6CCM338A Mathematical Finance II: Continuous Time. Summer 2021
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6CCM338A Mathematical Finance II: Continuous Time.
Summer 2021
1. a. Consider a random variable X on (0, 的) with cumulative distribution
function FX (x) = Φ(x)2 , where Φ(x) = 尸(北)& dz is the standard cu-
mula for simulating X . Does X have the same distribution as Z2 , where
Z 建 N (0, 1)? [30%]
Solution. F尸X1 (y) = Φ尸1 (^y) and set X = F尸X1 (U), where U 建 U [0, 1]. For final part, answer is no (cdf of Z2 is different to cdf of X, see Hwk1 q1)
2. Let (Wt )t一0 denote a standard Brownian motion throughout.
a. Compute the distribution function of Φ(Wt ) [30%]
Solution.
P(Φ(Wt ) x x) = P(Wt x Φ尸1 (x)) = Φ(Φ ^(尸1)x)) .
b. Let Xt = Wt + a. Compute E(XT(2)|Xt = x) [40%]
Solution Conditioned on Xt = x we see that XT = WT + a = Wt + a + WT _Wt = x+WT _Wt so XT 2 N(x, T _t), so E(XT(2)|Xt = x) = x2 +T _t
c. Let dXt = Xt(2)dWt . Compute the SDE satisfied by Yt = Xt(2) (you may assume Xt < 0). What happens to E(Ys Yt ) if W is replaced with _W?
[30%]
Solution. Set f(x, t) = x2 so fx (x, t) = 2x, fxx (x, t) = 2 and ft (x, t) = 0.
Then from Ito’s lemma
dYt = 2Xt dXt + _ 2Xt(4)dt = 2 ^3Yt Yt dWt + Yt2 dt
= 2Yt 2 dWt + Yt2 dt .
If W is changed to _W , Xt and Yt change but they have the same distri- bution because _W is also a Brownian motion and moreover Yt1 , ..., Yt亢 will have the same joint distribution for any t1 < t2 < ...tn , so E(Ys Yt ) remains unchanged
3. Let (Wt )t一0 denote a standard Brownian motion and recall the Black-Scholes model
dSt = St (µdt + σdWt )
for a Stock price process S under the physical measure P.
a. Compute the distribution function of log(St + a) for a < 0 under Q in terms of the distribution function FSt(S) of St and write this as an expectation [30%]
Solution.
P(log(St + a) x x) = P(St x ex _ a) = FSt(ex _ a) = E Q (1St′ez尸a )
b. Compute the no-arbitrage price at time zero and at a general time t ∈ [0, T] of a contract which pays ST(2) at time T. Now set r = 0 and suppose we delta-hedge this contract at time zero but not after time zero - do we want more or less variability of S over a small time interval [0, ∆T]?
[30%]
Solution. From many previous questions we know that
P (S, t) = e尸r(T尸t)E Q (ST(2) |St = S) = e尸r(T尸t)S2 e2(r尸 σ 2 )(T尸t)+2σ2 (T尸t) = e尸r(T尸t)S2 e2r(T尸t)+σ2 (T尸t) .
If r = 0, Greeks are given by
PSS (S, t) = 2eσ 2 (T尸t) > 0
Pt (S, t) = _σ S22 eσ 2 (T尸t) < 0
and recall from the Black-Scholes PDE they are intimately related: Pt (S, t)+ σ 2 S2 PSS (S, t) = 0, so for any option payoff Gamma and Theta must have opposite signs when r = 0. Over a small time interval ∆t, applying Tay- lor’s theorem at time zero we see that our profit/loss is approximately given by
Profit/Loss = ∆P _ PS (S0 , 0)∆S
~ PS (S0 , 0)∆S + PSS (S0 , 0)(∆S)2 + Pt (S0 , 0)∆t _ PS (S0 , 0)∆S = PSS (S0 , 0)(∆S)2 + Pt (S0 , 0)∆t
i.e. we have removed the PS (S0 , 0)∆S term with the initial delta hedge, and we are ignoring smaller higher order terms here which are O((∆t)2 ), O((∆S)3 ) and O(∆S∆t). The Gamma term PSS (S0 , 0) is positive, so we want |∆S| to be large to compensate for the negative theta term Pt (S0 , 0)
c. State whether each of these statements are true or false (no explanation is required).
i. S can tend to +o in finite time under P if µ _ σ 2 > 0
ii. The inverse maximum process (HB )B一S0 defined by HB := minut : St = B} is continuous for B < S0
iii. P(ST = S0 ) > 0
iv. put options is worth less than or equal to 1 [40%]
Solution. i) False, S cannot explode in finite time since St is continuous a.s.
ii) False, as for Brownian motion, the maximum process of S has flat periods, so its inverse has jumps. Moreover, HB jumps over any interval [b1 , b2]
iii) False since ST has a density
iv) True since (K _ ST )+ = (1 _ )+ < 1
4. Let (Wt )t一0 denote a standard Brownian motion on some probability space (Ω , e, P) with filtration et . Consider the Black-Scholes model
dSt = St (µdt + σdWt )
for a Stock price process S .
b. Consider a double barrier option which pays 1 at time HB if HB < min(HL , T) and pays zero at time T if HL < min(HB , T) and pays zero if T x min(HL , HB ), where L < S0 < B . Write down the boundary con- dition satisfied by the price P (S, t) of this option at time t ∈ [0, T] given that St = S . Write down a probabilistic representation for the price of the option in terms of indicator functions. What is the sign of the delta of this option? [40%]
Solution. P (L, t) = 0, P (U, t) = 1, P (S, T) = 0 for S ∈ (L, U). Price at time zero is e尸rTE Q (1HB<min(HL,T)). Delta is clearly positive since we want S to be closer to U rather than L.
c. Compute the price of a No-Touch option with barrier B < S0 at time zero which pays 1 at time T if S stays above B for t ∈ [0, T] and zero otherwise, in terms of the price of its semi-static hedge portfolio. Write down an expression for the delta of the option at time zero in terms of the density pST (S) of ST under Q, and comment on the sign of the delta.
[30%]
Solution. Let f(S) = 1S>B . Then f() = 1B2/S>B = 1S<B . Then from HwkExtra5 q4
P (S0 , 0) = e尸rTE Q (f(ST ) _ ( )2γf()) = e尸rTE Q (1ST>B _ ( )2γ1ST<B)
where γ = _ r/σ 2 . Can re-write this as
P (S0 , 0) = e尸rT 0 (1S>B _ ( B )2γ1S<B ) pST (S)dS
where pST (S) = e尸(log 毖(毖)0(T) 尸(r尸 σ 2 )T)2 /(2σ2 T) . The Delta is then com-
puted as
∂ & S ∂
∂S0 0 B ∂S0
Delta is positive since No Touch is worth more when S is further away from B
2022-07-28