is a martingale for each a real.

2

5. Suppose Mn is a martingale, Nn = Mn , and E Nn < ∞ for each n. Show

E [Nn+1 | Fn ] ≥ Nn for each n. Do not use Jensen™s inequality.

6. Suppose Mn is a martingale, Nn = |Mn |, and E Nn < ∞ for each n. Show

E [Nn+1 | Fn ] ≥ Nn for each n. Do not use Jensen™s inequality.

7. Suppose Xn is a martingale with respect to Gn and Fn = σ(X1 , . . . , Xn ). Show Xn is

a martingale with respect to Fn .

8. Show that if Xn and Yn are martingales with respect to {Fn } and Zn = max(Xn , Yn ),

then E [Zn+1 | Fn ] ≥ Zn .

2 2

9. Let Xn and Yn be martingales with E Xn < ∞ and E Yn < ∞. Show

n

E Xn Yn ’ E X0 Y0 = E (Xm ’ Xm’1 )(Ym ’ Ym’1 ).

m=1

1

10. Consider the binomial asset pricing model with n = 3, u = 3, d = 2 , r = 0.1, S0 = 20,

and K = 10. If V is a European call with strike price K and exercise date n, compute

explicitly the random variables V1 and V2 and calculate the value V0 .

11. In the same model as problem 1, compute the hedging strategy ∆0 , ∆1 , and ∆2 .

12. Show that in the binomial asset pricing model the value of the option V at time k is

Vk .

13. Suppose Xn is a submartingale. Show there exists a martingale Mn such that if

An = Xn ’ Mn , then A0 ¤ A1 ¤ A2 ¤ · · · and An is Fn’1 measurable for each n.

14. Suppose Xn is a submartingale and Xn = Mn + An = Mn + An , where both An and

An are Fn’1 measurable for each n, both M and M are martingales, both An and An

increase in n, and A0 = A0 . Show Mn = Mn for each n.

15. Suppose that S and T are stopping times. Show that max(S, T ) and min(S, T ) are

also stopping times.

104

16. Suppose that Sn is a stopping time for each n and S1 ¤ S2 ¤ · · ·. Show S = limn’∞ Sn

is also a stopping time. Show that if instead S1 ≥ S2 ≥ · · · and S = limn’∞ Sn , then S is

again a stopping time.

2

17. Let Wt be Brownian motion. Show that eiuWt +u t/2

can be written in the form

t

Hs dWs and give an explicit formula for Hs .

0

18. Suppose Mt is a continuous bounded martingale for which M is also bounded.

∞

Show that n

2 ’1

)2

(M i+1 ’ M i

2n

n 2

i=0

converges to M 1 as n ’ ∞.

[Hint: Show that Ito™s formula implies

(i+1)/2n

)2 =

(M i+1 ’ M (Ms ’ M ’M

)dMs + M .

i i i+1 i

2n 2n 2n

n 2n

2

i/2n

Then sum over i and show that the stochastic integral term goes to zero as n ’ ∞.]

1

19. Let fµ (0) = fµ (0) = 0 and fµ (x) = 2µ 1(’µ,µ) (x). You may assume that it is valid to

use Ito™s formula with the function fµ (note fµ ∈ C 2 ). Show that

/

t

1

1(’µ,µ) (Ws )ds

2µ 0

converges as µ ’ 0 to a continuous nondecreasing process that is not identically zero and

that increases only when Xt is at 0.

t

1

[Hint: Use Ito™s formula to rewrite 2µ 0 1(’µ,µ) (Ws )ds in terms of fµ (Wt ) ’ fµ (W0 ) plus a

stochastic integral term and take the limit in this formula.]

20. Let Xt be the solution to

dXt = σ(Xt )dWt + b(Xt )dt, X0 = x,

where Wt is Brownian motion and σ and b are bounded C ∞ functions and σ is bounded

below by a positive constant. Find a nonconstant function f such that f (Xt ) is a martin-

gale.

[Hint: Apply Ito™s formula to f (Xt ) and obtain an ordinary di¬erential equation that f

needs to satisfy.]

21. Suppose Xt = Wt + F (t), where F is a twice continuously di¬erentiable function,

F (0) = 0, and Wt is a Brownian motion under P. Find a probability measure Q under

105

which Xt is a Brownian motion and prove your statement. (You will need to use the

general Girsanov theorem.)

t

22. Suppose Xt = Wt ’ Xs ds. Show that

0

t

es’t dWs .

Xt =

0

23. Suppose we have a stock where σ = 2, K = 15, S0 = 10, r = 0.1, and T = 3. Suppose

we are in the continuous time model. Determine the price of the standard European call

using the Black-Scholes formula.

23. Let

ψ(t, x, y, µ) = P(sup(Ws + µs) = y for s ¤ t, Wt = x),

s¤t

where Wt is a Brownian motion. More precisely, for each A, B, C, D,

D B

P(A ¤ sup(Ws + µs) ¤ B, C ¤ Wt ¤ D) = ψ(t, x, y, µ)dy dx.

s¤t C A

(ψ has an explicit formula, but we don™t need that here.) Let the stock price St be given

by the standard geometric Brownian motion. Let V be the option that pays o¬ sups¤T Ss

at time T . Determine the price at time 0 of V as an expression in terms of ψ.

25. Suppose the interest rate is 0 and St is the standard geometric Brownian motion stock

price. Let A and B be ¬xed positive reals, and let V be the option that pays o¬ 1 at time

T if A ¤ ST ¤ B and 0 otherwise.

(a) Determine the price at time 0 of V .

(b) Find the hedging strategy that duplicates the claim V .

26. Let V be the standard European call that has strike price K and exercise date T . Let

r and σ be constants, as usual, but let µ(t) be a deterministic (i.e., nonrandom) function.

Suppose the stock price is given by

dSt = σSt dWt + µ(t)St dt,

where Wt is a Brownian motion. Find the price at time 0 of V .

106