Stochastic Modelling in Finance - Solutions 7 7.1 Consider the Black & Scholes Model with volatility a > 0, drift /i, interest rate r and initial stock price Sq. Take a put-option with strike /C > 0 and maturity T > 0. The value of this option is V(t, s) = EQ[(K- Sr)+e-r(r-f) St = s]. Without reference to the put-call parity, show that the value of the put option at time t>0 satisfies V(t, St) = tfe-r<t-"(l - *(<L(T - t, St))) - 5((1 - *(d+(r - t, St))) where <[> denotes the CDF of the Normal distribution and d±(ttx) are defined as This is pretty much identical to the calculation done in lecture: In the Black & Scholes model ST = exp((r - \a2)t + awt) where WT ~ JV(O, T) and St = St exp I f r -a2 ) (T t) + awr-t ) Hence, a) = Q[(tf - Sr)4"e"r(T- \St = s] C _ o p-r{t-t) J?\Q tt I C _ el where X ~ A^(0,1). For the final equality we calculate that {K > ST] = ik> 5texp ((r - j)(t - t) = { < exp (-(r - y)(t - t) = {in ( ) + (r - y)(t - t) < = {d.(t-t,st)<x} for -Wr-t/x/T3! = X ~ N{0,1) under Q. Hence, Eq{Ik>St \St = s] = EQ%t_(T-t,.)<x] = -^= / e"1'/2 dx = 1 - ^(d_(t - *, s)) V27T Jd-(T-t,s)
and» S, = s] = 1 f = s / e V27T Jd-{T-t,s) 1 r = s == / e V27T Jd-(T-t,s) = s f= / e~y ^2 c?2/ Hence V(«, St) = Ke'^-^il - $(d_(t - i, S«))) - St(l - $(^+(T - t, 50))- 7.2 Consider the Black &; Scholes Model with volatility a > 0, drift fi, interest rate r and initial stock price So- Take a 'power call-option' with power n > 1, strike K > 0 and maturity T > 0. The value of this option is V(t, s) = EQ [(5? - K)+e-riT-l) \ Show that the value of this option at time t = 0 satisfies 1/(0, So) = SJc(n-l)rTe2l!^ffaT$(d^(r,So) H-r where $ denotes the CDF of the standard Normal distribution and d^l{t^x) is defined as Let V(0, So) = e-rteq{s^>k] - Ke-rTEQ[Isf>K]. (0.1) Next we look more closely at the event {SJ > K} and write > exp {-awt - (r - ^2) t) } Note that -Wr ~ N(0,T) and hence X := -WT/\/T ~ W(0,1) so {in (J^j + (r- \a^ T > -owr} = {<P.(T,So) > X},
Using this we can rewrite the second term in (0.1) as Ke-rTEQ[lSn>K} = Ke-rrEQ[ = ^=- \ V^TT J- The first term is: = SSEQ [exp (nawt - ^cj2t) lsn>k\ ' n.(n-l)rt Eg [exp [-navfx - -a2t) r<r_{t,s0) exp (-nvvty - ^2t) e^2/2dy Next we need to 'complete the square' in this exponential, i.e. note that and thus 1= V27T / d-(t,st) Changing variables using z = y -\- no\/t we get qnp(n-l)rtpn(n-l)a2t/2 ^/2 (r, 5b) + nay/f). So we have derived that the price of the power call option is given by V(0, Sb) = SSe{n-l)rTe2il*2*aaT${<rL(T, So) + navt) - 7.3 It was shown in an example in Lecture that in the Black &, Scholes Model that the value of a call option with strike K > 0 and maturity T > 0 is = St*(d+(T - i, St)) - Ke-^T-l^(d_{T - t, 5t)). Suppose that u(t, s) is a solution to the PDE d d Id2 u(t, s) + rs u(t, s) + -crv u(, s) - ru(t, s) = 0 at as 2 us* subject to the terminal condition u(t, s) = (s K)+. 3
(i) Show that limt_>t V(Tt s) = (s - K) + Recall that so lim d±(t-i,s) = lim v 1 (, ^ T^7 = lim - t +00 s > K oo s < K. 0 s = K Hence lim$(d±(r-i,a))= i s> 0 s < and lim V(t, s) = li - t, ft)) - - t St))) s-k s>k 0 s<k o s = /r so we can conclude that \imt->t V(t, s) = (s /C)+ as required. (ii) Show that V{t,x) = u(tfx) by substituting the explicit expression for V(t,s) into the PDE determining u(tts). First we need to calculate the derivatives ^V(t, s), ^V(t,s) and ^?V(t, s). You can take these from chapter 3 of the background reading or directly calculate: ^V(t, s) = - - t,a)) - - t, a)),
Inserting these into the PDE gives we should have sa d 0 = - tt a)) - - t, s)) 1 say/t tdx - r{s$(d+{t - t, s)) - Canceling terms on the right-hand side validates that the PDE indeed holds. 7.4 The price of a put-option with strike price K > 0 in the Black & Scholes model is M) = EQ [(K - St = s] - tt St))) - St(l - - t, St))) Find explicit expressions for the following sensitivities: (i) The delta of this put option is denned as Explain how the delta of a option in the Black k, Scholes model can be used for hedging. This one looks complicated but once we have proven the statement (0.2) will be much more straightforward. the others say/t t We can calculate using the chain-rule that - t, s)) cf>(d+(t - tt s)) = - tt so\/t t ojt -t
where (f>(x) = -^${x). Furthermore, = -{d+{t - i, s))2 - a2{t - t) + 2d+{T - t, s)a = -(d+(t-t,s))2-a2(t-t) -^) +2r{T-t)+ a2(t - t) K tt s)) K 2r(T - t). so d -t s)- d+{t-tts)-oyjt-t dy -oo -1 1 exp{-(d+{t -t,s) - osjt - if ser(t-t) ser{t-t) 4(d+(T-tya)). Thus we have found that d d (0.2) and thus ds We saw in lecture that the price process associated with the put option can be decomposed as i = tt(p)o + / &udsu where (At)t>o is the replicating strategy. Applying the Ito formula to V(t, s) and using that (e~7't7r(c)i)t>o is a martingale we can see that Thus the delta of the option calculated in this question is the number of units of stock held in a replicating portfolio. (ii) The gamma of this put option, which is denned as
Differentiating the answer given in (i) gives S(J Vf where Six) := -j-^(x). (iii) The theta of this put option, which is denned as First we note that d_(t tts) - d+(t t, s) o\/t i, showing that Applying the chain rule we see that dd (T-ta)-±d lt-ta)+ " Ft ( *'5)-at+u t's' + IV(t, s) = (T - t, a))) - s(l - $(d+(t - t, s))) - t, s))) + scj>(d+(t - t, St)))^d+(T - i, a) - Ke-T{T-t]<t>{d-{T - t, a))^-d_(r - t, s) C(7 - t, a))) /// Q U'+\1 L> */ t Due to (0.2) the second term is zero so we have derived that ft C1* (iv) The vega of this put option, which is defined as First we note that cl(t t,s) = d+(t t,s) oy/t t, showing that -d.(t -tis) = d+(t - t, s) - 00 ocr
Applying the chain rule we see that = s4>(d+(t - i, St)))-^ ^d.{t - t} s) t, St))) - - tt s)))^-d+(t - t, s). Due to (0.2) the second term is zero so we have derived that d V{t,s) = (v) The rho of this put option, which is defined as p(t,s):=-v(t,s). First we differentiate d±(t t, s), showing that Applying the chain rule we see that -V(t, s) = -K(T - (scp(d+(t-t,st))- - t, Due to (0.2) the final term is zero so V{t,s) = -K(T- - t,