Problem 1. In North America, it is common to issue investment products with return guarantees. For instance, a product may offer the returns of the S&P 500 index except the investor is guaranteed a 0% return over the investment horizon and the maximum total return the investor will be credited over the investment horizon is 30%.

You are the pricing actuary for a large insurance company. Your company has decided to sell an investment product for which the investments will be credited a return equal to that on an index that will experience either a 20% gain or a 10% loss at the end of one period (i.e. the single-period binomial model). The continuously compounded interest-rate (force of interest) for the period is 0.08. The customer is guaranteed a 5% return and you are to set the maximum total return the customer will receive over the period (denoted a) so that your insurance company will break even on the product.

Determine a.

Problem 2. An equity securities market model follows a multi-period binomial model. At each node of the binomial tree the current stock price S will branch to uS in the upstate and dS in the downstate. You are given that the initial stock price is 10, u = 1.25, d = 0.85 and the interest-rate is 5% effective per period.

(i) (5 points) Compute the price of a European put option on the stock which expires in 4 periods and has a strike price of 8.5.

(ii) (5 points) Compute the price of an American put option on the stock which expires in 4 periods and has a strike price of 8. 5 and describe the optimal exercise policy for this American put option.

Problem 3. You are interested in pricing and hedging a European put option on a stock using a two-period binomial model with notation and set-up as in Chapter 10 of the text.

So = 10, u = 1.15, d = 0.92, h = 1, 6 = 0, and r = 0

The put option expires at time 2 and has a strike price of 12.

Compute the price of the put option at time 0 and compute the dynamic hedging strategy needed to replicate the put option payoffs at time 2.

[As you know, dynamic hedging strategy consists of the trading positions in stock and bond at time 0 and in each of the upstate and downstates at time 1 that are needed to get the put option payoffs at time 2. In class we used the notation (A, B), (Au, Bu) and (Ad, B". So long as you clearly indicate what node the trading strategy applies to any reasonable notation is acceptable.]

Problem 4. Consider a market maker in a call option written on a stock with the following information under the Black-Scholes framework (as per Chapter 13 of McDonald).

⑴ So = 35

(ii) a = 0.30

(iii) The continuously compounded risk-free interest rate is 3%

(iv) 6 = 0.01

(v) T = 90 days (Call option expires in 90 days, 365 days in a year.)

(vi) The market maker transacts in the option at the Black-Scholes prices.

The market maker sells 100 call options at the money (i.e. strike price K = 35) and delta hedges the position at daily frequency over the next two days, closing out her position at the end of the second day.

If the stock price at the end of the first day is 34.00 and the stock price at the end of the second day is 35.25, what is the market maker's profit or loss?

Problem 5. Assume the Black-Scholes framework. Consider a one-year at-the-money European put option on a nondividend-paying stock.

You are given.

(i) The ratio of the put option price to the stock price is less than 5%.

(ii) The Delta (mathematical notation △ for this greek) of the put option is -0.4364.

(iii) The continuously compounded risk-free interest rate is 1.2%

Which of the following is closest to the value of a?

(A) 0.12

(B) 0.14

(C) 0.16

(D) 0.18

(E) 0.20

Problem 6. Consider a Black-Scholes market, where we have written a European put option on a stock with a 1-year expiration date. The underlying stock and the put option have the following properties:

• So = 125

• Stock does not pay dividends

• r = 0.02 (continuously compounded)

• a = 0.11

• Strike K = 120

• Price for the 120-strike put is 2.425017

• Delta for the put option △ = -0.271618 (Remember that you are shorting a negative-delta put option)

• Gamma for the put option is r = 0.024119

You want to Delta-Gamma hedge this option with a 110-Strike call option. We know the following about the 110-strike call option:

• Price is 17.70675

• Delta is: △ = 0.91908

• Gamma is: r = 0.01091

a) Compute the Delta-Gamma hedge position for writing the 120-strike put option and hedging with the 110-strike call option.

b) In six months (1/2 year), you take another look at your hedge. Suppose the stock price has moved to 130. What is the profit or loss you have made on your hedged portfolio at this point?