# Project #5313 - Finance

Please download the option pricer posted on Blackboard to calculate Black & Scholes option prices (select Analytic European under Option Type).

Problem # 1

A pension fund plans to sell 1,000,000 shares of a given stock in one year and is concerned that its price may decline. The current stock price is \$ 40. The minimum price the manager would like to sell the stock for is \$ 30.

A financial institution (FI) offers the pension fund manager the following contract. If in a year the stock price is below \$ 30, the manager may sell the stock to the FI for \$ 30. If the stock price is above \$ X, the manager will sell the stock to the FI at the price of \$ X. If the stock price is between \$ 30 and \$ X, the manager will sell the stock at that price. The higher bound X is set such that initially, the value of the contract is zero (i.e. the manager pays nothing to enter the contract).

In addition, assume that the annualized volatility of the stock for the coming year will be 50 % and the annual risk-free rate is 1%.

1)             Identify the two implicit options. What are the financial institution position (i.e. which options are bought/sold) and its risk?

2)             Price the option with the strike price of \$ 30 (use the Black & Scholes formula).

3)             Determine the higher bound X.

4)             Plot the terminal payoff and profit of this strategy. When (at which levels of terminal stock price) the strategy is most unfavorable to the fund manager?

5)             Assume that the FI decides to trade the stock to hedge its Delta with respect to its position (the delta neutral portfolio). How many shares of stock must the FI trade today?

6)             What is the aggregate Gamma of the position? Is Convexity favorable to the FI? Explain. Hint: check Chapter 17 ppt.

Assume that six months have passed and the pension fund manager would like to exit the

contract prematurely. Further assume that the stock price rose to \$ 50 and at the same

time the implied volatility dropped to 40%. The annual risk-free rate is still 1%.

7)             Calculate that the terms that the FI may propose to the client to exit the contract (re-price the implicit options).

8)             What was the P&L for the FI? Decompose the performance in terms of implicit options and delta neutral hedging (assuming the delta neutral stock position initially taken was never rebalanced over the past 6 months)? What do you conclude?

Problem # 2

Today, you initiate the following option strategy (a call spread) on a given stock. You purchase the call \$ 30 and write the call \$ 32 with expiration date in 10 days (or about 10/252=0.04 of a year). The risk-free rate is 1%, the current stock price \$ 29.7 and the volatility 50 % (in addition you will assume that the stock pays no dividend).

1)   Price both options using the Black and Scholes formula.

2)   Plot your payoff and profit at maturity (in 10 days).

3)   What is the Theta of the call spread and what does it tell you? (Hint: check page 10 of Chapter 17 ppt). If nothing else changes how much would you expect to lose/gain in the next 5 trading days?

4)  Assume that you may trade the same call spread 30/32 but with expiration date in one year. What is the Theta of this spread? If nothing else changes how much would you expect to lose/gain in the next 5 trading days?

Which position is riskier in terms of passage of time (time decay)?

 Subject Business Due By (Pacific Time) 04/30/2013 05:30 pm
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