MGT 821/ECON 873 Volatility Smiles

1
MGT 821/ECON 873Volatility Smiles
Extension of Models
2
What is a Volatility Smile?

• 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

3
Why the Volatility Smile is the Same for Calls
and Put

• Put-call parity p S0e-qT c Ker T holds for
  market prices (pmkt and cmkt) and for
  Black-Scholes prices (pbs and cbs)
• It follows that pmkt-pbscmkt-cbs
• When pbspmkt, it must be true that cbscmkt
• 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

4
The Volatility Smile for Foreign Currency Options
5
Implied Distribution for Foreign Currency Options

• Both tails are heavier than the lognormal
  distribution
• It is also more peaked than the lognormal
  distribution

6
The Volatility Smile for Equity Options
Implied
Volatility
Strike
Price
7
Implied Distribution for Equity Options

• The left tail is heavier and the right tail is
  less heavy than the lognormal distribution

8
Other Volatility Smiles?

• 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

9
Ways of Characterizing the Volatility Smiles

• 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)

10
Possible Causes of Volatility Smile

• 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

11
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

12
Volatility Term Structure

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

13
Example of a Volatility Surface
14
Greek Letters

• If the Black-Scholes price, cBS is expressed as a
  function of the stock price, S, and the implied
  volatility, simp, the delta of a call is
• Is the delta higher or lower than

15
Three Alternatives to Geometric Brownian Motion

• Constant elasticity of variance (CEV)
• Mixed Jump diffusion
• Variance Gamma

16
CEV Model

• When a 1 the model is Black-Scholes
• When a gt 1 volatility rises as stock price rises
• When a lt 1 volatility falls as stock price rises
• European option can be value analytically in
  terms of the cumulative non-central chi square
  distribution

17
CEV Models Implied Volatilities
K
18
Mixed Jump Diffusion

• Merton produced a pricing formula when the asset
  price follows a diffusion process overlaid with
  random jumps
• dp is the random jump
• k is the expected size of the jump
• l dt is the probability that a jump occurs in the
  next interval of length dt

19
Jumps and the Smile

• Jumps have a big effect on the implied volatility
  of short term options
• They have a much smaller effect on the implied
  volatility of long term options

20
The Variance-Gamma Model

• Define g as change over time T in a variable that
  follows a gamma process. This is a process where
  small jumps occur frequently and there are
  occasional large jumps
• Conditional on g, ln ST is normal. Its variance
  proportional to g
• There are 3 parameters
• v, the variance rate of the gamma process
• s2, the average variance rate of ln S per unit
  time
• q, a parameter defining skewness

21
Understanding the Variance-Gamma Model

• g defines the rate at which information arrives
  during time T (g is sometimes referred to as
  measuring economic time)
• If g is large the change in ln S has a relatively
  large mean and variance
• If g is small relatively little information
  arrives and the change in ln S has a relatively
  small mean and variance

22
Time Varying Volatility

• Suppose the volatility is s1 for the first year
  and s2 for the second and third
• Total accumulated variance at the end of three
  years is s12 2s22
• The 3-year average volatility is

23
Stochastic Volatility Models

• When V and S are uncorrelated a European option
  price is the Black-Scholes price integrated over
  the distribution of the average variance

24
Stochastic Volatility Models continued

• When V and S are negatively correlated we obtain
  a downward sloping volatility skew similar to
  that observed in the market for equities
• When V and S are positively correlated the skew
  is upward sloping. (This pattern is sometimes
  observed for commodities)

25
The IVF Model
26
The Volatility Function

• The volatility function that leads to the
  model matching all European option prices is

27
Strengths and Weaknesses of the IVF Model

• The model matches the probability distribution of
  asset prices assumed by the market at each future
  time
• The models does not necessarily get the joint
  probability distribution of asset prices at two
  or more times correct