Dynamic Hedging and Equilibrium PDEs

1. Dynamic Hedging and Equilibrium PDEs Note: There are several differences between the notation and equation numbers used here and those found in the text)

2. 1. We want to show that the Black-Scholes option pricing model can be derived from a hedge portfolio and no arbitrage. Assume a stock that pays no dividends with price S(t). Its price follows the diffusion process If an amount B(t) is invested in the risk free asset, its value follows the process Now consider a call option that matures at date T

3. What process does c(S(T),T) follow prior to maturity? By applying Itô’s lemma, we find that the call option c(S(T),T) exhibits the same risk as the stock, that risk is dz.

4. The Hedge Portfolio Now form a portfolio by selling one call option and hedging that call liability by purchasing theunderlying stock and borrowing at the risk-free rate. • This portfolio has a zero net investment • It is self-financing (any surplus or deficit of funds is made up by either borrowing or lending at the risk-free rate) If the number of shares purchased is w(t), the zero net investment restriction implies that the amount invested in the risk-free asset must satisfy, B(t) = c(t) – w(t)*S(t).

5. The value of the hedge portfolio, H(t) Substituting for dS from (9.1) and for dc from (9.4) into (9.5), we get Assume the investor chooses to hold w(t) = ∂c/∂S units of stock. Call ∂c/∂S the hedge ratio.

6. Dynamic trading leads to complete markets and the pricing of contingent claims Since c(S,t) is a nonlinear function of S and t, w(t) will vary as S and t change, the portfolio will have to be continuously rebalanced to maintain w(t) = ∂c/∂S. Now substitute this position into (9.6) to get a portfolio that has a riskless rate of return - - the portfolio is riskless since both μ (drift) and dz (volatility) drop out of the equation.

7. No Arbitrage Implies the Black-Scholes PDE To avoid arbitrage, the instantaneous rate of return on the hedge portfolio must equal r (the risk-free rate of return). In addition, a zero net investment requires that H(t=0) = 0.

8. The call option value must satisfy the PDE subject to the boundary condition in (9.10) The solution to (9.9) and (9.10) is the Black-Scholes call option pricing equation where N(·) is the standard normal distribution function ~ N(0,1) and

9. By put-call parity the European put option price is Taking partial derivatives of (9.11) and (9.13) with respect to S(t), we can derive the hedge ratios. Thus, (0 < ∂c/∂S < 1) and (-1 < ∂p/∂S < 0).

10. 2. We want to show that an equilibrium term structure model can be used to price bonds. A “one-factor” bond pricing model - - • assume that one underlying (state) variable affects the prices of all bonds. • assume a continuous time stochastic process, continuous trading, and no arbitrage • derive the equilibrium relationship between bonds with different maturities

11. Vasicek model Assume that the single factor is the instantaneous yield on a short maturity bond, r(t). Let P(t, τ) be the price of a pure discount (zero-coupon) bond that pays $1 at τ periods in the future. So, 0< P(t, τ) ≤ 1.

12. the instantaneous yield follows a process

13. Ornstein-Uhlenbeck process

14. Bond price process If r(t) is the current yield, we can write the bond’s price as P(r(t),τ), and apply Itô’s lemma. Recall the bond yield process is: The resulting bond price process exhibits GBM:

15. The Hedge Portfolio (dzr drops out)

16. Portfolio instantaneous rate of return is riskless Using (9.18) and (9.19) and the approach we used for stock, we can write the hedge portfolio’s instantaneous rate of return as The portfolio return is riskless, so the absence of arbitrage implies that the expected rate of return must equal the instantaneous riskless rate, r(t).

17. Rewriting (9.20) with E[μp( )] = r(t) Equating terms on the first and second lines in (9.21) and rearranging terms, the implication is that in equilibrium the market interest rate risk premium can be written as

18. Bond prices and the interest rate risk premium To derive equilibrium prices of bonds we need to specify the form of the risk premium. */ Initially, assume that the market price of bond risk is a constant over time and equal to q, so that for any bond with maturity (τ) the return can be derived from as */ Cox, Ingersoll and Ross (“A Theory of Term Structure,” Econometrica 1985) show that the bond risk premium can be derived from individual preferences and technology variables.

19. or

20. Solving (9.27) subject to the boundary condition that at τ = 0 the bond price equals $1,we get where

21. Using (9.28) it is possible to derive values for q from the implied bond yield curve: define R( ) as the continuously compounded YTM with maturity =