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Financial Engineering The Valuation of Derivative Securities Zvi Wiener [email protected] tel: 02-588-3049 Zvi Wiener ContTimeFin - 4 slide 1 Derivative Security A derivative security is one whose value depends exclusively on a fixed set of asset values and time. Derivatives on traded securities can be priced in an arbitrage setting. Derivatives on non traded securities can be priced in an equilibrium setting. Zvi Wiener ContTimeFin - 4 slide 2 Derivative Security Black-Scholes, Merton 1973 Options, Forwards, Futures, Swaps Real Options Zvi Wiener ContTimeFin - 4 slide 3 Derivative Security - the proportion of the value paid in cash. Pure options: = 1. Pure Forwards: = 0. No arbitrage assumption. Free tradability of the underling asset. Otherwise one have to find the equilibrium. Zvi Wiener ContTimeFin - 4 slide 4 Arbitrage Valuation Primary security X: dX = (X,t)dt + (X,t) dZ Derivative security V = V(X,t): dV = VxdX + 0.5Vxx(dX)2 - Vdt Zvi Wiener ContTimeFin - 4 slide 5 Arbitrage Valuation How we pay for a derivative security? A proportion is paid now (deposited in a margin account). If securities can be deposited in margin account, then = 0. If paid in full, = 1. Zvi Wiener ContTimeFin - 4 slide 6 Arbitrage Valuation Arbitrage portfolio: P = V + hX. dP = dV + h dX dP = (Vx+h) dX + 0.5Vxx(dX)2- Vdt In order to completely eliminate the risk, we should choose (Vx+h) = 0. Such a portfolio has no risk, thus it must earn the risk free interest. Important assumption: X is traded. Zvi Wiener ContTimeFin - 4 slide 7 Arbitrage Valuation Set h = -Vx. dP must be proportional to the investment in the portfolio P. This investment is V-Xh = V-XVx Thus dP = rPdt = r(V-XVx) dt Zvi Wiener ContTimeFin - 4 slide 8 Arbitrage Valuation dP = rPdt = r(V-XVx) dt 0.5Vxx(dX)2- Vdt = r(V-XVx) dt 0.5 2Vxx+ rXVx - rV - V = 0 the general valuation for derivatives Zvi Wiener ContTimeFin - 4 slide 9 Arbitrage Valuation 0.5 2Vxx+ rXVx - rV - V = 0 Note that (X,t) does NOT enter the equation! In addition to the equation one has to determine the boundary conditions, and then to solve it. Zvi Wiener ContTimeFin - 4 slide 10 The Forward Contract Agreement between two parties to buy/sell a security in the future at a specified price. No payment is made now (forward), thus =0. Let X be the price of the underlying asset. Assume that there are no carrying costs (dividends, convenience yield, etc.) Zvi Wiener ContTimeFin - 4 slide 11 The Forward Contract Assume that X follows GBM: (X,t) = X (X,t) = X The boundary conditions are: V(X,0)=X immediate purchase V(0, ) = 0 zero is an absorbing boundary Vx(X, ) < the hedge ratio is finite Zvi Wiener ContTimeFin - 4 slide 12 The Forward Contract 0.5 2X2Vxx+ rXVx - rV - V = 0 V(X,0) = X This equation was described in Chapter 2. a = 0.52 b=r c = - r d=0 e=0 m=1 n=0 Zvi Wiener ContTimeFin - 4 slide 13 The Forward Contract 0.5 2X2Vxx + rXVx - rV - V = 0 V(X,0) = X The Laplace transform is equal X/(s-(1- )r). The inverse Laplace transform is V(X,)=Xer(1-). As soon as <1, the forward price is higher than the spot price. Zvi Wiener ContTimeFin - 4 slide 14 The Forward Contract The hedge ratio is Vx = V/X 1. A perfectly hedged position holds one forward contract and is short V/X units of the spot commodity. Zvi Wiener ContTimeFin - 4 slide 15 The European Call Option Strike E. Time to maturity . Value of the option at maturity is: Max(X-E,0). V X E Zvi Wiener ContTimeFin - 4 slide 16 The European Call Option V(X,0) = Max(X-E,0) V(0, ) = 0 Vx(X, ) < Normally the price is paid in full, = 1. The PDE becomes: 0.5 2X2Vxx+ rXVx - rV - V = 0 V(X,0) = Max(X-E,0) Zvi Wiener ContTimeFin - 4 slide 17 The European Call Option 0.52X2Vxx+ rXVx - rV - V = 0 V(X,0) = Max(X-E,0) Can be solved with the Laplace transform. Zvi Wiener ContTimeFin - 4 slide 18 The European Call Option V ( X , ) XN (d1 ) Ee r N (d 2 ) X 1 2 ln r E 2 d1 d 2 d1 Zvi Wiener ContTimeFin - 4 slide 19 Normal Distribution N(x) x Zvi Wiener ContTimeFin - 4 slide 20 Put Call Parity 0.52X2Vxx+ rXVx - rV - V = 0 V(X,0) = Max(E-X,0) V Put Option X E Zvi Wiener ContTimeFin - 4 slide 21 Put Call Parity Underlying V Call Put E Zvi Wiener ContTimeFin - 4 X slide 22 Put Call Parity Underlying V Call-Put E Zvi Wiener ContTimeFin - 4 X slide 23 Put Call Parity Underlying = V Call-Put+Bond Bond = Ee-r E Zvi Wiener ContTimeFin - 4 X slide 24 Put Call Parity X = Call - Put + Ee-r Synthetic market portfolio Zvi Wiener ContTimeFin - 4 slide 25 Hedging X - h*Call - riskless What is h? 1 C h X Zvi Wiener ContTimeFin - 4 slide 26 Hedging X d X C C X C 1 C C 2 dX dX dX 2 C X 2 X X 2 1 C C 2 dt 2 2 X X 2 Zvi Wiener Riskless if volatility does not change. ContTimeFin - 4 slide 27 Greeks Delta of an option is C X Gamma of an option is Theta of an option is C t Vega of an option is Rho of an option is Zvi Wiener C r ContTimeFin - 4 C 2 X 2 C slide 28 BMS Formula and BMS Equation Delta of an option is C N (d1 ) X Gamma of an option is equal to vega. When = (t) the BMS can be modified by T t T 2 ( )d t Zvi Wiener ContTimeFin - 4 slide 29 Implied Volatility The value of volatility that makes the BMS formula to be equal to the observed price. Volatility smile. Confirms that the BMS formula is more general than the BMS formula. Zvi Wiener ContTimeFin - 4 slide 30 Equilibrium Valuation This corresponds to the case when the underlying security does not earn the risk-free rate r. Example: dividends are paid (continuously or discrete) it is not traded cost-of-carry (storage, maintenance, spoilage costs) convenience yield from liquid assets Zvi Wiener ContTimeFin - 4 slide 31 Equilibrium Valuation If the rate of return on X is below the equilibrium rate, i.e. dX = (-)Xdt + XdZ 0.52X2Vxx+ (r-)XVx - rV - V = 0 Can be solved by a substitution and change of a numeraire. Y = Xe- Zvi Wiener V(X, ) = W(Y, ) ContTimeFin - 4 slide 32 The American Option dX = (-)Xdt + XdZ While the option is alive it satisfies the PDE: 0.52X2Vxx+ (r-)XVx - rV - V = 0 Optimal exercise boundary: Q() high contact condition = smooth pasting condition Zvi Wiener ContTimeFin - 4 slide 33 The American Option When X < Q, the equilibrium equation: 0.52X2Vxx+ (r-)XVx - rV - V = 0 When X > Q, the following equation: 0.52X2Vxx+ (r-)XVx - rV - V = rE- X is derived by substituting V = X-E in the lhs. V and Vx are continuous at X=Q. 0.52X2Vxx- V is discontinuous at X=Q. Zvi Wiener ContTimeFin - 4 slide 34 Exercise 3.1 V is a forward contract on X. X follows a GBM. Assume that there are no carrying costs, convenience yield, or dividends. Let the rate of return on the cash commodity (X) be = r+(M-r) a. Find the expected future cash price. b. Relationship between the forward price and the expected cash price. c. Under what conditions the expectation hypothesis is correct? Zvi Wiener ContTimeFin - 4 slide 35 Solution 3.1 dX Xdt XdZ 1 1 1 2 d ln X dX (dX ) 2 X 2 X d (ln X ) dt 2 2 dt dZ 2 t Z t ln X t ln X 0 a 2 X t X 0e Zvi Wiener ( 0.5 2 ) t Z t ContTimeFin - 4 slide 36 Solution 3.1 a. E(Xt)=X0et. b. F0= E(Xt)e (r-)t. c. r = , or = 0. Zvi Wiener ContTimeFin - 4 slide 37 Exercise 3.2 What are the effects of carrying costs, convenience yields, and dividends? Zvi Wiener ContTimeFin - 4 slide 38 Solution 3.2 E ( X t ) X 0e ( r c d y )t r - risk free rate, c - carrying cost, d - dividend yield, y - convenience yield. All variables represent proportions of costs or benefits incurred continuously. Zvi Wiener ContTimeFin - 4 slide 39 Exercise 3.3 Suppose that an underlying commodity’s price follows an ABM with drift and volatility . What economic problems will it cause? What is the value of a forward contract assuming that a proportion of the price, , is kept in a zero-interest margin account? Zvi Wiener ContTimeFin - 4 slide 40 Exercise 3.4 Suppose that the value of X follows a mean reverting process: dX = (-X)dt+XdZ When this situation can be used? Value a forward contract on value of X in periods. Zvi Wiener ContTimeFin - 4 slide 41 Exercise 3.8 Value a European option on an underlying index X, that follows a mean-reverting square root process: dX = ( - X)dt+XdZ When this situation can be used? Value a forward contract on value of X in periods. Zvi Wiener ContTimeFin - 4 slide 42 Home Assignment X follows an ABM. Calculate Et(Xs). X follows a GBM. Calculate Et(Xs). Zvi Wiener ContTimeFin - 4 slide 43