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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.

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.

Tag: finance

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