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第三章 期权定价的离散模型--------二叉树方法 Chapter 3 Binomial Tree Methods ------ Discrete Models of Option Pricing An Example S $45 u T S0 $40 STd $35 Question: When t=0, buying a call option of the stock at with strike price $40 and 1 month maturity. If the risk-free annual interest rate is 12% throughout the period [0, T], how much should the premium for the call option(看涨期权) be? Example cont.1 c ( S K ) T (到期日收益)payoff = T S $45, cT (45 40) $5 u T S0 $40 Consider STd $35, cT (35 40) $0 a portfolio(投资组合) S 2c Example cont.2 When t=T, 45 2*5 35, if S , VT () $35 35 2*0 35, if S . has fixed value $35, no matter S is up or down Example cont.3 If risk free interest r =12%, a bank deposit of B=35/(1+0.01) after 1 month 35 1 VT ( B) (1 12%) 35 VT ( ) 1 0.01 12 By arbitrage-free principle V0 ( B) V0 (). Example cont.4 That is S0 2c0 B0 Then This 35 34.65 1 0.01 40 34.65 c0 2.695 2 is the investor should pay $2.695 for this stock option. Analysis of the Example ① ② the idea of hedging: it is possible to construct an investment portfolio with S and c such that it is risk-free. The option price thus determined (c_0=$2.695) has nothing to do with any individual investor's expectation on the future stock price. One-Period & Two-State One-period: assets are traded at t=0 & t=T only, hence the term one period. Two-state: at t=T the risky asset S has two possible values (states): STu & STd , with their probabilities satisfying 0 Prop ST STu , Prop ST STd 1 Prop ST S u T Prop S T S d T 1 One-Period & Two-State Model The model is the simplest model. Consider a market consisting of two assets: a risky S and a risk-free B If: risky asset St and risk free asset Bt known S0 , B0, when t=0, t=T, 2 possibilities S S0u , ST : d u d. ST S0 d , Option Price at t=0? (for strike price K, expired time T) u T Analysis of the Model St - Stock Price, is a stochastic variable S S 0u Up, with probability p STd S0 d Down, with probability 1-p u T S0 V ( S0u K ) u T V0 VTd ( S0 d K ) where Vt is a stochastic variable. Question & Analysis If known VT ( ST ) at t=T, how to find out V0 when t=0? Assume the risky asset to be a stock. Since the stock option price is a random variable, the seller of the option is faced with a risk in selling it. However, the seller can manage the risk by buying certain shares (denoted asΔ) of the stocks to hedge the risk in the option. This is the idea! Δ- Hedging Definition Definition: for a given option V, trade Δ shares of the underlying asset S in the opposite direction, so that the portfolio V S is risk-free. Analysis of Δ- Hedging free asset BT B0 , 1 rT If Π is risk free, then, on t=T, risk T VT ST is risk free. i.e. T 0 so that VT ST 0 Analysis of Δ- Hedging cont. VT , ST are random variables, when t=T, both of them have 2 possible values u VT S0u (V0 S0 ), V S0 d (V0 S0 ), d T where &V0 are unknown, solve them: VTu VTd S0 (u d ) Analysis of Δ- Hedging (Probability Measure) 1 d u u d V0 T S0 VT VT u d ud 1 Define a new Probability Measure d u qu ProbQ ST ST ud u d qd ProbQ ST ST ud Obviously , 0 qu , qd 1, qu qd 1. Solution of Premium From the discussion above, 1 Q V0 E (VT ), Q E (VT ) denotes the expectation of the where random variable VT under the probability measure Q. Definition of Discounted Price Let U be a certain risky asset, and B a risk-free asset, then U t / Bt is called the discounted price贴现价格 (also known as the relative price相对价格) of the risky asset U at time t. V V E . B0 BT 0 Q T Theorem 3.1 Under the probability measure Q, an option's discounted price is its expectation on the expiration date. i.e. Q E ( S K ) / B , call T T V0 B0 E Q ( K ST ) / BT , put Remark In order to examine the meaning of the probability measure Q, consider S is an underlying risky asset. Calculate 1 Q ST u d E qu ST qd ST BT B0 1 d u S0 S 0u S0 d B0 u d ud B0 Risk-Neutral World Under the probability measure Q, the expected return of a risky asset S at t=T is the same as the return of a risk-free bond. A financial market possessing this property is called a Risk-Neutral World In a risk-neutral world, no investor demands any compensation for risks, and the expected return of any security is the riskfree interest rate. Definitions the probability measure Q defined by qu ProbQ ST S u T d ud , u qd ProbQ ST S ud d T is called by risk-neutral measure. The option price given under the riskneutral measure is called the risk-neutral price. Definition of Replication In a market consisted of a risky asset S and a riskfree asset B, if there exists a portfolio S B (where α,β are constants, Φ, V are both random variables) such that the value of the portfolio Φ is equal to the value of the option V at t=T, ST BT VT then Φ is called a replicating portfolio of the option V, then option price V0 0 S0 B0 Theorem 3.2 In a market consisted of a risky asset S and a risk-free asset B, d<ρ<u is true if and only if the market is arbitrage-free. Proof of Theorem 3.2 (1st dir.) 1) arbitrage-free d<ρ<u Suppose ρ>= u, consider the following portfolio: S (S0 / B0 ) B Its values at t=0 and at t=T are: 0 S0 (S0 / B0 ) B0 0 T ST (S0 / B0 ) BT Proof of Theorem 3.2 (1st dir.) cont. T is a random variable with two possible values: u S0 T S0u B0 ( u ) S0 0, u B0 for ST ST T d S d S 0 B ( d ) S 0 0 0 0 T B 0 d for ST ST Proof of Theorem 3.2 (1st dir.) cont. So that, for the portfolio Φ T 0, & Prob T 0 Prob ST STd 0. That shows that there exists arbitrage opportunity for portfolio Φ, contradiction! to that the market is arbitrage-free. Same to ρ<=d. Proof of Theorem 3.2 (2nd dir.) If market is arbitrage-free, for any portfolio S B If 0 S0 B0 0, T ST BT 0, then T ST BT 0. In fact, define a risk-neutral measure Q qu ProbQ ST S u T d ud , u qd ProbQ ST S ud d T Proof of Theorem 3.2 (2nd dir.) cont. Then Consider 0 qu , qd 1, qu qd 1. the expectation of the random variable T Q u d E (T ) qu T qd T According to the definition of the risk-neutral measure Q, d u Q E ( T ) ( S0u B0 ) ud ( S0 B0 ) 0 0. ud ( S0 d B0 ) Proof of Theorem 3.2 (2nd dir.) cont. That qu qd 0. u T is, But Then i.e. There d T Tu 0, Td 0 0. u T d T Prob T 0 0. exists no arbitrage opportunity. Theorem 3.2 If the market is arbitrage-free, then there exists a risk-neutral measure Q defined by qu ProbQ ST S u T d ud , u qd ProbQ ST S ud d T such that Q E ( S K ) / BT , call V0 T B0 E Q ( K ST ) / BT , put Binomial Tree Method Divide the option lifetime [0, T] into N intervals: 0 t0 t1 ... t N T . Suppose the price change of the underlying asset S in each interval [tn , tn1 ](0 n N 1) can be described by the one-period twostate model, then the random movement of S in [0, T] forms a binomial tree Binomial Tree Method cont. This means that if at the initial S time price of the Sthe 0 underlyingS asset is , then at T t=T, will have N+1N possible values S u 0 d 0,1,... N Take call option as example, VT ( ST K ) the option value at t=T, is also a random variable, with corresponding possible values N ( S0u d K ) 0,1,... N Binomial Tree Method Notation Denote S S u n n d , V V (S , tn ), n n (0 n N , 0 N ), ˆ max | S u N d K 0, 0 N Binomial Tree S 0u S0 S0 d S 0u 2 …… S0ud S0 d …… 2 …… …… S0u N S 0u N 1d … S0u N d … S0ud N 1 S0 d N Possible Values of Option at t=T V0N S0u N K S0N K , V S0u N N ˆ 1 V N N V 0, 0. N ˆ d K S K, N ˆ Problem Option Pricing by BTM V (0 N ) N If are given, how can we determine N 1 V in particular 0 0 (0 N 1) V V ( S 0 , t0 ) ? Answer to the Problem With the one period and two-state model, and using backward induction, we can determine N h V step by step. Induction Steps When to VN (0 N ) N 1 V are given, (0 N 1) find consider the following one period and twoSN 1u SN state model. SN 1 and VN 1 SN 1d SN1 VN ( SN K ) VN1 ( SN1 K ) Induction Steps cont. Define a risk-neutral measure Q d u qu q, qd 1 q ud ud N 1 V Then, So 1 [qVN (1 q)VN1 ] (0 N 1). that for any ˆ h(1 h N ) h h l V h q (1 q)l ( SNl K ) l 0 l N h ˆ when , V 0. N h 1 Induction Steps cont. But Denote SNl SN hu h l d l , qu (1 q )d qˆ uq / Thus VN h d (1 q) 1 qˆ N h ˆ h h l l S q (1 q ) l 0 l K ˆ h h l h q (1 q)l , ˆ , l 0 l 0, ˆ . European call option valuation formula m m l l (n, m, p) p (1 p) l 0 l n Denote Then the European call option valuation K formula isN h N h V S ( , h, qˆ ) h (ˆ , h, q) Especially, h=N, α=0, V ( S0 , 0) S0( , N , qˆ ) K N (ˆ , N , q) Discount Factor Discount Factor BT e r (T t ) satisfies dBt rBt , (0 t T ), dt The BT 1. financial meaning of the discount factor: to have $1 at t=T (including continuous compound interests), B one needs to deposit t in bank at t (t<T). Discount Factor in BTM in the binomial tree method, ttrading tn (0 occurs n N) at discrete times the compound interests should also be calculated for the discrete case. Bn (n 0,1,...N ) Let denote the discrete discount factors. They satisfy the difference equations Bn1 Bn r tBn , (0 n N 1), BN 1. Discount Factor in BTM cont. That is 1 1 B Bn 1 1 r t 1 r t n N n N B 1 N n where ρis the [growth t , t t ] of the risk-free bond in i.e. Bn 1 Bn Call---Put Parity in discrete form for the binomial tree method, the call---put parity (in discrete form) becomes N h c K / p h N h N h S European put option valuation formula Using European call option valuation formula and put---call party, we have N h p K h SN h ( , h, q) SN h (ˆ , h, qˆ ) 0 h N , 0 N h. Investment vs. Gambling Game Investing in options can be compared to a gambling game. U0 Initial stake be . After U T one game, the stake becomes . UT is a random variable. If the expectation E (UT ) U 0 then the gamble is said to be fair Fair Gambling Game - the bet at n-th game, U n 1 - the next bet. If under the condition that complete information of all the previous n-games are available, the expectation of U n 1 equals the previous stake U n i.e., Un E (U n1 | (U1 U n )) U n then we say the gamble is fair. σ-Algebra (U1 Un ) denotes information of U1 complete Un the bets up to n-th game, E( X | Y ) and denotes the conditional expectation of X under condition Y. (U1 U n ) In mathematics, is called σalgebra in stochastic theory Martingale Martingale is often used to refer to a fair gamble. Un : 0 n N The bet sequence that satisfies condition E (U n1 | (U1 U n )) U n as a discrete random process, is called a Martingale. Mathematical Definition of Martingale A sequence Y Yn : n 0 is a Martingale with respect to sequence X X n : n 0 if for all n ≥0 E | Yn | E (Yn1 | X 0 , X1 X n ) Yn Risk-neutral measure vs. Martingale Under the risk-neutral measure Q, the discount prices S of an underlying , (n 0,1 N ) asset S, as a discrete B t random process, satisfy the equation: S Q S E ( S0 Sn ) , (0 n N ) B B tn tn 1 n Martingale Measure Hence the discount price sequence of an underlying asset is a martingale. The risk-neutral measure Q is called the martingale measure Q equivalent to the probability measure P. Definition of Equivalence Probability measure P and probability measure Q are said to be equivalent if and only if for any probability event A (set) there is Prob P ( A ) 0 ProbQ ( A ) 0 i.e. the probability measures P and Q have the same null set. European option under Martingale The European option valuation formula under the sense of equivalent Martingale measure Q, can be written as S Q S E ( S0 , B t N h B t N or VtN h E Especially h Q S V0 , St N h ) K | ( S0 , N N E Q S N K , StN h ) Relation of the arbitrage-free principle & Martingale measure What is the relation between the arbitragefree principle and the existence of equivalent Martingale measure? Arbitrage-free principle d< ρ <u existence of equivalent Martingale measure Q European option pricing in a risk-neutral world Theorem 3.3 - the fundamental theorem of asset pricing If an underlying asset price moves as a binomial tree, there exists an equivalent Martingale measure if and only if the market is arbitrage-free. Proof of Theorem 3.3 a risky asset S and a risk-free asset B, “sufficiency” by Theorem 3.2. “necessity”- A portfolio if 0 0, t * 0, s.t. ProbP (t* 0) 1 then what we need to prove is that there must be Prob P ( t* 0) 0 where $P$ denotes an objective measure. Proof of Theorem 3.3 cont. In fact, let Q- equi. Mart. Meas. of P, then / B 0 E Q / B t * E (t* ) 0 Bt* / B0 0 Since P Q , it implies ProbQ ( t* 0) 1 Thus Q we have ProbQ ( t 0) 1 therefore ProbQ ( t 0) 0 Since measure P and measure Q are equivalent, this means Prob P ( t* 0) 0 * * Dividend-Paying An underlying asset pays dividends in two ways: 1. Pay dividends discretely at certain times in a year; 2. Pay dividends continuously at a certain rate. This section, the continuous model is considered only Reason for Studying the Continuous Model 1 Asset -- foreign currency. exchange rate changes randomly the foreign currency is a risky asset . If it is deposited in a bank in its native country, it would accrue interests according to the local int.rate The interest be regards dividend of the "security" this dividend is paid continuously. Therefore, the "dividend rate" is the risk-free interest rate of the foreign currency in its native country. Reason for Studying the Continuous Model 2 Suppose the underlying asset is a portfolio of a large number of risky assets. Since each risky asset in the portfolio pays dividend at a certain rate at certain times, the number of dividend payments for the portfolio would be large, and we can approximate it as continuous payment (dividend rate can be timedependent). Example A company needs to buy M Euro at time t=T to pay a German company. To avoid any loss if Euro goes up, the company buys a call option of M Euro with expiration date t=T at rate K. How much premium should the company pay? Example cont. Over the same period, due to the risk-free interest ("dividend"), 1Euro in the local bank can grow to 1Euro Euro 1 qt , where q is the risk-free interest rate in a German bank. Example cont. [t , t t ] Therefore the value of 1 Euro in changes as S u ( RMB / Euro) St ( RMB / Euro) t St d ( RMB / Euro) Let B be a risk-free [t , t Bank t ] of China bond. Its change in is Bt ( RMB) Bt ( RMB) 1 r t where and r is the risk-free interest rate in BOC bank. Example cont.-[t , t t ] each interval , apply Δ-hedging strategy, i.e. to construct a portfolio In V S and select Δ, such that is risk-free. t t Example cont.-- t t t t Solve V S u Vt t St t d d V S t t t t d t (Vt St ) the system, u t t u t t d Vt ut Vt 1 d t , Vt [quVt ut q dVt t] (u d ) St Example cont.-- / d u / qu , qd ud ud We assume dη<ρ<uη, so that 0 qu , q d 1, qu q d 1 Example cont.--- Since the price of the option at t=T ( in RMB) is VT M ( ST K ) M ( S0u N d K ) ,(0 N ) where M is the required amount of Euro, and K is the agreed exchange rate. let M=1, similar to before, using backward induction, we can get: Option Pricing (Dividend, call) The pricing formula for dividend-paying European call option: N h V SN h h K ~ ˆ ( , h, q ) h ( , h, q ) ^ Where q ~ / d ud , q ^ uq ~ Option Pricing (Dividend, put) The pricing formula for dividend-paying European put option: N h p K h ( , h, q ^ ) SN h h (ˆ , h, q ~ ) Binomial Tree Method of American Option Pricing American option pricing is different from European option pricing. At each node SN h (1 h N , 0 N h) for American option, the price must satisfy the constraint N h V N h ( K S ) Backward Induction - American option pricing Therefore for American option pricing (taking put option as example), its backward induction process is: VN ( K SN ) , 0 N n=N n=N-1 N 1 V 1 N N N max [qV (1 q)V 1 ], ( K S ) , 0 N Backward Induction cont. N h (0 N 1) is given, then If V 1 N h N h N h 1 V max [qV (1 q)V 1 ], ( K S ) A B (0 N h 1) d q ud where N h 1 in each step, after A is calculated, it must be compared with the payoff function B, VN h 1 be the larger of the two, and so on, until V 0 0 is arrived at. Another View of American Option Suppose ud=1 the underlying n nasset price S S0u d (0 n) can For be nwritten jas S j S0u ( j n, n 2, S0 1. simplicity, let we construct a grid: , n 2, n) In the plane (S,t) American Option Grid 0 S j S j 1 0 t0 where tn tN T Sj u j, j 0, 1, tn nt (t T / N ), n 0,1, , , N. V V (S j , tn ). n j Notice Then American put option pricing: 1 n 1 n V max qV j 1 (1 q)V j 1 , ( K S j ) n j Theorem 3.4 If V jn (n 0,1, N , j 0, 1, ) option price, then V V , V n j n j 1 is American put n 1 j V n j Theorem 3.5 tn (0 n N 1), j jn when j jn 1, V j n j when j jn , V j n j when j jn 1, V j n j s.t. Optimal Exercise Boundary – Free Boundary t t=T Stopping Region 2 Continuation Region 2 S Optimal Exercise Boundary Continuation & Stopping Region In region Σ1, the option value is greater than the payoff from exercising the option, the option holder should continue to hold the contract rather than early exercising it. Therefore Σ1 is called the continuation region. In region Σ2, since V 1/ [qV n j n 1 j 1 (1 q)V n 1 j 1 ] which means the option's expected return is less than the risk-free interest rate, the holder should stop the contract, i.e. early exercise the option immediately. Therefore Σ2 is called the stopping region. Optimal Exercise Boundary S S (t ) is of great importance in finance, as the interface of the continuation and stopping, and is called the optimal exercise boundary. Theoretically, American option holder should choose a suitable exercise strategy according to the above analysis to avoid loss. American Call-Put Symmetry Call-put parity does not hold for American options. One naturally asks whether there exists other kind of relation between American call and put options. American Call-Put Symmetry Example An option as a contract gives its holder the right to exchange cash for stock (call option), or to exchange stock for cash (put option), at the strike price on the expiration date. American Call-Put Symmetry Example We may regard the cash as a risk-free bond earning interests according to the risk-free interest rate, and regard the stock as a risky bond earning risk-free interests according to the dividend rate. Then we can see a certain symmetry exists between the call and put options: C ( S , K ; , ; t ) P( K , S ; , ; t ) i.e. for options (European or American) with the same expiration date, if the positions of S and K, and the positions of η and ρ are both swapped, the call option price and put option price should be equal. Theorem 3.6 (call-put symmetry) If ud=1, then for American options with the same expiration date, relation C ( S , K ; , ; t ) P( K , S ; , ; t ) is true, where t tn ,(0 n N ). Theorem 3.7 For American options with the same expiration date, let Sc ( S p ) & K c ( K p ) denote the underlying asset price and strike price for the call (put) option respectively. If K p / S p Sc / K c then C ( Sc , K c ; , ; t ) P( S p , K p ; , ; t ) Sc K c where t tn ,(0 n N ). SpKp Summary 1. Have Introduced a discrete model---BTM to describe the underlying asset price movement, and have priced its derivatives (options) using this model. 2. Based on the arbitrage-free principle, using the Δhedging technique, have introduced a risk-neutral equivalent martingale measure. The BTM of option pricing has produced a fair valuation that is independent of any individual investor's risk preference. Summary cont. 3. Using the BTM of American option, we have shown that there exist two regions for American put option: the continuation region and the stopping region, which are separated by the optimal exercise boundary. 4. For American options, although there is no call--put parity, there exists call-put symmetry, as for European options. 作业:P22、1,2,3,4