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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

X_j = 1 if the jth toss results in a head

      -1 if the jth toss results in a tail.

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

M_n = S_j=1ⁿ X_j,  n ³ 1.

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

S_n = S₀e^{Mn log u}

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

Y_n = S_n/(1 + r)ⁿ

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

b. Show that the discounted stock value Y_n is a martingale with respect to F_n = s(S₁, S₂, ..., S_n).

 

2) 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.

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) = e^z + 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 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 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 ò^y_-¥ 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[e^uX+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.

P_X(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 = 2^W. 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 = 2^W 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[e^W(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 Y_n be a martingale such that Y₀ = 0. Show that

a. E[Y_n] = 0.

b. Cov(Y_n+1, Y_n)

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

Y_n = E[X|F_n]

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

11. If X(t) = e^W(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) = e^aW(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 F_t, 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(W²_t, W²_s)

14. If X(t) = e^W(t), compute the following

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

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

c. Show that

E[e^W(s)+W(t)] = e^t+s/2 e^{min{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

X_j = 1 if the jth toss results in a head

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

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

M_n = S_j=1ⁿ X_j,      n ³ 1.

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

S_n = S₀e^{Mn log u}.

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

Y_n = S_n/(1+r)ⁿ

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

18. Let (W, F, P) be a probability space. Let {F_n}_n³1 be a filtration. Let M_n denote the symmetric random walk with Z_n = M_n — M_n-1

i. Calculate g(l) = E[e^lZn].

il. Let

                                 Y_n = exp{lM_n — n log g(l)}.

Show that E[Y_n+1|F_n] = Y_n.

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

iii. Is Y_n = M²_n —n a martingale?

iv. Is Y_n = M³_n — 3nM_n a martingale?

v. Suppose q Î R and let

                         Y_n = (sechq)ⁿ e^qMn           n = 0,1,2,....

Is Y_n a martingale?. (Recall that sech(z) = 2/e^z+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 F_t, 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) = e^x+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 ò^y_-¥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

Lim_n®¥ S^n-1_i=0(W(t_i+1) – W(t_i))^2m.s.= T.

Where t_i = iT/n, 0 = t₀ < t₁ < t₂ < … < t_n-1 < t_n = 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

Lim_n®¥ S^n-1_i=0 (W(t_i+1) – W(t_i))(t_i+1-t_i) =^m.s. 0

Lim_n®¥ S^n-1_i=0(t_i+1 – t_i)² = 0

Where t_i = iT/n, 0 = t₀ < t₁ < t₂< … <t_n-1 < t_n = 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 S^n-1_i=0(In(S(t_i+1)/S(t_i)))²

Where

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

and t_i = T₁ + i(T₂ – T₁)/n.