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MA6629 – Advanced Stochastic Analysis in Finance

Midterm exam

This is an open-book exam. A non-programmable calculator is allowed.

The answer sheet should be uploaded to Canvas by 10:10pm. Late submissions will not be accepted.

This exam consists of 5 independent exercises (120 points at total). Each answer needs to be justiﬁed carefully, with a detailed proof (unless explicitly speciﬁed). Numerical results should be given with an accuracy of 3 decimal digits.

In the whole exam, we consider a standard Brownian Motion (wt)t>0, and we denote by (ft)t>0 the associated ﬁltration.

You may use (without proving it) the following formula for the Gaussian distribution. For x ~

N(u, 72) (u e R, 7 > 0), E[ex ] = eu+ .

1. (24 points) In this exercise, we consider the random variable x = (w1 )2 + (w3 )2 - 2(w4 )2 .

(i) (4 points) Compute the expectation of x .

(ii) (4 points) Let Z ~ N(0, 1). Compute E[Z4] by using successive integrations by parts. (iii) (8 points) Compute the variance of x .

(iv) (8 points) Determine the conditional expectation E[x|f2]. Explain in detail which prop- erties are used.

2. (16 points) Consider two standard Brownian Motions x = (xt)t>0 and x/ = (x )t>0. Wet(/) assume that x and x/ are independent. We also let yt = xt - xt(/) , t > 0.

(i) (4 points) For any s, t > 0, compute the distribution of xt - xs(/) . (ii) (4 points) Let a > 0. For t > 0, what is the distribution of ya.t ?

(iii) (8 points) Show that there exists a unique constant a > 0 such that (Zt)t>0 = (ya.t)t>0 is a standard Brownian Motion (i.e. ﬁnd a explicitly, and prove that for this value, (ya.t)t>0 is indeed a standard Brownian Motion). A detailed proof is expected.

3. (20 points)

(i) (10 points) Compute d(s2 . ws) by applying the product rule. (ii) (10 points) Determine the distribution of 0(t)(s . ws)ds.

4. (20 points) Recall that for any real number z, |z| denotes its integer part, i.e. the largest integer which is < z. Hence, for example: |3.27| = |3.99| = |3| = 3.

(i) (10 points) Determine the distribution of

|s|dws .

(ii) (10 points) Compute

cos ╱w||s│ 、dws .

5. (40 points) Let a e R and 7 > 0 be given. For all t > 0, let yt = at + 7wt .

(i) (4 points) Compute dyt (in terms of dt and dwt).

(ii) (4 points) For all t > 0, compute E[eyt].

(iii) (4 points) For all t > 0, let Zt = yteyt, and u(t) = E[Zt]. Compute u(0). (iv) (6 points) Compute dZt (in terms of dt and dwt).

(v) (8 points) Find an ordinary diﬀerential equation (ODE) satisﬁed by u(t).

(vi) (6 points) Determine u(t) for all t > 0 (hint: look for a solution to the ODE of the form u(t) = eAtv(t), for some well-chosen A).

(vii) (8 points) We consider the Black-Scholes model with drift u, volatility 7 > 0, and initial price 50. The interest rate is r. For T > 0, compute E[f (5T)], where f (z) = z lnz.