Black-Scholes-Merton Derivation (Part 3: Equating Portfolio Value & Option Value Evolutions)

This is a continuation of Black-Scholes-Merton Derivation (Part 2: Option Value Evolution).

A hedging portfolio starts with initial capital X(0) and invests in the stock and money market account so that the portfolio value X(t) at each time t\in [0,T] is equal to c(t,S(t)). This is equivalent to the condition that e^{-rt}X(t)=e^{-rt}c(t,S(t)) for all t.

One way to impose this equality is to ensure that

\displaystyle d(e^{-rt}X(t))=d(e^{-rt}c(t,S(t))) for all t\in [0,T) and X(0)=c(0,S(0)).

Comparing the expressions d(e^{-rt}X(t)) and d(e^{-rt}c(t,S(t))) calculated previously in Part 1 and Part 2 respectively, we see that we require the following to hold:

\displaystyle\begin{aligned}&\Delta(t)(\alpha-r)S(t)\, dt+\Delta(t)\sigma S(t)\, dW(t)\\&=\left[-rc(t,S(t))+c_t(t,S(t))+\alpha S(t)c_x(t,S(t))+\frac{1}{2}\sigma^2 S^2(t)c_{xx}(t,S(t))\right]\, dt\\&\quad+\sigma S(t)c_x(t,S(t))\, dW(t)\end{aligned}

 

Comparing the dW(t) terms, we get \Delta(t)=c_x(t,S(t)). This is known as the delta-hedging rule.

Next, comparing the dt terms, we have

\displaystyle\begin{aligned}&(\alpha-r)S(t)c_x(t,S(t))\\&=-rc(t,S(t))+c_t(t,S(t))+\alpha S(t)c_x(t,S(t))+\frac{1}{2}\sigma^2 S^2(t)c_{xx}(t,S(t))\end{aligned}

 

Simplifying and rearranging, we get \displaystyle rc(t,S(t))=c_t(t,S(t))+rS(t)c_x(t,S(t))+\frac{1}{2}\sigma^2S^2(t)c_{xx}(t,S(t))

Let x=S(t), then we have derived the Black-Scholes-Merton partial differential equation:

\boxed{\displaystyle c_t(t,x)+rxc_x(t,x)+\frac{1}{2}\sigma^2x^2 c_{xx}(t,x)=rc(t,x)} for all t\in [0,T), x\geq 0.

 

The terminal condition to be satisfied is c(T,x)=(x-K)^+, where K is the strike price of the call option.

Reference

Shreve, Steven E. Stochastic calculus for finance II: Continuous-time models. Vol. 11. New York: Springer, 2004.

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