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
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
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.
The formulae given for
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
Note that we don’t need to worry about
We now work towards proving that (15.10) holds. From Theorem 15.2.5 we know that
In particular, this implies that
Since
Substituting in (15.1) and (15.2), this becomes
Next: recalling that
where we have suppressed the
Equating the
With this in hand, equating the
so as
Using (15.14) to substitute in for
Next we use Lemma 14.1.3, which tells us that the solution to be Black-Scholes equation can be written as
Note that here we take expectation in the risk neutral world
To finish the proof, we use that
Setting
as required. ∎
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Corollary 15.3.3 The replicating portfolio given in Theorem 15.3.1 for
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. ∎
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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
.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:
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• 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.
, 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.
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