- Financial derivatives are financial instruments that are written as contracts and their values depend on some underlying assets (bonds, stocks, commodities, currencies, etc) and so they are named derivatives.
- Examples of financial derivatives are options, futures, forwards and swaps contracts
- A financial option is a contract between two parties for exchanging risk with return and it gives one party, the holder of the option, the right but not the obligation to exercise the option under certain conditions on or at any time until a prescribed date known as maturity T.
- An option that gives its holder the right to buy an underlying security is called a call option, while the option that gives its holder the right to sell an underlying security is called a put option.
- European call option (c) is an option that gives its holder the right but not the obligation to buy the underlying security at maturity date T and for a pre-determined exercise price K.
- The price of a European call option at maturity date, T, has therefore a pay-off of \(c(T) = max(S(T)−K, 0)\), where \(S(T)\) is the price of the underlying security at date T (See Figure 1).
- European put option (p) is an option that gives its holder the right but not the obligation to sell the underlying security at maturity date T and for a pre-determined exercise price K.
- The price of a European put option at maturity date, T, has therefore a pay-off of \(p(T) = max(K-S(T), 0)\), where \(S(T)\) is the price of the underlying security at date T.
- Asian option has a pay-off at the maturity date that depends on the average price of the underlying security over a prescribed period. This reduces the possibility of price manipulation to the pay-off when using European options that depend on the price of the security only at the maturity date T.
- The average of the share price can be arithmetic or geometric, and it is calculated over a prescribed fixed period of time, which ends at the maturity date.
- The price of an Asian call option at maturity date, T, has therefore a value of \(c(T) = max(A(T)-K, 0)\), where \(A(T)\) is the average of the share price of the underlying security over the period \([t_0, T]\).
- The price of an Asian put option at maturity date, T, has therefore a value of \(p(T) = max(K − A(T), 0)\), where \(A(T)\) is the average of the share price of the underlying security over the period \([t_0, T]\).
- An Asian call option costs less than a European call option, provided both contracts have the same conditions.
- An Asian put option costs more than a European put option, provided both contracts have the same conditions.
- American call option (C) is a type of options that gives its holder the right but not the obligation to buy the underlying security at anytime until the maturity date T and for a pre-determined exercise price K.
- American put option (P) is a type of options that gives its holder the right but not the obligation to sell the underlying security at anytime until the maturity date T and for a pre-determined exercise price K.
- Matching the conditions in the two contracts, an American option costs more than a European option, since the American option gives more choices to its holder in addition to the choice that the European option gives (See Figure 5).
- The relation between put and call options is called put-call parity
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| Figure 1: The pay-off of a European call option with exercise price K=10 |
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| Figure 2: Pay-off of investment strategy that is based on a call option with initial price S0=9 and K=8 |
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| Figure 3: Pay-off of investment strategy that is based on a put option with initial price S0=9 and K=8 |
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| Figure 4: Straddle: an investment strategy that is based on holding call and put option with initial price S0=9 and K=8 |
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| Figure 5: American and European put options priced with K=8 σ=0.2, r=0.04 and T=1 |
References
- Wilmott P, Howison S and Dewunne J, The Mathematics of Financial Derivatives: A Student Introduction, (1995).
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