volatility smiles. what is a volatility smile? it is the relationship between implied volatility and...
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Volatility Smiles
What is a Volatility Smile?
It is the relationship between implied volatility and strike price for options with a certain maturity
Why the volatility smile is the same for calls and puts
When Put-call parity p +S0e-qT = c +K e–r T holds 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
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%
Foreign currency options
Impliedvolatility
Strike price
Figure 1
Volatility smile for foreign currency options
Implied and lognormal distribution for foreign currency options
Implied
Lognormal
K1 K2
Figure 2
σ( 波動率 )
S( 匯價 )
Empirical Results
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)
Table 1
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 :
1. The volatility of the asset is constant
2. The price of the asset changes smoothly with no jump
Equity options
Implied
Strike
Volatility smile for equities
Figure 3
volatility
Implied and lognormal distribution for equity options
Implied
Lognormal
K1 K2
Figure 4
σ( 波動率 )
s( 股價 )
The reason for the smile in equity options One possible explanation for the smile in equi
ty options concerns leverage Another explanation is “crashophobia”
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)
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
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
Table 2 Volatility surface
K/S0 0.90 0.95 1.00 1.05 1.10
1month 14.2 13.0 12.0 13.1 14.5
3month 14.0 13.0 12.0 13.1 14.2
6month 14.1 13.3 12.5 13.4 14.3
1 year 14.7 14.0 13.5 14.0 14.8
2 year 15.0 14.4 14.0 14.5 15.1
5 year 14.8 14.6 14.4 14.7 15.0
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
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
Greek letters
Delta of a call option is given by
impBS BS
imp
c c
S S
Where cBS is the Black-Scholes price of the option expressed as a function of the asset price S and the implied volatility σimp
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 S
imp
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 $8 . The probability distribution the stock price in 1 month might consist of a mixture of two lognormal distributions, the first corresponding to favorable news, the second to unfavorable news . The situation is illustrated in Figure 5.
When a single large jump is anticipated
Figure5
Stock price
Effect of a single large jump. The solid line is the true distribution; the dashed line is the lognormal distribution
When a single large jump is anticipated Suppose further that the risk-free rate is 12%
per annum. The situation is illustrated in Figure 6. Options can be valued using the binomial model from Chapter 11. In this case u=1.16, d=0.84, a=1.0101, and p=0.5314
The results from valuing a range of different options are shown in Table 3
When a single large jump is anticipated
●
●
●
50
58
42
Change in stock price in 1 month
Figure 6
Table 3 Implied volatilities in situation where true distribution is binomial
Strike price ($)
Call price ($)
Put price ($)
Implied volatility ($)
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
Figure 7 Volatility smile for situation in Table 3
90
70
80
60
50
40
30
20
10
0
44
46
48
50
52
54
56
Impliedvolatility
Strike price
It is actually a “frown” with volatilities declining as we move out of or into the money