Szymon Borak Wolfgang Karl Härdle - Statistics of Financial Markets: Exercises and Solutions

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Part 1
Option Pricing

Szymon Borak , Wolfgang Karl Hrdle and Brenda Lpez-Cabrera Universitext Statistics of Financial Markets 2nd ed. 2013 Exercises and Solutions 10.1007/978-3-642-33929-5_1

Springer-Verlag Berlin Heidelberg 2013

1. Derivatives

Szymon Borak 1, Wolfgang Karl Hrdle 1 and Brenda Lpez-Cabrera 1

(1)

Humboldt-Universitt zu Berlin Ladislaus von Bortkiewicz Chair of Statistics, C.A.S.E. Centre for Applied Statistics and Economics School of Business and Economics, Berlin, Germany

Abstract

A derivative (derivative security or contingent claim) is a financial instrument whose value depends on the value of others, more basic underlying variables. Options, future contracts, forward contracts, and swaps are examples of derivatives. The aim of this chapter is to present and discuss various options strategies. The exercises emphasize the differences of the strategies through an intuitive approach using payoff graphs.

Dont put all eggs in one basket

A derivative (derivative security or contingent claim) is a financial instrument whose value depends on the value of others, more basic underlying variables. Options, future contracts, forward contracts, and swaps are examples of derivatives. The aim of this chapter is to present and discuss various options strategies. The exercises emphasize the differences of the strategies through an intuitive approach using payoff graphs.

Exercise 1.1 (Butterfly strategy).

Consider a butterfly strategy: a long call option with an exercise price of 100 USD, a second long call option with an exercise price of 120 USD and two short calls with an exercise price of 110 USD. Give the payoff table for different stock values. When will this strategy be preferred?

The payoff table for different stock values:

This strategy is preferred when the stock price fluctuates slightly around 110 USD.

Exercise 1.2 (Risk of a strategy).

Consider a simple strategy: an investor buys one stock, one European put with an exercise price K, sells one European call with an exercise price K. Calculate the payoff and explain the risk of this strategy.

This is a risk-free strategy. The value of portfolio at time T is the exercise price K , which is not dependent on the stock price at expiration date.

Exercise 1.3 (Bull call spread).

One of the most popular types of the spreads is a bull spread. A bull-call-price spread can be made by buying a call option with a certain exercise price and selling a call option on the same stock with a higher exercise price. Both call options have the same expiration date. Consider a European call with an exercise price of K 1 and a second European call with an exercise price of K 2 . Draw the payoff table and payoff graph for this strategy.

Suppose that a trader buys a call for 12 USD with a strike price of K 1=100 USD and sells a call for 8 USD with a strike price of K 2=120 USD. If the stock price is above 120 USD, the payoff from this strategy is 16 USD (8 USD from short call, 8 USD from long call). The cost of this strategy is 4 USD (buying a call for 12 USD, selling a call for 8 USD). If the stock price is between 100 and 120 USD, the payoff is S T 104. The bull spread strategy limits the traders upside as well as downside risk. The payoff graph for the bull call spread strategy is shown in Fig..

Fig. 1.1

Bull call spread SFSbullspreadcall

Strategy
A long call at 100		S T 100
A long call at 120
Two short calls at 110
Total

Strategy
Buy a stock	S T	S T
Buy a put	K S T
Sell a call
Total	K	K

Strategy
A long call at K 1
A short call at K 2

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