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Financial Engineering

This article explores risk-neutral pricing and equivalent martingale measure in financial engineering, with a focus on replication, pricing by arbitrage, and the binomial model.

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Financial Engineering

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  1. Financial Engineering Risk Neutral Pricing Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049 ContTimeFin - 9

  2. Equivalent Martingale Measure and Risk-Neutral Pricing Rangarajan K. Sundaram New York University Journal of Derivatives ContTimeFin - 9

  3. Replication A contingent claim is replicable if it is possible to construct a portfolio of other securities with two properties: • The value of the portfolio at maturity is identical in all circumstances to the value of the contingent claim. • Once the portfolio is set up, there are no other cash flows (self-financing). ContTimeFin - 9

  4. Pricing by Arbitrage In a complete financial market one can price all securities from prices of a small set of securities used in replication. Otherwise there would be an arbitrage. Assumption: no taxes, transactions costs or short restrictions. ContTimeFin - 9

  5. Settings Bond - default free money market account. Risk-free rate is the return on this bond. A contingent claim X with a known payoff at maturity depending on other securities. ContTimeFin - 9

  6. r q q uS 1-q 1-q r dS The Binomial Model There are two assets: bond and stock: 1 S q + (1-q)=1, 0 < q < 1. d < r < u Arbitrage? ContTimeFin - 9

  7. r q q uS Xu q X 1-q 1-q 1-q r dS Xd The Binomial Model assets: bond, stock, and an option 1 S Note that here r is what we usually denote by 1+r. ContTimeFin - 9

  8. Pricing by Arbitrage The Binomial model is an example of a complete market. All claims can be priced by arbitrage. Fix any contingent claim (i.e. fix Xu and Xd). Consider a portfolio consisting of units of a stock units of the bond ContTimeFin - 9

  9. Pricing by Arbitrage Value of this portfolio at maturity is ContTimeFin - 9

  10. Replication in a Binomial Model Replicating portfolio consists of  shares and a bonds. Thus the price of the contingent claim must be equal: Note that we did not use probabilities of up and down (they are hidden in prices already). ContTimeFin - 9

  11. Risk-Neutral Probabilities 1. Identify a new probability measure, called risk-neutral probability, or an equivalent martingale measure. 2. Compute the expected discounted payoff from the contingent claims, where the expectations are taken under the risk-neutral measure. ContTimeFin - 9

  12. Equivalent Measures Two probability measures are equivalent if and only if any event with positive probability under one measure has positive probability under the second measure (and vice versa). ContTimeFin - 9

  13. Martingale A stochastic process is martingale if its expected change is always zero. A more precise definition is ContTimeFin - 9

  14. Discounting Discounting is necessary, since otherwise the bond is not risky, but grows (at the risk free rate). ContTimeFin - 9

  15. Existence and Uniqueness There is no risk-neutral probability measure if and only if there exists an arbitrage*. Multiple risk-neutral probability measures can occur if and only if there are contingent claims that can not be replicated. However they predict the same prices for all replicable claims. ContTimeFin - 9

  16. 1 S r q q uS 1-q 1-q r dS Binomial Case Since d < r < u, the probability p is 0 < p <1 ContTimeFin - 9

  17. The Risk-Neutral Price of a Claim Consider a contingent claim paying Xu and Xd. Then its price can be found as Arbitrage free price coincide with the risk-neutral price. ContTimeFin - 9

  18. K Arrow Securities – State Prices A contingent claim that pays $1 if and only if a particular state of the world occurs. State price = discounted probability of the state. ContTimeFin - 9

  19. State Prices and Risk-Neutral Probabilities ContTimeFin - 9

  20. u1S1 u2S2 S1 S2 d1S1 d2S2 Example Binomial model with two risky assets and a bond. Let S1 and S2 denote the initial prices of the risky assets. Let the possible prices be ContTimeFin - 9

  21. Example We must have Assume that this equation does not hold. ContTimeFin - 9

  22. Example Consider a portfolio: $a in bonds $b in S1 $c in S2 its cost now is a + b + c. At maturity it gives: If the equality does not hold there is an arbitrage. ContTimeFin - 9

  23. r qu qm 1 r qd r Completeness uS 3 variables, 2 equations, there are many solutions. S mS dS ContTimeFin - 9

  24. Continuous Time Models Note that the first equation describes a bond: B(t)=B0ert. The second equation will correspond to a stock. ContTimeFin - 9

  25. The Discounted Price Process Denote by Q the discounted price process ContTimeFin - 9

  26. Girsanov’s Theorem If we wish to change the drift of the process to . • Define a new process Z using , , and . • Redefine Q using the new process Z. ContTimeFin - 9

  27. Step 1 Define  by the solution to  –  =  Note that  is not constant. This equation can always be solved if -1 exists. Define a new process Z such that dZ = dt + dW ContTimeFin - 9

  28. By definition Step 2 ContTimeFin - 9

  29. Black-Merton-Scholes Model Try There is NO solution. ContTimeFin - 9

  30. Black-Merton-Scholes Model Try There is a solution! ContTimeFin - 9

  31. Financial Engineering Interest Rate Derivatives Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049 following Hull and White ContTimeFin - 9 Hull and White

  32. Black’s Model • Similar to the Black-Scholes model. • Assumes that the future value of interest rate, a bond price or some other variable is lognormal. • The mean of the probability distribution is the forward value of the variable. • The standard deviation is defined by a volatility as in BS. ContTimeFin - 9 Hull and White

  33. Black’s Model for Caps The value of a caplet corresponds to the time period between t1 and t2 is F - forward IR for t1,t2 X - the cap rate R - risk free yield to t2 A - principal  - forward rate volatility ContTimeFin - 9 Hull and White

  34. Swap Options and Bond Options • An IR swap can be regarded as an exchange of a fixed-rate bond for a floating rate bond. • A swaption is an option to exchange these two bonds. • Floating rate bond is worth par, so the swaption is an option to exchange a fixed rate bond for par. • An option on a swap where fixed is paid and floating is received is a put option on the bond with a strike price of par. ContTimeFin - 9 Hull and White

  35. Assumptions of Black’s Model • Assume bond price is lognormal • Assume bond yield is lognormal ContTimeFin - 9 Hull and White

  36. If Bond Prices Are Lognormal A European call on a bond is priced with B - price of the forward bond B* - price of a discount bond maturing at T B - volatility of the forward bond price ContTimeFin - 9 Hull and White

  37. Using Duration to Convert YieldVolatilities to Price Volatilities D - is the modified duration ContTimeFin - 9 Hull and White

  38. If Bond Yield Is Lognormal max[B-X,0] = max[B,0]  max[XD(YX-YF),0] YF - the forward bond yield YX - yield at which B=X ContTimeFin - 9 Hull and White

  39. Drawbacks of Black’s Model • Can be used when derivative depends on a single interest rate observed at a single time. • Provides no linkage between different interest rates and their volatilities. • Cannot be used for valuing long-dated American options and other complex derivatives. ContTimeFin - 9 Hull and White

  40. Yield Curve Based Models A no-arbitrage yield-curve-based model designed in such a way that it is automatically consistent with the current term structure and permits no arbitrage opportunities. ContTimeFin - 9 Hull and White

  41. Risk-Neutral Valuation The risk-neutral valuation for equity assumes that we get the right answer if we: • Assume that the expected return on the equity is the risk free rate. • Discount payoffs at the risk free rate. ContTimeFin - 9 Hull and White

  42. Risk-Neutral Valuation The risk-neutral valuation principle can be extended to value interest rate derivatives. We assume that the expected return on all bond prices is the risk-free rate and discount payoffs at the risk free rate. ContTimeFin - 9 Hull and White

  43. Risk-Neutral World or Real World In the two worlds variables have the same volatilities, but different drifts. RN - is a rough approximation, this is a world where there is no liquidity premium, so that forward rates equal expected future spot rates. All our models are valid in RN world only! ContTimeFin - 9 Hull and White

  44. Valuation A derivative security paying off fT at maturity is worth today Here * means in the risk-neutral world. ContTimeFin - 9 Hull and White

  45. Valuation of Bonds A discount bond maturing at time T is: Here * means in the risk-neutral world. ContTimeFin - 9 Hull and White

  46. Approaches • Model Bond Prices: Ho and Lee 1986. • Model Forward Rates: Heath, Jarrow and Morton 1987. • Model Short Rate: Black, Derman and Toy 1990, Hull and White 1990. ContTimeFin - 9 Hull and White

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