In the following we will see how the results of the previous section can be used to derive the Black-Scholes formula – this time in a rigorous way, using the tools developed in the previous section.
8.3 A Rigorous Path to the Black-Scholes Formula 295
8.3.1 Derivation of the Black-Scholes Formula for Call Options We describe an underlying assetS(a stock) by ageometric Brownian motion with drift, i.e.
dS.t/DS.t/dtCS.t/dB.t/;
whereS.0/DS0is given. Such a process is often calledlog-normal, since log.S.t//
is normally distributed for everyt. The processis called thevolatility.
We consider a second asset, a bond, with fixed interest rate that hence follows the price process
ˇ.t/Dˇ0ert; (8.5)
whereˇ0> 0is its initial price andris the fixed interest rate. This gives rise to the It¯o process
dˇ.t/Drˇ.t/dt: (8.6)
We can describe (8.6) as a differential equation with solution (8.5).
We call atrading strategywith a portfolio that containsa.t/shares of stocks and b.t/shares of bonds at timet(witha;b2L2)self-financingif for allt:
a.t/S.t/Cb.t/ˇ.t/Da.0/S0Cb.0/ˇ0C Z t
0 a.s/dS.s/C Z t
0 b.s/dˇ.s/:
This condition ensures that at each time the current value of the portfolio corre- sponds to the initial value plus accumulated gains and losses.
We want to price an option based on the stockS. Let us consider as an example again a European call option, i.e., the right to buy at maturityTthe stock at a given priceK. The payoff of this call option at maturity is therefore max.S.T/K; 0/, as we have seen in Sect.8.1.
We want to find a self-financing strategy.a;b/that replicates the payoff structure at maturity, i.e.
a.T/S.T/Cb.T/ˇ.T/Dmax.S.T/K; 0/:
Given the existence of such a strategy, the price of the option at timethas to be a.t/S.t/Cb.t/ˇ.t/, since otherwise we would have an arbitrage opportunity. Thus, once the trading strategy has been determined, the pricing of the option is done.
Let us assume first that the price of the option at timetequals some function V.S.t/;t/and thatV is twice differentiable (we will see later that this is the case).
We can apply the It¯o formula forVand 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/: (8.7)
We have seen this formula in Sect.8.1, but this time a rigorous derivation led us here. Our goal is now to directly derive a hedging strategy. To this aim we assume the existence of this self-financing trading strategy.a;b/with
dV.S;t/Da.t/dS.t/Cb.t/dˇ.t/: (8.8) Inserting the expressions forS.t/andˇ.t/, we obtain
dV.S;t/D.a.t/S.t/Cb.t/ˇ.t/r/dtCa.t/S.t/dB.t/: (8.9) Can we constructaandbfrom (8.9) and (8.7)? We can: all we have to do is to match the coefficients in the two expressions, i.e., equal the terms with dtand with dB.t/
separately. Let us do this for dB.t/and we get S.t/@V.S;t/
@S Da.t/S.t/: (8.10)
From this equation we obtain
a.t/D @V.S;t/
@S ; which we insert into (8.8) to get
dV.S;t/D @V.S;t/
@S dS.t/Cb.t/dˇ.t/:
Solving this forb.t/we obtain b.t/D 1
ˇ.t/
V.S;t/@V.S;t/
@S S.t/
:
Now let us match the coefficients of dtin (8.9) and (8.7):
rV.S;t/C@V.S;t/
@t CrS.t/@V.S;t/
@S 1
22@2V.S;t/
@S2 D0:
8.3 A Rigorous Path to the Black-Scholes Formula 297 Thus, we arrive again at the Black-Scholes equation. Considering the boundary condition att D T, namely thatV.S;T/ D max.0;SK/, we can check that the following theorem gives a solution to this equation:
Theorem 8.7 (Black-Scholes formula) The value of a European call option with strike K, maturity T and underlying asset S (described by a geometric Brownian motion with driftand volatility) is given by
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 variables d1and d2are given by
d1D ln.S=K/C.rC2=2/.Tt/
p
Tt ;
d2Dd1p Tt:
To be rigorous, we need to double check that the assumptions made about the existence of the hedging strategy are in fact satisfied, i.e., we need to prove that a different price would indeed allow for arbitrage strategies. For simplicity, we prove this fortD0, but the proof carries over to all timestT.
Suppose that the price of the option at time zero is larger thanV.S0; 0/. Consider the trading strategy.1;a;b/in option, stock and bond witha.t/andb.t/as given above. We havea.T/S.T/Cb.T/ˇ.T/D max.S.T/K; 0/which is the value of the option at timeT, thus we have made a riskless profit (an arbitrage), namely the difference between the price for which we sold the option andV.S0; 0/.
In the same way, if the price of the option at time zero is smaller thanV.S0; 0/, the strategy.1;a;b/is an arbitrage. In other words, the no-arbitrage condition implies indeed that the price isV.S0; 0/.
The Black-Scholes model can be easily extended to higher dimensional situ- ations, i.e., toN 1 underlying assets. To model this, we define a vectorS.t/ D .S1.t/; : : : ;SN.t//and aD-dimensional Brownian motionB.t/D.B1.t/; : : : ;BD.t//.
Moreover, we define adrift vector2RN and avolatility matrix 2RD N. Then the asset process for then-th asset can be written as
dSn.t/DnSn.t/dtCSn.t/dB.t/:
We will encounter this model again briefly in Sect.8.6. For details on such multi- dimensional processes we refer to [Duf96] or other textbooks on mathematical finance.
8.3.2 Put-Call Parity
Can we use the result for call options derived above to price put options? A put option with strikeKand maturityTon an underlyingSgives the right to sell a share ofS at timeT for the priceK. The payoff of a put option at maturity is therefore max.KS; 0/.
The put-call parity will provide us with a way to price put options, when we know the price for a call option and vice versa, provided that both have the same underlying, the same strike and the same maturity. To derive this parity, we consider the following two portfolios:
• One put option and one share.
• One call option andKbonds that pay each1at maturity.
Computing the payoff of these portfolios at maturity, we notice that both payKif SKandSifSK, therefore both portfolios have the same value – also at times t<T. (Otherwise we would have a natural arbitrage opportunity.)
Let us denote the value of the put option at timetbyP.t/, the value of the call option byC.t/and the value of stock and bond byS.t/andR.t/, respectively. Then the following relationship holds:
C.t/CKR.t/DP.t/CS.t/:
This relation is calledcall-put parity.
If the bond pays a constant interest rater, thenR.t/Der.Tt/, thus the value of a put option is
P.t/DC.t/S.t/CKer.Tt/:
Similarly, we could compute the value of a call option from the value of a put option, always provided we knowS.t/andr.