P = -5 (- ) + ~" (-( - aVx)). (2.1b)

Therefore, we have the following relationship, referred to as put-call parity:

C-P=S-Ke~r\ (2.2)

Intuitively, the left-hand side is the value of a bought call and a sold put with the same strike, maturity and volatility. Now either S > Ke~rz, in which case the put would be worth nothing if exercised now but the call would be worth S - Ke~n in present value terms. Or 5 < Ke~rx, in which case the call would be worth nothing if exercised now but the put has worth -(S - Ke~").

Some people speak of call options being in the money (ITM) if the current price is above the discounted value of the strike; out of the money (OTM) if the current price is below the discounted value of the strike; and at the money (ATM) if the current price is equal to the discounted value of the strike. However, it is more natural to define the moneyness of an option so that options are ITM, OTM or ATM according as moneyness is positive, negative or zero. A number of slightly different measures of moneyness are used (and so there are corresponding differences in the definition of the ATM strike). Many people use x itself for the moneyness; this is done in Figure 2.1. Thus an option is ITM, ATM or OTM according as x > 0, x - 0, or x < 0. Under constant volatility this definition of the moneyness is approximately the same as A - \, where is the option delta (§2.3.3). However, there are other less formal definitions of moneyness that have slipped into common use.6

As is obvious from formulae (2.1), the Black-Scholes price of an option has a non-linear relationship with volatility. Some graphs of a call price (left-hand scale) and the moneyness measure x (right-hand scale) against volatility are shown in Figure 2.1. The reader may generate their own figures using the implied volatility spreadsheet on the CD. These figures show that Black-Scholes call option prices increase monotonically with volatility. Intuitively, the greater the volatility, the more uncertainty and the greater the value of the option.7 Note that ATM call option prices are approximately linear with respect to volatility, but this is not the case for ITM and OTM call option prices. The sensitivity of an option price to changes in volatility is commonly called the option vega.8 ATM options have the greatest sensitivity to volatility because they can so easily move ITM or OTM even with low levels of volatility (§2.3.3 and §5.4).

6Other definitions include: S - Ab~rT; the same without discounting the strike; the ratio S/K (with or without discounting K); and ln(S/K) with or without discounting the strike. With these definitions the moneyness is not comparable between different options because it will vary with the price of the underlying. The reason why x is often the preferred definition for moneyness is that in the first term of x the numerator, \n(S/Ke~rT), measures the deviation between the underlying and the discounted strike and the denominator, , is the -period volatility. Therefore, in x the deviation is normalized by the volatility, which makes x comparable between different options.

7Without uncertainty, options would have no intrinsic value.

8This is a term that was invented by Americans, and intended to sound like a Greek letter. Some people call it the option zeta, and at least in this case there is no confusion about the letter of the alphabet that should be used in the notation.

Black-Scholes call option prices increase monotonically with volatility

Figure 2.1 Call price and moneyness versus volatility: S = and ( ) = 95; (b) = 100; ( ) = 105.

100, = 0.012,

Exactly the same implied volatility should be implicit in the prices of all options of the same type on the same underlying asset. However, this assumes that the pricing model is correct, and generally it is not

If the market price of an option can be observed, along with the current price of the underlying S, then every variable in the Black-Scholes option pricing model has a known value - except the volatility. Therefore, the pricing model may be used with the known quantities , r and x, and the observed quantities and S, to back out the volatility. In fact implied volatility is really just an inverse option price in the sense of the inverse function theorem.

There is no closed-form solution for implied volatility, even when the option pricing model has an analytic form. We know that implied volatility is usually increasing with option prices, but there is no simple formula that relates implied volatility to the market price of the option, the underlying price and the other variables. Instead one has to use numerical methods to solve for volatility as an implicit function of the known quantities:9

Implied Volatility =f(C, K, S, x, r).

Within the Black-Scholes assumptions there is a constant volatility for the underlying process, so all options on the same underlying should give the same implied volatility. Put another way, exactly the same implied volatility should be implicit in the prices of all options of the same type on the same underlying asset. However, this assumes that the pricing model is correct, and generally it is not. When the Black-Scholes model is used, one obtains different implied volatilities for different strikes and maturities on the same underlying asset.

Rather than changing the model, a quick fix is to change the only unknown factor-the volatility

Therefore either the market information is not accurate or the market does not believe in the assumptions of the Black-Scholes model: in particular, the constant volatility and the normality of returns may be questioned by the market. Market prices often reflect properties of the price process that are not assumed in the Black-Scholes model. Rather than changing the model, a quick fix is to change the only unknown factor - the volatility. There is nothing else that appears to be unknown so the Black-Scholes model uses different implied volatilities for different strikes and maturities.

2.1.2 Call and Put Implied Volatilities

In a simple European option, if the pricing model is applied correctly then it should not matter whether one uses a call price or a put price to back out implied volatility. If a put and a call of the same maturity are both available for a given strike, the implied volatility in the call should be the same as the implied volatility in the put. Even though one is ITM and the other is OTM, the same constant volatility is implicit in both option prices.10 However, some sources

Often Newton-Raphson iteration is the preferred method, much faster than the simple method of bisections (Chriss, 1997). See the implied volatility spreadsheet on the CD.

10Also, in the put-call parity relationship (2.2) the right-hand side is independent of volatility. Therefore, if the price of a call of a fixed strike goes up due to an increase in volatility, the price of the put of the same strike must go up by an equal amount.