derivation of Black-Scholes formula in martingale form

Equation (1) can be used in practice to calculate $V_{t}$ for all times $t$ , because from the specification of a financial contract, the value of the portfolio at time $T$ , or in other words, its pay-off at time $T$ , will be a known function. Mathematically speaking, $V_{T}$ gives the terminal condition for the solution of a stochastic differential equation.

0.1 Assumptions

0.1.1 Asset price

The asset or stock price $X_{t}$ is to be modelled by the stochastic differential equation:

\displaystyle dX_{t}=\mu X_{t}\,dt+\sigma X_{t}\,dW_{t}\,,

(2)

where $\mu$ and $\sigma>0$ are constants.

The stochastic process $W_{t}$ is a standard Brownian motion adapted to the filtration $\{\mathcal{F}_{t}\}$ .

See the main article on the Black-Scholes formula (http://planetmath.org/BlackScholesFormula) for an explanation and justification of this modelling assumption.

0.1.2 Money-market account

The money-market account accumulates interest compounded continuously at a rate of $r$ . It satisfies the stochastic differential equation:

\displaystyle dM_{t}=rM_{t}\,dt\,.

(3)

This happens to take the same form as an ordinary differential equation, for the process $M_{t}$ has no randomness in it at all, under the assumption of a fixed interest rate $r$ .

The solution is to equation (3) with initial condition $M_{0}$ is $M_{t}=M_{0}\,e^{rt}$ .

0.1.3 Portfolio process

The price of the option is derived by following a replicating portfolio consisting of $\Delta_{t}$ units of the stock $X_{t}$ and $\Theta_{t}$ units of the money-market account. If $V_{t}$ denotes the value of this portfolio at time $t$ , then

\displaystyle V_{t}=\Delta_{t}\,X_{t}+\Theta_{t}\,M_{t}\,.

(4)

A certain “self-financing condition” on the portfolio requires that $V_{t}$ also satisfy the stochastic differential equation:

\displaystyle dV_{t}=\Delta_{t}\,dX_{t}+\Theta_{t}\,dM_{t}\,.

(5)

This condition essentially says that we cannot input extra amounts of money out of thin air into our portfolio; we must start with what we have.

Equation (5) is not a mathematically proven statement, but another modelling assumption, justified by an analogous equation governing trading in discretized time periods.

0.2 Derivation

We first manipulate the stochastic differential equation (4) for the portfolio process $V_{t}$ , to express it in terms of the Brownian motion $W_{t}$ .

$\displaystyle dV_{t}$	$\displaystyle=\Delta_{t}\,dX_{t}+\Theta_{t}\,rM_{t}\,dt$	from eq. (5) and (3)
	$\displaystyle=\Delta_{t}\,dX_{t}+r\bigl{(}V_{t}-\Delta_{t}X_{t}\bigr{)}\,dt$	from eq. (4)
	$\displaystyle=\Delta_{t}\,\bigl{(}\mu X_{t}\,dt+\sigma X_{t}\,dW_{t}\bigr{)}$
	$\displaystyle\qquad+r\bigl{(}V_{t}-\Delta_{t}X_{t}\bigr{)}\,dt$	from eq. (2)
	$\displaystyle=rV_{t}\,dt+\Delta_{t}X_{t}\,\bigl{(}(\mu-r)\,dt+\sigma\,dW_{t}% \bigr{)}$	rearrangement

0.2.1 Change of probability measure

Define the Brownian motion with drift $\lambda$ :

\displaystyle\widetilde{W}_{t}=\lambda t+W_{t}\,,\quad\lambda=\frac{\mu-r}{% \sigma}\,;

(6)

so that $d\widetilde{W}_{t}=\lambda\,dt+dW_{t}$ , and

\displaystyle dV_{t}=rV_{t}\,dt+\sigma\,\Delta_{t}X_{t}\,d\widetilde{W}_{t}\,.

(7)

The introduction of the process $\widetilde{W}_{t}$ is not merely for notational convenience but is mathematically meaningful. If the probability space we are working in is $(\Omega,\mathcal{F}_{T},\mathbb{P})$ , and $W_{t}$ , for $0\leq t\leq T$ , is a standard Wiener process on $(\Omega,\mathcal{F}_{T},\mathbb{P})$ , then $\widetilde{W}_{t}$ will not be a standard Wiener process on $(\Omega,\mathcal{F}_{T},\mathbb{P})$ , but it will be a standard Wiener process under $(\Omega,\mathcal{F}_{T},\mathbb{Q})$ with a different probability measure $\mathbb{Q}$ .

The probability measure $\mathbb{Q}$ is obtained by Girsanov’s theorem. The exact form for $\mathbb{Q}$ can be calculated, but it will not be needed in this derivation.

In finance, $\mathbb{Q}$ is known as the risk-neutral measure, and the quantity $\lambda$ is the market price of risk.

0.2.2 Discounted portfolio process is a martingale

From equation (7), we see that the value of the portfolio grows at the risk-free interest rate of $r$ , apart from the randomness associated due to the stochastic differential $d\widetilde{W}_{t}$ .

It is thus reasonable to expect that, if we normalize the portfolio value amount by the amount that cash grows due to accumulation of risk-free interest, the resulting process, $V_{t}/M_{t}$ , should have a zero growth rate. That this is indeed the case can be verified by a computation with Itô’s formula — more specifically, the for Itô integrals:

$\displaystyle d\left(\frac{V_{t}}{M_{t}}\right)$	$\displaystyle=d\left(V_{t}\cdot\frac{1}{M_{t}}\right)$
	$\displaystyle=\bigl{(}dV_{t}\bigr{)}\,\frac{1}{M_{t}}+V_{t}\,d\left(\frac{1}{M% _{t}}\right)$
	$\displaystyle=r\frac{V_{t}}{M_{t}}\,dt+\sigma\Delta_{t}\frac{X_{t}}{M_{t}}\,d% \widetilde{W}_{t}+V_{t}\,d\left(\frac{1}{M_{t}}\right)$	from eq. (7)
	$\displaystyle=r\frac{V_{t}}{M_{t}}\,dt+\sigma\Delta_{t}\frac{X_{t}}{M_{t}}\,d% \widetilde{W}_{t}+\frac{V_{t}}{M_{t}}\,dt$	from $\frac{1}{M_{t}}=\frac{e^{-rt}}{M_{0}}$ .

Thus,

\displaystyle d\left(\frac{V_{t}}{M_{t}}\right)=\sigma\Delta_{t}\,\frac{X_{t}}% {M_{t}}\,d\widetilde{W}_{t}\,.

Or, in integral form:

\displaystyle\frac{V_{t_{1}}}{M_{t_{1}}}=\frac{V_{t_{0}}}{M_{t_{0}}}+\int_{t_{% 0}}^{t_{1}}\,\sigma\Delta_{t}\,\frac{X_{t}}{M_{t}}\,d\widetilde{W}_{t}\,,\quad 0% \leq t_{0}\leq t_{1}\leq T\,.

(8)

Assuming $\Delta_{t}$ is a $\mathcal{F}_{t}$ -adapted process — where $\{\mathcal{F}_{t}\}$ is the filtration generated by the Brownian motion $W_{t}$ (or equivalently $\widetilde{W}_{t}$ ) — the Itô integral in equation (8) is a martingle under the probability space $(\Omega,\mathcal{F}_{T},\mathbb{Q})$ .

0.2.3 Portfolio process as a conditional expectation

Then by the definition of a martingale, we have

\frac{V_{t_{0}}}{M_{t_{0}}}=\mathbb{E}^{\mathbb{Q}}\left[\frac{V_{t_{1}}}{M_{t% _{1}}}\mid\mathcal{F}_{t_{0}}\right]\,,\quad 0\leq t_{0}\leq t_{1}\leq T\,,

where $\mathbb{E}^{\mathbb{Q}}[\cdot\mid\mathcal{F}_{t}]$ denotes the conditional expectation, of a random variable on the measurable space $(\Omega,\mathcal{F}_{T})$ , under the probability measure $\mathbb{Q}$ .

In particular, setting $t_{0}=t\leq T$ and $t_{1}=T$ , and rearranging the factors of $M_{t}=e^{rt}$ , we obtain the desired result, equation (1).

0.3 Existence of solutions

So far, we have derived the form of the solution for the portfolio value process $V_{t}$ , assuming that it exists. Actually, if we were to take only equations (4) and (5) as the problem to solve mathematically, without any reference to the financial motivations, it is possible to work backwards and deduce the existence of the solution.

0.3.1 Proposed construction

Let $\mathbb{Q}$ be the risk-neutral probability measure, and let $U$ be any given $\mathbf{L}^{1}(\Omega,\mathcal{F}_{T},\mathbb{Q})$ random variable, representing the terminal condition. Define the family of random variables dependent on time,

\displaystyle V_{t}=e^{-r(T-t)}\,\mathbb{E}^{\mathbb{Q}}[U\mid\mathcal{F}_{t}]% \,,\quad 0\leq t\leq T\,.

(9)

It is easy to verify that, for any $U$ , the process $V_{t}\,e^{-rt}$ is a martingale with respect to $\mathcal{F}_{t}$ , the filtration generated by the Wiener process $\widetilde{W}_{t}$ under the probability measure $\mathbb{Q}$ .

0.3.2 Verification

We now invoke the martingale representation theorem for Itô processes: for any martingale $Z_{t}$ , with respect to $\mathcal{F}_{t}$ under the probability measure $\mathbb{Q}$ , there exists a $\mathcal{F}_{t}$ -adapted process $G_{t}$ such that $Z_{t}$ has the representation:

\displaystyle Z_{t_{1}}-Z_{t_{0}}=\int_{t_{0}}^{t_{1}}G_{t}\,d\widetilde{W}_{t% }\,.

Letting $Z_{t}=V_{t}\,e^{-rt}$ and comparing with equations (8) and (4), we are motivated to define the $\mathcal{F}_{t}$ -adapted processes:

\Delta_{t}=\frac{G_{t}\,e^{rt}}{\sigma X_{t}}\,,\quad\Theta_{t}=\frac{V_{t}-% \Delta_{t}X_{t}}{M_{t}}=\frac{Z_{t}-G_{t}/\sigma}{M_{0}}\,.

Then the process $V_{t}$ constructed by equation (9) trivially satisfies equation (4). And it is a simple matter to check that equation (5) holds as well:

$\displaystyle dV_{t}=d\bigl{(}Z_{t}\,e^{rt}\bigr{)}$	$\displaystyle=e^{rt}\,dZ_{t}+re^{rt}\,Z_{t}\,dt$	Itô’s product rule
	$\displaystyle=e^{rt}\,G_{t}\,d\widetilde{W}_{t}+rV_{t}\,dt$
	$\displaystyle=\sigma\Delta_{t}X_{t}\,d\widetilde{W}_{t}+rV_{t}\,dt$
	$\displaystyle=\Delta_{t}\,X_{t}(r\,dt+\sigma d\widetilde{W}_{t})$	add and subtract
	$\displaystyle\qquad+r\bigl{(}V_{t}-\Delta_{t}X_{t}\bigr{)}$	the $d t$ term
	$\displaystyle=\Delta_{t}\,dX_{t}+r\,\Theta_{t}\,M_{t}\,dt\,,$

where in the last equality we have used the SDE for $X_{t}$ in terms of $d\widetilde{W}_{t}$ in place of $dW_{t}$ :

\displaystyle dX_{t}=rX_{t}\,dt+\sigma X_{t}\,d\widetilde{W}_{t}\,,

obtained by substituting in equation (2), the differential of equation (6).

References

1 Bernt Øksendal. Stochastic Differential Equations, An Introduction with Applications, 5th edition. Springer, 1998.
2 Steven E. Shreve. Stochastic Calculus for Finance II: Continuous-Time Models. Springer, 2004.

Title	derivation of Black-Scholes formula in martingale form
Canonical name	DerivationOfBlackScholesFormulaInMartingaleForm
Date of creation	2013-03-22 17:20:37
Last modified on	2013-03-22 17:20:37
Owner	stevecheng (10074)
Last modified by	stevecheng (10074)
Numerical id	5
Author	stevecheng (10074)
Entry type	Derivation
Classification	msc 60H10
Classification	msc 91B28
Related topic	BlackScholesFormula