derivation of Black-Scholes formula in martingale form
- 0.1 Assumptions
- 0.2 Derivation
- 0.3 Existence of solutions
The portfolio process representing a stock option will be shown to satisfy:
Equation (1) can be used in practice to calculate for all times , because from the specification of a financial contract, the value of the portfolio at time , or in other words, its pay-off at time , will be a known function. Mathematically speaking, gives the terminal condition for the solution of a stochastic differential equation.
0.1.1 Asset price
The asset or stock price is to be modelled by the stochastic differential equation:
where and are constants.
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
0.1.3 Portfolio process
The price of the option is derived by following a replicating portfolio consisting of units of the stock and units of the money-market account. If denotes the value of this portfolio at time , then
A certain “self-financing condition” on the portfolio requires that also satisfy the stochastic differential equation:
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.
We first manipulate the stochastic differential equation (4) for the portfolio process , to express it in terms of the Brownian motion .
|from eq. (5) and (3)|
|from eq. (4)|
|from eq. (2)|
0.2.1 Change of probability measure
Define the Brownian motion with drift :
so that , and
The introduction of the process is not merely for notational convenience but is mathematically meaningful. If the probability space we are working in is , and , for , is a standard Wiener process on , then will not be a standard Wiener process on , but it will be a standard Wiener process under with a different probability measure .
The probability measure is obtained by Girsanov’s theorem. The exact form for can be calculated, but it will not be needed in this derivation.
In finance, is known as the risk-neutral measure, and the quantity 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 , apart from the randomness associated due to the stochastic differential .
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, , 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:
|from eq. (7)|
0.2.3 Portfolio process as a conditional expectation
Then by the definition of a martingale, we have
In particular, setting and , and rearranging the factors of , 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 , 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 be the risk-neutral probability measure, and let be any given random variable, representing the terminal condition. Define the family of random variables dependent on time,
It is easy to verify that, for any , the process is a martingale with respect to , the filtration generated by the Wiener process under the probability measure .
We now invoke the martingale representation theorem for Itô processes: for any martingale , with respect to under the probability measure , there exists a -adapted process such that has the representation:
|Itô’s product rule|
|add and subtract|
where in the last equality we have used the SDE for in terms of in place of :
- 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|
|Date of creation||2013-03-22 17:20:37|
|Last modified on||2013-03-22 17:20:37|
|Last modified by||stevecheng (10074)|