Derivatives
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 t
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Derivatives
Don’t 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: Strategy
ST 100
100 < ST 110
110 < ST 120
120 < ST
A long call at 100 A long call at 120 Two short calls at 110 Total
0 0 0 0
ST 100 0 0 ST 100
ST 100 0 2.110 ST / 120 ST
ST 100 ST 120 2.110 ST / 0
This strategy is preferred when the stock price fluctuates slightly around 110 USD.
S. Borak et al., Statistics of Financial Markets, Universitext, DOI 10.1007/978-3-642-33929-5 1, © Springer-Verlag Berlin Heidelberg 2013
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1 Derivatives
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. Strategy Buy a stock Buy a put Sell a call Total
ST K ST K ST 0 K
ST > K ST 0 .ST K/ K
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 K1 and a second European call with an exercise price of K2 . Draw the payoff table and payoff graph for this strategy. Strategy A long call at K1 A short call at K2 Total
ST K1 0 0 0
K1 < ST K2 ST K1 0 ST K1
K2 < ST ST K1 K2 ST K2 K1
Suppose that a trader buys a call for 12 USD with a strike price of K1 D 100 USD and sells a call for 8 USD with a strike price of K2 D 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 ST 104. The bull spread strategy limits the trader’s upside as well as downside risk. The payoff graph for the bull call spread strategy is shown in Fig. 1.1. Exercise 1.4 (Straddle). Consider a strategy of buying a call and
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