derivation of Black-Scholes formula in martingale form

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

The portfolio process $V_t$ representing a stock optionwill be shown to satisfy:

$V_t=e^{-r(T-t)}{𝔼}^{\mathbb{Q}}[V_T\mid {ℱ}_t].$(1)

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

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 conditionfor 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 stochasticdifferential equation:

$d{X}_t=\mu X_{t}dt+\sigma X_{t}d{W}_t,$(2)

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

The stochastic process $W_t$ is a standard Brownian motionadapted to the filtration $\{{ℱ}_t\}$.

See the main articleon 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 continuouslyat a rate of $r$.It satisfies the stochastic differential equation:

$d{M}_t=r{M}_{t}dt.$(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 portfolioconsisting of ${\mathrm{\Delta }}_t$ units of the stock $X_t$and ${\mathrm{\Theta }}_t$ units of the money-marketaccount. If $V_t$ denotes the value of this portfolio at time $t$,then

$V_t={\mathrm{\Delta }}_t{X}_t+{\mathrm{\Theta }}_t{M}_t.$(4)

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

$d{V}_t={\mathrm{\Delta }}_{t}d{X}_t+{\mathrm{\Theta }}_{t}d{M}_t.$(5)

This condition essentially says that we cannot input extra amountsof 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 equationgoverning trading in discretized time periods.

0.2 Derivation

We first manipulate thestochastic differential equation (4)for the portfolio process $V_t$,to express it in terms of the Brownian motion $W_t$.

$d{V}_t$$={\mathrm{\Delta }}_{t}d{X}_t+{\mathrm{\Theta }}_{t}r{M}_{t}dt$from eq. (5) and (3)

$={\mathrm{\Delta }}_{t}d{X}_t+r\left(V_t-{\mathrm{\Delta }}_t{X}_t\right)dt$from eq. (4)

$={\mathrm{\Delta }}_t\left(\mu X_{t}dt+\sigma X_{t}d{W}_t\right)$

$\mathrm{\hspace{1em}\hspace{1em}}+r\left(V_t-{\mathrm{\Delta }}_t{X}_t\right)dt$from eq. (2)

$=r{V}_{t}dt+{\mathrm{\Delta }}_t{X}_t\left((\mu -r)dt+\sigma d{W}_t\right)$rearrangement

0.2.1 Change of probability measure

Define the Brownian motion with drift $\lambda$:

${\tilde{W}}_t=\lambda t+W_t,\lambda ={\displaystyle \frac{\mu -r}{\sigma }};$(6)

so that $d{\tilde{W}}_t=\lambda dt+d{W}_t$, and

$d{V}_t=r{V}_{t}dt+\sigma {\mathrm{\Delta }}_t{X}_{t}d{\tilde{W}}_t.$(7)

The introduction of the process ${\tilde{W}}_t$is not merely for notational convenience but is mathematicallymeaningful. If the probability space we are working inis $(\mathrm{\Omega },{ℱ}_T,\mathbb{P})$,and $W_t$, for $0\le t\le T$, is astandard Wiener process on $(\mathrm{\Omega },{ℱ}_T,\mathbb{P})$,then ${\tilde{W}}_t$ will not be a standard Wiener processon $(\mathrm{\Omega },{ℱ}_T,\mathbb{P})$, but it will be a standard Wiener processunder $(\mathrm{\Omega },{ℱ}_T,\mathbb{Q})$ 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-freeinterest rate of $r$,apart from the randomness associated due to the stochasticdifferential $d{\tilde{W}}_t$.

It is thus reasonable to expect that,if we normalize the portfolio value amountby the amount that cash grows due to accumulation of risk-freeinterest,the resulting process, $V_t/M_t$, should have a zero growth rate.That this is indeed the case can be verifiedby a computation with Itô’s formula— more specifically, the for Itô integrals:

$d\left({\displaystyle \frac{V_t}{M_t}}\right)$$=d\left(V_t\cdot {\displaystyle \frac{1}{M_t}}\right)$

$=\left(d{V}_t\right){\displaystyle \frac{1}{M_t}}+V_{t}d\left({\displaystyle \frac{1}{M_t}}\right)$

$=r{\displaystyle \frac{V_t}{M_t}}dt+\sigma {\mathrm{\Delta }}_t{\displaystyle \frac{X_t}{M_t}}d{\tilde{W}}_t+V_{t}d\left({\displaystyle \frac{1}{M_t}}\right)$from eq. (7)

$=r{\displaystyle \frac{V_t}{M_t}}dt+\sigma {\mathrm{\Delta }}_t{\displaystyle \frac{X_t}{M_t}}d{\tilde{W}}_t+{\displaystyle \frac{V_t}{M_t}}dt$from $\frac{1}{M_t}=\frac{e^{-rt}}{M_0}$.

Thus,

$d\left({\displaystyle \frac{V_t}{M_t}}\right)=\sigma {\mathrm{\Delta }}_t{\displaystyle \frac{X_t}{M_t}}d{\tilde{W}}_t.$

Or, in integral form:

${\displaystyle \frac{V_{t_1}}{M_{t_1}}}={\displaystyle \frac{V_{t_0}}{M_{t_0}}}+{\displaystyle {\int }_{t_0}^{t_1}}\sigma {\mathrm{\Delta }}_t{\displaystyle \frac{X_t}{M_t}}𝑑{\tilde{W}}_t,0\le t_0\le t_1\le T.$(8)

Assuming ${\mathrm{\Delta }}_t$ is a ${ℱ}_t$-adapted process— where $\{{ℱ}_t\}$ is the filtration generated by the Brownianmotion $W_t$ (or equivalently ${\tilde{W}}_t$) —the Itô integral in equation (8)is a martingle under the probability space $(\mathrm{\Omega },{ℱ}_T,\mathbb{Q})$.

0.2.3 Portfolio process as a conditional expectation

Then by the definition of a martingale,we have

$$\frac{V_{t_0}}{M_{t_0}}={𝔼}^{\mathbb{Q}}[\frac{V_{t_1}}{M_{t_1}}\mid {ℱ}_{t_0}],0\le t_0\le t_1\le T,$$

where ${𝔼}^{\mathbb{Q}}[\cdot \mid {ℱ}_t]$ denotes the conditionalexpectation, of a random variableon the measurable space $(\mathrm{\Omega },{ℱ}_T)$,underthe probability measure $\mathbb{Q}$.

In particular, setting $t_0=t\le T$ 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 onlyequations (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 existenceof the solution.

0.3.1 Proposed construction

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

$V_t=e^{-r(T-t)}{𝔼}^{\mathbb{Q}}[U\mid {ℱ}_t],0\le t\le T.$(9)

It is easy to verify that, for any $U$,the process $V_t{e}^{-rt}$ is a martingalewith respect to ${ℱ}_t$, the filtration generatedby the Wiener process ${\tilde{W}}_t$ under the probabilitymeasure $\mathbb{Q}$.

0.3.2 Verification

We now invoke the martingale representation theorem for Itô processes:for any martingale $Z_t$, with respect to ${ℱ}_t$ underthe probability measure $\mathbb{Q}$,there exists a ${ℱ}_t$-adapted process $G_t$such that $Z_t$ has the representation:

$Z_{t_1}-Z_{t_0}={\displaystyle {\int }_{t_0}^{t_1}}{G}_t𝑑{\tilde{W}}_t.$

Letting $Z_t=V_t{e}^{-rt}$ andcomparing with equations (8)and (4),we are motivated to definethe ${ℱ}_t$-adapted processes:

$${\mathrm{\Delta }}_t=\frac{G_t{e}^{rt}}{\sigma X_t},{\mathrm{\Theta }}_t=\frac{V_t-{\mathrm{\Delta }}_t{X}_t}{M_t}=\frac{Z_t-G_t/\sigma }{M_0}.$$

Then the process $V_t$ constructedby equation (9)trivially satisfies equation (4).And it is a simple matter to check thatequation (5) holds as well:

$d{V}_t=d\left(Z_t{e}^{rt}\right)$$=e^{rt}d{Z}_t+r{e}^{rt}{Z}_{t}dt$Itô’s product rule

$=e^{rt}{G}_{t}d{\tilde{W}}_t+r{V}_{t}dt$

$=\sigma {\mathrm{\Delta }}_t{X}_{t}d{\tilde{W}}_t+r{V}_{t}dt$

$={\mathrm{\Delta }}_t{X}_t(rdt+\sigma d{\tilde{W}}_t)$add and subtract

$\mathrm{\hspace{1em}\hspace{1em}}+r\left(V_t-{\mathrm{\Delta }}_t{X}_t\right)$the $dt$ term

$={\mathrm{\Delta }}_{t}d{X}_t+r{\mathrm{\Theta }}_t{M}_{t}dt,$

where in the last equality we have used the SDE for $X_t$in terms of $d{\tilde{W}}_t$ in place of $d{W}_t$:

$d{X}_t=r{X}_{t}dt+\sigma X_{t}d{\tilde{W}}_t,$

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