As a final comment, we note that although the time-T call value is a nonsmooth hockey stick, (8.16), the functionC(S,t) is smooth at all times 0≤t<T; this phenomenon of ‘instant smoothing’ is typical of diffusion PDEs like (8.15).
E X E R C I S E S
8.1. Show that (8.2) leads to the approximate relation (8.3). [Hint: use Exer- cise 3.7.]
8.2. Show that (8.21) can be replaced by (8.22).
8.3. Confirm thatC(S,t)in (8.19) satisfies (8.16), (8.17) and (8.18). [Hint:
to deal with (8.16), take the limitt→T−, to deal with (8.17) take the limit S →0+and to deal with (8.18) take the limitS → ∞.]
8.4. Use the argument in Section 2.5 to obtain the general put–call parity relation (8.23).
8.5. Confirm thatP(S,t)in (8.24) satisfies (8.25)–(8.27).
8.6. It is intuitively obvious that call and put options are linear – the value of two options is twice the value of one option. Show how this follows from the Black–Scholes formulas (8.19) and (8.24).
8.7. Show that limE→0C(S,t)=S in (8.19) and limE→0P(S,t)=0 in (8.24), and give a financial interpretation of the results.
8.8. Write down a PDE and final time/boundary conditions for the value of a butterfly spread, as described in Exercise 1.3.
8.9. Verify that
V(S,t)= e(σ2−2r)(T−t) S
is a solution of the Black–Scholes PDE (8.15). What is the practical impli- cation of this result?
8.10. Verify thatS ander t are solutions of the Black–Scholes PDE (8.15) and give an accompanying financial explanation.
8.11. Consider the problem posed in Exercise 2.6 of finding a fair value for a forward contract. Use Exercise 8.7 above to confirm that F =S(0)er T.
8.7 Program of Chapter 8 and walkthrough
Unlike the previous seven cases, our code for this chapter, which is listed in Figure 8.2, is a MATLAB function. This means that it must be supplied withinput argumentsand it will returnoutput argu- ments. The input argumentsS,E,r,sigmaandtaurepresent, respectively, the asset price at timet, the exercise price, the interest rate, the volatility and the time to expiry,T−t. It is assumed thattau is non-negative.
function [C, Cdelta, P, Pdelta] = ch08(S,E,r,sigma,tau)
% Program for Chapter 8
% This is a MATLAB function
%
% Input arguments: S = asset price at time t
% E = Exercise price
% r = interest rate
% sigma = volatility
% tau = time to expiry (T-t)
%
% Output arguments: C = call value, Cdelta = delta value of call
% P = Put value, Pdelta = delta value of put
%
% function [C, Cdelta, P, Pdelta] = ch08(S,E,r,sigma,tau) if tau > 0
d1 = (log(S/E) + (r + 0.5*sigmaˆ2)*(tau))/(sigma*sqrt(tau));
d2 = d1 - sigma*sqrt(tau);
N1 = 0.5*(1+erf(d1/sqrt(2)));
N2 = 0.5*(1+erf(d2/sqrt(2)));
C = S*N1-E*exp(-r*(tau))*N2;
Cdelta = N1;
P = C + E*exp(-r*tau) - S;
Pdelta = Cdelta - 1;
else
C = max(S-E,0);
Cdelta = 0.5*(sign(S-E) + 1);
P = max(E-S,0);
Pdelta = Cdelta - 1;
end
Fig. 8.2. Program of Chapter 8:ch08.m.
The output argumentsC,Cdelta,PandPdeltarepresent, respectively, the European call, call delta, put and put delta values.
The lines of code betweenif tau > 0andelseare executed in the case wheretau, the time to expiry, is positive. In this case we are evaluating the Black–Scholes values given by (8.19), (8.24), and also the deltas (9.1) and (9.2) that are introduced in Chapter 9, usingerfas a means to obtain N(x), as described in Exercise 4.1.
The lines of code betweenelseandendare executed in the remaining case, wheretauis zero.
Here, we are at expiry and to avoiddivision by zeroerrors in (8.20) and (8.22), we revert to the expressions (8.16), (8.25), along with (9.7) and (9.8) from Chapter 9. We make use of the signum function,sign, which is defined by
sign(x)=
1, ifx>0, 0, ifx=0,
−1, ifx<0.
8.7 Program of Chapter 8 and walkthrough 85 An example of the function in use is
>> S = 2; E = 2.5; r = 0.03; sigma = 0.25; tau = 1;
>> [C, Cdelta, P, Pdelta] = ch08(S,E,r,sigma,tau) which outputs
C = 0.0691 Cdelta = 0.2586 P = 0.4953 Pdelta = -0.7414
P R O G R A M M I N G E X E R C I S E S
P8.1. Use ch08.m to produce graphs illustrating the limits limt→T−C(S,t)= max(S(T)−E,0)and limS→∞C(S,t)=Sestablished in Exercise 8.3.
P8.2. Write a program that illustrates (8.4) in the style of Figure 8.1.
Quotes
Stephen Belloti:‘Myron, what do you have more of – money or brains?’
Myron Scholes:‘Brains, but it’s getting close.’
Source(Lowenstein, 2001) In the early 1970s, Merton tackled a problem
that had been partially solved by two other economists, Fischer Black and Myron S. Scholes:
deriving a formula for the ‘correct’ price of a stock option.
Grasping the intimate relation between an option and the underlying stock, Merton completed the puzzle with an elegantly mathematical flourish.
Then he graciously waited to publish until after his peers did;
thus the formula would ever be known as the Black–Scholes model.
Few people would have cared given that no active market for options existed.
But coincidentally, a month before the formula appeared,
the Chicago Board Options Exchange had begun to list stock options for trading.
Soon, Texas Instruments was advertising inThe Wall Street Journal,
‘Now you can find the Black–Scholes value using our . . . calculator.’
This was the true beginning of the derivatives revolution.
Never before had professors made such an impact on Wall Street.
R O G E R L O W E N S T E I N(Lowenstein, 2001) In 1975 I crammed the Black–Scholes formula into a TI-52 handheld calculator,
which was capable of giving me one option price in about thirteen seconds.
It was pretty crude,
but in the land of the blind I was the guy with one eye.
J O E R I T C H I E,option trader, source(Bass, 1999)
To someone who came out of graduate school in the mid-eighties, the decade spanning roughly 1969–79
seems like a golden age of dynamic asset pricing theory . . . The Black–Scholes model now seems to be, by far,
the most important single breakthrough of this ‘golden decade’ . . .
Theoretical developments in the period since 1979, with relatively few exceptions, have been a mopping-up operation.
D A R R E L L D U F F I E(Duffie, 2001)
9
More on hedging
O U T L I N E
• practical illustration of hedging
• behaviour of delta near expiry
• Long-Term Capital Management