last updated: May 1, 2025

Stochastic Processes and Financial Mathematics
(part two)

15.3 The Black-Scholes equation

We now look at how a replicating portfolio can be found explicitly, for a given contingent claim Φ(ST). We will, from now on, assume that our model is free of arbitrage.

In the case of the binomial model we found ourselves solving pairs of linear equations for each time-step; now, in continuous time, we will instead find ourselves solving a partial differential equation. This is natural – our linear equations told us how things changed across a single time-step, and PDEs can describe the changing state of a system in continuous time.

Let Φ(ST) be a contingent claim in the Black-Scholes market (with parameters r,μ,σ), and suppose that F(t,s) is a (suitably differentiable) function such that F(t,St) denotes the value of the contingent claim Φ(ST) at time t[0,T]. Then, as we will show in Theorem 15.3.1, for all s>0 and t[0,T] we have

(15.10)Ft(t,s)+rsFs(t,s)+12s2σ22Fs2(t,s)rF(t,s)=0,(15.11)F(T,s)=Φ(s). This is known as the Black-Scholes equation. It dates from a now famous research article by Fischer Black and Myron Scholes1, published in 1973. The rigorous mathematical basis for the model was provided, also in 1973, by Robert C. Merton2 In 1997 Merton and Scholes received the Nobel prize in economics, in recognition of their contributions (Black died in 1995, and Nobel prizes are only awarded to the living).

1 Black, Fischer; Myron Scholes (1973). ”The Pricing of Options and Corporate Liabilities”. Journal of Political Economy. 81 (3): 637–654.

2 Merton, Robert C. (1973). ”Theory of Rational Option Pricing”. Bell Journal of Economics and Management Science. The RAND Corporation.

  • Theorem 15.3.1 Let Φ(ST) be a contingent claim such that EQ[Φ(ST)]<. Then a replicating portfolio for Φ(ST) is given by

    xt=1rCt(Ft(t,St)+12σ2St22Fs2(t,St))yt=Fs(t,St) where F(t,s) is the solution to (15.10),(15.11). The value of this portfolio at time t is equal

    (15.12)F(t,St)=er(Tt)EQ[Φ(ST)|Ft],

    and in particular its value at time 0 is

    (15.13)erTEQ[Φ(ST)].

The formulae given for xt,yt are a big improvement on Theorem 15.2.5, because a computer can simulate solutions to (15.10), as well as their partial derivatives, without (much) difficulty. This allows computation of the replicating portfolio in real-time. You are not expected to memorize these formulae for ht=(xt,yt). The other formula in Theorem 15.3.1 can be found on the formula sheet, in Appendix E.

Of course, (15.13) is known as the risk-neutral valuation formula, in direct analogy to the discreet time version we found in Proposition 5.2.6.

We will most of this section proving the claims in Theorem 15.3.1.

Proof: First let us deduce that (15.11). Equation (15.11) is known as the boundary condition, because it relates to the exercise time T. It holds because, by definition of F(t,St), at the exercise time T the value of Φ(ST) must be equal F(T,ST) i.e. Φ(ST)=F(T,ST). Since ST may take any positive value we simply replace ST with a general s>0.

Note that we don’t need to worry about s<0 because St is a geometric Brownian motion, which (from Section 13.2) is always positive.

We now work towards proving that (15.10) holds. From Theorem 15.2.5 we know that Φ(ST) can be replicated by a self-financing portfolio strategy ht=(xt,yt). Therefore, because we assume that our model is free of arbitrage, the value of this portfolio strategy at time t must be equal to F(t,St):

(15.14)F(t,St)=Vth=xtCt+ytSt

In particular, this implies that dF(t,St)=dVth. We plan to calculate both these stochastic differentials, written out in full, and then use Lemma 12.4.5 to equate the dt and dBt coefficients. It will turn out that this leads to precisely (15.10), and along the way we will discover formulae for xt and yt.

Since (ht) is self-financing we have

dVth=xtdCt+ytdSt.

Substituting in (15.1) and (15.2), this becomes

(15.15)dVth=(xtrCt+ytμSt)dt+ytσStdBt.

Next: recalling that dSt=μStdt+σStdBt, by Ito’s formula we have

(15.16)dF(t,St)=(Ft+μStFs+12σ2St22Fs2)dt+σStFsdBt

where we have suppressed the (t,St) arguments of the partial derivatives of F.

  • Remark 15.3.2 Here, we assume, without justifying ourselves, that F(t,s) is differentiable (once in t and twice in s). This is a minor issue that can be dealt with using appropriate results from analysis, but it is beyond the scope of our course.

Equating the dBt coefficients between (15.15) and (15.16) gives us that

yt=Fs.

With this in hand, equating the dt coefficients gives us

(15.17)Ft+μStFs+12σ2St22Fs2=xtrCt+FsμSt and the terms μStFs cancel giving

Ft+12σ2St22Fs2=xtrCt.

so as

xt=Ft+12σ2St22Fs2rCt.

Using (15.14) to substitute in for Ct in (15.17), we obtain

Ft+12σ2St22Fs2=xtr(FytStxt)=rFrStFs. This is equation (15.10), but with St in place of s. Equation (15.10) follows, because we can see from (13.9) that St takes values on the whole of the positive reals (0,). Note that we have also discovered formulae for xt,yt along the way, which must satisfy F(t,St)=Vth because they were derived to satisfy (15.14).

Next we use Lemma 14.1.3, which tells us that the solution to be Black-Scholes equation can be written as

F(t,s)=er(Tt)Et,sQ[Φ(ST)].

Note that here we take expectation in the risk neutral world Q, in which St follows dSt=rStdt+σStdBt (in the notation of Lemma 14.1.3 we have α(t,s)=rs and β(t,s)=σs). We deduce that the value at time t of the replicating portfolio is

F(t,St)=er(Tt)Et,StQ[Φ(ST)]

To finish the proof, we use that St is a Markov process. By the Markov property at time t, from Lemma 14.2.1, we have

Et,StQ[Φ(ST)]=EQ[Φ(ST)|Ft].

Setting t=0, and recalling that F0 is the trivial σ-field (containing no information) we have

EQ[Φ(ST)|F0]=EQ[Φ(ST)],

as required.   ∎

  • Corollary 15.3.3 The replicating portfolio given in Theorem 15.3.1 for Φ(ST) is unique.

Proof: The calculations in the proof of Theorem 15.3.1 showed that any replicating portfolio (consisting solely of stocks and cash) could be written in terms of the solution to the Black-Scholes equation, as in the statement of Theorem 15.3.1. It is known from the world of PDEs that the Black-Scholes equation has a unique solution, so the replicating portfolio is also unique.   ∎

  • Remark 15.3.4 We have now seen that the Feynman-Kac formula is important to mathematical finance; it is the key to establishing the risk-neutral valuation formula (15.13). This formula links arbitrage free pricing theory to Brownian motion, through the stochastic differential equation dSt=μStdt+σStdBt.

    The Feynman-Kac formula has much wider applications too, alongside other ways of connecting stochastic processes to PDEs. We have already mentioned heat diffusion and movements of particles within fluids, which is an example of a different type of connection known as ‘duality’ that we won’t discuss in detail here. We list two further examples here:

    • The Keynmann-Kac formula was originally developed to describe solutions to the Schrödinger equation, a PDE which is the equivalent in quantum mechanics of Newton’s second law (i.e. F=ma, the ‘law of motion’).

    • The method of duality is used, in a variety of SDE based models, by mathematicians trying to model evolution, to analyse the positions and/or proportions of genes within a population.

    There are many others – Brownian motion sits right at the heart of the physical world.