Building the Model. The Lognormal Density The following assumptions and their mathematical interpretations are key in under- standing the fact that S follows the lognormal law. The first assumption is that all new information is instantaneously reacted to by the market. This relates to some kind of market efficiency, closest to semi-strong form efficiency, where all public in- formation is included in the share price. This instantaneous reaction is necessary to interpret the problem as continuous. The second assumption is that price is memory- less. In this sense, the current price holds all information inherently in that number. This assumption ultimately frames the problem such that changes in S, the under- lying, can be interpreted as a random walk, Brownian motion with drift. Brownian motion is conceptually derived from the process of random particles drifting into each other in a liquid, but the model has vast applications in math, economics, and physics [13]. Thus S can be represented as a combination of its drift over time, denoted by µ, and its Brownian motion. This leads to the equation: dS = µdt + σdB (2.3)
Building the Model. The Equation Let us consider further assumptions in order to build (2.1). We must assume that 1. volatility is constant and known
Building the Model. The Boundary and “Final” Conditions Boundary and initial conditions are a key component of solving a PDE. With B&S we have a final condition rather than an initial condition. The general form of a PDE of this order, for example the heat equation, has a negative diffusion term with an initial condition. Since the B&S equation has a positive space derivative term and a final condition it is still well posed as we will be able to use a variable change to alter this. Boundary conditions and the final condition are going to be dependent upon whether solving a put option or a call option. For a put, when the stock price is zero, the put is worth the present value of the exercise price. As S approaches infinity the holder would not exercise the option and thus it has a value of zero. At the date of maturity, time T , the option has no time value, so the value should just be its intrinsic value; that is the value of the put should be the exercise price minus the stock price, or zero if this is negative. Therefore we have: Figure 2.2: Well Posed Conditions P (0, t) = Ee−r(T−t) P (S1, t) = 0 S1 → +∞ P (S, T ) = max(E − S, 0) (2.18) Similarly, with a call option we know that if the stock price is zero, then it will not be exercised and is very unlikely to become in the money so it has a value of zero. As S approaches infinity, the value of the call should approach infinity. At the final time T , the call should have the value of the stock price minus the exercise price, or zero if this is negative. Thus we have: C(0, t) = 0 C(S1, t) − (S − Ee−r(T−t)) → 0 S1 → 0 C(S, T ) = max(S − E, 0) (2.19) For the reasons stated above, we must now use a variable change to make the problem well posed. In order to solve with a change of variable we take: 2 x = log S E τ = 1 σ2(T − t) V w(x, τ ) = E (Eex, T 2τ σ2 Σ Σ −
