- and holdingvalue functions of an American put option. 7. Use another graph, and plot the value function of this American put option. 8. Provide an intuitive explanation why an American put option can always be exercised optimally before maturity. 9. Characterize the optimal exercise policy of an American put option at time t 2 [0, T ], as a function of a threshold at , which is itself a function of the option’s parameters. 10. Provide an intuitive explanation for the behaviour of at over time, from the start to maturity. 128 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Exercise 4 (20%): Consider a …rm with a very simple capital structure, that is, a share of stock and a pure discount bond with maturity T and face value FV. The …rm’s value process is indicated by Vt, the bond’s value process by Bt, and the stock price by St, for t 2 [0, T ]. The continuously compounded risk-free rate is indicated by r (in % per year). Examine the …gures, and brie‡y explain each one. Exercise 5 (20%): Brie‡y comment on the business article. 8.2 Second Midterm Exam Exercise 1 (20%): Consider a three-period binomial tree for a non-dividend-paying stock with the following 129 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 parameters: S0 = $100, u = 1:25, d = 0:8, r = 7% (per period), and p = 1+r d . u d All option contracts considered here are of European style, have an exercise price of X = $100, and expire at time t3 = T . 1. Draw the three-period binomial tree for the stock. 2. Evaluate the asset-or-nothing call option. 3. Evaluate the lookback call option. 4. Evaluate the up-and-in call option with a barrier of b = $150. 130 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Exercise 2 (20%): Merton extended the Black and Scholes model to a stock that continuously pays dividends at a …xed rate (in % per year). The present value of all dividends paid over a time window is proportional to the initial stock price, dividend rate, and time period. This model, known as the Black, Scholes, and Merton model, allows one to evaluate European vanilla options in a closed form by adjusting downward the stock price as follows: S0 = S0 e T, and using the Black and Scholes formula. For example, the European call-option value on a dividend-paying stock is Xe rT N (d2) , c0 = S0 N (d1) where d1 d2 ln S0 =X + r + 2=2 T p = T p = d1 T. 131 and Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 1. Show that the adjusted stock price S0 is nothing else than the stock price S0 net of all dividends paid over the option’s life. 2. Explain why a foreign currency can be interpreted as a …nancial asset that continuously pays dividends. Give Black, Scholes, and Merton’s formula for a European call option on a foreign currency. 3. Give the de…nition of a European call option on a futures contract on a …nancial asset. 4. Explain why a call option on a futures contract on a …nancial asset can be evaluated using Black, Scholes, and Merton’s formula. This result was established by Black. 5. *** The futures price under Black’s model is assumed to follow a lognormal process, as it is the case 132 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 for the stock in Black and Scholes’model. Explain why Black’s formula is totally unacceptable for a call option on the CBOT long-term T-bond futures contract, though frequently used. Use the convergence principle at the futures contract’s maturity and its underlying T-bonds’maturities. Exercise 3 (20%): Consider an instalment option in the Black and Scholes model. Let t1; : : : ; tN be the sequence of decision dates, and let 1; : : : ; N 1 2 (0, T ) be a sequence of instalments. The up-front payment is indicated by 0. 1. De…ne a European instalment option. 2. List the pros of instalment options as hedging tools. 3. Show that an instalment call option with only one instalment 1 to be paid at t1 2 (0, T ) can be interpreted as a compound call option. 133 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 4. Brie‡y explain how compound options can be used to analyse portfolios of corporate bonds. Exercise 4 (30%): Consider a …rm with a simple capital structure, that is, a common stock and a discount bond with principal amount P and maturity T . Let Vt, Bt, and St be the …rm’s value, the bond’s (present) value, and the stock price at time t, respectively. The …rm plays the role of the underlying asset in the Black and Scholes model, with a known initial value V0 and volatility V . 1. Interpret the stock and the bond as derivative contracts on the …rm. 2. Interpret the expression P (VT BT ) , where P is the physical probability measure. What happens if the risk-neutral probability measure P were used. 134 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 3. Prove that Bt, for t 2 [0, T ] . Vt When can the company go bankrupt? 4. Consider now a …rm with a simple capital structure, that is, a common stock, a short-term discount bond with principal amount P1 and maturity t1, and a long-term discount bond with principal amount P and maturity T . Explain how the stock and the corporate debt can be evaluated using compound options, as functions of the …rm’s value V0 and volatility V . Now, Bt refers to the present value of the bond portfolio at time t. Explain how and when the company can go bankrupt, and interpret the expressions 1 = P Vt1 Bt1 and T BT j Vt1 > Bt1 . = P VT 135 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 5. This is a numerical investigation that reports the stock price, the present value of the debt, and the term-structure of default probabilities, as functions of the …rm’s value V0 and volatility V . The parameters are as follows: = 4% (per year), r = 4% (per year), t1 = 1 year, P1 = $50, t5 = T = 5 years, and PT = $100. The parameter is the (instantaneous) rate of return on the …rm under the physical probability measure, and r is the rate of return on the …rm under the risk-neutral probability measure. (a) For each table, comment on the quality of the corporate debt. (b) Table 1–Table 3 contain several zeros. Brie‡y explain. (c) *** How does the default probability 1 behave as a function of the volatility V ? Explain. (d) In Table 1, for V = 20%, one has 1 = 90:3% and 5 = 13:4%. Interpret this result. 136 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Table 1: S0, B0, and Default Prob. for V0 = $100 V t t0 t0 t1 t2 t3 t4 t5 S0 B0 1 2 3 4 T 0 :1 0:01599 99:98401 0:99609 0 0 0 0:00956 0 :2 1:17348 98:8265 0:90344 0 0 0 0:1342 0 :3 4:63836 95:36164 0:78278 0 0 0 0:26961 0 :4 9:61668 90:38332 0:69246 0 0 0 0:37824 Table 2: S0, B0, and Default Prob. for V0 = $150 V t t0 t0 t1 t2 t3 t4 t5 S0 B0 1 2 3 4 T 0:1 20:57029 129:42971 0:081431 0 0 0 0:002756 0 :2 24:75861 125:24139 0:233911 0 0 0 0:084375 137 0 :3 31:47386 118:52614 0:284145 0 0 0 0:206394 0 :4 39:17126 110:82874 0:304564 0 0 0 0:316672 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 Table 3: S0, B0, and Default Prob. for V0 = $200 V t t0 t0 t1 t2 t3 t4 t5 S0 B0 1 2 3 4 T 0:1 70:08769 129:91231 0:00002 0 0 0 0:00005 0 :2 70:67994 129:32016 0:01521 0 0 0 0:03499 0 :3 74:29304 125:70696 0:06292 0 0 0 0:14066 0 :4 80:54610 119:45398 0:10908 0 0 0 0:25391 Exercise 5 (15%): This exercise deals with the CBOT long-term T-bond futures contract. 1. De…ne the options embedded in this futures contract. 2. Is the cheapest to deliver known in advance? Brie‡y explain. 138 Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17 3. Ignore the margin system and the timing option, and assume a ‡at term-structure of interest rates. The (fair) value of the T-bond (c, M ) at time t is indicated by pt (c; M; r), where r is the level of interest rates. (a) Is the cheapest to deliver known in advance? Brie‡y explain. (b) Suppose that only the reference T-bond is available for delivery. What is the futures’ price at the start? (c) *** Suppose that only T-bonds (c1, M1) and (c2, M2) are available for delivery. Give a criterion that identi…es the cheapest to deliver, and determines the (fair) futures’price at the start. 139