1. Zvi WienerContTimeFin - 4 slide 1 Financial Engineering The Valuation of Derivative Securities Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049

2. Zvi WienerContTimeFin - 4 slide 2 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.

3. Zvi WienerContTimeFin - 4 slide 3 Derivative Security Black-Scholes, Merton 1973 Options, Forwards, Futures, Swaps Real Options

4. Zvi WienerContTimeFin - 4 slide 4 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.

5. Zvi WienerContTimeFin - 4 slide 5 Arbitrage Valuation Primary security X: dX =  (X,t)dt +  (X,t) dZ Derivative security V = V(X,t): dV = V x dX + 0.5V xx (dX) 2 - V  dt

6. Zvi WienerContTimeFin - 4 slide 6 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.

7. Zvi WienerContTimeFin - 4 slide 7 Arbitrage Valuation Arbitrage portfolio: P = V + hX. dP = dV + h dX dP = (V x +h) dX + 0.5V xx (dX) 2 - V  dt In order to completely eliminate the risk, we should choose (V x +h) = 0. Such a portfolio has no risk, thus it must earn the risk free interest. Important assumption: X is traded.

8. Zvi WienerContTimeFin - 4 slide 8 Arbitrage Valuation Set h = -V x. dP must be proportional to the investment in the portfolio P. This investment is  V-Xh =  V-XV x Thus dP = rPdt = r(  V-XV x ) dt

9. Zvi WienerContTimeFin - 4 slide 9 Arbitrage Valuation dP = rPdt = r(  V-XV x ) dt 0.5V xx (dX) 2 - V  dt = r(  V-XV x ) dt 0.5  2 V xx + rXV x - r  V - V  = 0 the general valuation for derivatives

10. Zvi WienerContTimeFin - 4 slide 10 Arbitrage Valuation 0.5  2 V xx + rXV x - r  V - 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.

11. Zvi WienerContTimeFin - 4 slide 11 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.)

12. Zvi WienerContTimeFin - 4 slide 12 The Forward Contract Assume that X follows GBM:  (X,t) =  X  (X,t) =  X The boundary conditions are: V(X,0)=Ximmediate purchase V(0,  ) = 0zero is an absorbing boundary V x (X,  ) <  the hedge ratio is finite

13. Zvi WienerContTimeFin - 4 slide 13 The Forward Contract 0.5  2 X 2 V xx + rXV x - r  V - V  = 0 V(X,0) = X This equation was described in Chapter 2. a = 0.5  2 b = rc = -  r d = 0e = 0m = 1 n = 0

14. Zvi WienerContTimeFin - 4 slide 14 The Forward Contract 0.5  2 X 2 V xx + rXV x - r  V - V  = 0 V(X,0) = X The Laplace transform is equal X/(s-(1-  )r). The inverse Laplace transform is V(X,  )=Xe r(1-  ) . As soon as  <1, the forward price is higher than the spot price.

15. Zvi WienerContTimeFin - 4 slide 15 The Forward Contract The hedge ratio is V x = V/X  1. A perfectly hedged position holds one forward contract and is short V/X units of the spot commodity.

16. Zvi WienerContTimeFin - 4 slide 16 The European Call Option Strike E. Time to maturity . Value of the option at maturity is: Max(X-E,0). X V E

17. Zvi WienerContTimeFin - 4 slide 17 The European Call Option V(X,0) = Max(X-E,0) V(0,  ) = 0 V x (X,  ) <  Normally the price is paid in full,  = 1. The PDE becomes: 0.5  2 X 2 V xx + rXV x - rV - V  = 0 V(X,0) = Max(X-E,0)

18. Zvi WienerContTimeFin - 4 slide 18 The European Call Option 0.5  2 X 2 V xx + rXV x - rV - V  = 0 V(X,0) = Max(X-E,0) Can be solved with the Laplace transform.

19. Zvi WienerContTimeFin - 4 slide 19 The European Call Option

20. Zvi WienerContTimeFin - 4 slide 20 Normal Distribution x N(x)

21. Zvi WienerContTimeFin - 4 slide 21 Put Call Parity 0.5  2 X 2 V xx + rXV x - rV - V  = 0 V(X,0) = Max(E-X,0) E X V Put Option

22. Zvi WienerContTimeFin - 4 slide 22 Put Call Parity E X V Call Put Underlying

23. Zvi WienerContTimeFin - 4 slide 23 Put Call Parity Call-Put E X V Underlying

24. Zvi WienerContTimeFin - 4 slide 24 Put Call Parity E X V Bond = Ee -r  Underlying = Call-Put+Bond

25. Zvi WienerContTimeFin - 4 slide 25 Put Call Parity X = Call - Put + Ee -r  Synthetic market portfolio

26. Zvi WienerContTimeFin - 4 slide 26 Hedging X - h*Call - riskless What is h?

27. Zvi WienerContTimeFin - 4 slide 27 Hedging Riskless if volatility does not change.

28. Zvi WienerContTimeFin - 4 slide 28 Greeks Delta of an option is Gamma of an option is Theta of an option is Rho of an option is Vega of an option is

29. Zvi WienerContTimeFin - 4 slide 29 BMS Formula and BMS Equation Delta of an option is Gamma of an option is equal to vega. When  =  (t) the BMS can be modified by

30. Zvi WienerContTimeFin - 4 slide 30 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.

31. Zvi WienerContTimeFin - 4 slide 31 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

32. Zvi WienerContTimeFin - 4 slide 32 Equilibrium Valuation If the rate of return on X is below the equilibrium rate, i.e. dX = (  -  )Xdt +  XdZ 0.5  2 X 2 V xx + (r-  )XV x - r  V - V  = 0 Can be solved by a substitution and change of a numeraire. Y = Xe -  V(X,  ) = W(Y,  )

33. Zvi WienerContTimeFin - 4 slide 33 The American Option dX = (  -  )Xdt +  XdZ While the option is alive it satisfies the PDE: 0.5  2 X 2 V xx + (r-  )XV x - rV - V  = 0 Optimal exercise boundary: Q(  ) high contact condition = smooth pasting condition

34. Zvi WienerContTimeFin - 4 slide 34 The American Option When X < Q, the equilibrium equation: 0.5  2 X 2 V xx + (r-  )XV x - rV - V  = 0 When X > Q, the following equation: 0.5  2 X 2 V xx + (r-  )XV x - rV - V  = rE-  X is derived by substituting V = X-E in the lhs. V and V x are continuous at X=Q. 0.5  2 X 2 V xx - V  is discontinuous at X=Q.

35. Zvi WienerContTimeFin - 4 slide 35 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?

36. Zvi WienerContTimeFin - 4 slide 36 Solution 3.1

37. Zvi WienerContTimeFin - 4 slide 37 Solution 3.1 a. E(X t )=X 0 e  t. b. F 0 = E(X t )e (r-  )t. c. r = , or  = 0.

38. Zvi WienerContTimeFin - 4 slide 38 Exercise 3.2 What are the effects of carrying costs, convenience yields, and dividends?

39. Zvi WienerContTimeFin - 4 slide 39 Solution 3.2 r - risk free rate, c - carrying cost, d - dividend yield, y - convenience yield. All variables represent proportions of costs or benefits incurred continuously.

40. Zvi WienerContTimeFin - 4 slide 40 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?

41. Zvi WienerContTimeFin - 4 slide 41 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.

42. Zvi WienerContTimeFin - 4 slide 42 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.

43. Zvi WienerContTimeFin - 4 slide 43 Home Assignment X follows an ABM. Calculate E t (X s ). X follows a GBM. Calculate E t (X s ).