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Homework answers / question archive / 1) Let (W, F, P) be a probability space on which the following random variables are defined

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 M_{0}, M_{1}, M_{2},... defined by M_{0} = 0

M_{n} = S_{j=1}^{n} X_{j}, n ³ 1.

Suppose the price S_{n} of a risky asset at time n is given by

S_{n} = S_{0}e^{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(S_{1}, S_{2}, ..., 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) = W^{3}(t) — 3tW(t). Is X(t) a martingale?

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

X(t) = e^{-1/2}^{l2t} 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-1/2}^{s2)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 — 1/2s^{2}) and

p(t, x, y) = 1/(syÖ2pt) e – (log(y/x) – vt)^{2}/2s^{2}(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-1/2}^{s2)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^{-rT}F (d_(T, S(0)))

where

d_{+}(T, S(0)) = 1/sÖT [In (S(0)/K) + (r + ½ s^{2}) T]

d_{-}(T, S(0)) = 1/sÖT [In (S(0)/K) + (r – ½ s^{2}) /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^{-1/2z2}dz.

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[X^{2}Y^{2}]

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 = X^{2}.

a. Show that X and Y are not independent.

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

3. Let X = (X_{1}, X_{2}) 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}(x_{1}, x_{2}) = ¼ if x_{1}= -1 and x_{2} = 1

= ½ if x_{1} = 0 and x_{2} = 0

= ¼ if x_{1} = 1 and x_{2} = 1

= 0 otherwise

a. Show that X_{1} and X_{2} are not independent.

b. Show that E[X_{1}X_{2}] = E[X_{1}] . E[X_{2}].

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 W^{2}(t) a martingale?

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

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

a martingale for 0 __<__ s __<__ t.

9. Let Y_{n} be a martingale such that Y_{0} = 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/2}X(t) is a martingale.

c. Show Y(t) = e^{aW(t)-1/2}^{a2t} 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^{-1/2}^{l2t} sinh(lW(t))

are martingales.

13. a. Compute E[W^{3}(t)] and E[/W^{4}(t)|

b. Find E[(W^{2}(t)—t)(W^{2}(s) —s)]. (Hint: use the fact that of W^{2}(t) - t is a martingale).

c. Compute E[W^{2}(t)W^{2}(s)].

d. Compute Cov(W^{2}(t), W^{2}(s)).

e. Find Corr(W^{2}_{t}, W^{2}_{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[Y^{2}(t)).

c. Find Var[Y^{2}(t)].

16. Define a new process by Y(t) = tW(1/t), t > 0, and define Y_{0} = 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 M_{0}, M_{1}, M_{2},... defined by M_{0} = 0

M_{n} = S_{j=1}^{n} X_{j}, n ³ 1.

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

S_{n} = S_{0}e^{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(S_{1}, S_{2}, ..., 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}}_{n}_{³1} 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} = M^{2}_{n} —n a martingale?

iv. Is Y_{n} = M^{3}_{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) = W^{2}(t) — t is a martingale.

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

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

is a martingale. (Hint: Use the identity cosh(z) = e^{x}+e^{-z}/2)

c. Show that X(t) = W^{3}(t) — 3tW(t) is a martingale.

d. Find Corr(W^{2}(t), W^{2}(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-}^{m}^{t-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-1/2}^{s2)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 – 1/2s^{2}) and

P(t, x, y) = 1/syÖ2pt e^{-1/2 (log(y/x)-vt)2/2}^{s2(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-1/2}^{s2)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^{-rT}f (d_(T, S(0)))

Where

d_{+}(T, S(0)) = 1/sÖT[In (S(0)/K) + (r+1/2s^{2})T]

d_{-}(T,S(0)) = 1/sÖT [In(S(0)/K) + (r-1/2s^{2})T]

and f is the cumulative standard normal distribution function

f(y) = 1/Ö2p ò^{y}_{-}_{¥}e^{-1/2z2}dz = 1/Ö2pò^{¥}_{-y}e^{-1/2z2}dz.

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 = t_{0} < t_{1} < t_{2} < … < 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 = t_{0} < t_{1} < t_{2}< … <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-1/2}^{s2)t+}^{sW(t)}.

Show that

s^{2} » 1/T_{2}-T_{1} S^{n-1}_{i=0}(In(S(t_{i+1})/S(t_{i})))^{2}

Where

T_{1} = t_{0} < t_{1} < t_{2} <---<t_{n-1} < t_{n} =T_{2}

and t_{i} = T_{1} + i(T_{2} – T_{1})/n.