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The Implied Volatility Surface

Liuren Wu

Zicklin School of Business, Baruch College

Options Markets

Liuren Wu (Baruch) Implied Volatility Options Markets 1 / 24
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Implied volatilityRecall the BMS formula:

c(S, t,K,T) =er(T

t)

[Ft,TN(d1) KN(d2)] , d1,2 = ln

Ft,TK

12

2(T

t)

T t

The BMS model has only one free parameter, the asset return volatility .

Call and put option values increase monotonically with increasing under

BMS.

Given the contract specifications (K,T) and the current marketobservations (St,Ft, r), the mapping between the option price and is auniqueone-to-one mapping.

The input into the BMS formula that generates the market observedoption price(or sometimes a model-generated price) is referred to as theimplied volatility (IV).

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

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The relation between option price and under BMS

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

45

Calloptionvalue,

ct

Volatility,

K=80

K=100

K=120

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

45

50

Putoption

value,

pt

Volatility,

K=80

K=100

K=120

An option value has two components: Intrinsic value: the value of the option if the underlying price does not

move (or if the future price = the current forward). Time value: the value generated from the underlying price movement.

Since options give the holder only rights but no obligation, larger movesgenerate bigger opportunities but no extra risk Higher volatility increasesthe options time value.

At-the-money option price is approximately linear in implied volatility.

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Implied volatility versus

If the real world behaved just like BMS, would be a constant.

In this BMS world, we could use one input to match market quoteson options at all days, all strikes, and all maturities.

Implied volatility is the same as the securitys return volatility (standarddeviation).

In reality, the BMS assumptions are violated. With one input, the BMS

model can only match one market quote at a specific date, strike, andmaturity.

The IVs at different (t,K,T) are usually different direct evidencethat the BMS assumptions do not match reality.

IV no longer has the meaning of return volatility. IV still reflects the time value of the option. The intrinsic value of the option is model independent (e.g.,

er(Tt)(FK)+ for call), modelers should only pay attention to timevalue.

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Implied volatility at (t,K,T)

At each date t, strike K, and expiry date T, there can be two Europeanoptions: one is a call and the other is a put.

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

Recall put-call parity: c p=er(Tt)(F K). The difference between the call and the put at the same (t,K, T) is

the forward value. 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 theintrinsic value and the value of the out-of-the-money option:

IfF>K, call is ITM with intrinsic value er(Tt)(F

K), put is OTM.

Hence, c=er(Tt)(F K) +p. IfF

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Why so entrenched in implied volatility?

The implied volatility is calculated from the BMS model.

The fact that practitioners use the BMS model to quote options does not

mean they agree with the BMS assumptions. On the contrary, the very fact that they quote/vary/twist/model

implied volatility shows that BMS assumptions are violated.


1 Informational: It is much easier to gauge/express views in terms ofimplied volatility than in terms of option prices. IV is unitless; option prices are not units are not views (remember

my scaling). IV does not depend on intrinsic value; option prices do intrinsic has

no informational value.

IV has a normal return distribution (BMS model) benchmark. Deviation from a flat line (across strike) reveals return deviationfrom normality. A higher IV for OTM puts (low strikes) than for OTM calls (highstrikes) says that the left tail is heavier than the right tail. Higher IVs for OTM options than for ATM options suggests fatter

tails (leptokurtosis).Liuren Wu (Baruch) Implied Volatility Options Markets 6 / 24
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1 Informational:

2 No arbitrage:

Merton (1973): model-free bounds on option prices based onno-arbitrage arguments:

Type I: No-arbitrage between options and the underlying and cash:call/put prices intrinsic;call prices (dividend discounted) stock price;

put prices (present value of the) strike price.Type II: No-arbitrage between options of different strikes and maturities:

put-call parity;bull, bear, and calendar spreads can never be priced negatively;butterfly spreads can never be priced negatively.

Hodges (1996): These bounds can be expressed in implied volatilities.

Type I: Implied volatility is positive.Type II: Put-call parity One IV per (K,T)

If market makers quote options in terms of an implied volatility surface,many no-arbitrage conditions are automatically guaranteed.

It makes it much easier to balance order flow with no-arbitrage.

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Put-call parity, revisited

On the exchange, calls and puts at the same maturity and strike arequoted/traded separately. It is possible to observe put-call parity being

violated at some times. Violations can happen when there are market frictions such as

short-sale constraints. For American options, there only exists a put-call inequality. The effective maturities of the put and call American options with

same strike and expiry dates can be different. We take forward as given, but it is not: Assumptions on interest

rates, borrow/rebate rates, and dividend schedules can be wrong.

Put-call violations can predict future spot price movements.

One can use the call and put option prices at the same strike to

compute an option implied spot price: Sot = (ct pt+e

r

K) eq

. The difference between the implied spot price and the price from the

stock market can contain predictive information: St Sot.Cremers & Weinbaum, 2007, Deviations from Put-Call Parity and Stock Returns, wp.

Compute implied forward for option pricing model estimation. Predict dividend from options.

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Implied volatility quotes on OTC currency options

At each date tand each fixed time-to-maturity = (T t), OTC currencyoptions are quoted in terms of

1 Delta-neutral straddle implied volatility (ATMV):A straddle is a portfolio of a call & a put at the same strike. The strikehere is set to make the portfolio delta-neutral:eq(Tt)N(d1) eq(Tt)N(d1) = 0 N(d1) = 12 d1 = 0.

2 25-delta risk reversal: RR25 =IV(c= 25) IV(p= 25).

3 25-delta butterfly spreads:BF25 = (IV(c= 25) +IV(p= 25))/2 ATMV.

The three types of quotes reflect views on three distinct dimensions:1 Straddle volatility level.2 Risk reversal (slope)skewness.3 Butterfly spreads (curvature)

kurtosis.

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From delta to strikes

Given these quotes, we can compute the IV at the three deltas:

IV = ATMV at d 1 = 0IV = BF25+ATMV+RR25/2 at c= 25%IV = BF25+ATMV RR25/2 at p= 25%

The three strikes at the three deltas can be inverted as follows:

K = Fexp

12 IV2

at d1 = 0

K = Fexp

12 IV

2 N1(ceq)IV

at c= 25%K = Fexp

12

IV2+N1(|p|eq)IV

at p= 25%

Put-call parity is guaranteed in the OTC quotes: one implied volatility at

each delta/strike.

These are not exactly right... They are simplified versions of the much

messier real industry convention.

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Example

On 9/29/2004, I obtain the following quotes on USDJPY: St= 110.87,

ATMV = 8.73, 8.5, 8.66, RR25 = 0.53,0.7,0.98, andBF25 = 0.24, 0.26, 0.31 at 3 fixed maturities of 1, 3, and 2 months. (Thequotes are in percentages).

The USD interest rates at 3 maturities are: 1.82688, 1.9, 2.31. The JPYrates are 0.03625, 0.04688, 0.08125. (Assume that they are continuously

compounding). All rates are in percentages.

Compute the implied volatility at three moneyness levels at each of the threematurities.

Compute the corresponding strike prices.

Compute the invoice prices for call and put options at these strikes andmaturities.

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The information content of the implied volatility surface

At each time t, we observe options across many strikes Kand maturities=T t.When we plot the implied volatility against strike and maturity, we obtain animplied volatility surface.

If the BMS model assumptions hold in reality, the BMS model should beable to match all options with one input.

The implied volatilities are the same across all K and. The surface is flat.

We can use the shape of the implied volatility surface to determine whatBMS assumptions are violated and how to build new models to account forthese violations. For the plots, do not use K, K

F, K/For even ln K/Fas the

moneyness measure. Instead, use a standardized measure, such asln K/F

ATMV

, d2, d1, or delta.

Using standardized measure makes it easy to compare the figuresacross maturities and assets.

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Return non-normalities and implied volatility smiles/skews

BMS assumes that the security returns (continuously compounding) arenormally distributed. ln ST/St N

( 122), 2

.

= r qunder risk-neutral probabilities.A smile implies that actual OTM option prices are more expensive thanBMS model values.

The probability of reaching the tails of the distribution is higherthan that from a normal distribution.

Fat tails, or (formally)leptokurtosis.A negative skew implies that option values at low strikes are more expensivethan BMS model values.

The probability of downward movements is higher than that from anormal distribution.

Negativeskewnessin the distribution.Implied volatility smiles and skews indicate that the underlying securityreturn distribution is not normally distributed (under the risk-neutralmeasure We are talking about cross-sectional behaviors, not time series).

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Quantifying the linkage

IV(d)

ATMV1 +

Skew.

6

d+Kurt.

24

d2 , d= ln K/F

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

The curvature measures the excess kurtosis of the distribution.

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

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

The currency option quotes: Risk reversals measure slope/skewness,butterfly spreads measure curvature/kurtosis.Check the VOLC function on Bloomberg.

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Implied volatility smiles & skews on a stock

3 2.5 2 1.5 1 0.5 0 0.5 1 1.5 20.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

AMD: 17Jan2006

Moneyness=ln(K/F)

ImpliedVolatility

Shortterm smile

Longterm skew

Maturities: 32 95 186 368 732

For single name stocks (AMD), the short-term return distribution is highly

fat-tailed. The long-term distribution is highly negativelyskewed.Liuren Wu (Baruch) Implied Volatility Options Markets 15 / 24
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Implied volatility skews on a stock index (SPX)

3 2.5 2 1.5 1 0.5 0 0.5 1 1.5 20.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

SPX: 17Jan2006

Moneyness=ln(K/F)

ImpliedVolatility

More skews than smiles

Maturities: 32 60 151 242 333 704

For stock indexes (SPX), the distributions are negatively skewed at both short

and long horizons.Liuren Wu (Baruch) Implied Volatility Options Markets 16 / 24
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Average implied volatility smiles on currencies

10 20 30 40 50 60 70 80 9011

11.5

12

12.5

13

13.5

14

Put delta

Averag

eimpliedvolatility

JPYUSD

10 20 30 40 50 60 70 80 908.2

8.4

8.6

8.8

9

9.2

9.4

9.6

9.8

Put delta

Averag

eimpliedvolatility

GBPUSD

Maturities: 1m (solid), 3m (dashed), 1y (dash-dotted)For currency options, the average distribution has positive butterfly spreads (fattails).


S
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Stochastic volatility time series on stock indexes

96 97 98 99 00 01 02 030.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Im

pliedVolatility

SPX: Implied Volatility Level

96 97 98 99 00 01 02 030.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Im

pliedVolatility

FTS: Implied Volatility Level

At-the-money implied volatilities at fixed time-to-maturities from 1 month to 5years. Equity index return volatilities vary strongly over time.


S h i l ili i
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Stochastic volatility on currencies

1997 1998 1999 2000 2001 2002 2003 2004

8

10

12

14

16

18

20

22

24

26

28

Impliedvolatility

JPYUSD

1997 1998 1999 2000 2001 2002 2003 2004

5

6

7

8

9

10

11

12

Impliedvolatility

GBPUSD

Three-month delta-neutral straddle implied volatility. Currency return volatilities also vary strongly over time.


S h i k k i d
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Stochastic skewness on stock indexes

96 97 98 99 00 01 02 030.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

ImpliedVolatilityDifference,

80%

120%

SPX: Implied Volatility Skew

96 97 98 99 00 01 02 030

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

ImpliedVolatilityDifference,

80%

120%

FTS: Implied Volatility Skew

Implied volatility spread between 80% and 120% strikes at fixedtime-to-maturities from 1 month to 5 years. Equity index return skewness varies strongly over time, but stays negative allthe time.


St h ti k i
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Stochastic skewness on currencies

1997 1998 1999 2000 2001 2002 2003 2004

20

10

0

10

20

30

40

50

R

R10andBF10

JPYUSD

1997 1998 1999 2000 2001 2002 2003 2004

15

10

5

0

5

10

R

R10andBF10

GBPUSD

Three-month 10-delta risk reversal (blue lines) and butterfly spread (red lines). Currency return skewness also varies strongly over time, with possible(frequent) sign switches.


Wh t d th i li d l tilit l t t ll ?
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What do the implied volatility plots tell us?

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

They all have fatter tails than normal (most of the time). The distribution is also skewed, mostly negative for stock indexes (and

sometimes single name stocks), but can be of either direction (positiveor negative) for currencies.

Return non-normality does not decline (but also increases) as option

maturity (horizon) increasesViolations of central limit theorem?The return distribution is not constant over time, but varies strongly.

The volatility of the distribution is not constant. Even higher moments (skewness, kurtosis) of the distribution are not

constant, either.

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


Wei d i lied olatilit sha es
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Weird implied volatility shapes

Sometimes, the implied volatility plot against moneyness can show weird shapes:

Implied volatility frown This does not happen often, but it does happen.

It is more difficult to model than a smile.

The implied volatility shape around corporate actions such as mergers,takeovers, etc.

Although most of our models are time homogeneous, calendar day effectsare important considerations in practice. How to reconcile the two?

Build a business calendar and modify the option time to maturity basedon business activities: Treat non-trading days as zero (or a small fraction) of one day.

Treat each earnings announcement date as multiple days. Pre-process the data as much as possible before applying to a model.

Most deterministic components should be pre-processed.


Implied volatility surface behavior documentation
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Implied volatility surface behavior documentation

Equities

Carr, Wu: Finite Moment Log Stable Process and Option Pricing, JF, 2003, 58(2), 753.

Foresi, Wu: Crash-O-Phobia: A Domestic Fear or A Worldwide Concern? Journal of Derivatives, 2005, 13(2), 8.

Wu: Dampened Power Law: Reconciling the Tail Behavior of Financial Security Returns, Journal of Business,

2006, 79(3), 1445.

Carr, Wu: Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions: Disentangling the

Multi-Dimensional Variations in S&P 500 Index Options, wp.

Currencies:

Carr, Wu: Stochastic Skew in Currency Options, JFE, 2007, 86(1), 213-247.

Cross-market linkages between implied volatilities and credits:

Carr, Wu: Theory and Evidence on the Dynamic Interactions Between Sovereign Credit Default Swaps and

Currency Options, Journal of Banking and Finance, 2007, 31(8), 2383.

Carr, Wu: Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation, Journal

of Financial Econometrics, forthcoming.

Carr, Wu: A Simple Robust Link Between American Puts and Credit Protection, RFS, forthcoming.

Most evidence documentation leads to new model development/estimation.Liuren Wu (Baruch) Implied Volatility Options Markets 24 / 24
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890 IV Surface