73323319-16-Volatility-Smiles - V olatility Stniles How...

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Volatility Stniles How close are the market prices of options to those predicted by Black-Scholes? Do traders really use Black-Scholes when determining a price for an option? Are the probability distributions of asset prices really lognormal? In this chapter we answer these questions. We explain that traders do use the Black-Scholes model-but not in exactly the way that Black and Scholes originally intended. This is because they allow the volatility used to price an option to depend on its strike price and time to maturity. A plot of the implied volatility ofan option as a function ofits strike price is known as a volatility smile. In this chapter we describe the volatility smiles that traders use in equity and foreign currency markets. We explain the relationship between a volatility smile and the risk-neutral probability distribution being assumed for the future asset price. We also discuss how option traders allow volatility to be a function of option maturity and how they use volatility surfaces as pricing tools. 16.1 PUT-CAll PARITY REVISITED Put-<:all parity, which we explained in Chapter 9, provides a good starting point for understanding volatility smiles. It is an important relationship between the price c of a European call and the price P of a European put: (16.1) The call and the put have the same strike price, K, and time to maturity, T. The variable So is the price of the underlying asset today, r is the risk-free interest rate for maturity T, and q is the yield on the asset. A key feature of the put-<:all parity relationship is that it is based on a relatively simple no-arbitrage argument. does not require any assumption about the probability distribution of the asset price in the future. is true both when the asset price distribution is lognormal and when it is not lognormal. Suppose that, for a particular value of the volatility, PBS CBS are the values of European put and call options calculated using the Black-Scholes model. Suppose 375
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CHAPTER 16 further that Pmkt and Cmkt are the market values of these options. Because put-eall parity holds for the Black-Scholes model, we must have o -qT K -rT PBS + "oe = CBS + e In the absence of arbitrage opportunities, it also holds for the market prices, so that + Soe- qT = + Ke- rT Subtracting these two equations, we get - = CBS - (16.2) This shows that the dollar pricing error when-the Black-Scholes model is used to price a European put option should be exactly the same as the dollar pricing error when it is used to price a European call option with the same strike price and time to maturity. Suppose that the implied volatility of the put option is 22%. This means that = when a volatility of 22% is used in the Black-Scholes model. From equa- tion (16.2), it follows that CBS = when this volatility is used. The implied volatility of the call is, therefore, also 22%. This argument shows that the implied volatility of a European call option is always the same as the implied volatility of a European put option when the two have the same strike price and maturity date. To put this another
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This note was uploaded on 01/30/2012 for the course MATH 174 taught by Professor Donblasius during the Spring '11 term at UCLA.

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73323319-16-Volatility-Smiles - V olatility Stniles How...

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