It is the relationship between implied volatility and strike price for options with a certain maturity

Chapter 18

• Chapter 18

It is the relationship between implied volatility and strike price for options with a certain maturity

• It is the relationship between implied volatility and strike price for options with a certain maturity

• The volatility smile for European call options should be exactly the same as that for European put options

• The same is at least approximately true for American options

Put-call parity p +S0e-qT = c +Ke–r T holds for market prices (pmkt and cmkt) and for Black-Scholes prices (pbs and cbs)

• Put-call parity p +S0e-qT = c +Ke–r T holds for market prices (pmkt and cmkt) and for Black-Scholes prices (pbs and cbs)

• It follows that pmkt−pbs=cmkt−cbs

• When pbs=pmkt, it must be true that cbs=cmkt

• It follows that the implied volatility calculated from a European call option should be the same as that calculated from a European put option when both have the same strike price and maturity

Both tails are heavier than the lognormal distribution

• Both tails are heavier than the lognormal distribution

• It is also “more peaked” than the lognormal distribution

What is the volatility smile if

• What is the volatility smile if

• True distribution has a less heavy left tail and heavier right tail

• True distribution has both a less heavy left tail and a less heavy right tail

Plot implied volatility against K/S0 (The volatility smile is then more stable)

• Plot implied volatility against K/S0 (The volatility smile is then more stable)

• Plot implied volatility against K/F0 (Traders usually define an option as at-the-money when K equals the forward price, F0, not when it equals the spot price S0)

• Plot implied volatility against delta of the option (This approach allows the volatility smile to be applied to some non-standard options. At-the money is defined as a call with a delta of 0.5 or a put with a delta of −0.5. These are referred to as 50-delta options)

Asset price exhibits jumps rather than continuous changes

• Asset price exhibits jumps rather than continuous changes

• Volatility for asset price is stochastic

• In the case of an exchange rate volatility is not heavily correlated with the exchange rate. The effect of a stochastic volatility is to create a symmetrical smile
• In the case of equities volatility is negatively related to stock prices because of the impact of leverage. This is consistent with the skew that is observed in practice

In addition to calculating a volatility smile, traders also calculate a volatility term structure

• In addition to calculating a volatility smile, traders also calculate a volatility term structure

• This shows the variation of implied volatility with the time to maturity of the option

The volatility term structure tends to be downward sloping when volatility is high and upward sloping when it is low

• The volatility term structure tends to be downward sloping when volatility is high and upward sloping when it is low

If the Black-Scholes price, cBS is expressed as a function of the stock price, S, and the implied volatility, imp, the delta of a call is

• If the Black-Scholes price, cBS is expressed as a function of the stock price, S, and the implied volatility, imp, the delta of a call is

• Is the delta higher or lower than