1 Volatility Smiles Chapter 18

2 Put-Call Parity Arguments Put-call parity p +S 0 e -qT = c +X e –r T holds regardless of the assumptions made about the stock price distribution Put-call parity p +S 0 e -qT = c +X e –r T holds regardless of the assumptions made about the stock price distribution It follows that It follows that p mkt -p bs =c mkt -c bs

3 Implied Volatilities 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 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 The same is approximately true of American options The same is approximately true of American options

4 Volatility Smile A volatility smile shows the variation of the implied volatility with the strike price A volatility smile shows the variation of the implied volatility with the strike price The volatility smile should be the same whether calculated from call options or put options The volatility smile should be the same whether calculated from call options or put options

5 The Volatility Smile for Foreign Currency Options Implied Volatility Strike Price

6 Implied Distribution for Foreign Currency Options The implied distribution is heavier in both tails than for the lognormal distribution. The implied distribution is heavier in both tails than for the lognormal distribution. It is also “more peaked” than the lognormal distribution It is also “more peaked” than the lognormal distribution

7 The Volatility Smile for Equity Options Implied Volatility Strike Price

8 Implied Distribution for Equity Options The right tail is less heavy and the left tail is heavier than the lognormal distribution

9 Other Volatility Smiles? What is the volatility smile if True distribution has a less heavy left tail and heavier right tail 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 True distribution has both a less heavy left tail and a less heavy right tail

10 Possible Causes of Volatility Smile Asset price exhibiting jumps rather than continuous change Asset price exhibiting jumps rather than continuous change Volatility for asset price being stochastic Volatility for asset price being stochastic (One reason for a stochastic volatility in the case of equities is the relationship between volatility and leverage) (One reason for a stochastic volatility in the case of equities is the relationship between volatility and leverage)

11 Volatility Term Structure 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 This shows the variation of implied volatility with the time to maturity of the option

12 Volatility Term Structure The volatility term structure tends to be downward sloping when volatility is high and upward sloping when it is low (mean reversion)

13 Example of a Volatility Surface Strike Price mnth mnth mnth year year year

14 Brief Review

15 Consider a 3-month put (European) on a stock with K = $20. Over the next 3 months the stock is expected to either rise by 10% or drop by 10%. The risk-free rate is 5%. Consider a 3-month put (European) on a stock with K = $20. Over the next 3 months the stock is expected to either rise by 10% or drop by 10%. The risk-free rate is 5%. Q1: What is the risk-neutral probability of the up-move? Q1: What is the risk-neutral probability of the up-move? Q2: what if the stock pays a dividend yield of 3% per year? Q2: what if the stock pays a dividend yield of 3% per year?

16 Q3: What position in the stock is necessary to hedge a long position in 1 put option (assume no dividends)? Q3: What position in the stock is necessary to hedge a long position in 1 put option (assume no dividends)? Buy 0.5 shares Buy 0.5 shares Q4: How would the answer to Q3 change if the stock paid a dividend yield of 3%? Q4: How would the answer to Q3 change if the stock paid a dividend yield of 3%? Q5: What is the value of the put? Q5: What is the value of the put?

17 Assume now that the put has 6 months to maturity and there are 2 periods on the tree (each period is 3-moths long). Everything else is the same. Assume now that the put has 6 months to maturity and there are 2 periods on the tree (each period is 3-moths long). Everything else is the same. Q6: Compute the value of the put option. Q6: Compute the value of the put option. Q7: Answer question 6 for an American put. Q7: Answer question 6 for an American put. Q8: The cost of hedging which option is higher – American or European put? Compute it for both. Compute Gamma of the European put option. Can you hedge your Gamma-exposure by trading futures? Q8: The cost of hedging which option is higher – American or European put? Compute it for both. Compute Gamma of the European put option. Can you hedge your Gamma-exposure by trading futures?

18 Consider a European call option on FX. The exchange rate is $/FX, the strike price is , the T is 1 year, the domestic risk-free rate is 5% the foreign risk-free rate is 3%. If the call is selling for 0.05 $/FX, is there arbitrage and, if so, how would you exploit it? Consider a European call option on FX. The exchange rate is $/FX, the strike price is , the T is 1 year, the domestic risk-free rate is 5% the foreign risk-free rate is 3%. If the call is selling for 0.05 $/FX, is there arbitrage and, if so, how would you exploit it? Hence, borrow and sell units of FX, invest the PV of K and buy the call today. Hence, borrow and sell units of FX, invest the PV of K and buy the call today. At maturity, you have $K which is more than enough to buy the currency. At maturity, you have $K which is more than enough to buy the currency.

19 A trader in the US has a portfolio of derivatives on the AUD with the delta of 450. The USD and AUD risk-free rates are 5% and 7%. A trader in the US has a portfolio of derivatives on the AUD with the delta of 450. The USD and AUD risk-free rates are 5% and 7%. Q1: What position in the AUD creates a delta- neutral position? Q1: What position in the AUD creates a delta- neutral position? Short 450 AUD Short 450 AUD Q2: What position in 1-year futures contract on the AUD creates a delta-neutral position? Q2: What position in 1-year futures contract on the AUD creates a delta-neutral position? Redo Q2 for the case of forwards. Redo Q2 for the case of forwards.

20 A portfilio of derivatives on a stock has a delta of 2000 and a gamma of A portfilio of derivatives on a stock has a delta of 2000 and a gamma of Q1: What position in the stock would create a delta-neutral position? Q1: What position in the stock would create a delta-neutral position? Short 2,000 shares Short 2,000 shares Q2: If an option on the stock with a delta of 0.6 and a gamma of 0.04 can be traded, what position in the option and the stock creates a portfolio that is both gamma and delta neutral? Q2: If an option on the stock with a delta of 0.6 and a gamma of 0.04 can be traded, what position in the option and the stock creates a portfolio that is both gamma and delta neutral? Short 3,500 shares and buy 2,500 options Short 3,500 shares and buy 2,500 options