Volatility Smiles

1. Volatility Smiles

2. What is a Volatility Smile? • It is the relationship between implied volatility and strike price for options with a certain maturity

3. Why the volatility smile is the same for calls and puts • When Put-call parityp +S0e-qT = c +K e–r Tholds for the Black-Scholes model, we must have pBS +S0e-qT = cBS +K e–r T it also holds for the market prices pmkt +S0e-qT = cmkt +K e–r T subtracting these two equations, we get pBS －pmkt = cBS－ cmkt • It shows that the implied volatility of a European call option is always the same as the implied volatility of European put option when both have the same strike price and maturity date

4. Example • The value of the Australian dollar: $0.6(S0) Risk-free interest rate in US(per annum):5% Risk-free interest rate in Australia(per annum):10% The market price of European call option on the Australia dollar with a maturity of 1 year and a strike price of $0.59 is 0.0236.Implied volatility of the call is 14.5% The European put option with a strike price of $0.59 and maturity of 1 year therefore satisfies p +0.60e-0.10x1 = 0.0236 +0.59e-0.05x 1 so that p=$0.0419 , volatility is also 14.5%

5. Foreign currency options Implied volatility Strike price Figure 1 Volatility smile for foreign currency options

6. Implied and lognormal distribution for foreign currency options Implied Figure 2 σ(波動率) Lognormal K1 K2 S(匯價)

7. Empirical Results Table 1 Real word Lognormal model >1 SD >2 SD >3 SD >4 SD >5 SD >6 SD 25.04 5.27 1.34 0.29 0.08 0.03 31.73 4.55 0.27 0.01 0.00 0.00 Percentage of days when daily exchange rate moves are greater than one, two,… ,six standard deviations (SD=Standard deviation of daily change)

8. Reasons for the smile in foreign currency options • Why are exchange rates not lognormally distributed ? Two of the conditions for an asset price to have a lognormal distribution are : • The volatility of the asset is constant • The price of the asset changes smoothly with no jump

9. Equity options Implied Figure 3 volatility Strike Volatility smile for equities

10. Implied and lognormal distribution for equity options Implied Figure 4 σ(波動率) Lognormal s(股價) K1 K2

11. The reason for the smile in equity options • One possible explanation for the smile in equity options concerns leverage • Another explanation is “crashophobia”

12. Alternative ways of characterizing the volatility smile • 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)

13. The volatility term structure • In addition to a volatility smile, traders use a volatility term structure when pricing options • It means that the volatility used to price an at-the-money option depends on the maturity of the option

14. The volatility surfaces • Volatility surfaces combine volatility smiles with the volatility term structure to tabulate the volatilities appropriate for pricing an option with any strike price and any maturity

15. Table 2 Volatility surface

16. The volatility surfaces • The shape of the volatility smile depends on the option maturity .As illustrated in Table 2, the smile tends to become less pronounced as the option maturity increases

17. Greek letters • The volatility smile complicate the calculation of Greek letters • Assume that the relationship between the implied volatility and K/S for an option with a certain time to maturity remains the same

18. Delta of a call option is given by Greek letters Where cBS is the Black-Scholes price of the option expressed as a function of the asset price S and the implied volatility σimp

19. Greek letters • Consider the impact of this formula on the delta of an equity call option . Volatility is a decreasing function of K/S . This means that the implied volatility increases as the asset price increases , so that >0 As a result , delta is higher than that given by the Black-scholes assumptions

20. When a single large jump is anticipated • Suppose that a stock price is currently $50 and an important news announcement due in a few days is expected either to increase the stock price by $8 or to reduce it by $8 . The probability distribution the stock price in 1 month might consist of a mixture of two lognormal distributions, the first corresponding to favorable news, the second to unfavorable news . The situation is illustrated in Figure 5.

21. When a single large jump is anticipated Figure5 Stock price Effect of a single large jump. The solid line is the true distribution; the dashed line is the lognormal distribution

22. When a single large jump is anticipated • Suppose further that the risk-free rate is 12% per annum. The situation is illustrated in Figure 6. Options can be valued using the binomial model from Chapter 11. In this case u=1.16, d=0.84, a=1.0101, and p=0.5314 The results from valuing a range of different options are shown in Table 3

23. When a single large jump is anticipated Figure 6 58 ● 50 ● ● 42 Change in stock price in 1 month

24. Table 3 Implied volatilities in situation where true distribution is binomial Strike price ($) Put price ($) Implied volatility ($) Call price ($) 42 44 46 48 50 52 54 56 58 8.42 7.37 6.31 5.26 4.21 3.16 2.10 1.05 0.00 0.00 0.93 1.86 2.78 3.71 4.64 5.57 6.50 7.42 0.0 58.8 66.6 69.5 69.2 66.1 60.0 49.0 0.0

25. Figure 7 Volatility smile for situation in Table 3 Implied volatility 90 80 70 60 50 40 30 20 Strike price 10 0 44 46 48 50 52 54 56 It is actually a “frown” with volatilities declining as we move out of or into the money