OPT-3/4 Implied Volatility & Alternative Models

Implied Volatility

Definition

Suppose

S,r,X

, and

T-t

are already observable, so the Black-Scholes price of any option can be expressed as a function of

\sigma

c^{BS}(S,\sigma,X,r,T-t)=c^{BS}(\sigma)

The Black-Scholes implied volatility

\sigma^{IV}

is the

\sigma

that equates the B-S price of an option with the option’s price observed in the market, i.e.

c^{BS}(\sigma^{IV})=c^{Market}

Remark:

$c^{BS}$ is non-linear function of $\sigma$ . We can use Solver to compute $\sigma^{IV}$

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

c_H^{Market}

is overpriced relative to

c_L^{Market}

then we want to form a portfolio

-c_H+nc_L+Tbills

Sell the overpriced option $H$ and buy the underpriced option $L$

To be delta neutral, set position $n$ so that: $-\Delta_H+n\Delta_L=0$ ⇒ $n=\Delta_H/\Delta_L$

To be self-financing, the Tbills position should be $c_H-\Delta_H/\Delta_L c_L$

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

The Volatility Smile for Foreign Currency Options

IV Distribution 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 $K-\Delta K$

Buy 1 call with strike $K+\Delta K$

Sell 2 calls with strike $K$

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

P

as the risk-neutral probability density. Since

\Delta K

is small, we can assume

P

is constant over

K-\Delta K<S_T<K+\Delta K

The area under the spike is

\Delta K^2

. The cost of the butterfly spread is

\text{Price} = C(K-\Delta K,T)-2C(K,T)+C(K+\Delta K, T)

therefore

e^{-rT}P(K)\Delta K^2=C(K-\Delta K,T)-2C(K,T)+C(K+\Delta K,T)

and thus

P(K)=e^{rT}\frac{C(K-\Delta K,T)-2C(K,T)+C(K+\Delta K,T)}{\Delta K^2}

3_Example Vol Surface.xlsb

56.3KB

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: $dS_t/S_t=\mu dt + \sigma dW_t$
LV: $dS_t/S_t=\mu dt + v(S_t,t)dW_t$

Generalization of BS PDE:

\begin{aligned} &BS:\frac{\partial c}{\partial t}+\frac{\sigma^2 S_t^2}{2}\frac{\partial^2c}{\partial S^2}=r\big[c_t-\frac{\partial c}{\partial S}S_t\big]\\ &LV: \frac{\partial c}{\partial t}+\frac{v(S_t,t)^2S_t^2}{2}\frac{\partial^2 c}{\partial S^2}=r\big[c_t-\frac{\partial c}{\partial S}S_t\big] \end{aligned}

Generalized dynamic replication:

For both BS and LV: $d c_t=\frac{\partial c}{\partial S}dS_t+\big[c_t-\frac{\partial c}{\partial S}S_t\big]rdt$

LV models have different specific formulas for $\frac{\partial c}{\partial S}$ than the GBM based BS delta

3_Example mcs_4vols.xlsx

255.4KB

There are three alternatives to Geometric Brownian Motion:

Constant elasticity of variance (CEV)

Jump diffusion

Variance Gamma

CEV Models

dS=rSdt+\sigma S^\beta dz

Assume that the stock price volatility is

\sigma S^\beta

when $\beta=1$ , it is the Black-Scholes case

when $\beta<1$ , 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 $\beta > 1$ , 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

4_Example CEV Model.xlsb

47.4KB

Mixed Jump Diffusion Models

\frac{dS}{S}=(r-\lambda k)dt + \sigma dz + dq

where

$dq$ is the random jump (jumps generated by Poisson process)

$k$ is the expected size of the jump

$\lambda dt$ is the probability that a jump occurs in the next interval of length $dt$

When no jump occurs,

dS/S=(r-\lambda k)dt+\sigma dz

and thus the future stock price is lognormal but the expected stock price is a little lower than usual

E(S_T)=S_0e^{(r-\lambda k)T}

If one jump occurs, effect of jump is to multiply stock price by some number.

S_{T,1}=S_T\times J_1

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.

4_Example Jump Model.xlsb

45.5KB

Variance-Gamma Model

There are 3 parameters

v: the variance rate of the gamma process

$\sigma^2$ : the average variance rate of $\ln S$ per unit time

$\theta$ : a parameter defining skewness

g

is change over time

T

in a variable that follows a gamma process.

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, the change in $\ln S$ has a relatively small mean and variance

The mean and variance of

\ln S_T

depend on

g

. For any value of

g

S_T

is lognormally distributed.

4_Example Variance Gamma.xlsb

109.7KB