Hull, John - Volatility Surfaces Rules of Thumb

8/13/2019 Hull, John - Volatility Surfaces Rules of Thumb

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Volatility Surfaces:

Theory, Rules of Thumb, and Empirical Evidence

Toby Daglish

Rotman School of Management

University of Toronto

Email: [email protected]

John Hull

Rotman School of Management

University of Toronto


Wulin Suo

School of Business

Queens University


First version: March 2001This version: December 2002

Abstract

Implied volatilities are frequently used to quote the prices of options. The impliedvolatility of a European option on a particular asset as a function of strike price andtime to maturity is known as the assets volatility surface. Traders monitor movementsin volatility surfaces closely. In this paper we develop a no-arbitrage condition for theevolution of a volatility surface. We examine a number of rules of thumb used bytraders to manage the volatility surface and test whether they are consistent with theno-arbitrage condition and with data on the trading of options on the S&P 500 takenfrom the over-the-counter market.


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1 Introduction

Option traders and brokers in over-the-counter markets frequently quote option prices usingimplied volatilities calculated from Black and Scholes (1973) and other similar models. Put

call parity implies that, in the absence of arbitrage, the implied volatility for a European calloption is the same as that for a European put option when the two options have the samestrike price and time to maturity. This is convenient. When quoting an implied volatilityfor a European option with a particular strike price and maturity date, a trader does notneed to specify whether a call or a put is being considered.

The implied volatility of European options on a particular asset as a function of strikeprice and time to maturity is known as the volatility surface. Every day traders and brokersestimate volatility surfaces for a range of different underlying assets from the market prices ofoptions. Some points on a volatility surface for a particular asset can be estimated directlybecause they correspond to actively traded options. The rest of the volatility surface istypically determined by interpolating between these points.

If the assumptions underlying BlackScholes held for an asset, its volatility surfacewould be flat and unchanging. In practice the volatility surfaces for most assets are not flatand change stochastically. Consider for example equities and foreign currencies. Authorssuch as Rubinstein (1994) and Jackwerth and Rubinstein (1996) show that the impliedvolatilities of stock and stock index options exhibit a pronounced skew (that is, theimplied volatility is a decreasing function of strike price). For foreign currencies this skewbecomes a smile (that is, the implied volatility is a U-shaped function of strike price).For both types of assets, the implied volatility can be an increasing or decreasing functionof the time to maturity. The volatility surface changes through time, but the general shapeof the relationship between volatility and strike price tends to be preserved.

Traders use a volatility surface as a tool to value a European option when its price isnot directly observable in the market. Providing there are a reasonable number of actively

traded European options and these span the full range of the strike prices and times tomaturity that are encountered, this approach ensures that traders price all European optionsconsistently with the market. However, as pointed out by Hull and Suo (2002), there isno easy way to extend the approach to price path-dependent exotic options such as barrieroptions, compound options, and Asian options. As a result there is liable to be some modelrisk when these options are priced.

Traders also use the volatility surface in an ad hoc way for hedging. They attempt tohedge against potential changes in the volatility surface as well as against changes in theasset price. As described in Derman (1999) one popular approach to hedging against assetprice movements is the volatility-by-strike or sticky strike rule. This assumes that theimplied volatility for an option with a given strike price and maturity will be unaffected

by changes in the underlying asset price. Another popular approach is the volatility-by-moneyness or sticky delta rule. This assumes that the volatility for a particular maturitydepends only on the moneyness (that is, the ratio of the price of an asset to the strike price).

In this paper we develop a general diffusion model for the evolution of a volatilitysurface. We derive the restriction on the specification of the model necessary for it to be ano-arbitrage model. The first attempts to model the volatility surface were by Rubinstein(1994), Derman and Kani (1994), and Dupire (1994). These authors show how a one-factormodel for an asset price, known as the implied volatility function (IVF) model, can bedeveloped so that it is exactly consistent with the current volatility surface. Unfortunately,the evolution of the volatility surface under the IVF model can be unrealistic. The volatility

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surface given by the model at a future time is liable to be quite different from the initialvolatility surface. For example, in the case of a foreign currency the initial U-shapedrelationship between implied volatility and strike price is liable to evolve to one where thevolatility is a monotonic increasing or decreasing function of strike price. Dumas, Fleming,

and Whaley (1997) have shown that the IVF model does not capture the dynamics ofmarket prices well. Hull and Suo (2002) have shown that it can be dangerous to use themodel for the relative pricing of barrier options and plain vanilla options. Other researchthat has independently followed a similar theoretical approach to our own in attemptingto generate a general model of the volatility surface is Ledoit and Santa Clara (1998),Schonbucher (1999), and Brace et al (2001). Another paper that explores a different aspectof the problem we do is BrittenJones and Neuberger (2000), who produce some interestingresults characterizing the set of all continuous price processes that are consistent with agiven set of option prices.

The rest of this paper is organized as follows. Section 2 proposes a general model for theevolution of a volatility surface and derives the no-arbitrage condition. Section 3 discusses

the implications of the no-arbitrage condition. Section 4 examines a number of specialcases of the model. Section 5 considers three rules of thumb used by traders and examineswhether they are consistent with the no-arbitrage condition. Section 6 uses the rules ofthumb to develop and test a number of hypotheses on the evolution of a volatility surfaceusing data from the over-the-counter market. Conclusions are in Section 7.

2 The Dynamics of the Implied Volatility

We suppose that the risk-neutral process followed by the price of an asset, S, is

dS

S = [r(t) q(t)]dt+ dz, (1)

where r(t) is the risk-free rate, q(t) is the yield provided by the asset, is the assetsvolatility, and z is a Wiener process. We suppose that r(t) and q(t) are deterministicfunctions of time and that follows a diffusion process. Our model includes the IVF modeland stochastic volatility models such as Hull and White (1987), Stein and Stein (1991) andHeston (1993) as special cases.

Most stochastic volatility models specify the process for directly. We instead specifythe processes for all implied volatilities. DefineTK(t, S) as the implied volatility at timetof an option with strike price Kand maturity Twhen the asset price is Sand VTK(t, S)as the implied variance of this option (t < T) so that

VTK(t, S) = [TK(t, S)]2

Suppose that the process followed by VTKin a risk-neutral world is

dVTK=TK dt+VTK

Ni=1

TKi dzi (2)

where z1, , zNare Wiener processes driving the volatility surface. Without loss of gen-erality, we assume that these Wiener processes are uncorrelated. The zi may be correlatedwith the Wiener process, z, driving the asset price in equation (1). We define i as thecorrelation between z and zi. The initial volatility surface is TK(0, S0) where S0 is the

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initial asset price. This volatility surface can be estimated from the current (t= 0) pricesof European call or put options and is assumed to be known.

The family of processes in equation (2) defines the multi-factor dynamics of the volatilitysurface. The parameter TKi measures the sensitivity ofVTKto the Wiener process,zi. In

the most general form of the model the parameters TKandTKi (1 i N) may dependon past and present values ofS, past and present values ofVTK, and time.

There is clearly a relationship between the instantaneous volatility(t) and the volatilitysurfaceTK(t, S). The appendix shows that the instantaneous volatility is the limit of theimplied volatility of an at-the-money option as its time to maturity approaches zero. Forthis purpose an at-the-money option is defined as an option where the strike price equalsthe forward asset price.1 Formally:

limTt

TF(t, S) =(t). (3)

where Fis the forward price of the asset at time t for a contract maturing at time T. Ingeneral the process for is non-Markov. This is true even when the processes in equation(2) defining the volatility surface are Markov.2

Definec(S, VTK, t; K, T) as the price of a European call option with strike price Kandmaturity T when the asset price, S, follows the process in equations (1) and (2). Fromthe definition of implied volatility and the results in Black and Scholes (1973) and Merton(1973) it follows that:

c(S, VTK, t; K, T) =e

T

t q() dSN(d1) e

T

t r() dKN(d2)

where

d1 = ln(S/K) +Tt [r() q()]d

VTK(T t)+1

2

VTK(T t)

d2 = ln(S/K) +

Tt [r() q()]d

VTK(T t) 1

2

VTK(T t)

Using Itos lemma equations (1) and (2) imply that the drift ofc in a risk-neutral worldis:

c

t+(rq)Sc

S+

1

22S2

2c

S2+TK

c

VTK+

1

2V2TK

2c

V2TK

Ni=1

(TKi)2+SVTK

2c

SVTK

Ni=1

TKii

In the most general form of the model i, the correlation betweenz and zi, is a function ofpast and present values ofS, past and present values ofVTKand time. For there to be no

1Note that the result is not necessarily true if we define an at-the-money option as an option where thestrike price equals the asset price. For example, when

TK(t, S) = a+b 1

T tln(F/K)

with a and b constants, limTtTF(t, S) is not the same as limTtTS(t, S)2There is an analogy here to the Heath, Jarrow, and Morton (1992) model. When each forward rate

follows a Markov process the instantaneous short rate does not in general do so.

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arbitrage the process followed by c must provide an expected return ofr in a risk-neutralworld. It follows that

c

t

+ (r

q)S

c

S

+1

2

2S22c

S2

+TKc

VTK

+1

2

V2TK2c

V2

TK

N

i=1

(TKi)2

+SVTK 2c

SVTK

Ni=1

TKii= rc

When VTK is held constant, c satisfies the Black-Scholes (1973) and Merton (1973)differential equation. As a result

c

t+ (r q)Sc

S =rc 1

2VTKS

22c

S2

It follows that

12

S22

cS2

(2VTK)+TK cVTK

+ 12

V2TK 2

cV2TK

Ni=1

(TKi)2+SVTK 2

cSVTK

Ni=1

TKii= 0

or

TK = 1

2c/VTK

S2

2c

S2(2 VTK) +

2c

V2TKV2TK

Ni=1

(TKi)2

+2SVTK 2c

SVTK

Ni=1

TKii

(4)

The partial derivatives of c with respect to S and VTKare the same as those for theBlackScholes model:

c

S = e

T

t q() dN(d1)

2c

S2 =

(d1)e

T

t q() d

S

VTK(T t)c

VTK=

SeT

t q() d(d1)

T t

2

VTK

2c

V2TK=

SeT

t q() d(d1)

T t

4V3/2TK

(d1d2 1)

2c

SVTK = e

T

t q() d(d1)d2

2VTK

where is the density function of the standard normal distribution:

(x) = 1

2exp

x

2

2

, < x < .

Substituting these relationships into equation (4) and simplifying we obtain

TK = 1

T t (VTK 2) VTK(d1d2 1)

4

Ni=1

(TKi)2 +d2

VTKT t

Ni=1

TKii (5)

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Equation (5) provides an expression for the risk-neutral drift of an implied variance interms of its volatility. The first term on the right hand side is the drift arising from thedifference between the implied variance and the instantaneous variance. The second termarises from the part of the uncertainty about future volatility that is uncorrelated with

the asset price. The third term arises from the correlation between the asset price and itsvolatility.

The first term can be understood by considering the situation where the instantaneousvariance, 2, is a deterministic function of time. The variableVTK is then also a functionof time and

VTK= 1

T t

Tt

()2d

Differentiating with respect to time we get

dVTKdt

= 1

T t [VTK (t)2]

This is the first term.The analysis can be simplified slightly by considering the variable VTK instead of V

where VTK= (T t)VTK. Because

dVTK= VTKdt+ (T t)dVTK

it follows that

dVTK=

2 VTK(d1d2 1)

4

Ni=1

(TKi)2 +d2

VTK

Ni=1

TKii

dt +VTK

Ni=1

TKidzi

3 Implications of the No-Arbitrage ConditionEquation (5) provides a no-arbitrage condition for the drift of the implied variance as afunction of its volatility. In this section we examine the implications of this no-arbitragecondition. In the general case where VTKis nondeterministic the first term in equation (5)is mean fleeing; that is it provides negative mean reversion. This negative mean reversionbecomes more pronounced as the option approaches maturity. For a viable model theTKi must be complex functions that in some way offset this negative mean reversion.Determining the nature of these functions is not easy. However, it is possible to make somegeneral statements about the volatility smiles that are consistent with stable models.

The Zero Correlation Case

Consider first the case where all the i are zero so that the third term in equation (5)disappears. As before we define Fas the forward value of the asset for a contract maturingat time T so that

F =SeT

t [r()q()]d

From equation (5) the drift ofVTK VTF is

1

T t (VTK VTF)VTK(d1d2 1)

4

Ni=1

(TKi)2 VTF[1 +VTF(T t)/4]

4

Ni=1

(TFi)2

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Suppose thatd1d21> 0 andVTK< VTF. In this case each term in the drift ofVTKVTFis negative. As a resultVTK VTFtends to get progressively more negative and the modelis unstable with negative values ofVTKbeing possible. We deduce from this thatVTKmustbe greater than VTF when d1d2 > 1. The condition d1d2>1 is satisfied for very large and

very small values ofK. It follows that the case where the i are zero can be consistent withthe U-shaped volatility smile. It cannot be consistent with a upward or downward slopingsmile because in these cases VTK< VTFfor either very high or very low values ofK.

Our finding is consistent with a result in Hull and White (1987). These authors showthat when the instantaneous volatility is independent of the asset price, the price of aEuropean option is the BlackScholes price integrated over the distribution of the averagevariance. They demonstrate that when d1d2> 1 a stochastic volatility tends to increase anoptions price.

A U-shaped volatility smile is commonly observed for options on a foreign currency. Ouranalysis shows that this is consistent with the empirical result that the correlation betweenimplied volatilities and the exchange rate is close to zero (see, for example, Bates (1996)).

Volatility Skews

Consider next the situation where the volatility is a declining function of the strike price.The variance rateVTKis greater thanVTF whenKis very small and less thanVTF whenKis very large. When we do not make the zero-correlation assumption the drift ofVTKVTFis

1

T t (VTK VTF)VTK(d1d2 1)

4

Ni=1

(TKi)2 VTF(1 +VTF(T t)/4)

4

Ni=1

(TFi)2

+d2

VTKT t

Ni=1

TKii+ VTF

2Ni=1

TFii

WhenK > F, VTK VTF is negative and the effect of the first three terms is to provide anegative drift as before. For a stable model we require the last two terms to give a positivedrift. AsKincreases and we approach option maturity the first of the two terms dominatesthe second. Because d2< 0 we must have

Ni=1

TKii< 0 (6)

whenKis very large. The instantaneous covariance the asset price and its variance is

Ni=1

TKii

Because >0 it follows that when Kis large the asset price must be negatively correlatedwith its variance.

Equities provide an example of a situation where there is a volatility skew of the sortwe are considering. As has been well documented by authors such as Christie (1982), thevolatility of an equity price tends to be negatively correlated with the equity price. This isconsistent with the result we have just presented.

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Other Results

Consider the situation where volatility is an increasing function of the strike price. (Thisis the case for options on some commodity futures.) A similar argument to that just givenshows that the no-arbitrage relationship implies that the volatility of the variable shouldbe positively correlated with the level of the variable.

Another possibility for the volatility smile is an inverted-U-shaped pattern. In this casea similar analysis to that given above shows that for a stable model

Ni=1

TKii

must be less than zero when Kis large and greater than zero when Kis small. It is difficultto see how this can be so without the stochastic terms in the processes for the VTK havinga form that quickly destroys the inverted-U-shaped pattern.

4 Special Cases

In this section we consider a number of special cases of the model developed in Section 2.

Case 1: VTK is a deterministic function only of t, T, and K

In this situation TKi = 0 for all i 1 so that from equation (2)

dVTK=TK dt,

Also from equation (5)

TK= 1

T t

VTK 2so that

dVTK = 1

T t

VTK 2

dt

which can be written as(T t)dVTK VTKdt= 2 dt

or

2 =

d [(T t)VTK]

dt

(7)

This shows that is a deterministic function of time. The only model that is consistentwithVTKbeing a function only oft,T, andKis therefore the model where the instantaneousvolatility,, is a function only of time. This is Mertons (1973) model.

In the particular case where VTK depends only on T and K, equation (7) shows thatVTK=

2 and we get the Black-Scholes constant-volatility model.

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Case 2: VTK is independent of the asset price, S

In this situation i is zero and equation (5) becomes

TK=

1

T t (VTK 2

)VTK(d1d2

1)

4

Ni=1

(TKi)2

Both TK and the TKi must be independent ofS. Because d1 and d2 depend on S wemust have TKi zero for all i. Case 2 therefore reduces to Case 1. The only model that isconsistent with VTKbeing independent ofS is therefore Mertons (1973) model where theinstantaneous volatility is a function only of time.

Case 3: VTK is a deterministic function oft, T, and K/S

In this situation

VTK=GT,t,KF ,

whereGis a deterministic function and as beforeFis the forward price ofS. From equation(3) the spot instantaneous volatility, , is given by

2 =G (t,t, 1) ,

This is a deterministic function of time. It follows that, yet again, the model reduces toMertons (1973) deterministic volatility model.

Case 4: VTK is a deterministic function oft, T, S and K

In this situation we can write

VTK=G(T,t,F,K ), (8)

where G is a deterministic function. From equation (3), the instantaneous volatility, isgiven by

2 = limTt

G(T,t,F,F)

This shows that the instantaneous volatility is a deterministic function ofF and t. Equiv-alently it is a deterministic function of underlying asset price, S, and t. It follows that themodel reduces to the IVF model. Writing as (S, t), Dupire (1994) and Andersen andBrothertonRatcliffe (1998) show that

[(K, T)]2 = 2c/T+q(T)c+K[r(T) q(T)]c/K

K2

(2

c/K2

)where c is here regarded as a function of S, K and T for the purposes of taking partialderivatives.

5 Rules of Thumb

A number of rules of thumb have been proposed about volatility surfaces. Some of these aredesigned to assist in the calculation of hedge ratios such as delta and gamma. Others aredesigned to provide a relationship between the volatilities of options with different strikeprices and times to maturity. In this section we consider three of these rules.

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The Sticky Strike Rule

The sticky strike rule assumes that the implied volatility of an option is independent of theasset price. This is an appealing assumption because it implies that the sensitivity of theprice of an option to S is

c

S

where for the purposes of calculating the partial derivative the option price,c, is consideredto be a function ofS, VTK, and t. The assumption enables the BlackScholes formulas tobe used to calculate delta with the volatility parameter set equal to the options impliedvolatility. The same is true of gamma.

Under the most basic form of the sticky strike rule the implied volatility of an option isassumed to remain the same for the whole of its life. The variance VTK is a function onlyofK and T. The analysis in Case 1 of the previous section shows that the only version ofthis model that is internally consistent is the model where the volatilities of all options are

the same and constant. This is the original Black and Scholes (1973) model.A rather more sophisticated version of the sticky strike rule is where VTKis independent

of S, but possibly dependent on other stochastic variables. As shown in Case 2 of theprevious section, the only version of this model that is internally consistent is the modelwhere the instantaneous volatility of the asset price is a function only of time. This isMertons (1973) model.

When the instantaneous volatility of the asset price is a function only of time, all Euro-pean options with the same maturity have the same implied volatility. We conclude that allversions of the sticky strike rule are inconsistent with any type of volatility smile or volatil-ity skew. If a trader prices options using different implied volatilities and the volatilitiesare independent of the asset price, there must be arbitrage opportunities.

The Sticky Delta Rule

An alternative to the sticky strike rule is the sticky delta rule. This assumes that the impliedvolatility of an option depends on S and K through its dependence on the moneynessvariable, K/S. The delta of a European option in a stochastic volatility model is

= c

S+

c

VTK

VTKS

Again, for the purposes of calculating partial derivatives the option price c is considered tobe a function ofS,VTK, andt. The first term in this expression is the delta calculated usingBlack-Scholes with the volatility parameter set equal to implied volatility. In the second

term, c/VTK is positive. It follows that, if VTK is a declining (increasing) function ofthe strike price, it is an increasing (declining) function of S and is greater than (lessthan) that given by Black-Scholes. For equities VTK is a declining function ofK and sothe Black-Scholes delta understates the true delta. For an asset with a U-shaped volatilitysmile Black-Scholes understates delta for low strike prices and overstates it for high strikeprices.

In the most basic form of the sticky delta rule the implied volatility is assumed to bea deterministic function ofK/Sand T t. Case 3 in the previous section shows that theonly version of this model that is internally consistent is Mertons (1973) model where theinstantaneous volatility of the asset price is a function only of time. Again we find that the

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model is inconsistent with any type of volatility smile or volatility skew. For no arbitrage,implied volatilities must be independent ofSand K.

A more general version of the sticky delta rule is where the process for VTK dependson K, S, T, and t only through its dependence on K/S and T t. We will refer to thisas the generalized sticky delta model. Models of this type can be consistent with theno-arbitrage condition. This is because equation (5) shows that if eachTKi depends onK,S,T, and t only through a dependence on K/Sand T t, the same is true ofTK.

Many traders argue that a better measure of moneyness than K/S is K/F where asbefore F is the forward value of S for a contract maturing at time T. A version on thesticky delta rule often used by traders is that TK(S, t) TF(S, t) is a function only ofK/FandTt. Here it is the excess of the volatility over the at-the-money volatility, ratherthan the volatility itself, which is assumed to be a deterministic function of the moneynessvariable, K/F. This form of the sticky delta rule allows the overall level of volatility tochange through time and the shape of the volatility term structure to change, but whenmeasured relative to the at-the-money volatility, the volatility is dependent only on K/S

and T t. We will refer to this model as the relative sticky delta model. If the at-the-money volatility is stochastic, but independent ofS, the model is a particular case of thegeneralized sticky delta model just considered.

The Square Root of Time Rule

A rule that is sometimes used by traders is what we will refer to as the square root oftime rule. This is described in Natenberg (1994) and Hull (2002). It provides a specificrelationship between the volatilities of options with different strike prices and times tomaturity at a particular time. One version of rule is

TK(S, t)

TF(S, t)

= ln(K/F)T t where is a function. An alternative that we will use is

TK(S, t) TF(S, t) =

ln(K/F)T t

(9)

We will refer version of the rule where does not change through time as the stationarysquare root of time modeland the version of the rule where the form of the function changes stochastically as thestochastic square root of time model.

The square root of time model (whether stationary or stochastic) simplifies the specifi-cation of the volatility surface. If we know

1. The volatility smile for options that mature at one particular time T, and2. At-the-money volatilities for other maturities

we can compute the complete volatility surface. Suppose that F is the forward price of theasset for a contract maturing at T. We can compute the volatility smile at time T fromthat at time T using the result that

TK(S, t) TF(S, t) =TK(S, t) TF(S, t)

whereln(K/F)

T t =ln(K/F)

T t

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or

K =F

K

F

(Tt)/(Tt)If the at-the-money volatility is assumed to be stochastic, but independent of S, the

stationary square root of time model is a particular case of the relative sticky strike modeland the stochastic square root of time model is a particular case of the generalized stickystrike model.

6 Empirical Tests

In this section we use implied volatilities for over-the-counter options on the S&P 500 totest which of the rules of thumb in Section 5 give the best fit to market data.

Data

The data we use are monthly volatility surfaces for 47 months (June 1998 to April 2002).The data for June 1998 is shown in Table 1. Six maturities are considered ranging from sixmonths to five years. Seven values ofK/Sare considered ranging from 80 to 120. A total of42 points on the volatility surface are therefore provided each month and the total numberof volatilities in our data set is 4247 = 1974. As illustrated in Table 1, implied volatilitiesfor the S&P 500 exhibit a volatility skew with TK(t, S) being a decreasing function ofK.

The data was supplied to us by Totem Market Valuations Limited with the kind per-mission of a selection of Totems major bank clients. Totem collects implied volatility datain the form shown in Table 1 from a large number of dealers each month. These dealersare market markets in the over-the-counter market. Totem uses the data in conjunctionwith appropriate averaging procedures to produce an estimate of the mid-market implied

volatility for each cell of the table and returns these estimates to the dealers. This enablesdealers to check whether their valuations are in line with the market. Our data consists ofthe estimated mid-market volatilities returned to dealers. The data produced by Totem areconsidered by market participants to be more accurate than either the volatility surfacesproduced by brokers or those produced by any one individual bank.

Tests

We first used the data to test the sticky strike rule. The most basic version of the stickystrike rule, where the implied volatility is a function only ofK and T, may be plausible inthe exchange-traded market where the exchange defines a handful of options that trade andtraders anchor on the volatility they first use for any one of these options. Our data comes

from the over-the-counter market. It is difficult to see how the basic version of the stickystrike rule can apply in that market because there is continual trading in options with manydifferent strike prices and times to maturity. A more plausible version of the rule in theover-the-counter market is that the implied volatility is a function only ofK andT t.

We tested this version of the sticky strike rule using

TK(t, S) =a0+a1K+ a2K2 +a3(T t) +a4(T t)2 +a5K(T t) + (10)

where the ai are constant and is a normally distributed error term. The terms on theright hand side of this equation can be thought of as the first few terms in a Taylor series

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expansion of a general function ofKand Tt. As shown in Table 2, the model is supportedby the data, but has an R2 of only 27%.

We now move on to test the sticky delta rule. The version of the rule we consider is therelative sticky strike model where TK(t, S) TF(t, S) is a function ofK/F and T t.The model we test is

TK(t, S) TF(t, S) = b0+b1ln

K

F

+b2

ln

K

F

2+b3(T t)

+b4(T t)2 +b5ln

K

F

(T t) + (11)

where thebiare constants andis a normally distributed error term. The model is analogousto the one used to test the sticky strike rule. The terms on the right hand side of thisequation can be thought of as the first few terms in a Taylor series expansion of a generalfunction of ln(K/F) and T t. The results are shown in Table 3. In this case the R2 ismuch higher at 94.93%.

If two models have equal explanatory power, then the observed ratio of the two modelssquared errors should be distributed F(N1, N2) whereN1and N2are the number of degreesof freedom in the two models. When comparing the sticky strike and relative sticky deltamodels using a two tailed test, this statistic must be greater than 1.12 or less than 0.89for significance at the 1% level. The value of the statistic is 32.6 indicating that we canoverwhelmingly reject the hypothesis that the models have equal explanatory power. Therelative sticky delta model in equation (11) can explain the volatility surfaces in our datamuch better than the sticky strike model in equation (10).

The third model we test is the version of the stationary square root of time rule wherethe function in equation (9) does not change through time so that TK(t, S) TF(t, S)is a known function of ln(K/S)/

T t. Using a similar Taylor Series expansion to the

other models we test

TK(t, S) TF(t, S) =c1ln(K/F)

T t +c2[ln(K/F)]2

T t + (12)

wherec1and c2are constants and is a normally distributed error term. The results for thismodel are shown in Table 4. In this case theR2 is 97.12%. This is somewhat better thanthe R2 for model in (11) even though the model in equation (12) involves two parametersand the the one in equation (11) involves six parameters.

We can calculate a ratio of sums of squared errors to compare the stationary squareroot of time model in equation (12) to the relative sticky delta model in equation (11). Inthis case, the ratio is 1.11. When a two tailed test is used it is not quite possible to rejectthe hypothesis that the two models have equal explanatory power at the 1% level, but itis possible to reject this hypothesis at the 5% level. We conclude that equation (12) is animprovement over equation (11).

In the stochastic square root of time rule, the functional relationship between TK(t, S)TF(t, S) and

ln(K/F)T t

changes stochastically through time. A model capturing this is

TK(t, S) TF(t, S) =c1(t)ln(K/F)

T t +c2(t)[ln(K/F)]2

T t + (13)

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wherec1(t) andc2(t) are stochastic. To provide a test of this model we fitted the model inequation (12) to the data on a month by month basis. When the model in equation (13) iscompared to the model in equation (12) the ratio of sums of squared errors statistic is 2.01indicating that the stochastic square root of time model does provide a significantly better

fit to the data that the stationary square root of time model at the 1% level.The coefficient,c1, in the monthly tests of the square root of time rule is always signifi-

cantly different from zero with a very high level of confidence. Interestingly, in 20 of the 47months it was not possible to reject the hypothesis that c2 = 0 at the 5% level suggestingan approximately linear relationship between TK(t, S) TF(t, S) and3

ln(K/F)T t

Figure 1 shows the level of c1 and the S&P 500 for the period covered by our data. Thecoefficient of correlation between changes in c1 and changes in the level of the S&P 500 is21.5 percent. This is statistically significantly different from zero at the 10% confidence level.

An increase (reduction) inc1corresponds to a decrease (increase) in the skew. The positivecorrelation can be viewed as an extension of the crashophobia phenomenon identified byRubinstein (1994). When the level of the S&P 500 decreases (increases) investors becomemore concerned about the possibility of a crash and the volatility skew becomes morepronounced (less pronounced).

7 Summary

It is a common practice in the over-the-counter markets to quote option prices using theirBlackScholes implied volatilities. In this paper we have developed a model of the evolutionof implied volatilities and produced a no-arbitrage condition that must be satisfied by the

volatilities. Our model is exactly consistent with the initial volatility surface, but moregeneral than the IVF model of Rubinstein (1994), Derman and Kani (1994), and Dupire(1994). The no-arbitrage condition leads to the conclusions that a) when the volatility isindependent of the asset price there must be a U-shaped volatility smile and b) when theimplied volatility is a decreasing (increasing) function of the asset price there must be anegative (positive) correlation between the volatility and the asset price.

A number of rules of thumb have been proposed for how traders manage the volatilitysurface. These are the sticky strike, sticky delta, and square root of time rules. Someversions of these rules are clearly inconsistent with the no-arbitrage condition; for otherversions of the rules the no-arbitrage condition can in principle be satisfied.

Our empirical tests of the rules of thumb using 47 months of volatility surfaces for the

S&P 500 show that the relative sticky delta model (where the excess of the implied volatilityof an option over the corresponding at-the-money volatility is a function of moneyness) out-performs the sticky strike rule. Also, the stochastic square root of time model outperformsthe relative sticky delta rule.

3The approximate linearity of the volatility skew for S&P 500 options has been mentioned by a numberof researchers including Derman (1999).

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Appendix

Proof of Equation (3)

For simplicity of notation, we assume that r and q are constants. DefineF() as the

forward price at time for a contract maturing at time t + t so that

F(t) =S(t)e(rq)t

The price at time t of a call option with strike price F(t) and maturity t + t is given by

c(S(t), Vt+t,F(t), t; F(t), t+ t) =eqtS(t)[N(d1)N(d2)]

where in this case

d1= d2=

t

2 t+t,F(t)(t, S(t))

The call price can be written

c(S(t), Vt+t,F(t), t; F(t), t+ t) =eqtS(t)

d1

d1

(x) dx

where

(x) = 1

2exp

x

2

2

For some x d1

d1

(x) dx = 2d1(x)

and the call price is therefore given by

c(S(t), Vt+t,F(t), t; F(t), t+ t) = 2e

qt

S(t)d1(x)or

c(S, Vt+t,F(t), t; F(t), t+ t) =eqtS(t)(x)t+t,F(t)(t, S(t))

t (14)

The process followed by S is

dS

S = (r q) dt+ dz

Using Itos lemmadF =F dz

When terms of order higher that (t)1/2 are ignored

F(t+ t) F(t) =(t)F(t)[z(t+ t) z(t)]

It follows that

limt0

1t

E[F(t+ t) F(t)]+ = limt0

1t

(t)F(t)E[z(t+ t) z(t)]+

whereE denotes expectations under the risk-neutral measure.Because

E[F(t+ t) F(t)]+ =ertc(S(t), Vt+t,F(t), t; F(t), t+ t)

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and

limt0

1t

E[z(t+t) z(t)]+ = 12

it follows that

limt0

1t

ertc(S(t), Vt+t,F(t), t; F(t), t+ t) =(t)F(t)

2

Substituting from equation (14)

limt0

erteqtS(t)(x)t+t,F(t)(t, S(t)) =(t)F(t)

2

As t tends to zero, (x) tends to 1/

2 so that

limt0

t+t,F(t)(t, S(t)) =(t)

This is the required result.

References

[1] Black, F. and M.S. Scholes (1973), The pricing of options and corporate liabilities,Journal of Political Economy, 81, pp637-659

[2] Bates, D. S. (1996), Jumps and Stochastic Volatility: Exchange Rate Process Implicitin Deutsche Mark Options, Review of Financial Studies, 9, 1, 69-107.

[3] Brace, A., B. Goldys, F. Klebaner, and R. Womersley (2001), Market model of

stochastic volatility with applications to the BGM model, Working paper S01-1, Deptof Statistics, University of New South Wales.

[4] BrittenJones, M., and A. Neuberger (2000) Option Prices, Implied Price Processes,and Stochastic Volatility. Journal of Finance, 55, 2 (2000), 83966.

[5] Christie, A. A. (1982), The stochastic behavior of common stock variances: Value,leverage, and interest rate effects, Journal of Financial Economics, 10, 4,

[6] Derman, E. (1999), Regimes of volatility, Quantitative Strategies Research Notes,Goldman Sachs, New York, NY. Also in Risk Magazine, April, 1999

[7] Derman, E. and I. Kani (1994a), The volatility smile and its implied tree,Quantitative

Strategies Research Notes, Goldman Sachs, New York, NY

[8] Dumas, B., J. Fleming and R.E. Whaley (1997), Implied volatility functions: empiricaltests, Journal of Finance 6, pp2059-2106

[9] Dupire, B. (1994), Pricing with a smile,Risk, 7, pp18-20

[10] Heath, D., R. Jarrow, and A. Morton (1992), Bond Pricing and the Term Structureof the Interest Rates: A New Methodology, Econometrica, 60, 1, 77105.

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[11] Heston, S.L. (1993), A closed-form solution for options with stochastic volatility appli-cations to bond and currency options, Review

of Financial Studies, 6, pp327-343

[12] Hull, J. (2002) Options, Futures, and Other Derivatives 5th Edition, Prentice Hall,Englewood Cliffs, NJ.

[13] Hull, J. and W. Suo (2002), A methodology for assessing model risk and its applicationto the implied volatility function model, Journal of Financial and QuantitativeAnalysis, 37, 2, pp297318.

[14] Hull, J. and A. White (1987), The pricing of options with stochastic volatility,Journalof Finance, 42, pp281-300

[15] Jackwerth, J.C. and M. Rubinstein (1996), Recovering probabilities from option prices,Journal of Finance, 51, pp1611-1631

[16] LeDoit O. and P. Santa Clara Relative pricing of options with stochastic volatilityWorking Paper, Anderson Graduate School of Management, University of California,Los Angeles, 1998.

[17] Merton, R. C. (1973), Theory of rational option pricing, Bell Journal of Economicsand Management Science, 4, pp141-83.

[18] Natenberg S. (1994) Option Pricing and Volatility: Advanced Trading Strategiesand Techniques, 2nd ed. McGraw-Hill.

[19] Rubinstein, M. (1994), Implied binomial trees,Journal of Finance, 49, pp771-818

[20] Schonbucher, P.J. (1999), A market model of stochastic implied volatility, Philosoph-ical Transactions of the Royal Society, Series A, 357, pp.2071-2092

[21] Stein, E. and C. Stein (1991), Stock price distributions with stochastic volatilities: Ananalytical approach, Review of Financial Studies, 4, pp727-752

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1999 2000 2001 20021000

1100

1200

1300

1400

1500

1600

1999 2000 2001 20020.3

0.28

0.26

0.24

0.22

0.2

0.18

Figure 1: Plot of the level of the S&P 500 index (unmarked line, with scale on the leftaxis) and the estimates ofb1(line with diamond symbols, with scale on the right axis) - ourmeasure of skew in the volatility surface. Note the positive correlations between the twoseries.

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Time to Maturity (months)

Strike 6 12 24 36 48 60

120 15.91% 18.49% 20.20% 21.03% 21.44% 21.64%110 18.13% 20.33% 21.48% 21.98% 22.18% 22.30%105 19.50% 21.26% 22.21% 22.52% 22.59% 22.70%100 20.94% 22.38% 23.06% 23.14% 23.11% 23.07%

95 22.73% 23.71% 23.92% 23.73% 23.58% 23.53%90 24.63% 24.99% 24.78% 24.40% 24.10% 23.96%80 28.41% 27.71% 26.66% 25.83% 25.23% 24.83%

Table 1: Volatility matrix for June 1998. Time to maturity is measured in months whilestrike is in percentage terms, relative to the level of the S&P 500 index.

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Variable Estimate Standard Error t-statistic

a0 0.4438616 0.0238337 18.62a1 -0.0001944 0.000036 -5.40a2 1.98e-8 1.38e-8 1.44a3 -0.0262681 0.0033577 -7.82a4 -0.0006589 0.0003454 -1.91a5 0.000029 2.18e-6 13.30

R2 0.2672Standard errorof residuals 0.03274

Table 2: Estimates for the version of the sticky strike model in equation (10) where volatilitydepends on strike price and time to maturity.

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b0 0.005848 0.0003801 15.39b1 -0.2884075 0.0019565 -147.41b2 0.0322727 0.0067576 4.78b3 -0.007574 0.0003487 -21.72b4 0.0015705 0.0000701 22.42b5 0.0414902 0.000918 45.20

R2 0.9493Standard error

of residuals 0.00573

Table 3: Estimates for the version of the relative sticky delta model in equation (11).

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c1 -0.2486061 0.0010002 -248.56c2 0.0023856 0.0031146 0.77

R2 0.9712Standard errorof residuals 0.00543

Table 4: Estimates for the stationary square root of time rule in equation (12).

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Test Statistic

Equation (11) vs equation (10) 32.65Equation (12) vs equation (11) 1.11Equation (13) vs equation (12) 2.01

Table 5: Comparison of Models using the ratio of sums of squared errors statistic. In atwo-tailed test the statistic must be greater than 1.123 (1.111) to reach the conclusion that

the first equation to provide a better explanation of the data than the second equation atthe 1% (5%) level

Hull, John - Volatility Surfaces Rules of Thumb

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