1 Math 797 FM. Homework I. Due Oct. 1, 2013

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1 The first part is homework which you need to turn in. The second part is exercises that will not be graded, but you need to turn it in together with the take-home final exam. 1 Math 797 FM. Homework I. Due Oct. 1, Assume you guess that the AAPL stock is going up for the next half year, (1) You buy 100 European call option for AAPL with K = $600, T = Jan. 18, What is your price today? (Search on google finance, and label your date of purchasing). If AAPL is $640 at t = T, what is your payoff? If AAPL is $590 on that day, how much do you lose? (2) What else can you do? Describe a strategy in terms of European put option, with K = $500, T = Jan. 18, If AAPL is $510 at t = T, what is your profit? If AAPL is $490 at t = T, what do you need to do? Remark: Never try to trade options yourself unless you are working for a trading firm, it is very risky! 2. Prove that for a European put option with maturity T, for t [0, T ] Kp(t, T ) P ut(t) max{kp(t, T ) S(t); 0} where S(t) =stock price at t, K is the strike price, p(t, T ) is the price of a zero-coupon bound with maturity T and face value $1. Hint: Let S(t) = (p(t, T ), P ut(t), S(t)), and a strategy φ(t) = (K, 1, 1). Find the value of this strategy V φ (t) and the payoff matrix X(T ), as well as Xφ. Then use the arbitrage-free assumption. 3. Let (R, P, B) be a probability space with the Boreal σ-algebra. Assume G is the smallest σ-algebra generated by all half open half closed interval [n, n + 1), for all integers n. (1) What are typical events in G look like? List two typical (nontrivial) events. (2) Let ξ(x) be a real valued, measurable function on R. Give the formula for E(ξ G). (3) Let ξ(x) = x for any x R. Is ξ measurable with respect to G? Why? If not what is the smallest σ-algebra such that ξ is measurable? 1

2 4. Show that if a filtration {F t } t 0 is right continuous, then any option time τ is also stopping time. Hint: Use the fact that (τ t) = ε>0 (τ < t + ε). Part II. Turn in at the end of the term. 1. Give an example that a optional time is not a stopping time. 2. Give an example that X is a modification of Y on (Ω, F, P), but they are not indistinguishable. 2

3 2 Math 797 FM. Homework II. Due Oct. 22, 2013 Part I. 1. Prove that a standard Brownian motion is a Martingale with respect to any filtration for the Brownian motion. 2. Prove that the process Z(t) = e σw (t) 1 2 σ2t is a Martingale with respect to any filtration for the Brownian motion, where σ > 0 is a constant. 3. Use the Ito s formula to show that R(t) = e t R(0) + r(1 e t ) + σ t 0 es t dw (s) is a solution for the SDE dr(t) = (r R(t))dt + σdw (t) Here r > 0, σ > 0 are constants. Moreover show that R(r) is normally distributed, find its mean and variance. 4. For the BSM equation together with its terminal c(t, x) and boundary conditions at x = 0,, we have given the solution using BSM(s, x; K, r, σ). (1) Verify the equation Ke r(t t) N (d ) = xn (d + ); (2) Show that c x = N(d + ). this is the delta of the option. Hint: d + = d + (x). (3) Show that the theta of the option is: c t = rke r(t t) N(d ) σx 2 T t N (d + ) (4) Show that for x > K, lim t T d ± =, but for 0 < x < K, lim t T d ± =. 5. Derive the BSM formula (PDE). You need to write your own derivation, can not copy from the note or the book directly. Part II. 1. Calculate the integral T 0 (t)dw (T ) by using the middle point ( (t i) + (t i+1 ))/2 instead of (t i ) on the interval [t i, t i+1 ) when evaluating the integral. What formula do you get? Try to recalculate the integral T 0 W (t) dw (t) using your new formula. What is the main difference between your result and the result by using the Ito s integral? Indeed this new integral formula 3

4 is called the Stratonovich integral, which is widely used in fields other than finance. 2. Goto and find the Greeks of AAPL call option expiring on Jan. 17, 2014 with strike price K = $500, compare with that of call option for MCD expiring on Jan. 17, 2014 with strike price K = $100. What conclusion do you draw about these Greeks, what do they tell about these two options? 4

5 3 Math 797 FM. Homework III. Due Nov. 26, 2013 Part I. 1. Exercise 5.3 on page Exercise 5.5 on page 253. Part II. Choose one of the following problem to turn in at the end. 1. Exercise 4.9 on page Exercise 5.8 (William Hu) 3. Exercise Exercise Exercise on page 293 (Jinchao Feng). 6. Exercise 6.7. on page 288 (Zijing Zhang). 7. Exercise 6.9 on page