MATHEMATICAL FINANCE EXAM 2003/4 RP s solutions & comments

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1 MATHEMATICAL FINANCE EXAM 23/4 RP s solutions & comments This version is from January 8, 24, ie. after the exam and after I ve gone through your answers. Since I m not the one taking the exam, I can be as sketchy and arrogant as I like. And I am not being punished for the typos or God forbid worse errors that remain. EXERCISE ; STOCHASTIC CALCULUS a [%] We can write Xt = atbt, where dbt = btdt + ctdw t. Because b and c are deterministic, Björk s Lemma 3.5 tells us that B is normally distributed. Since a is also deterministic, then Xt is also normal. We have that dat = a tdt so by using the Ito product rule dxt = a tbtdt + atdbt + = a txt/at + atbtdt + atctdw t, where we have used that B = X/a. b[%] The specific X process fits the bill with x =, at = t so a t =, bt =, and ct = t. Some would say that this just shows that X is a solution to the SDE, but how do we know that there aren t others? We don t care. But if you insist, uniqueness goes something like this: Pick some arbirarily small ɛ; work on [ : ɛ]; verify that a unique solution exists on this interval though we haven t really discussed diffusion on intervals by checking the Lipschitz condition 4.6, page 52 in Björk OK with K = /ɛ. Since for any t < we have t / s2 ds <, Björk s Proposition 3.4 shows that EXt = for all t <.

2 No, X is not a martingale. This is perhaps most easily seen from the dx-expression: It has a dt-term, and hence by Björk s Lemma 3.9 isn t a martingale. Now consider s t < and look at s dw u t covxs, Xt = t se u dw v because of -mean v s = t se dw u u s dw v v + t s dw v v The last term produces something with mean : Condition on F s, use the tower law/ iterated expectations and Björk s Proposition 3.7. Many of you use independence instead of martingality to make the last term go away. The factors are indeed independent because increments of BM are, but this hinges on the integrand being deterministic. It is not generally true that s gudw u is independent of t gudw u. To make a counterexample take g simply to be an F s -measurable random variable independent of W and compare the mean of the product of the squared integrals to the product of the variances of the integrals. But this is a small detail, so as long as you know & explain what you re doing... Anyway, it suffices to look at s dw u s E u dw v s = E v u du by the Ito Isometry, Björk = = s [ v ] When collecting the terms this shows that s v dv = 2 = s v dv 2 + s covxs, Xt = s t for s t <. v = u = s s. Notice that varxt = t t for t, or in other words since EXt Xt L2, so it s natural to define X to be. c[%] i It is clear from the properties of Brownian motion that Y is a -mean Gaussian process. A Gaussian process is one where Y t,..., Y t n is an n-dimensional normal variable for any n and vector of time points. This is ever-so-slightly stronger that Y t being normal for any t, but any well-argued; it s clear works nicely statement of the latter gives you full credit. 2

3 If we remember that covw u, W v = minu, v, then for s t we can also readily find the covariance ii Note first that covy s, Y t = covw s sw, W t tw Xs, Xt, X N 3, = covw s, W t covw s, tw covsw, W t + covsw, tw = s ts st + ts = s t. s s s s t t s t for s t. Using standard results about conditioning in the normal distribution fx Sætning.39 i Ernsts Stat A-noter we get that Z = Xs, Xt X = is twodimensionally normally distributed with mean and covariance-matrix s s s t s t and that gives the result. s t = s s s t s 2 st st t 2 = s s s t s t t t, EXERCISE 2; BLACK/SCHOLES OPTION PRICING 2a [%] The arbitrage-free price is π SQ t = e rt t E Q t S 2 T = e rt t E Q t St expr σ 2 /2T t + σw Q T W Q t 2 = S 2 t expr σ 2 T te Q t exp N, 4σ 2 T t = S 2 t expr + σ 2 T t, where we have used the martingale property of arbitrage-free prices, the well-known BGM solution, standard rules for conditional expectation and the fact that if Z Nm, v then EexpZ = expm + v/2. You should know this, but using Exercise 4. from the weekly notes remind yourself is OK. It is a simple claim, so from Björk s Theorem 7.5 it can be -hedged in the usual way, ie. by holding with usual sloppy notation dπ SQ ds = 2St expr + σ2 T t 3

4 units of the stock and keeping the self-financing condition via the bank-account. Finding relative weights is of course fine too. 2b [%] The pay-off at chooser option expiry T is π chooser T = maxcallt, putt = maxcallt, callt + K exp rt 2 T ST = callt + maxk exp rt 2 T ST,, where the 2. equality follows from the put/call-parity Björk s Proposition 8.2. The first component is the time T value of an expiry-t 2 strike-k call. The second is the pay-of of an expiry-t, strike-k exp rt 2 T put. The desired result follow be linearity of the martingale pricing rule, π chooser t = e rt t E Q t callt + K exp rt 2 T ST + = callt expiry = T 2, strike = K + putt expiry = T, strike = K exp rt 2 T, and both call- and put-prices can be found by plugging into the Black/Scholes formulas. Or what formula or observed price we might have; the analysis does not hinge on the stock price being GBM. With different expiry dates and strikes we still have that callt = BScallT, ST T c, K c, where BScall denotes the call-formula in the B/S-model; a well- know function and likewise for put. Therefore π chooser t = e rt t maxbscallt, x T c, K c, BSputT, x T p, K p φx, T t, S t dx where x φx, T t, y is the lognormal density of S T conditional on S t = y. This can then be calculated by simulation, numerical integration, a complicated closedform solution, or by solving a PDE the chooser option is still a simple claim. The PDE-method started from maxt c, T p would also work in some non-b/s-cases. A reference to the exercise you did about extendible options 9.3 on Weekly Note #9 makes a good impression. EXERCISE 3; OPTION PRICING WITH TWISTS 3a [%] 4

5 Conditional on time-t information, ST K Q NSt K, σ 2 T t, and we can express the arbitrage-free price as π call, Bach t = E Q t ST K + = E Q t ST K ST K. Now apply the stated result with µ = St K, σ 2 = σ 2 T t, l =, h =, and note that Φ x = Φx and φx = φ x because the standard normal distribution is symmetric around. To obtain the -formula differentiate wrt. St. Remember the chain rule, gf = f g f and that Φ = φ by definition and φ x = xφx by the chain rule. 3b [%] From Björk s Proposition 8.5 we know that the RHS is lns/k + r + σ 2 Φ BS /2T σ BS T specializing to r = is OK. Under the S-numeraire martingale measure we have from Björk s Theorem 9.8 ds = Sr + σ 2 BS dt + σ BSdW QS or in other words use Björk s Proposition 4.2 with α = r + σ 2 BS ST = S expr + σ 2 BS/2T + σ BS W QS T Hence divide, take logs, move over & use symmetry lns/k + r + σ Q S 2 ST K = Φ BS /2T σ BS T Some version of or precise reference to the calculations that Björk does around page works too. A third way is to plug back into the B/S-formula and arrive at S KΦd 2 = SQ S ST K KQST K. Since standard calculations tell us that QST K = Φd 2, the desired result follows. This was not the way of answering I had thought of, but it is quite elegant. When done correctly, that is. You need the last bit of the argument Q = Φ, otherwise you are kind of claiming that a + b = c + d a = c and b = d. 5

6 Implied volatility skew in the Bachelier model implied volatility strike 3c [%] i The implied volatilities in the Bachelier model as defined in in Björk & with the parameters specified in the exercise look like this: We see a skew pattern: Low strike options ie. OTM puts have high implied volatilities, and high strikes have low implied volatilities. At least the low strike - effect is something we are familiar with from data. This has lead people to re-discover the Bachelier model and use it, despite the unrealistic property that stock prices can go negative. The skew is not surprising. In the B/S model the price volatility what is in front of dw in the ds expression is proportional to the stock price itself, ie. it goes down when the stock price goes down whereas the return volatility is constant. In the Bachelier model, the term in front of dw in the ds-expression is constant, so compared to the B/S model prices are more volatile when they are low in the Bachelier model a price change of is equally likely whether the stock price is 5 or, in the B/S model large absolute price changes are unlikely when stock prices are small, and therefore put options that pay-off for low stock prices ie. when volatility is high are worth more, because higher volatility in some sense makes puts and calls worth more. Vega is positive. It could be argued that volatility-sensitivity calculations are typically specific to the B/S-model. But the qualitative result is a consequence of the convexity of the pay-off. It the pay-off is not convex, different things can happen, as seen for digital options in the 2/2 exam. ii Note that QST K = Φ S K σ. From the first part this is also the of the T 6

7 call. Isolating Q S in the general formula and using the Bachelier formula we get Q S ST K = S KΦ... + σ T φ... + KΦ... S Φ... =. = Φ... + σ T φ... S Arguably, this is not the best counter-example, because the use of S that can be and negative as numeraire is suspect. But it still shows that possibly forgotten terms cancel explicitly because we re in the B/S-model. Incidentally, they also cancel in Merton s jump-diffusion model, which you don t know what is but never mind EXERCISE 4; INTEREST RATES 4a [%] The first part follows directly from Exercise 4.4 in Björk with with A = κ and b = κθ Q. ZCB-prices and the A and B functions in the following are given in Björk s Proposition 7.3, with a = κ and b = κθ Q. You should probably write up what they are; I don t have to. The forward rates are ln P, T f, T = = A T, T + B T, T r, T where with subscripts denoting differentiation B T ; T = exp κt and A T is a bit more involved. You can do many or few calculations on this as you like. In any case, you will find that no, forward rates and Q-expected future short rates are not equal when interest rates are stochastic. You can plug in numbers, do the algebraic manipulation or refer to the discussion in in Björk. But they are pretty close. 4b [%] ZC yields are y, T = ln P, T /T. The long-maturity yields are θ Q minus some small σ 2 -term. Easily deduced from Vasicek s ZCB formula. This level is the same irrespective of which value the short rate takes. The short end is r. When r = θ Q we get a yield curve that is slightly downward sloping, but almost flat. When r = θ P < θ Q we get and upward sloping yield curve; a normal term structure, where the difference between long and short rates is almost λ. Talking about the yield curve under Q/P is non-sense. I m not pedantic, though, so mostly I understand what you mean to say. The yield curve is what is. What we can talk about is the Q/P -typical shape, that corresponds to a short rate around the 7

8 ftt Forward rates, ZCB yields & E^Q short rate: Microscopic view E^Qrtau ZCB yield forward rate tau Forward rates, ZCB yields & E^Q short rate: Bird s eye view ftt forward E^Qrtau ZCB yield rate tau θ-parameter that the short rate mean-reverts to. Very good answers will illustrate an understanding of this; good answers will just refrain from making non-sensical statements. It is a stylized empirical fact that yield curves are mostly positively sloping that s why it s called normal is then a clear indication that Q P. If they were equal, we d mostly see ever-so-slightly inverse yield curves. That Q P and λ positive corresponds to risk-averse investors. This should mean that risky assets have higher expected rates of return. The risky assets in this world are long bonds, and the positively sloping yield curve tells us ignoring some small non-linearity/jensen-effects that this is the case. So the pieces fit together nicely. 8

9 yield curves with r=theta^p... and r=theta^q logzcb/tau tau 9