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Unformatted text preview: MA3245: Financial Mathematics I Notes 2: Basic option theory (a) Dr. Oliver Chen ochen@math.nus.edu.sg Department of Mathematics, National University of Singapore 1 Basics 2 Properties of options 3 Summary MA3245 Notes 2: Introduction 2 / 93 Basics European option definitions European option definitions A European call (put) is a contract that gives the right (without the obligation) to purchase (sell) the underlying asset at a predetermined price and future date. The buyer of an option pays the seller a fee for the option, which is called the premium . If the option is invoked (the asset is bought (sold)) then it is said to be exercised . The predetermined date is called the maturity date or exercise date. The predetermined price is called the strike date or exercise price. MA3245 Notes 2: Introduction 4 / 93 Basics European option definitions European options (cont.) If the holder exercises the option, then the writer must sell the asset to the holder for a call, or buy the asset from the holder for a put. The payoff function is the value of the option at maturity as a function of the price of the underlying asset. MA3245 Notes 2: Introduction 5 / 93 Basics European option definitions European options (cont.) Exercise: Draw the payoff function for: (i) a call that has been shorted; and (ii) a put that has been shorted. The intrinsic value is the payoff that would be received if the underlying asset is at its current value when the option expires. An option is: inthemoney if the intrinsic value is positive. atthemoney if the asset is at the strike value. outofthemoney if the the asset is not at the strike value and the intrinsic value is zero. If an options premium is greater than its intrinsic value, then the difference is called the time value . The time value is due to the uncertainty in the asset price at maturity. MA3245 Notes 2: Introduction 6 / 93 Basics A naive model A naive model Lets look at a naive model for a stock price. This model will turn out to be incorrect , but for the moment it is instructive. Suppose that ZYX stock has a price of $100 today, 22/01/08 and that interest rates are constant at 5%. Suppose that in one year, 22/01/09, there are only two possibilities. One possibility is for ZYX to have a price of $130, and the other possibility is for it to have a price of $80. Finally, suppose that both possibilities have a 1 2 chance of occurring. MA3245 Notes 2: Introduction 7 / 93 Basics A naive model A naive model (cont.) What is the price of a European call struck at $100 and maturing on 22/01/09 under this model? MA3245 Notes 2: Introduction 8 / 93 Basics A naive model A naive model (cont.) If the stock has price $130 on 22/01/09, then the holder of the call can (and will) buy the stock for $100 and can sell it for $130, making a profit of $30. In this case the option has value $30 on 22/01/09....
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