## Implied Volatility Surface

Liuren Wu

Zicklin School of Business, Baruch College

Options Markets

## Implied volatility

Recall the BSM formula:

c(S,t,K,T) =e−r(T−t)[Ft,TN(d1)−KN(d2)],d1,2=

lnFt,T

K ±

1

2σ

2_{(}_{T}_{−}_{t}_{)}

σ√T −t

The BSM model has only one free parameter, the asset return volatilityσ. Call and put option values increase monotonically with increasingσunder BSM.

Given the contract specifications (K,T) and the current market

observations (St,Ft,r), the mapping between the option price and σis a

uniqueone-to-one mapping.

Theσinput into the BSM formula that generates themarket observed option priceis referred to as theimplied volatility (IV).

Practitioners often quote/monitor implied volatility for each option contract instead of the option invoice price.

## The relation between option price and

## σ

## under BSM

0 0.2 0.4 0.6 0.8 1

0 5 10 15 20 25 30 35 40 45

Call option value, c

t

Volatility, σ

K=80 K=100 K=120

0 0.2 0.4 0.6 0.8 1

0 5 10 15 20 25 30 35 40 45 50

Put option value, p

t

Volatility, σ

K=80 K=100 K=120

An option value has two components:

I Intrinsic value: the value of the option if the underlying price does not move (or if the future price = the current forward).

I Time value: the value generated from the underlying price movement. Since options give the holder only rights but no obligation, larger moves generate bigger opportunities but no extra risk — Higher volatility increases the option’s time value.

## Implied volatility versus

## σ

If the real world behaved just like BSM,σwould be a constant.

I In this BSM world, we could use oneσinput to match market quotes on options at all days, all strikes, and all maturities.

I Implied volatility is the same as the security’s return volatility (standard deviation).

In reality, the BSM assumptions are violated. With oneσinput, the BSM model can only match one market quote at a specific date, strike, and maturity.

I The IVs at different (t,K,T) are usually different — direct evidence that the BSM assumptions do not match reality.

I IV no longer has the meaning of return volatility, but is still closely related to volatility:

F The IV for at-the-money option is very close to the “expected return volatility” over the horizon of the option maturity.

F A particular weighted average of allIV2 across different moneyness is very close to expected return variance over the horizon of the option maturity.

## Implied volatility at (

## t

## ,

## K

## ,

## T

## )

At each datet, strikeK, and expiry date T, there can be two European options: one is a call and the other is a put.

The two options should generate the same implied volatility value to exclude arbitrage.

I Recall put-call parity: c−p=er(T−t)_{(}_{F}_{−}_{K}_{).}

I The difference between the call and the put at the same (t,K,T) is the forward value.

I The forward value does not depend on (i) model assumptions, (ii) time value, or (iii) implied volatility.

At each (t,K,T), we can write the in-the-money option as the sum of the intrinsic value and the value of the out-of-the-money option:

I _{If}_{F}_{>}_{K}_{, call is ITM with intrinsic value}_{e}r(T−t)_{(}_{F}_{−}_{K}_{), put is OTM.}

Hence,c=er(T−t)_{(}_{F}_{−}_{K}_{) +}_{p}_{.}

I IfF<K, put is ITM with intrinsic value er(T−t)_{(}_{K} _{−}_{F}_{), call is OTM.}

Hence,p=c+er(T−t)_{(}_{K} _{−}_{F}_{).}

I IfF=K, both are ATM (forward), intrinsic value is zero for both options. Hence,c=p.

## The information content of the implied volatility surface

At each timet, we observe options across many strikesK and maturities

τ=T −t.

When we plot the implied volatility against strike and maturity, we obtain an implied volatility surface.

If the BSM model assumptions hold in reality, the BSM model should be able to match all options with oneσinput.

I The implied volatilities are the same across allK andτ.

I The surface is flat.

We can use the shape of the implied volatility surface to determine what BSM assumptions are violated and how to build new models to account for these violations.

I _{For the plots, do not use}_{K}_{,}_{K}_{−}_{F}_{,}_{K}_{/}_{F} _{or even ln}_{K}_{/}_{F} _{as the}

moneyness measure. Instead, use a standardized measure, such as

lnK/F

ATMV√τ,d2,d1, or delta.

I Using standardized measure makes it easy to compare the figures across maturities and assets.

## Implied volatility smiles & skews on a stock

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0.4

0.45 0.5 0.55 0.6 0.65 0.7 0.75

AMD: 17−Jan−2006

Moneyness=ln(_{σ}K/F√_{τ} )

Implied Volatility

* Short−term smile*

* Long−term skew*

## Implied volatility skews on a stock index (SPX)

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0.08

0.1 0.12 0.14 0.16 0.18 0.2 0.22

SPX: 17−Jan−2006

Moneyness=ln(_{σ}K/F√_{τ} )

Implied Volatility

* More skews than smiles*

## Average implied volatility smiles on currencies

10 20 30 40 50 60 70 80 90 11

11.5 12 12.5 13 13.5 14

Put delta

Average implied volatility

JPYUSD

10 20 30 40 50 60 70 80 90 8.2

8.4 8.6 8.8 9 9.2 9.4 9.6 9.8

Put delta

Average implied volatility

GBPUSD

## Return non-normalities and implied volatility smiles/skews

BSM assumes that the security returns (continuously compounding) are
normally distributed. lnST/St∼N (µ−1_{2}σ2)τ, σ2τ

.

I µ=r−qunder risk-neutral probabilities.

A smile implies that actual OTM option prices are more expensive than BSM model values.

I _{⇒}The probability of reaching the tails of the distribution is higher
than that from a normal distribution.

I ⇒Fat tails, or (formally)leptokurtosis.

A negative skew implies that option values at low strikes are more expensive than BSM model values.

I _{⇒}The probability of downward movements is higher than that from a
normal distribution.

I ⇒Negativeskewnessin the distribution.

Implied volatility smiles and skews indicate that the underlying security return distribution is not normally distributed (under the risk-neutral measure —We are talking about cross-sectional behaviors, not time series).

## Quantifying the linkage

IV(d)≈ATMV

1 +Skew. 6 d+

Kurt. 24 d

2

,d= lnK/F

σ√τ

If we fit a quadratic function to the smile, the slope reflects the skewness of the underlying return distribution.

The curvature measures the excess kurtosis of the distribution.

A normal distribution has zero skewness (it is symmetric) and zero excess kurtosis.

This equation is just an approximation, based on expansions of the normal density (Read “Accounting for Biases in Black-Scholes.”)

The currency option quotes: Risk reversals measure slope/skewness, butterfly spreads measure curvature/kurtosis.

## Revisit the implied volatility smile graphics

For single name stocks (AMD), the short-term return distribution is highly fat-tailed. The long-term distribution is highly negatively skewed.

For stock indexes (SPX), the distributions are negatively skewed at both short and long horizons.

For currency options, the average distribution has positive butterfly spreads (fat tails).

Normal distribution assumption does not work well.

Another assumption of BSM is that the return volatility (σ) is constant — Check the evidence in the next few pages.

## Stochastic volatility on stock indexes

96 97 98 99 00 01 02 03 0.1

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Implied Volatility

SPX: Implied Volatility Level

96 97 98 99 00 01 02 03 0.05

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

Implied Volatility

FTS: Implied Volatility Level

At-the-money implied volatilities at fixed time-to-maturities from 1 month to 5 years.

## Stochastic volatility on currencies

1997 1998 1999 2000 2001 2002 2003 2004 8

10 12 14 16 18 20 22 24 26 28

Implied volatility

JPYUSD

1997 1998 1999 2000 2001 2002 2003 2004 5

6 7 8 9 10 11 12

Implied volatility

GBPUSD

## Stochastic skewness on stock indexes

96 97 98 99 00 01 02 03 0.05

0.1 0.15 0.2 0.25 0.3 0.35 0.4

Implied Volatility Difference, 80%−120%

SPX: Implied Volatility Skew

96 97 98 99 00 01 02 03 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Implied Volatility Difference, 80%−120%

FTS: Implied Volatility Skew

Implied volatility spread between 80% and 120% strikes at fixed time-to-maturities from 1 month to 5 years.

## Stochastic skewness on currencies

1997 1998 1999 2000 2001 2002 2003 2004 −20

−10 0 10 20 30 40 50

RR10 and BF10

JPYUSD

1997 1998 1999 2000 2001 2002 2003 2004 −15

−10 −5 0 5 10

RR10 and BF10

GBPUSD

## What do the implied volatility plots tell us?

Returns on financial securities (stocks, indexes, currencies) are not normally distributed.

I They all have fatter tails than normal (on average).

I The distribution is also skewed, mostly negative for stock indexes (and sometimes single name stocks), but can be either direction (positive or negative) for currencies.

The return distribution is not constant over time, but varies strongly.

I The volatility of the distribution is not constant.

I Even higher moments (skewness, kurtosis) of the distribution are not constant, either.

A good option pricing model should account for return non-normality and its stochastic (time-varying) feature.

## A new, simple, fun model, directly on implied volatility

Black-Merton-Scholes: dSt/St =µdt+σdWt, with constant volatilityσ.

Based on the evidence we observed earlier, we allow σt to be stochastic

(vary randomly over time)

Normally, one would specify howσt varies and derive option pricing values

under the new specification.

I This can get complicated, normally involvesnumerical integration/Fourier transformeven in the most tractable case. We do not specify howσt varies, but instead specify how the implied

volatility of each contract (K,T) will vary accordingly:

dIt(K,T)/It(K,T) =mtdt+wtdWt2,dWt is correlated withdWt with

correlation ρt.

Then, the whole implied volatility surfaceIt(k, τ) becomes the solution to a

quadratic equation,

0 = 1 4w

2 tτ

2

It(k, τ)4+(1−2mtτ−wtρtσtτ)It(k, τ)2− σt2+ 2wtρtσtk+wt2k 2

.

wherek = lnK/F andτ =T−t.