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Unformatted text preview: 4-1CHAPTER 4: OPTION PRICING MODELSEND-OF-CHAPTER QUESTIONS AND PROBLEMS1.When we price an option according to its boundary conditions, we do not find an exact price for the option. Weprovide only limits on the maximum and minimum price of an option or group of options. In the case ofoptions differing by exercise prices or in the case of put-call parity, we can price only the relationship ordifference between the option prices. We cannot price each option individually without an option pricingmodel; however, an option pricing model must provide prices that conform to the boundary conditions.Because boundary condition rules require fewer assumptions, we can say that they are more generallyapplicable and are more likely to hold in practice. However, they are incomplete in the sense that they do nottell us exactly what the option price should be, which is what an option pricing model does tell us.2.In the one-period model, there are only two possible outcomes at expiration. In the two-period model, there arethree possible outcomes at expiration. As we extend the number of periods, the number of possible outcomesat expiration increases. There can be an infinite number of periods in which the length of each period can bemade infinitesimally small. At this point the model comes close to approximately the assumptions of theBlack-Scholes model. As a practical matter the binomial price tends to converge to the Black-Scholes price at50-100 time periods.3.A binomial option pricing model enables us to see the relationship between the stock price and the call price.The model shows, in a simple framework, how to construct a riskless portfolio by appropriately weighting thestock against the option. By noting that the riskless portfolio should return the risk-free rate, we can see whatthe call price must be. We can also understand the forces that bring the call option price in line if it is notpriced according to the model. In addition, the model illustrates the importance of revising the hedge ratio. Finally, the model is probably the best way to handle the problem of pricing an American option.4.The variables in the binomial model are S (the stock price), E (the exercise price), r (the discrete risk-freerate), u (the return on the stock if it goes up), d (the return on the stock if it goes down) and n, the number oftime periods. The variables in the Black-Scholes model are S (the stock price), E (the exercise price), rc(thecontinuously compounded risk-free rate), (the standard deviation of the continuously compounded return onthe stock) and T (the time to expiration). The variables S and E are the same in both models. In the binomialmodel, r is the discrete interest rate per period. In the Black-Scholes model, the interest rate must be expressedin continuously compounded form. The annual discrete interest rate (r) is related to the annual continuouslycompounded rate (rc) by the formula, rc= ln(1 + r). Then the binomial rate per period is found as (1 + r)T/n- 1.- 1....
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- Fall '08