Dynamic Hedging and Equilibrium PDEs. Note: There are several differences between the notation and equation numbers used here and those found in the text). 1. We want to show that the Black-Scholes option pricing model can be derived from a hedge portfolio and no arbitrage.
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Note: There are several differences between the notation and equation numbers used here and those found in the text)
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
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.
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.
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).
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.
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.
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.
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
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).
A “one-factor” bond pricing 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.
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:
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).
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
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
*/ 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.
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 =