Implied Volatility
Definition
Suppose , and are already observable, so the Black-Scholes price of any option can be expressed as a function of :
The Black-Scholes implied volatility is the that equates the B-S price of an option with the option’s price observed in the market, i.e. .
Remark:
- is non-linear function of . We can use Solver to compute
- IVs can be used to
- calibrate models to price exotic option consistently with market prices of traded options
- compute Greeks for dynamic hedging & Greek-based portfolio structuring
Volatility Arbitrage
Suppose Black-Scholes is right and the market prices are wrong. If there is a pricing mistake in the market, where is overpriced relative to then we want to form a portfolio
- Sell the overpriced option and buy the underpriced option
- To be delta neutral, set position so that: ⇒
- To be self-financing, the Tbills position should be
Volatility Smile
If the BS model were correct, all European options maturing at the same time would have the same implied volatility.
However, all options do not have the same implied volatility. Volatility smile shows the variation of the implied volatility with the strike price. Volatility smile reflects nonlognormality.
Because of the Put-Call parity, the volatility smile should be the same whether calculated from call options or put options.
Implied Volatility for Equity Options
The left tail (in the histogram) is heavier than the lognormal distribution. The right tail is less heavy than the lognormal distribution
Reasons for Smile in Equity Options:
There is a negative correlation between equity prices and volatility. Possible reasons include: leverage, volatility feedback, and Crashophobia. This means, when price decreases (increases), volatility tend to increase (decrease), making further decreases (increases) more (less) likely.
Implied Volatility for Foreign Currency Options
In the histograms, both tails of IV are heavier than the lognormal distribution. IV is also “more peaked” than the lognormal distribution.
Possible reasons:
- Exchange rate exhibits jumps rather than continuous changes
- Volatility of exchange rate is stochastic
Implied (Risk-Neutral) Probabilities
Suppose a butterfly spread:
- Buy 1 call with strike
- Buy 1 call with strike
- Sell 2 calls with strike
The value of this position can be calculated by integrating the payoff over the risk-neutral probability density and discounting at risk-free rate. Denote as the risk-neutral probability density. Since is small, we can assume is constant over .
The area under the spike is . The cost of the butterfly spread is
therefore
and thus
Alternative Models
BS may be wrong because
- maybe volatility for asset price should be stochastic
- asset prices exhibit jumps rather than continuous change
- maybe the stock follows a heteroskedatic Local Vol (LV) process rather than a homoskedatic GBM
- BS:
- LV:
Generalization of BS PDE:
Generalized dynamic replication:
- For both BS and LV:
- LV models have different specific formulas for than the GBM based BS delta
There are three alternatives to Geometric Brownian Motion:
- Constant elasticity of variance (CEV)
- Jump diffusion
- Variance Gamma
CEV Models
Assume that the stock price volatility is
- when , it is the Black-Scholes case
- when , volatility falls as stock price rises
- this corresponds to a volatility smile where the IVs for low strike options (in-the-money calls / out-of-the-money puts) are higher
- corresponding probability distribution has a heavy left tail and a less heavy right tail
- when , volatility rises as stock price rises
- this corresponds to a volatility smile where the IVs for high strike options (out-of-the-money calls / in-the-money puts) are higher (sometimes observed for options on futures)
- corresponding probability distribution has a heavy right tail and a less heavy left tail
Mixed Jump Diffusion Models
where
- is the random jump (jumps generated by Poisson process)
- is the expected size of the jump
- is the probability that a jump occurs in the next interval of length
When no jump occurs, and thus the future stock price is lognormal but the expected stock price is a little lower than usual
If one jump occurs, effect of jump is to multiply stock price by some number.
If the jump size if lognormal then the new stock price is lognormal. The jump affects the mean and the standard deviation of the lognormal distribution.
Variance-Gamma Model
There are 3 parameters
- v: the variance rate of the gamma process
- : the average variance rate of per unit time
- : a parameter defining skewness
is change over time in a variable that follows a gamma process. defines the rate at which information arrives during time ( is sometimes referred to as measuring economic time),
- if is large, the change in has a relatively large mean and variance
- if is small, the change in has a relatively small mean and variance
The mean and variance of depend on . For any value of , is lognormally distributed.
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