Tag: finance

Black-Scholes-Merton Derivation (Part 1: Portfolio Value Evolution)

First, we define the following variables.

X(t): The value of the portfolio at time t. The portfolio invests in a money market account paying a constant interest rate r and in a stock modeled by geometric Brownian motion

dS(t)=\alpha S(t)\, dt+\sigma S(t)\, dW(t).

\Delta(t): At time t, the portfolio holds \Delta(t) shares of stock.

The remainder of the portfolio value X(t)-\Delta(t)S(t) is invested in the money market account.

The differential dX(t) is due to two factors, the capital gain \Delta(t)\, dS(t) on the stock position and the interest earnings r(X(t)-\Delta(t)S(t))\, dt on the cash position.

Mathematically, we have

\begin{aligned} dX(t)&=\Delta(t)\, dS(t)+r(X(t)-\Delta(t)\, S(t))\, dt\\&=\Delta(t)(\alpha S(t)\, dt+\sigma S(t)\, dW(t))+r(X(t)-\Delta (t)S(t))\, dt\\&=rX(t)\, dt+\Delta(t)(\alpha-r)S(t)\, dt+\Delta(t)\sigma S(t)\, dW(t).\end{aligned}

 

The interpretation of the three terms in the last line above is as follows:

rX(t)\, dt: reflects an average underlying rate of return r on the portfolio

\Delta(t)(\alpha-r)S(t)\, dt: reflects a risk premium \alpha-r for investing in the stock

\Delta(t)\sigma S(t)\, dW(t): a volatility term proportional to the size of the stock investment

Next, we consider the discounted stock price e^{-rt}S(t) and the discounted portfolio value e^{-rt}X(t). We let f(t,x)=e^{-rt}x, and apply the Ito-Doeblin formula (for an Ito process) to get the following.

Differential of the discounted stock price:

\begin{aligned}d(e^{-rt}S(t))&=df(t,S(t))\\&=f_t(t,S(t))\, dt+f_x(t,S(t))\, dS(t)+\frac{1}{2}f_{xx}(t,S(t))\, dS(t)\, dS(t)\\&=-re^{-rt}S(t)\, dt+e^{-rt}\, dS(t)+0\\&=-re^{-rt}S(t)\, dt+e^{-rt}[\alpha S(t)\, dt+\sigma S(t)\, dW(t)]\\&=(\alpha-r)e^{-rt}S(t)\, dt+\sigma e^{-rt}S(t)\, dW(t)\end{aligned}

 

Differential of the discounted portfolio value:

\begin{aligned}d(e^{-rt}X(t))&=df(t,X(t))\\&=f_t(t,X(t))\, dt+f_x(t,X(t))\, dX(t)+\frac{1}{2}f_{xx}(t,X(t))\, dX(t)\, dX(t)\\&=-re^{-rt}X(t)\, dt+e^{-rt}\, dX(t)+0\\&=-re^{-rt}X(t)\, dt+e^{-rt}[rX(t)\, dt+\Delta(t)(\alpha-r)S(t)\, dt+\Delta(t)\sigma S(t)\, dW(t)]\\&=\Delta(t)(\alpha-r)e^{-rt}S(t)\, dt+\Delta(t)\sigma e^{-rt}S(t)\, dW(t)\\&=\Delta(t)\, d(e^{-rt}S(t))\end{aligned}

 

The last line shows that change in the discounted portfolio value is solely due to change in the discounted stock price (note that the term reflecting underlying rate of return r has vanished in the second-last line).

Reference

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

Ito-Doeblin Formula

The Ito-Doeblin Formula is an important formula in stochastic calculus, analogous to the chain rule in ordinary calculus. It has many applications in finance, including the derivation of the Black-Scholes-Merton equation.

Let f(x) be a differentiable function and W(t) be a Brownian motion.

The Ito-Doeblin formula in differential form is:

\displaystyle\boxed{df(W(t))=f'(W(t))\, dW(t)+\frac{1}{2}f''(W(t))\, dt}

 

The main difference (compared to ordinary calculus) is that there is an extra term due to the fact that W has nonzero quadratic variation.

Integrating this expression, we get the Ito-Doeblin formula in integral form:

\displaystyle\boxed{f(W(t))-f(W(0))=\int_0^t f'(W(u))\, dW(u)+\frac{1}{2}\int_0^t f''(W(u))\, du}

 

There is also a slightly more generalized version that allows f to be a function of both t and x.

Ito-Doeblin Formula for Brownian Motion

Let f(t,x) be a function where the partial derivatives f_t(t,x), f_x(t,x), and f_{xx}(t,x) are continuous, and let W(t) be a Brownian motion. Then, for all T\geq 0,

\displaystyle\boxed{\begin{aligned}f(T,W(T))=&f(0,W(0))+\int_0^T f_t(t,W(t))\, dt\\&+\int_0^T f_x(t,W(t))\, dW(t)+\frac{1}{2}\int_0^T f_{xx}(t,W(t))\, dt\end{aligned}}

 

The Ito-Doeblin formula in differential form is:

\displaystyle\boxed{df(t,W(t))=f_t(t,W(t))\, dt+f_x(t,W(t))\, dW(t)+\frac{1}{2}f_{xx}(t,W(t))\, dt}

Reference

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