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Volatility Smiles

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Page 1: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

Volatility Smiles

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

What is a Volatility Smile?

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

Page 3: 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

Page 4: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

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%

Page 5: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

Foreign currency options

Impliedvolatility

Strike price

Figure 1

Volatility smile for foreign currency options

Page 6: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

Implied and lognormal distribution for foreign currency options

Implied

Lognormal

K1 K2

Figure 2

σ( 波動率 )

S( 匯價 )

Page 7: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

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

Page 8: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

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

Page 9: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

Equity options

Implied

Strike

Volatility smile for equities

Figure 3

volatility

Page 10: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

Implied and lognormal distribution for equity options

Implied

Lognormal

K1 K2

Figure 4

σ( 波動率 )

s( 股價 )

Page 11: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

The reason for the smile in equity options One possible explanation for the smile in equi

ty options concerns leverage Another explanation is “crashophobia”

Page 12: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

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)

Page 13: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

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

Page 14: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

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

Page 15: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain 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

Page 16: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

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

Page 17: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

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

Page 18: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

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

Page 19: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

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

Page 20: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

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.

Page 21: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

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

Page 22: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

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

Page 23: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

When a single large jump is anticipated

50

58

42

Change in stock price in 1 month

Figure 6

Page 24: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

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

Page 25: Volatility Smiles. What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity

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