STOCHASTIC METHODS IN FINANCE 2021–22 STAT0013
Exercises 4 – Combining Options
The aim of this Exercise Sheet is to help you develop a deeper understand- ing of the no-arbitrage principle. In particular, to help you learn how to build and value portfolios, and how to use portfolios together with the no- arbitrage principal to determine pricing results. In all questions the interest rate is constant for all maturities and equal to r% (if not specified) per an- num with continuous compounding.
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1. What is the relationship (at time 0) between the value of a European call option and the value of a European put option on the same under- lying asset with the same strike price and maturity date T ? When are the two values the same?
(Hint: Consider two portfolios A and B involving call and put options, cash and shares, such that A and B have the same value at T.)
2. A binary call option is an option that pays £1 if the stock price ST at time T is greater than X (exercise price) and 0 otherwise. A binary put option is an option that pays £1 if the stock price ST at time T is less than X and 0 otherwise. What is the relationship between the values of a call and a put European binary options at time 0?
(Hint: Consider two portfolios A and B involving binary call and put options and cash such that A and B have the same value at T.)
3. Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively. How can the options be used to create: (a) a bull spread and (b) a bear spread? For both spreads, construct a table that shows the profit and payoff, and draw a graph of the profit.
4. A call with a strike price of $60 costs $6. A put with the same strike price and expiration date costs $4. Construct a table that shows the profit from a straddle, and draw a graph of the profit. For what range of stock prices would the straddle lead to a loss?
5. Suppose that c1, c2 and c3 are the prices of European call options with strike prices X1, X2 and X3, respectively, where X3 > X2 > X1 and X3 − X2 = X2 − X1. All options have the same maturity. Show that c2 ≤ 0.5(c1 + c3).
(Hint: Consider a portfolio that is long one option with strike price X1, long one option with strike price X3, and short two options with strike price X2.)
6. Suppose that p1, p2 and p3 are the prices of European put options with strike prices X1, X2 and X3, respectively, where X3 > X2 > X1 and X3 − X2 = X2 − X1. All options have the same maturity. Show that p2 ≤ 0.5(p1 + p3).
(Hint: Consider a portfolio that is long one option with strike price X1, long one option with strike price X3, and short two options with strike price X2.)
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