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Unformatted text preview: EC372 Bond and Derivatives Markets Topic #6: Options II: Price determination R. E. Bailey Department of Economics University of Essex Outline Contents 1 Option prices: fundamentals 1 2 Twostate optionpricing model 3 2.1 Numerical example: call option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Multiple time periods to expiry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 The BlackScholes model 5 3.1 Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 Contingent claims analysis 7 Reading: Economics of Financial Markets , chapter 19 1 Option prices: fundamentals Option prices: fundamentals Goal: a formula for the option price: c = f ( S, X, , R, ) c = European call option premium S = underlying asset (stock) price X = exercise price = time to expiry ( T t ) R = interest factor, typically written R = e r = volatility of the underlying assets rate of return The formula links c with S : it provides a partial theory Assumptions: Frictionless markets 1. Zero transactions costs 2. Unlimited borrowing and lending at riskfree interest rate 1 3. No restrictions on shortselling 4. Perfect divisibility of underlying asset Options contracts 1. Stock pays no dividends and is protected against stock splits 2. Option is European 6 6 X/R X/R X/R S S c p Call option Put option Figure 1: Call and put option prices as a function of the asset price S . The solid line in each panel depicts the option premium as a function of the underlying asset price (for given values of the other determinants of the option premium). Markets are assumed to be in equilibrium if and only if arbitrage opportunities are absent. Hence the heavy lines appear in the regions (delimited with dashed lines) for which arbitrage profits are zero. Three approaches to the theory of option prices are commonly proposed. All are equivalent, and begin by supposing that investors choose portfolios comprising three securities: (a) options; (b) the asset underlying the asset (typical a companys ordinary shares); and (c) borrowing or lending at the riskfree rate (which could be represented by issuing or purchasing zerocoupon bonds). The three approaches are as follows: 1. It is possible to construct a portfolio of the underlying asset and options in such a way that the portfolios payoff is the same for every state. The purchase of such a portfolio will re quire a nonzero initial outlay of capital, the payoff is riskfree (its the same in every state). Hence, in the absence of arbitrage opportunities, the rate of return on the portfolio (of the asset and options) must equal the riskfree rate (for borrowing/lending). From this equality, and assumptions about the probability distribution of the underlying assets future price, it is pos sible in principle to obtain a formula for the option price. But be aware that derivation of the formula is a mathematical challenge formulae are known only for a small set of probability...
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This note was uploaded on 03/15/2012 for the course EC 372 taught by Professor R.e.bailey during the Spring '12 term at Uni. Essex.
 Spring '12
 R.E.Bailey
 Economics

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