Financial Engineering Risk Neutral Pricing Zvi Wiener mswienermscc

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

Equivalent Martingale Measure and Risk-Neutral Pricing Rangarajan K. Sundaram New York University Journal of Derivatives Zvi Wiener Cont. Time. Fin - 9 2

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

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. Zvi Wiener Cont. Time. Fin - 9 4

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. Zvi Wiener Cont. Time. Fin - 9 5

The Binomial Model There are two assets: bond and stock: r q 1 q S 1 -q r d. S q + (1 -q)=1, 0 < q < 1. d<r<u Zvi Wiener u. S Cont. Time. Fin - 9 Arbitrage? 6

The Binomial Model assets: bond, stock, and an option q 1 r q S 1 -q r u. S q Xu X 1 -q d. S Xd Note that here r is what we usually denote by 1+r. Zvi Wiener Cont. Time. Fin - 9 7

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 the bond Zvi Wiener units of a stock Cont. Time. Fin - 9 8

Pricing by Arbitrage Value of this portfolio at maturity is Zvi Wiener Cont. Time. Fin - 9 9

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). Zvi Wiener Cont. Time. Fin - 9 10

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. Zvi Wiener Cont. Time. Fin - 9 11

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). Zvi Wiener Cont. Time. Fin - 9 12

Martingale A stochastic process is martingale if its expected change is always zero. A more precise definition is Zvi Wiener Cont. Time. Fin - 9 13

Discounting is necessary, since otherwise the bond is not risky, but grows (at the risk free rate). Zvi Wiener Cont. Time. Fin - 9 14

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. Zvi Wiener Cont. Time. Fin - 9 15

Binomial Case q 1 r q S 1 -q r u. S 1 -q d. S Since d < r < u, the probability p is 0 < p <1 Zvi Wiener Cont. Time. Fin - 9 16

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 riskneutral price. Zvi Wiener Cont. Time. Fin - 9 17

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

State Prices and Risk-Neutral Probabilities Zvi Wiener Cont. Time. Fin - 9 19

Example Binomial model with two risky assets and a bond. Let S 1 and S 2 denote the initial prices of the risky assets. Let the possible prices be u 1 S 1 u 2 S 2 d 1 S 1 Zvi Wiener d 2 S 2 Cont. Time. Fin - 9 20

Example We must have Assume that this equation does not hold. Zvi Wiener Cont. Time. Fin - 9 21

Example Consider a portfolio: $a in bonds $b in S 1 $c in S 2 its cost now is a + b + c. At maturity it gives: If the equality does not hold there is an arbitrage. Zvi Wiener Cont. Time. Fin - 9 22

Completeness S qu qm qd r u. S m. S 1 r r d. S 3 variables, 2 equations, there are many solutions. Zvi Wiener Cont. Time. Fin - 9 23

Continuous Time Models Note that the first equation describes a bond: B(t)=B 0 ert. The second equation will correspond to a stock. Zvi Wiener Cont. Time. Fin - 9 24

The Discounted Price Process Denote by Q the discounted price process Zvi Wiener Cont. Time. Fin - 9 25

Girsanov’s Theorem If we wish to change the drift of the process to . F Define a new process Z using , , and . F Redefine Q using the new process Z. Zvi Wiener Cont. Time. Fin - 9 26

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 d. Z = dt + d. W Zvi Wiener Cont. Time. Fin - 9 27

By definition Zvi Wiener Step 2 Cont. Time. Fin - 9 28

Black-Merton-Scholes Model Try There is NO solution. Zvi Wiener Cont. Time. Fin - 9 29

Black-Merton-Scholes Model Try There is a solution! Zvi Wiener Cont. Time. Fin - 9 30

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

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

Black’s Model for Caps The value of a caplet corresponds to the time period between t 1 and t 2 is F - forward IR for t 1, t 2 X - the cap rate R - risk free yield to t 2 A - principal - forward rate volatility Hull. Zvi and. Wiener White Cont. Time. Fin - 9 33

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

Assumptions of Black’s Model F Assume bond price is lognormal F Assume bond yield is lognormal Hull. Zvi and. Wiener White Cont. Time. Fin - 9 35

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 Hull. Zvi and. Wiener White Cont. Time. Fin - 9 36

Using Duration to Convert Yield Volatilities to Price Volatilities D - is the modified duration Hull. Zvi and. Wiener White Cont. Time. Fin - 9 37

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 Hull. Zvi and. Wiener White Cont. Time. Fin - 9 38

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

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. Hull. Zvi and. Wiener White Cont. Time. Fin - 9 40

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. F F Discount payoffs at the risk free rate. Hull. Zvi and. Wiener White Cont. Time. Fin - 9 41

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. Hull. Zvi and. Wiener White Cont. Time. Fin - 9 42

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! Hull. Zvi and. Wiener White Cont. Time. Fin - 9 43

Valuation A derivative security paying off f. T at maturity is worth today Here * means in the risk-neutral world. Hull. Zvi and. Wiener White Cont. Time. Fin - 9 44

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

Approaches F Model Bond Prices: Ho and Lee 1986. Model Forward Rates: Heath, Jarrow and Morton 1987. F Model Short Rate: Black, Derman and Toy 1990, Hull and White 1990. F Hull. Zvi and. Wiener White Cont. Time. Fin - 9 46