derivation of Black-Scholes formula in martingale form

This entry derives the Black-Scholes formula in martingaleMathworldPlanetmath form.

The portfolio process Vt representing a stock option will be shown to satisfy:

Vt=e-r(T-t)𝔼[VTt]. (1)

(The quantities appearing here are defined precisely, in the section on “AssumptionsPlanetmathPlanetmath” below.)

Equation (1) can be used in practice to calculate Vt 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, VT 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 Xt is to be modelled by the stochastic differential equation:

dXt=μXtdt+σXtdWt, (2)

where μ and σ>0 are constants.

The stochastic processMathworldPlanetmath Wt is a standard Brownian motionMathworldPlanetmath adapted to the filtrationPlanetmathPlanetmath {t}.

See the main article on the Black-Scholes formula ( 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:

dMt=rMtdt. (3)

This happens to take the same form as an ordinary differential equationMathworldPlanetmath, for the process Mt has no randomness in it at all, under the assumption of a fixed interest rate r.

The solution is to equation (3) with initial conditionMathworldPlanetmath M0 is Mt=M0ert.

0.1.3 Portfolio process

The price of the option is derived by following a replicating portfolio consisting of Δt units of the stock Xt and Θt units of the money-market account. If Vt denotes the value of this portfolio at time t, then

Vt=ΔtXt+ΘtMt. (4)

A certain “self-financing condition” on the portfolio requires that Vt also satisfy the stochastic differential equation:

dVt=ΔtdXt+ΘtdMt. (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 Vt, to express it in terms of the Brownian motion Wt.

dVt =ΔtdXt+ΘtrMtdt from eq. (5) and (3)
=ΔtdXt+r(Vt-ΔtXt)dt from eq. (4)
  +r(Vt-ΔtXt)dt from eq. (2)
=rVtdt+ΔtXt((μ-r)dt+σdWt) rearrangement

0.2.1 Change of probability measure

Define the Brownian motion with drift λ:

W~t=λt+Wt,λ=μ-rσ; (6)

so that dW~t=λdt+dWt, and

dVt=rVtdt+σΔtXtdW~t. (7)

The introduction of the process W~t is not merely for notational convenience but is mathematically meaningful. If the probability spaceMathworldPlanetmath we are working in is (Ω,T,), and Wt, for 0tT, is a standard Wiener process on (Ω,T,), then W~t will not be a standard Wiener process on (Ω,T,), but it will be a standard Wiener process under (Ω,T,) with a different probability measure .

The probability measure is obtained by Girsanov’s theoremMathworldPlanetmath. 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 λ 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 dW~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, Vt/Mt, should have a zero growth rate. That this is indeed the case can be verified by a computation with Itô’s formulaMathworldPlanetmath — more specifically, the for Itô integrals:

d(VtMt) =d(Vt1Mt)
=rVtMtdt+σΔtXtMtdW~t+Vtd(1Mt) from eq. (7)
=rVtMtdt+σΔtXtMtdW~t+VtMtdt from 1Mt=e-rtM0.



Or, in integral form:

Vt1Mt1=Vt0Mt0+t0t1σΔtXtMt𝑑W~t,0t0t1T. (8)

Assuming Δt is a t-adapted process — where {t} is the filtration generated by the Brownian motion Wt (or equivalently W~t) — the Itô integral in equation (8) is a martingle under the probability space (Ω,T,).

0.2.3 Portfolio process as a conditional expectation

Then by the definition of a martingale, we have


where 𝔼[t] denotes the conditional expectation, of a random variableMathworldPlanetmath on the measurable spaceMathworldPlanetmathPlanetmath (Ω,T), under the probability measure .

In particular, setting t0=tT and t1=T, and rearranging the factors of Mt=ert, 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 Vt , 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 U be any given 𝐋1(Ω,T,) random variable, representing the terminal condition. Define the family of random variables dependent on time,

Vt=e-r(T-t)𝔼[Ut],0tT. (9)

It is easy to verify that, for any U, the process Vte-rt is a martingale with respect to t, the filtration generated by the Wiener process W~t under the probability measure .

0.3.2 Verification

We now invoke the martingale representation theorem for Itô processes: for any martingale Zt, with respect to t under the probability measure , there exists a t-adapted process Gt such that Zt has the representation:


Letting Zt=Vte-rt and comparing with equations (8) and (4), we are motivated to define the t-adapted processes:


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

dVt=d(Ztert) =ertdZt+rertZtdt Itô’s product rule
=ΔtXt(rdt+σdW~t) add and subtract
  +r(Vt-ΔtXt) the dt term

where in the last equality we have used the SDE for Xt in terms of dW~t in place of dWt:


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


  • 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