Before we start to develop the basic ideas of mathematical finance systematically, we would like to give a short overview on the derivation of the Black-Scholes formula. To do this in a concise way, we have to “walk over some dead bodies”, i.e., be a little ruthless regarding mathematical precision. The concepts we use in this section will all be introduced rigorously in the sequel. For the moment, we ask you to trust us that the thin mathematical ice on which we walk is in fact stable enough for a proof. If you are afraid of drowning, just continue with the next section, and you will be safe and fine. A solid derivation of the Black-Scholes formula is given there.
The first assumption that we make to derive the Black-Scholes formula for the price of a derivative based on an asset is an assumption on the priceS.t/of this underlying asset. According to the information hypothesis we assume that the price of the asset changes when new information reaches the market. The information is assumed to be random, more precisely its influence on the return of the asset is assumed to be normally distributed. Moreover, there is some fundamental increase in the value of the asset which is predictable, but overlaid with the randomness of the information-driven price movements.
We can write this in the following form:
dS.t/DS.t/dtCS.t/dB.t/: (8.1)
where and are mean and standard deviation of the asset price andB.t/is a Brownian motion, which is (roughly) a continuous random process that has zero mean, is always independent of its past evolution and generates normally distributed returns.
The second fundamental assumption that we will make is familiar to us from our studies of time-discrete models (see Chaps.3,4, and5), namely the no-arbitrage principle: we assume that there is no arbitrage opportunity. What this means precisely in the time-continuous setting will be explained in the next section. For the moment we just state that there is no trading strategy that yields riskless excess returns over the risk-free asset.
We want to price an option on the assetS. We denote the value of this option at timetbyV.S;t/. It depends obviously on the timetand the price of the assetS.1At
1With this innocent looking assumption we have excluded many derivatives that are path dependent, i.e., their value does not only depend onSandt, but also on the previous prices of the underlying asset. We will come back to this later when we talk about numerical methods for the computation of asset prices.
8.1 A Rough Path to the Black-Scholes Formula 289 maturity the value of the option is known. Our goal is to derive a partial differential equation for the functionVand to solve this using the boundary conditions imposed particularly by our knowledge of the price at maturity.
We start by computing dV, the incremental change ofVusing a formula (called ItNo. formula) which we derive as Lemma (8.6). We obtain
dV.S;t/D
S.t/@V.S;t/
@S C @V.S;t/
@t C 1
22S.t/2@2V.S;t/
@S2
dt CS.t/@V.S;t/
@S dB.t/: (8.2)
There are now several possibilities to proceed. The probably easiest one (that makes you wonder how to find it, though) is to define a trading strategy that will turn out to replicate exactly the option. This trading strategy is defined as follows:
at every timet hold one option and @V.S;t/=@S stocks. The strategy is called delta hedge strategy. The value of this delta hedge portfolio at timetisV.S;t/ S.t/@V.S;t/=@S. The incremental profit (or loss)dRthat we make by following this strategy is then
dR.t/DdV@V.S;t/
@S dS: We insert (8.2) and (8.1) into this formula and get
dR.t/D
@V.S;t/
@t C1
22S.t/2@2V.S;t/
@S.t/2
dt:
We notice that we have lost the stochastic terms (the ones withB) here, in other words, the incremental returns of the delta hedge portfolio are not stochastic, but deterministic and follow the above formula. But if these returns are deterministic, they are risk-free, and this implies that they should not be different from the returns of the risk-free asset, since otherwise we would violate the No-arbitrage Principle.
Therefore, we can equal dRwith the incremental return of the risk-free asset of the same amount (which isV.S;t/S.t/@V.S;t/@S), i.e.
r
V.S;t/S.t/@V.S;t/
@S.t/
dtD
@V.S;t/
@t C 1
22S.t/2@2V.S;t/
@S.t/2
dt: Dividing by dt, we obtain the Black-Scholes equation, a partial differential equation in the two variablesSandt:
@V.S;t/
@t C 1
22S.t/2@2V.S;t/
@S.t/2 CrS.t/@V.S;t/
@S rV.S;t/D0: (8.3)
Partial differential equations (PDEs) typically have infinitely many solutions, so this alone would not be of much use. However, we have boundary condition, i.e., constraints that apply on the boundary of the set whereSandtare defined.
What are our boundary conditions? This depends on the option we want to price.
Let us price a simple call option with exercise priceK. This option allows at maturity T to buy a share for the priceK. Our first boundary condition is nowV.0;t/D 0 for allt, since the stochastic process (8.1) is constant zero if it is zero at any point in time, but a call option would not be exercised if the underlying asset is worthless, thus the option is worthless as well. On the other hand, if the asset price is extremely high, the value of the option is close to the value of the asset which gives the second boundary condition (actually, it is an asymptotic condition, i.e., the boundary is “at infinity”):V.S;t/=S!1asS! 1. Finally, we have the condition for the value of the option at maturity which isV.S;T/ D SK ifS > Kand zero otherwise. In summary we have the following boundary value problem:
8ˆ ˆˆ
<
ˆˆ ˆ:
@V.S;t/
@t C122S.t/2 @2@VS..tS/;2t/CrS.t/@V@.SS;t/rV.S;t/D0;
V.0;t/D0 for allt;
V.S;t/=S!1 asS! 1; for allt;
V.S;T/Dmax.SK; 0/ for allS:
This problem can in fact be solved in the following way: apply a suitable variable transformation to bring equation (8.3) to the form of a diffusion equation. Then use a standard solution ansatz and finally transfer back to the original variables.2The final result is:
V.S;t/DS.d1/Ker.Tt/.d2/;
whereis the normal cumulative distribution function, i.e., .x/WD
Z x 1
1 p
2 exp .y/2 22
! dy;
and the auxiliary variablesd1andd2are given by
d1D ln.S=K/C.rC2=2/.Tt/
p
Tt ;
d2Dd1p Tt:
2Anansatzis a specific functional form which we assume the solution to have in order to compute it. This assumption is a posteriori justified if we obtain a solution that is indeed of the assumed form. In the appendix more ideas on how to solve PDEs and further references are given.