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.
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
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.
The Hedge Portfolio
Now form a portfolio by selling one call option and hedging that call
liability by purchasing the underlying stock and borrowing at the riskfree 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).
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.
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.
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.
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
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).
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
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.
the instantaneous yield follows a process
Ornstein-Uhlenbeck process
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:
The Hedge Portfolio (dzr drops out)
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).
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
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.
or
Solving (9.27) subject to the boundary condition
that at τ = 0 the bond price equals $1, we get
where
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 =