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I Implied Volatility and Correlation

What is the correct price of an option? If a model price is not the same as the market price, which is the right price?

In game theory, the concept of a solution is determined by the rationality of the players; in finance theory the concept of the correct price is determined by the nature of the modeller. The British, being practical and empirical, might say that the market is right and their model is wrong. The French-rationalists and theoreticians - might say that their model is right and the market is wrong. However, the Americans, pragmatic and diplomatic as they are, would most likely say that both the market and their model are wrong. But, to be serious, investors will attempt to trade on market prices that are close to their model price; on the other hand, quantitative analysts will calibrate an option pricing model by training it on market price. These two types of market participants are always trying to catch up with each other, and this is one of the ways in which we achieve market efficiency, if not necessarily rationality.

The observed market prices of options may have resulted from the use of several different models. However, most simple options are priced using models of the Black-Scholes type, which are based on two basic assumptions: first, that markets are complete and efficient (arbitrage-free), so there is a perfect hedge for any financial asset; and second, that underlying prices S(t) are governed by a geometric Brownian motion (GBM) diffusion process with constant volatility rj:

dS(t)/S(t) = rdt + adZ(t),

where r is the risk-free rate of return and Z is a Wiener process.1 These two assumptions are simple and powerful, but most people would agree that the Black-Scholes model is only a crude approximation of reality. In fact, the assumptions underlying the Black-Scholes model are incorrect.

This chapter will focus on some of the consequences of using the wrong model to price options. Section 2.1 shows that although there can only be one true

That is. increments dZ are independent and normally distributed with mean zero and variance di. The assumption that prices are governed by a GBM diffusion process implies that log returns are independent!) and normally distributed with constant volatility (§4.4.2).

The volatility smile is a result of using an

over-simplistic model and would not be found if options were priced using

an appropriate model

volatility for the underlying price process, many different Black-Scholes volatilities are found to be implicit in the market prices of options on the same underlying price. If one is willing to accept these volatilities (rather than invent better models that are based on the observed empirical qualities of the underlying price), then their behaviour can be described by certain patterns. First, the volatility smile pattern found in options of different strikes on the same underlying is described and explained in §2.2. The volatility smile is a result of using an over-simplistic model and would not be found if options were priced using an appropriate model. This section also looks at the term structure of implied volatilities and the shape of the whole volatility smile surface in different types of markets.

Section 2.3 looks at the relationship between underlying price changes and changes in the implied volatility of an at-the-money option. For equity indices, empirical observation on the correlation in this relationship supports the use of different volatility assumptions for pricing and hedging, depending on the current market conditions. At the end of §2.3 some joint distributions for price and volatility changes are derived and their applications to probabilistic scenario analysis are discussed. The chapter concludes with a brief account of implied correlation and its use in pricing and hedging options.

Implied volatility is the volatility of the underlying asset price process that is implicit in the market price of an option

2.1 Understanding Implied Volatility

A simple option pricing model will give a theoretical price for an option as a function of a constant volatility for the underlying price process (§1.3) and other known values such as interest rates, time to maturity, exercise prices and so on. Option writers might use a statistical model to forecast some value for the volatility of the underlying process, and then substitute this volatility into the pricing model to obtain the theoretical or model price of the option. Of course, if the option is traded, the market price may not be the same as the model price. In that case one might ask, which volatility forecast does one have to use in the model so that the model price and the market price are the same? This is the implied volatility. In a constant volatility framework implied volatility is the volatility of the underlying asset price process that is implicit in the market price of an option according to a particular model. It is a volatility forecast, not an estimate of volatility, with horizon given by the maturity of the option.2

Implied volatility is a forecast of the process volatility. If process volatility is stochastic, implied volatility may be thought of as the average volatility of the underlying asset price process that is implicit in the market price of an option. Now, whatever the assumption made about the process volatility, it is very

2Note that in an option pricing model there may be other factors with no known value, such as dividends or average tax effects. Unfortunately, volatility is often the only parameter that is adjusted to match model prices to market prices.

likely that different options on the same underlying asset will give different implied volatilities. When real-world data are used to infer parameter values, different data will give different inferences. In a way, one might view the differences between different implied volatilities for the same underlying asset as a form of sampling error.

However, there is a problem which time and time again will lead to unresolved questions and irreconcilable differences. And that is, that the generic geometric Brownian motion model for the price process is wrong.

2.1.1 Volatility in a Black-Scholes World

It is not always possible to compute the volatility that is implicit in the market price of an option according to a certain model.3 However, most implied volatilities are based on the Black-Scholes formula for European options (Black and Scholes, 1973).4 In this formula the price of a call option with strike price and time to maturity x, on an underlying asset with no dividend and current price S and volatility a, is given by

- 5 ( ) - ~ ( - / ). (2.1 )

Here denotes the risk-free rate of interest, used to discount the strike into present value terms, and (-) is the normal distribution function.

The terms ( ) and ( - ) allow uncertainty in the price process to be accounted for in the option price. The quantity x provides a measure of the moneyness of the option (as described in more detail below); it is given by

x = ln(S/Keri)/a/x + o/x/2.

The process volatility is and x is the maturity of the option in years, so the term rj/x is the x-maturity standard deviation of returns under the assumption of constant volatility (§3.3).5 The first term in x measures the divergence between the current price and the discounted value of the strike, relative to the standard deviation. The second term is there because of the volatility dependence of C.

The price of a put option on the same underlying with the same strike, maturity and volatility is

3It is also possible that a given market price could be justified with several different volatilities if the option in question is not European, or is in some other way exotic. Or there may be no volatility that equates the model price with the market price.

4A European option is one that can only be exercised on the maturity date; American options may be exercised before their maturity date and are much more difficult to price than European options.

5If volatility is quoted in annualized percentage form it should be divided by 1004 when calculating .v. Note that the number of trading days (or risk days) per year is usually taken for the conversion of a daily standard deviation into an annualized percentage; that is, often A = 250 or 252 in (1.2). But note, on the other hand, that the maturity of the option is used to discount values to today and therefore an option having h days to expiry will have = Zi/365 (or 366 in a leap year).