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#### 1) Let (W, F, P) be a probability space on which the following random variables are defined

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1) Let (W, F, P) be a probability space on which the following random variables are defined. Consider the binomial model with 0 < d < 1 + r < u. Let the risk-neutral probabilities be given by

P = 1 + r – d/u – d,     q = u – 1 – r/u – d.

Toss a coin repeatedly and assume the probability of head on each toss is p and probability of tail on each toss is q. Let

Xj = 1 if the jth toss results in a head

-1 if the jth toss results in a tail.

Where the Xj’s are i.i.d(independent and identically distributed). Consider the stochastic process M0, M1, M2,... defined by M0 = 0

Mn = Sj=1n Xj,  n ³ 1.

Suppose the price Sn of a risky asset at time n is given by

Sn = S0eMn log u

Where d = 1/u. The discounted stock value is defined by

Yn = Sn/(1 + r)n

a. Find E|Sn] and Var[Sn].

b. Show that the discounted stock value Yn is a martingale with respect to Fn = s(S1, S2, ..., Sn).

2) Let (W, F, P) be a probability space and let {W(t), t > 0} be Brownian motion with respect to the filtration Ft, t > 0.

a. Let X(t) = W3(t) — 3tW(t). Is X(t) a martingale?

b. For l Î R, show that the following hyperbolic processes:

X(t) = e-1/2l2t cosh(lW(t))

is a martingale. (Hint: cosh(z) = ez + e-z/2)

3) Let (W, F,P) be a probability space and let {W(t),t > 0} be Brownian motion with respect to the filtration Ft , t > 0. By considering the geometric Brownian motion

S(t) = S(0)e(r-1/2s2)t+sW(t) a Î R

where r is the mean rate of return, s > 0, S(0) > 0.

Show that the stock price S(t) is a Markov process. That is for any Borel-measurable function f(y), and for any 0 < s < ¢ the function

g(x) = ò¥0 f(y)p(t, x, y)dy

where t = t - s, v= (r — 1/2s2) and

p(t, x, y) = 1/(syÖ2pt) e – (log(y/x) – vt)2/2s2(t – s)  y > 0,

satisfies

E[f(S(t))|F(s)] = g(S(s)).

p(t, x, y) is the transition density for S(t).

4) (Black-Schole-Merton formula) Let (W, F, P) be a probability space. Suppose the price S(t) of a risky asset at time t follows the geometric Brownian motion

S(t) = S(0)e(r-1/2s2)t+sW(t)  t ³ 0

where r is the mean rate of return, S(0) > 0 is the initial asset price and W(t) is a Brownian motion. Let K > 0.

a. Find E[S(t)] and Var[S(t)].

b. Show that, for T > 0,

E [e-rT (S(T) — K)+] = S(0)F (d+(T, S(0))) — Ke-rTF (d_(T, S(0)))

where

d+(T, S(0)) = 1/sÖT [In (S(0)/K) + (r + ½ s2) T]

d-(T, S(0)) = 1/sÖT [In (S(0)/K) + (r – ½ s2) /T]

and F is the cumulative standard normal distribution function

F(y) = 1/Ö2p òy-¥ e-1/2 z2 dz = 1/Ö2p ò¥-y e-1/2z2dz.

1. Suppose X and Y are uniform random variables on |0, 1| and are independent.

a. Determine the value of Cov(X, Y + 2).

b. Determine the value of E[X2Y2]

c. Determine the value of E[euX+vY] where u and v are real numbers.

2. Let X be a standard normal random variable and let Y = X2.

a. Show that X and Y are not independent.

b. Show that E[XY] = E|X] . E[Y].

3. Let X = (X1, X2) be an ordered pair that takes the points (—1, 1), (0, 0), and (1, 1) with probabilities 1/4, 1/2, and 1/4.

PX(x1, x2) = ¼ if x1= -1 and x2 = 1

= ½ if x1 = 0 and x2 = 0

= ¼ if x1 = 1 and x2 = 1

= 0     otherwise

a. Show that X1 and X2 are not independent.

b. Show that E[X1X2] = E[X1] . E[X2].

4. Let W = {1,2,3,4,5,6}, F = 2W. Let P denotes a probability measure on (W, F) with P({1}) = P({2}) = 1/12 and P({3}) = P({4}) = 1/4. Find P({5}) and P({6}) if the events {1,3,4} and {1,2,3,5} are independent.

5. Suppose a random variable (X, Y, Z) is equally likely to take any one of the values (1,0,0), (0,1,0), (0,0,1), (1,0,1), (1, 1,1).

a. Find E(X|Y).

b. Find E(Y|X).

6. Let W = {a,b,c,d}, F = 2W and P({w}) = ¼ for w Î W. Define

X(w) = 0 if w = a, d

= -1 if w = b

= 1 if w =c

Y(w) = -1 if w = a, b

= 1 if w = c, d

a. List the sets in s(X) and s(Y).

b. Find E[X|Y].

c. Find E[Y|X].

7. a. For s < t, compute E[eW(t)-W(s)].

b. Show that Cov(W(s), W(t)) = min{s, t}

c. Show that Corr(W(s), W(t)) = Ömin{s,t}/max{s,t}.

8. Let W(t) be a Brownian motion and let F(t) be a filtration for the Brownian motion.

a. Show that W(t) is a martingale.

b. Is W2(t) a martingale?

c. Show that the process Z(t), t > 0 defined by

Z(t) = e-1/2s2t+sW(t)

a martingale for 0 < s < t.

9. Let Yn be a martingale such that Y0 = 0. Show that

a. E[Yn] = 0.

b. Cov(Yn+1, Yn)

10. Suppose X is random variable (measurable with respect to F) such that X is integrable. Let

Yn = E[X|Fn]

Show that Yn is a martingale with respect to Fn. (Hint Show that E[Yn+1|Fn = Yn).

11. If X(t) = eW(t), compute the following

a. Show that X(t) is not a martingale.

b. Show that e-t/2X(t) is a martingale.

c. Show Y(t) = eaW(t)-1/2a2t is a martingale for any real number a.

12. Let (W, F, P) be a probability space and let {W(t),t > 0} be Brownian motion with respect to the filtration Ft, t > 0. For l Î R show that

Y(t) = e-1/2l2t sinh(lW(t))

are martingales.

13. a. Compute E[W3(t)] and E[/W4(t)|

b. Find E[(W2(t)—t)(W2(s) —s)]. (Hint: use the fact that of W2(t) - t is a martingale).

c. Compute E[W2(t)W2(s)].

d. Compute Cov(W2(t), W2(s)).

e. Find Corr(W2t, W2s)

14. If X(t) = eW(t), compute the following

a. Cov(X(s), X(t)).

b. Corr(X(s), X (t))

c. Show that

E[eW(s)+W(t)] = et+s/2 emin{s, t}.

15. The process Y(t) = W(t) — tW(1) is called the Brownian bridge fixed at both 0 and 1.

a. What is the distribution of Y(t)?

b. Find E[Y2(t)).

c. Find Var[Y2(t)].

16. Define a new process by Y(t) = tW(1/t), t > 0, and define Y0 = 0.

a. Find the distribution of Y(t)

b. Find the probability density of Y(t)

c. Compute Cov(Y(s), Y(t))

d. Compute E[Y (t) — Y(s)] and Var[Y (t) — Y(s)].

17. Consider the binomial model with 0 < d < 1 + r < u. Let the risk-neutral probabilities be given by

P = 1 + r – d/u – d,                 q = u – 1 – r/u – d.

Toss a coin repeatedly. Assume the probability of head on each toss is p and probability of tail on each toss is q. Let

Xj = 1 if the jth toss results in a head

= -1 if the jth toss results in a tail.

where the Xi’s are i.i.d. Consider the stochastic process M0, M1, M2,... defined by M0 = 0

Mn = Sj=1n Xj,      n ³ 1.

Suppose the price S,, of a risky asset at time n is given by

Sn = S0eMn log u.

Where d = 1/u. Show that the discounted stock value

Yn = Sn/(1+r)n

is a martingale with respect to Fn = s(S1, S2, ..., Sn). (Hint: show that E[Yn+1|Fn] = Yn).

18. Let (W, F, P) be a probability space. Let {Fn}n³1 be a filtration. Let Mn denote the symmetric random walk with Zn = Mn — Mn-1

i. Calculate g(l) = E[elZn].

il. Let

Yn = exp{lMn — n log g(l)}.

Show that E[Yn+1|Fn] = Yn.

(Trick for discrete exponential martingale: is to consider E [Yn+1/Yn Yn|Fn])

iii. Is Yn = M2n —n a martingale?

iv. Is Yn = M3n — 3nMn a martingale?

v. Suppose q Î R and let

Yn = (sechq)n eqMn           n = 0,1,2,....

Is Yn a martingale?. (Recall that sech(z) = 2/ez+e-z)

19. Let (W, F, P) be a probability space and let {W(t),t > 0} be Brownian motion with respect to the filtration Ft, t ³ 0.

a. Show that X(t) = W2(t) — t is a martingale.

b. For l Î R, show that the following hyperbolic processes:

X(t) = e-1/2l2t cosh(lW(t))

is a martingale. (Hint: Use the identity cosh(z) = ex+e-z/2)

c. Show that X(t) = W3(t) — 3tW(t) is a martingale.

d. Find Corr(W2(t), W2(s)).

20. Let W(t) : t > 0 be a Brownian motion and let F(t),t > 0 be a filtration for this Brownian motion. Show that W(t) : t > 0 is a Markov process.

That is

E[f(W(t))|F(s)] = g(W(s))

for 0 < s < t, where f and g are Borel measurable functions.

21. Let (W, F, P) be a probability space and let {W(t),t > 0} be Brownian motion with respect to the filtration F(t), t > 0. By considering the process

X(t) = mt + cW(t)

where w Î R, c > 0.

Show that for any Borel-measurable function f(y), and for any 0 < s < t the function

g(x) = 1/cÖ2p(t-s) ò¥-¥ f(y)e[-1/2[(y-m)(t-s)-x)2/c2(t-s)]dy.

Satisfies E[f(X(t))|F(s)] = g(X(s)), and hence X(t) has a Markov property. We may write g(x) as

g(x) = ò¥-¥ f(y)p(t, x, y)dy, where t = t — s and

p(t, x, y) = 1/cÖ2pt e-1/2[(y-mt-x)2/e2(t-s)]dy.

is the transition density for X(t).

22. Let (W, F, P) be a probability space and let {W(t),t > 0} be Brownian motion with respect to the filtration F(t), t > 0. By considering the geometric Brownian motion

S(t) = S(0)e(r-1/2s2)t+sW(t)   a Î R

where a Î R, s > 0, S(0) > 0.

Show that the stock price S(t) is a Markov process. That is for any Borel-measurable function f(y), and for any 0 < s < t the function

g(x) = ò¥0 f(y)p(t, x, y)dy

Where t = t – s, v = (r – 1/2s2) and

P(t, x, y) = 1/syÖ2pt e-1/2 (log(y/x)-vt)2/2s2(t-s)   y > 0,

satisfies

E[f(S(t))|F(s)| = g(S(s)).

p(t, x, y) is the transition density for S(t).

23. (Black-Schole-Merton formula) Let (W, F, P) be a probability space. Suppose the price S(t) of a risky asset at time t follows the geometric Brownian motion

S(t) = S(0)e(r-1/2s2)t+sW(t)    t ³ 0

where r > 0 is the mean rate of return, S(0) > 0 is the initial asset price and W(t) is a Brownian motion. Let K > 0.

a. Find the probability density of S(t).

b. Find E[S(t)] and Var[S(t)].

c. Show that, for T > 0,

E [e-rT (S(T) — K)+] = \$(0)f (d+(T, \$(0))) — Ke-rTf (d_(T, S(0)))

Where

d+(T, S(0)) = 1/sÖT[In (S(0)/K) + (r+1/2s2)T]

d-(T,S(0)) = 1/sÖT [In(S(0)/K) + (r-1/2s2)T]

and f is the cumulative standard normal distribution function

f(y) = 1/Ö2p òy-¥e-1/2z2dz = 1/Ö2pò¥-ye-1/2z2dz.

24. Let (W, F, P) be a probability space and let {W(t),t > 0} be a Brownian motion process. Show that the quadratic variation of W up to time T > 0 is

Limn®¥ Sn-1i=0(W(ti+1) – W(ti))2m.s.= T.

Where ti = iT/n, 0 = t0 < t1 < t2 < … < tn-1 < tn = T, n Î N.

25. Let (W, F, P) be a probability space and let {W(t), t > 0} be a Brownian motion process. Show that the following cross-variation between W (t) and t, and the quadratic variation of t, are

Limn®¥ Sn-1i=0 (W(ti+1) – W(ti))(ti+1-ti) =m.s. 0

Limn®¥ Sn-1i=0(ti+1 – ti)2 = 0

Where ti = iT/n, 0 = t0 < t1 < t2< … <tn-1 < tn = T, n Î N.

26. Let a and s > 0 be constants, and define the geometric Brownian motion

S(t) = S(O)e(a-1/2s2)t+sW(t).

Show that

s2 » 1/T2-T1 Sn-1i=0(In(S(ti+1)/S(ti)))2

Where

T1 = t0 < t1 < t2 <---<tn-1 < tn =T2

and ti = T1 + i(T2 – T1)/n.