Option pricing

with realistic ARCH processes

Gilles Zumbach, Luis Fernandez

Swissquote BankCh. de la Cretaux 33Case Postale 319CH-1196 Gland, Switzerland

www.swissquote.ch

June 8, 2011

Abstract

Option pricing should be based on a realistic process for the underlying and on theconstruction of a risk-neutral measure as induced by a no-arbitrage replication strategy.This paper presents a realistic and complete “first principles” computation of optionprices. The underlying is modeled by a long-memory ARCH process, with relative returns,fat tailed innovations and multi-scale leverage. The process parameters are estimated onthe SP500 stock index (in the physical P measure) and allows to reproduce all empiricalstatistics, from 1 day to 1 year. For a given risk aversion function, the change of measurefrom P to the risk-neutral measure Q can be derived rigorously along each path drawnfrom the process, namely the Radon-Nikodym derivative dQ/dP can be constructed. Asmall δt expansion allows to compute explicitly the change of measure. Finally, theEuropean option price is obtained as the expectation in P of the discounted payoff with aweight given by the change of measure dQ/dP. This procedure is implemented in a MonteCarlo simulation, and allows to compute the option prices, without further adjustableparameters. The empirical study uses European put and call options on the SP500 from1996 to 2010. The computed implied volatility surfaces are compared with the empiricalsurfaces, in particular with respect to the level, smile, smirk and term structure. All themain characteristics of the implied volatility surfaces are correctly reproduced. Furtherpoints concern the respective role of the P andQmeasures, the distribution of the terminalprices in both measures, the negligible effect of the risk aversion and drift premium, andthe empirical validity of the no-arbitrage argument. Finally, simplifications of the presentexact scheme are suggested.

Keywords: Option pricing, implied volatility, ARCH process, Student innovations,long memory volatility, SP500 European options

1 Introduction

Option pricing is based on the seminal works of [Black and Scholes, 1973] and [Merton, 1973].The success of the Black-Scholes (B-S) approach is due to the correct capture of the basicfactors in the option pricing, namely the diffusive behavior of the underlying and the relationbetween arbitrage and the replication strategy. The underlying is described by the simplestprocess: a random walk for the logarithm of the price with a constant volatility and withnormal innovations. This choice leads to analytical tractability, but is also not very accurateas it fails to capture “modern” stylized facts like heteroskedasticity, fat-tail and leverageeffects. For these reasons, the B-S textbook formula is no longer used for option pricing,but only as a dictionary between prices and implied volatility. This mapping is useful as ittransforms option prices depending on the payoffs into a volatility surface that can be usedfor all option payoffs. But this change of role for the B-S formula —from pricing to an impliedvolatility definition— leaves open the question of a more accurate option pricing framework.

The deficiencies of the B-S formula compared to option prices obtained from the market areconveniently summarized using the implied volatility surface. Instead of a flat “constant”volatility surface, the market implied volatility shows a changing overall level, smiles andsmirks (i.e. a dependence on the option moneyness) and a term structure (i.e. a dependenceon the option maturity). These features are depending on time, namely the implied volatil-ity surface is time-dependent and somehow related to the dynamics of the underlying. Anaccurate option pricing framework should be able to derive the observed features from thebehavior of the underlying, and much work has been devoted towards this goal. One difficultyis that too many convenient properties hold in the B-S model, among them aggregation andinfinite divisibility, Ito calculus, mapping to a differential equation, and identical processesup to a change of drift in the P and Q measures (respectively the physical and risk-neutralmeasures). Essentially, these properties follow from the choice of normal innovations, andto a lesser extend from the constant volatility. In order to describe accurately the underly-ing, more complex processes should be used. The challenge for a good process is to includefat-tailed distributions, heteroskedasticity and leverage, at time scales ranging from 1 dayto 1 year, as observed in the empirical time series. Clearly, these will break the convenientproperties used to derive the B-S pricing equation, and some of them have to be abandonedin a more general option pricing framework. In order to generalize the B-S scheme, essentiallytwo paths have been followed in the option pricing literature. They are summarized in thenext two paragraphs.

The first path uses stochastic volatility processes, while the distribution for the innovationscould be either a more general parametric family of distributions, or made of a Gaussian partand a jump part. The key point is that the same process up to a re-parametrization shouldbe obtained in the P and Q measures. This allows to estimate the process parameters in the Pmeasure using the underlying, them to map the parameters in the Q measure to do the optionpricing. The unobserved volatility states is a nuisance in the parameter estimation, but can beused as degrees of freedom to fit the empirical implied volatility surface. Many contributionsfollow this route, and some recent papers are [Gruber et al., 2010], [Chorro et al., 2010]. Analternative is to use high-frequency data to compute the latent volatility states as donein [Corsi et al., 2010].

The second path uses GARCH volatility processes, while the distribution for the innovationscan be either a normal distribution or a Student distribution. As the GARCH processes do nothave an appropriate continuum limit, the option replication should be done with a finite pro-

1

cess increment. This path was initiated by [Duan, 1995], and then by [Duan and Zang, 2001,Heston and Nandi, 2000]. Furthermore, the research works of [Christoffersen et al., 2010]and [O’Neil and Zumbach, 2009] separate cleanly the process specification from the changeof measure, giving a large freedom in the choices for the process and innovation distributionwhile providing a systematic construction of the Q measure. In this scheme, the flexibility inthe GARCH specification can be used to capture accurately the behavior of the underlying,while maintaining a simple change of measure from P to Q. The central measure is the phys-ical P measure, and the option price is computed by an expectation in P which includes theRadon-Nikodym derivative dQ/dP. As the GARCH processes do not have latent states, theestimation of the parameters on the underlying and the subsequent option pricing is fairlysimple.

We find that the second path is better for several reasons, related both to the description ofthe underlying and to the option pricing scheme. The family of GARCH processes is flexibleenough to naturally accommodate the stylized facts observed in the empirical time series. Thelist is fairly long and includes the long memory for the heteroskedasticity [Zumbach, 2004],the market components observed in the volatility correlation [Lynch and Zumbach, 2003], thefat-tailed distributions for the returns and the innovations [Bollerslev, 1986, Zumbach, 2006],the small or zero skew for the relative returns [O’Neil and Zumbach, 2009], the time re-versal asymmetry [Zumbach, 2009], and finally the leverage effect for stocks and stock in-dexes [Glosten et al., 1993, Zumbach et al., 2010]. This last contribution proposes a parsi-monious multiple GARCH component process that can reproduce all the statistical featuresobserved in the stock time series from 1 day to 1 year. This process is used below to modelthe SP500 index and in the Monte Carlo simulations. Notice that some empirical featurescannot be reproduced by stochastic volatility processes, like the time reversal asymmetry.On the option pricing side, the approach of [Christoffersen et al., 2010] allows to constructthe Q measure systematically, regardless of the process specifications. The risk aversion iscaptured by a function, that can be chosen according to the user preferences. The approachwas modified in [O’Neil and Zumbach, 2009] in order to allow for a fat-tailed distributionfor the innovations by using relative returns instead of logarithmic returns. The scheme isformulated with a discrete time step, naturally suited for GARCH processes. Following thestandard replication and arbitrage argument, the option price is expressed as an expectationin the Q measure, or equivalently an expectation in the P measure with weights given bythe Radon-Nikodym derivative dQ/dP. Only the physical P measure is used for the processdescribing the underlying, and the change of measure induced by the replication is used onlyin the option pricing. This structure separates naturally the role of each measure for itsrespective task, and only the option side depends on the risk aversion through dQ/dP.

The cost for this general scheme is the evaluation of an expectation and an equation to besolved. But these need to be done for each step along each path for the underlying. Inthis form, the cost is prohibitive. In this contribution, an analytical expansion in

√δt is

performed. The small δt expansion allows to perform the expectation analytically, leaving ascalar equation to be solved (for each step along each path). This can be done easily withtoday’s computers, leading to a practical numerical scheme for European option pricing.Roughly, the scheme can be summarized as follows. Choose your favorite GARCH process,and estimate the parameters on the underlying. For a given pricing time t0, input thehistorical underlying prices up to t0 into the process so that the internal states are correctlybuilt. Starting from these states, use the process to simulate the underlying up to theoption’s expiry. Along each trajectory, compute the change of measure from P to Q using theequations below. With a sufficient number of trajectories, compute the European option priceas the expectation (in P) of the discounted terminal payoff with a weight given by dQ/dP. If

2

desired, map the option price into the implied volatility using the B-S dictionary. Therefore,this scheme reduces the option pricing to the numerical implementation of the process andthe change of measure, and no further analytical computations are required.

There are two difficulties along this straight route. First, the risk aversion function shouldbe specified. Following the B-S scheme, a simple exponential can be used. Yet, other choicescan lead to a better replication of the empirical implied volatility (IV) surfaces. Second,the drift µ(t) for the underlying should be evaluated. This is a notoriously difficult task,because the diffusion dominates the process dynamics up to a few years. At best, a decentestimate for the mean drift is obtained. The process drift enters into the computation of thechange of measure through the risk premium combination φ = (µ− rrf)/σ

2, where each termis time-dependent, rrf is the risk-free rate and σ is the (one step) volatility according to theprocess. The difficulty is to determine the value for µ(t) to put in the process and in therisk premium. One obvious possibility is to use the long-term mean drift as a constant drift.Another possibility is to rearrange the risk premium definition in the form µ = rrf + φσ2,where the drift appears as the risk-free rate plus a risk premium proportional to the variance.This form suggest to fix φ and to compute the value for the drift µ(t). Another possibilityis to set the drift depending on the market conditions, mainly as a function of the volatility.The selection of the risk aversion function and of the drift is investigated in this paper. Themain outcome is that the implied volatility surface depends on the process for the underlying,but independent of these choices.

The organization of the paper is as follows. The next section sets up the notations andsummarizes the main results of [Christoffersen et al., 2010, O’Neil and Zumbach, 2009]. Sec-tion 3 presents the small δt expansion and the formula required for a numerical valuation ofoption prices. The relation of the present scheme with the textbook Black-Scholes derivationis investigated in Section 4. Section 5 discusses the numerical implementation of the optionscheme, and the process used to model the SP500 is described in Sec. 6. The drift usedin the process is presented in Section 7, in relation with the risk premium and the marketprice of risk. The typical implied volatility surfaces are analyzed in Sec. 8, for processesincorporating successively more realistic features. Based on these computations, the actualrole of the risk aversion function is discussed in Section 9. The main characteristics of theIV surfaces are extracted using exponential kernels, as described in Section 10. The impliedvolatility surfaces for the SP500 options are compared in Sec. 11 with the computed values,with a detailed analysis of the levels, smirks, smiles and term structures. Section 12 returnsto the topic of the drift and risk aversion but using numerical evaluation on the SP500 data,preceding the conclusions.

2 The general option pricing framework

The valuation time is t0, and the option matures at T with a time-to-maturity given by∆T = T − t0. The time t denotes times between t0 and T . The time is discretized with astep δt corresponding to the time increment of the process. This discretization is used toindex the time-dependent variables by an integer index i corresponding to ti = t0+ i δt, with0 ≤ i ≤ n and the number of steps to reach maturity is given by n = ∆T/δt.

3

Under the physical measure P, the geometric price process Si for the stock is given by

ri+1 =Si+1 − Si

Si= µi + zi+1 − 1 (1)

zi+1 |Fi ∼ D(1, σ2i ), zi+1 ≥ 0

where Si is the stock price at time ti and σ2i is the conditional variance of the relative return

in the period (ti, ti + δt). The expected gross rate of return is µi, and E [ri+1] = µi. Thevariance σ2 and the rate of return µ are taken at the scale δt, namely both are proportionalto δt. They are related to the corresponding annualized quantities by the diffusion scalingσ2 = (δt/1 year)σ2a where σa is the annualized standard deviation of the return (and similarlyfor the drift). As the notation makes clear, µi and σ2i are Fi-measurable. The probabilitydistribution D(1, σ2i ) for the innovation zi has a mean of 1 and a variance σ2i , but is otherwisearbitrary. In particular, fat-tailed distributions are allowed as long as the first two momentsare finite.

The risk aversion function f over the positive real line is

f(z) : R+ → R+ (2)

with the normalization function

ψi(ν) = EPi [f(ν zi+1)] for ν > 0. (3)

The positivity conditions on z, ν and f ensure that ψi > 0. The function f encapsulates therisk aversion of the investor by weighting differently the events zi+1. In practice, this shouldbe a decreasing function, and it can also be viewed as the derivative of a utility function.

For a given risk aversion function f , a change of measure can be constructed, and this quantityis called a Radon-Nikodym derivative. Essentially, the random variable z has a distributionspecified in P, and its distribution in another measure Q is obtained by multiplying by thechange of measure from P to Q. For a predetermined sequence {νi} with νi > 0 , the quantity

dQdP

∣∣∣∣Fn =

n∏i=0

f(νi zi+1)

ψi(νi)(4)

is a Radon-Nikodym derivative. For the particular choice f(z) = exp(−z), the above termis called an Esscher transform. This construction is such that the distribution for z in themeasure Q is a proper probability distribution, namely it is positive and with a unit mass (foreach time ti, and for any choice of f). The conditioning by the filtration on the left hand sidemakes explicit that the realization for zi should be known up to ti, and in particular throughthe time-dependent variance σi and drift µi. Intuitively, this introduces a dependence on thepath followed by the stock Si up to T , but at each step the information required about zi+1

is in Fi. Notice that {νi} can be chosen arbitrarily at this point (with νi in Fi).

This Radon-Nikodym derivative can now be used to specify an equivalent martingale measure(EMM). A risk-free bond process Bi is introduced, with Bi+1 = Bi (1 + rrf,i) and where rrfis the risk-free rate of return and is Fi-measurable. This definition is consistent with theprocess definition as both use a multiplicative set-up. The price process is a martingale in

the measure Q defined by the Radon-Nikodym derivative (4) when EQ[

Si+1

Bi+1

∣∣∣Fi

]= Si

Bi, or

equivalently EQ[Si+1

Si

/Bi+1

Bi

∣∣∣Fi

]= 1. This condition leads to an equation for νi

EPi [zi+1 f(νi zi+1)]

EPi [f(νi zi+1)]

= 1− µi + rrf,i = 1− φi σ2i (5)

4

where the risk premium factor

φi =µi − rrf,i

σ2i(6)

has been introduced in the last equality (see [O’Neil and Zumbach, 2009] for the proof).Notice that the risk premium is dimensionless with respect to δt, whereas the market priceof risk

λi =µi − rrf,i

σi(7)

scales with√δt. Let us emphasize that νi depends on the path followed by the process, and

equation (5) should be solved along each trajectory.

Assuming µi > rrf,i (or equivalently the risk premium factor φi is positive), the rhs in (5)is smaller than 1. For a solution of the (5) to exist, the lhs should also be smaller than1, imposing some restrictions on f . Essentially, f should decay fast enough so that thenumerator is smaller that the denominator. An exponential decreasing risk aversion functionf(x) = exp(−x) seems to be a convenient choice, but other dependencies are possible. Thisfreedom in the choice for f shows that different EMM can be constructed, equivalent to achoice of an utility function (see [Christoffersen et al., 2010] for a discussion on this point). Inorder to have convenient Taylor expansion with fat-tailed distributions, the function f(x) =1/(1 + exp(x)) will be used below.

[Christoffersen et al., 2010] show that the price of an European contingent claim Ct at t = tiwith t0 ≤ ti ≤ T is given by the discounted expectation under the EMM measure Q of theterminal value

Ci = EQ[Cn(Sn)

Bi

Bn

∣∣∣∣ Fi

]= EP

[dQdP

Cn(Sn)Bi

Bn

∣∣∣∣ Fi

](8)

and the Radon-Nikodym derivative between ti and tn = T is given by

dQdP

=

dQdP

∣∣∣Fn

dQdP

∣∣∣Fi

=n−1∏j=i

f(νj zj+1)

Ej [f(νj zj+1)]. (9)

Because the process is known in the P measure, the expectation in P can be evaluated. Thesubtle point embedded in the term dQ/dP is a dependence on the price process through thevariance σ2i and the rate of return µi. Therefore, the change of measure in the expectationinvolves the realizations for the random paths between ti and tn = T .

This formula is important as it allows to price a European contingent claim for very generalprocesses of the underlying. In the present general setting, the expected discounted payoffis the proper generalization of the many B-S equivalent formulas, but the expectation andchange of measure are path-dependent. As such, the option pricing formula is exact andcomplete (and the derivation does not involve any small δt expansion). But it is still not avery practical scheme, as it involves integrals that should be computed and implicit equationsthat should be solved. The next step is therefore to perform a small δt expansion so as tocompute the expectations analytically.

For a deterministic interest rate and convex payoffs, Jensen’s inequality can be used to finda lower bound on the price. For a European call option with strike K, the bound is

Ci =Bi

BnEQ [ [Sn −K]+

∣∣ Fi

]≥ Bi

Bn[Fn −K]+ (10)

5

with the forward price given by Fn = EQ [Sn]. This simple application of Jensen’s inequalitydepends only on the fact that Q is a probability measure with a known expectation Fn forthe underlying price at maturity Sn. Therefore, the same lower bound occurs in a B-S set-up,with a different measure Q and a different distribution for Sn, but the same forward price. Asany process and risk aversion functions share the same lower bound with the B-S equation,the implied volatility equation can always be solved. This lower bound shows that the B-Sdictionary between prices and implied volatilities is well defined (at least for a convex payoff),regardless of the process, parameters and risk aversion.

3 Small δt expansion

In the previous derivations, the dependence in δt is hidden in σ ∼√δt and µ ∼ δt. Systematic

Taylor expansion can be done in√δt, neglecting terms of order δt3/2 and higher. With a

general distribution for z, an expansion of the integrals about the mean can be performed.When the covariance σ2 is small enough, the terms of order higher than σ2 in the expansioncan be neglected. In order to alleviate the notation in the following computations, the timeindexes are omitted whenever possible.

The δt expansion is more natural after using the change of variable z = 1+σ ε. The propertiesE [z] = 1 and E

[(z − 1)2

]= σ2 are equivalent to E [ε] = 0 and E

[ε2]= 1. Yet, a bit of

care is required as the probability distribution for ε decays slowly, and convergence needs tobe preserved. The idea is to expand f(z) using a second order Taylor polynomial with anexplicit remainder given in the Lagrange form, namely

f(x) = f(a) + f ′(a) (x− a) +1

2f ′′(a) (x− a)2 +

1

3!f (3)(ξ) (x− a)3 (11)

for some real value ξ between a and x. The function f(x) is required to be 3-times differ-entiable. As convergence of the integrals over ε is required, the third derivative needs to bebounded uniformly from above

f (3)(x) ≤M ∀x. (12)

and the probability distribution is such that its third moment E[ε3]is finite.

Expanding up to order δt, we have

f(ν z) = f (ν(1 + σ ε))

= f + σ ε ν f ′ +1

2(σ ε ν)2 f ′′ +

1

3!(σ ε ν)3 f (3)(ξ)

= f

(1 + σ ε ν

f ′

f+

1

2ε2 σ2 ν2

f ′′

f+O

(σ3))

' f

(1− ε σ f1 +

1

2ε2 σ2 f2

)(13)

where the following definitions are used

f1(ν) = −ν f′

f(14a)

f2(ν) = ν2f ′′

f(14b)

6

and where f , f ′ and f ′′ are evaluated at ν. The function f1(ν) is positive provided that f(ν)is a decreasing function. The term in O

(σ3)can be neglected for σ small as its coefficient is

bounded and the expectation of ε3 is finite. Taking the expectation

E [f(νz)] ' f

(1 +

1

2σ2 f2

)(15)

and

f(νz)

E [f(νz)]' 1− ε σ f1 +

1

2σ2(ε2 − 1

)f2. (16)

A similar computation gives

E [z f(νz)] ' f

(1 + σ2

(1

2f2 − f1

))(17)

At the same order, (5) becomes

Ei [zi+1 f(νi zi+1)]

Ei [f(νi zi+1)]= 1− σ2i f1,i = 1− (µi − rrf) (18)

and the risk premium equation to be solved for νi is

f1(νi) =µi − rrfσ2i

= φi. (19)

The lhs is a function of νi only while the rhs is a number depending on the process up to tthrough the variance σ2i and drift µi. The rhs is positive, and the lhs is positive providedthat f is decreasing. This equation should be solved for νi, but the value for f1 is known asf1,i = f1(νi) = φi. The value for νi is used only in order to obtain f2,i = f2(νi), and onlythis term depends on the choice for the risk aversion function f(ν). Because the terms withf2 are of order δt in the Radon-Nikodym derivative (see (23) below), the choice of the riskaversion function only weakly influences the implied volatility surface. This shows that theoption prices are fairly robust against the choice of the risk aversion function f , but also thatthe risk aversion is of limited use to improve the match between the theoretical and empiricalIV surface.

With a fat-tailed distribution, some care needs to be taken in the choice for the risk aver-sion function f . A Student distribution can be shifted (and two small constants added toenforce the normalization and variance conditions), so that the technical conditions on thedistribution for z are fulfilled. For the risk aversion, the choice of a simple exponentialf(z) = exp(−z) can lead to a problem in the above expansion for z. For ν large enough,the exponential function overtakes the decrease of the Student distribution for small z. As aconsequence, the integrant p(z)f(z) is no longer maximum around z = 1, but around somevalue for z close to zero. In such a case, the above expansion of the integrals about z = 1breaks down. In order to alleviate this issue, we mainly used a risk aversion function withslower variations around zero given by f(z) = 1/(1 + exp(z)). Assuming the expansion canbe performed safely, the solution for ν in Eq. (19) is close to φ, with φ between 0.5 and 5for real data (see Fig. 11). Therefore, ν is not very large, justifying the expansion aboutz = 1. In practice, the choice for f is mostly irrelevant, as further discussed in Sec. 9 usingnumerical computations.

7

The price of the underlying at expiry needs to be evaluated. Using the definition of the stockprice process (1), the price at T is:

Sn = S0

n−1∏i=0

(1 + µi + σi εi+1) (20a)

= Fn

n−1∏i=0

(1 + µi + σi εi+1

1 + rrf

)(20b)

with Fn = S(t0)(1 + rrf)n the forward price at time T . This formula can be used directly in

a numerical simulation.

Using a Taylor expansion at order O(δt), the price can be approximated by:

Sn = Fn

n−1∏i=0

(1 + µi − rrf + σi εi+1 +O(δt2)

)(21)

' Fn

1 +∑i

(µi − rrf) +∑i

σi εi+1 +∑i<j

σi εi+1 σj εj+1

' Fn

1 +∑i

σi εi+1 +1

2

(∑i

σiεi+1

)2

+∑i

(µi − rrf)−1

2

∑i

σ2i ε2i+1

' Fn

(1 +R+

1

2R2 +

∑i

(µi − rrf)−1

2y − 1

2σ2int

)with:

R =

n−1∑i=0

σi εi+1 (22a)

y =n−1∑i=0

σ2i{ε2i+1 − 1

}(22b)

σ2int =

n−1∑i=0

σ2i . (22c)

The stock path innovation R is of order√n δt =

√∆T while the term y and the integrated

variance σ2int are of order ∆T = n δt. This computation shows that the leading term is thetotal return R ∼

√∆T , but there are corrections of order ∆T compared to the Bachelier

random walk used in the B-S scheme.

Using the same expansion and (16), the Radon-Nikodym derivative (9) becomes:

dQdP

=

n−1∏i=0

f(νizi+1)

E [f(νizi+1)]

'n−1∏i=0

[1− f1,i σi εi+1 +

1

2f2,i σ

2i

{ε2i+1 − 1

}]. (23)

This formula can be directly used in a numerical simulation. Notice that at this order

EQ [1] = EP[dQdP

]= 1. (24)

8

As for the price above, dQ/dP can be expanded up to order δt to yield:

dQdP

' 1−n−1∑i=0

f1,i σi εi+1 +∑i<j

f1,i σi εi+1 f1,jσj εj+1

+1

2

∑i

f2,i σ2i

{ε2i+1 − 1

}' 1−

∑i

f1,i σi εi+1 +1

2

(∑i

f1,i σi εi+1

)2

(25)

+1

2

∑i

f2,i σ2i

(ε2i+1 − 1

)− 1

2

∑i

f21,i σ2i ε

2i+1

' 1− R+1

2R2 +

1

2y − 1

2σ2int

with:

R =

n−1∑i=0

f1,i σi εi+1 =

n−1∑i=0

λi εi+1 (26a)

y =

n−1∑i=0

f2,i σ2i

(ε2i+1 − 1

)(26b)

σ2int =n−1∑i=0

f21,i σ2i ε

2i+1. (26c)

The term R is of order√δt and depends on the market price of risk λi. This computation

shows that the change of measure is given at first order by dQ/dP ' exp(−R), similarly to

the B-S case. But as for the price, corrections of order ∆T enter the formula for the changeof measure, and R depends on the process path up to T through σi and f1,i.

Using the above expressions, it is simple to check that at order δt the following occurs:EP[Sn] = S0(1 +

∑µi), E

P[dQ/dP] = 1 and EQ[Sn] = EP[dQ/dP · Sn] = Fn = S0(1 +∑rrf). In these computations, the expectations of the terms of order

√δt are zero, and the

expectation of the terms of order δt are such that they mostly cancel. The difference betweenthe expectations of Sn in the P and Q measures is created by the cross-product R · R, whichessentially changes the drift from µ to rrf.

In the B-S set-up, the change of measure is determined by the exponential function f(x) =exp(−x). This particular choice ensures that the probability distribution in the measure Qis also a normal distribution, but with a different location. With this choice, the continuumlimit can be taken for the process in P and in Q, with the convenient Ito calculus that followsin both measures. In the present set-up, the process time increment δt is fixed, and theconstruction of a continuum limit is avoided. Since infinite divisibility is not required, morefreedom in the choice for f is allowed. With relative returns, the natural set-up is products ofpositive random variables, leading to the product in (4). Finally, the Ito calculus is “replaced”with a second order Taylor expansion in

√δt.

As shown in this section, the expectations over the distributions for the innovations ε can becomputed in a small δt expansion. This leads to a practical framework to price Europeanoptions on a fundamental basis, which can include heteroskedasticity, fat-tailed distributions

9

and possibly other stylized facts. The user can choose her favorite process for the underlying,validated against empirical data. Monte Carlo simulations are used to draw paths for theunderlying, according to the chosen process. The option pricing formula (8) is used as anexpectation over the paths, with the Radon-Nikodym derivative for each path given by (23).There is no dependence on the process equation in the computation of dQ/dP, and thereforethe scheme is generic. Although numerically intensive, such a procedure is easy with today’scomputational power.

4 Making contact with the B-S formula

The present general option pricing scheme is clearly more complex than the familiar B-Sscheme. In this section, a constant volatility process, a normal innovation distribution, andan exponential risk aversion function are used in order to see how the usual option pricingformula is retrieved, but for relative returns. For f(ν) = exp(−ν), Equations (14) lead tof1 = ν, f2 = ν2 and f21 − f2 = 0. The path-dependent Equation (19) for the EMM reducesto the simple explicit solution νi = ν = (µ− rrf)/σ

2 = φ. The price change (21) becomes

Sn = Sn(R, y) ' Fn exp

(R+ n (µ− rrf)−

1

2y − 1

2σ2int

)(27)

with:

R = σ

n−1∑i=0

εi+1 (28a)

y = σ2n−1∑i=0

{ε2i+1 − 1

}(28b)

σ2int = nσ2 . (28c)

The y term is new and originates in using relative returns. For a call option with payoff[Sn −K]+, the positivity condition derived from (27) is

R− 1

2y ≥ ln

(K

Fn

)− n

(µ− rrf −

1

2σ2). (29)

This condition differs from the usual B-S computation due to the y term.

The change of measure (25) simplifies to

dQdP

=dQdP

(R) ' exp

(−R− 1

2σ2int

)(30)

with:

R = φσ

n−1∑i=0

εi+1 = φR (31a)

σ2int = nφ2 σ2 = φ2 σ2int. (31b)

In the B-S model, the distribution of the innovations for one time step is defined in thephysical measure by a normal distribution

dε pP(ε) =dε√2π

exp

(−1

2ε2). (32)

10

The risk-neutral probability for one time step is

dε pQ(ε) = dε pP(ε)dQdP

=dε√2π

exp

(−1

2ε2 − εφσ − 1

2φ2σ2

)(33)

=dε√2π

exp

(−1

2(ε+ λ)2

)(34)

which is a normal distribution with a shift proportional to the market price of risk λ = φσ,as in the B-S computation. As ν, f1 and f2 have no time dependence, the change of measureand the change of price are directly related, and the distribution for the innovations in theQ measure can be obtained analytically.

In order to make further progress towards a closed form solution, the joint distributionpQ(R, y) should be computed. This distribution is required to replace the multiple integralsover all the εi by two integrals over R and y. The random variable R has a simple normaldistribution, but the joint distribution has no simple analytical form. Using characteristicfunctions, the joint distribution can be expressed as a two-dimensional integral with complexarguments, but these integrals are not elementary. A simpler computation is to approximatethe joint distribution by a normal distribution N~µ,Σ(R, y), which is good for n large enough.Using straightforward Gaussian integration in the Q measure, the mean and covariance are

~µR,y = E

[(Ry

)]' nσ2

(−φσ2 φ2

)= n

(−δµδµ2

)(35a)

ΣR,y ' nσ2(

1 −2φσ2

−2φσ2 2σ2

)= nσ2

(1 −2δµ

−2δµ 2σ2

)(35b)

with δµ = µ− rrf = φσ2. Putting all the pieces together, the call option price C0 at time t0is given by

C0 ' (1 + rf )−n

∫ ∫dR dy N~µ,Σ(R, y) [Sn(R, y)−K]+ . (36)

This formula gives the call price as a two-dimensional Gaussian integral over a domain spec-ified by the option payoff. It cannot be computed easily because of the coupling between theoff-diagonal covariance and the boundary specified by the payoff. In order to get an explicitformula, the procedure is 1) to shift the integral so that a centered Gaussian distribution isobtained, 2) to rotate the R and y axes so that the covariance becomes diagonal and 3) toapproximate the integral boundary as to decouple both integrations. The shift is the easiestpart, and leads to the option pricing formula

C0 ' (1 + rf )−n

∫ ∫dR dy N~0,Σ(R, y)

[Sn(R− n δµ, y + n δµ2)−K

]+. (37)

A rotation by the angle α ' 2 δµ diagonalizes the covariance at order δt. We denote by R′

and y′ the coordinates after the rotation, which are given by R′ ' R − αy and y′ ' y + αRat order δt. The covariance in the new coordinates is

ΣR′,y′ ' nσ2(1

2σ2

). (38)

In these new coordinates the call price reads

C0 ' (1 + rf )−n

∫dy′ N0,2nσ4(y′)

∫dR′ N0,nσ2(R′)

[S′n(R

′, y′)−K]+

(39)

11

with the underlying at maturity given by

S′n(R

′, y′) ' Fn exp

(R′(1 + δµ)− 1

2y′(1− 4 δµ)− 1

2nσ2

). (40)

The variance for R′ is given by nσ2 = (∆T/1 year)σ2a while the variance for y′ is 2nσ4 =(∆T/1 year)σ2a (δt/1 year)σ2a. This shows that the fluctuations for y′ are of order δt smallerthan the fluctuation for R′. The positivity of the option payoff leads to the equation for theboundary

R′ − y′

2(1− 5 δµ) ≥ B (41)

with the boundary given by

B = (1− δµ)

(ln

(K

Fn

)+nσ2

2

). (42)

Equation (41) specifies a line of slope '1/2 in the (R′, y′) plane. Because the fluctuations inthe y′ direction are of order δt smaller compared to the R′ direction, the boundary can beapproximated by R′ ≥ B. This approximation decouples both integrals, leading to

C0 ' (1 + rf )−n

∫Rdy′ N0,2nσ4(y′)

∫ ∞

BdR′ N0,nσ2(R′)

[S′n −K

](43)

Using the formula∫ ∞

BdR′ Nµ,σ2(R′) = Φ

(− B − µ

σ

)(44)

with Φ(x) the cumulative function of the Gaussian distribution, the price of the call optioncan be expressed as

C0 =(1 + rf )−n

{Fne

nσ4(1/4+φ)Φ

(1− δµ√nσ

[ln

(Fn

K

)+nσ2

2(1 + 2δµ)

])

−KΦ

(1− δµ√nσ

[ln

(Fn

K

)− nσ2

2

])}(45)

At leading order in√δt, the B-S formula is retrieved. The corrections are of order δt and

involve the expression δµ = µ − rrf = φσ2. This computation shows that the correctionsare dependent on µ − rrf, or on the risk premium parameter. These corrections introduce asmile of order δt in an implied volatility computation, even though the B-S process and riskaversion have been used.

5 Numerical implementation

The option price for a call option given by (8) is computed with Monte Carlo simulations ofprice paths starting at t = t0 with

C0 = EP[dQdP

Cn(Sn)B0

Bn

∣∣∣∣ F0

]=

1

N

∑α

dQdP

∣∣∣∣α

Cn(Sα,n)B0

Bn, (46)

12

and similarly for put options where one just needs to change the payoff function. The subscriptα indexes the realization of the price path Sα drawn according to the LM-ARCH process(in the physical measure P). The Monte Carlo sample size is N . The Radon-Nikodymderivatives depend on the price paths, and therefore also carry an α index. The filtration inthe mathematical formula is translated into the initial conditions for the process that shouldinclude the history up to t0.

In order to compute the change of measure efficiently along each trajectory, we need to solveEq. (19). For the function f(z) = 1/(1 + exp(z)), this can be done simply by rewriting (19)as a fixed point equation for ν

ν(n+1) = φ(1 + e−ν(n)

)(47)

which is solved in a few iterations.

With the solution for νi, the term f2(νi) can be computed, and then the Radon-Nikodymderivative (23). The option prices are evaluated using the simple average over the path pricesgiven by Eq. (8).

5.1 Day counting

Some care needs to be taken with respect to day counting, respectively for the diffusion anddiscounting terms. As the stock exchange is closed on the week-end, the empirical statisticson the underlying time series are computed using a business day count (i.e. 5 days per week,260 days per year). Accordingly, the process parameters are estimated using a business daycount. In the Monte Carlo simulations, the diffusion is done consistently with a business daycount. Yet, the discounting should be done based on a physical day count (i.e. 7 days per week,365 days per year). For aggregated equations like B-S involving only the terms r∆Tphysicaland σ2∆Tbusiness, the differences are small (except possibly for very short maturities). Thedifference is usually ignored by annualizing all coefficients and time intervals (with theirrespective day count conventions), as can be done for example in the B-S pricing formula.

Because the present option framework is set at the process increment δt = 1 day and notat an aggregated level, the day count issue should be tracked. In the theoretical derivation,a business day count must be used for the diffusion. A simple approach is to replace thediscounting in physical days by an equivalent discounting in business days. The bond processused for setting the equivalent martingale measure should be replaced by Bi+1 = Bi(1+ r

∗rf,i)

with the rates r∗rf the equivalent risk-free rates on a business day count basis. The rates arerelated by (1 + r∗rf,i)

5 = (1 + rrf,i)7, or at leading order r∗rf = 7/5 rrf. The dividend yield q

as computed by OptionMetrics1 is given on a physical day count basis, and therefore shouldalso be transformed to a business day count with q∗ = 7/5 q. After these changes, the riskpremium factor is evaluated with the process parameters given at a daily time scale

φi =µi + q∗i − r∗rf,i

σ2iall terms in the rhs at a daily scale. (48)

As all the terms in the rhs scale with ∆t, the equation can be rewritten with the annualizedquantities by multiplying by ∆t = 260 days. Because 7 · 260/5 ' 365, this leads to

φi =µi + qi − rrf,i

σ2iall terms in the rhs at a yearly scale. (49)

1http://www.optionmetrics.com

13

This shows that the day count issue can be ignored provided that the drift, dividend yield,interest rate and volatility are annualized when computing the risk premium factor. Onlyin the process equations, the drift and volatility need to be scaled at a daily level using abusiness day count. Therefore, up to these precautions in setting the parameters and usinga business day count, the day count issue can be ignored.

5.2 Moneyness

In the discussions hereafter and the figures below, the smiles are given as a function of themoneyness defined by

m =ln(Fn/K)√

n σ(50)

with Fn the forward price at the option’s maturity, and n the number of days up to maturity.Essentially, the moneyness measures the proximity of the forward price to the strike K, inunits of the diffusion and with a volatility corresponding to the expected cumulative volatilityup to maturity. Note that the same definition is used for both call and put options. Withthis definition, the moneyness is zero when the strike equals the forward price discounted atthe risk-free rate. The more liquid options are out-of-the-money, corresponding to a negativemoneyness for call options, and positive moneyness for put options. The following tableillustrates how the ratio Fn/K increases with the number of days up to maturity for a fixedm = 1. This moneyness corresponds to deep-in-the-money call options.

n 1 2 5 10 21 65 130 260 520

Fn/K 1.01 1.02 1.03 1.04 1.06 1.11 1.15 1.22 1.33

The variance forecast σ2 in (50) is computed using the weights given in [Zumbach, 2004]applied to the variance components σ2k in the process. The analytical derivation of theforecast weights does not include the leverage effect, but this effect is included in σk. Thisis a plausible approximation as the leverage is odd in the return, and the returns have anapproximately symmetric distribution. A numerical check of this variance forecast againstthe realized variance in Monte Carlo simulations shows that this approximation is good.

5.3 Monte Carlo pricing

The paths for the underlying are generated up to the longest time-to-maturity found on agiven option set. All the options are priced on the same set of paths, reducing the numericalnoise on the price and volatility surface. Moreover, an antithetic scheme is used, with pathsgenerated pairwise with the innovations {εi} and {−εi}, as to reduce the variance of the priceestimators.

The price estimator (46) leads to a numerical problem for deep in-the money options. Inthis region, the option’s time value is extremely small compared to the intrinsic value, andthe heteroskedasticity and fat-tail distribution for the innovations makes the numerical con-vergence slow. Consequently, the lower bound given by Eq. (10) could be broken and thesubsequent implied volatility is undefined.

14

We observe that this technical difficulty arises more often for put options than for call options,typically for contracts with moneyness below -2. At these moneynesses, the price is essentiallygiven by the intrinsic value with a tiny contribution due to the diffusion. With the selectedprocess and leverage, the time value is reduced and hence the corresponding implied volatility.

Our first attempt to cure the above issue was to use quasi Monte Carlo methods. We triedseveral low-discrepancy sequences, none of which yielded an improvement for the problemat hand. The cause for this failure is the large dimensionality of our problem induced bythe path-dependence of the simulation: we need to generate point sets of dimension equalto the number of time steps up to the longest time-to-maturity, which in our case can easilybe beyond 520. Moreover, low-discrepancy sequences generally endure from a breakdown ofuniformity in high dimensions.

In order to improve the convergence, the simple price estimator given by (46) is modified byadding the theoretical lower bound and subtracting the numerical lower bound

Ccorr0 =

1N

∑α

dQdP

∣∣∣αCn(Sα,n)

B0Bn

1N

∑α

dQdP

∣∣∣α

(51)

+ w(n,m)

−B0

Bn

1N

∑α

dQdP

∣∣∣αSα,n

1N

∑α

dQdP

∣∣∣α

−K

+

+B0

Bn[Fn −K]+

,

where w(n,m) is a weight function taking values in [0, 1] and depending on both the time-to-maturity and the moneyness. The corresponding formula for put options is straightforward.Note that taking the dividend yield q into account, the forward price with compoundedcompounding reads Fn = S0(1 + rrf)

n(1 + q)−n.

In (51), we subtract the numerical value of the option payoff computed at the numericalforward price, and add the corresponding theoretical value, so as to leave unchanged theoption price. Furthermore, the Monte Carlo expectations are normalized by the numericalexpectation of the change of measure. This correction exploits the idea that the numericalbare option price and the forward price have similar errors for deep-in-the-money options.Essentially, the term in bracket should be zero, up to numerical errors that are negativelyrelated to the numerical errors in the option price. Therefore, the numerical errors shouldcompensate in (51), resulting in a more accurate value for the in-the-money option price. Thisnumerical compensation is not effective for all moneyness and maturity, and a weight factorw(n,m) is used to lower the correction when inappropriate. Note that we do not change thedistribution law of the underlying process, thus preserving the martingale property of thediscounted asset.

Figures 1 and 2 investigate the efficiency of the different option price estimators. The op-tion price surfaces are computed with 10.000 Monte Carlo simulations. This computation isreplicated 100 times, and the variance of the numerical prices over the replications is com-puted. The price variance increases with the time-to-maturity as

√τ . The figures display the

variance of the numerical option prices, scaled by 1/√τ .

The top lines correspond to the simple estimator (46) without and with antithetic variates.For the in-the-money side and for short maturities, the variance reduction for the price isvery large —note the logarithmic scales— when using antithetic variates. But the reductiondeteriorates for long maturities, and is ineffective on the out-of-money side. Intuitively, the

15

scheme works when most pairs of antithetic paths are included in the option price, whichoccurs deep in-the-money.

The middle lines correspond to the corrected estimator (51) without weights (i.e. w = 1). Asexpected, the correction works remarkably well for deep-in-the-money options, and roughlyuniformly for all time-to-maturities. But for a moneyness around zero, the variance can belarger than with the simple estimator. In this region, the numerical forward price and thesimple price estimators are weakly correlated, and the addition of both term deteriorates theoverall precision. We also note that the reduction in variance with antithetic variates is lessevident when using the corrected estimator.

In order to tame the correction around m = 0, we introduce the weight functions w(n,m) forcall and put options. The idea is to have w = 0 for out-of-the-money up to m = 0, and then agradual increase up to w = 1 for deep in-the-money options where the correction is effective.The cross-over is adjusted as function of the moneyness and time-to-maturity for put andcall options, so as to reduce the variance everywhere. The bottom lines correspond to thecorrected estimator (51) with weights. Comparing with the middle graphs, the reduction ofvariance is clear in the small moneyness range. Comparing the left and right plots showsthat using antithetic variates can result in increasing or decreasing the variance, dependingon the moneyness and time-to-maturity. Clearly, using antithetic variates does not uniformlyimprove the estimator. As the use of this last estimator with antithetic variates is better formost options, we decided to use it for the subsequent computations.

6 Process for the SP500

The multi-scales long-memory ARCH process used for the underlying has been describedin [Zumbach et al., 2010]. The process parameters are calibrated on the SP500 time series,on the period from January 1996 to June 2010, using the heuristic procedure described inthe latter reference. Essentially, the parameters are adjusted as to reproduce at best a broadpanel of (robust) statistics computed with characteristic time intervals ranging from 1 dayto 1 year. Because the SP500 time series corresponds to a stock index, some clear differencesappear compared to individual stocks.

First, the mean volatility is lower, of the order of 20%. Second, the leverage effect is muchlarger. Indeed, the coupling constant for the leverage is taken close to the maximum possible,while still leading to a simulated leverage lower than the empirical leverage. Possibly, theprocess is misspecified with respect to this property. Third, the heteroskedasticity is maxi-mum for shorter characteristic time intervals, and the decay for the weights of the volatilitycomponents is adjusted accordingly. Fourth, option data are provided only when the marketis open and no Holidays should be included in the process. As a consequence, this reducesthe kurtosis (because the fractions of zero returns due to Holidays increase the kurtosis mea-sures for stock time series given on a strict 5 days per week basis). Fifth, the kurtosis issmaller compared to stocks, as it is plausible that the index experiences less extreme movesthan its components. This is accommodated by increasing the number of degree of freedomsof the Student distribution used for the innovations, and by increasing slightly the couplingconstant w∞ of the mean volatility. Sixth, the index level granularity can be ignored. Thisis due to the construction of the index from the individual stock prices, which reduces thecomposite granularity to a negligible level.

16

-3 -2 -1 0 1 2 3

5x10-2

10-1

2x10-1

5x10-1

100

2x100

Moneyness-3 -2 -1 0 1 2 3

5x10-2

10-1

2x10-1

5x10-1

100

2x100

Moneyness

-3 -2 -1 0 1 2 3

5x10-2

10-1

2x10-1

5x10-1

100

2x100

Moneyness-3 -2 -1 0 1 2 3

5x10-2

10-1

2x10-1

5x10-1

100

2x100

Moneyness

-3 -2 -1 0 1 2 3

5x10-2

10-1

2x10-1

5x10-1

100

2x100

Moneyness-3 -2 -1 0 1 2 3

5x10-2

10-1

2x10-1

5x10-1

100

2x100

Moneyness

Figure 1: The standard deviation of the call option price estimators, scaled with 1/√τ . The

left (right) column is computed without (with) antithetic variates. The top line is the simpleprice estimator (46), the middle line is the corrected price estimator (51) with w = 1, thebottom line includes the weight.

Given these differences, the parameters for the process given in [Zumbach et al., 2010] are:w∞ = 0.05 (coupling constant for the mean volatility), σ∞ = 0.20 (mean volatility), τ0 = 520days (decay of the coupling constants for the volatility), λlev = 0.95 (intensity for the leverageeffect) and λrange = 0.25 (range for the leverage effect, in units of the volatility). The numberof degrees of freedom for the Student distribution of the innovations is ν = 7. The peculiar

17

-3 -2 -1 0 1 2 3

5x10-2

10-1

2x10-1

5x10-1

100

2x100

Moneyness-3 -2 -1 0 1 2 3

5x10-2

10-1

2x10-1

5x10-1

100

2x100

Moneyness

-3 -2 -1 0 1 2 3

5x10-2

10-1

2x10-1

5x10-1

100

2x100

Moneyness-3 -2 -1 0 1 2 3

5x10-2

10-1

2x10-1

5x10-1

100

2x100

Moneyness

-3 -2 -1 0 1 2 3

5x10-2

10-1

2x10-1

5x10-1

100

2x100

Moneyness-3 -2 -1 0 1 2 3

5x10-2

10-1

2x10-1

5x10-1

100

2x100

Moneyness

Figure 2: As for Figure 1, but for the put options.

case of the drift is discussed in the next section. Let us emphasize that these six parameters forthe process are sufficient to reproduce the empirical statistics on the SP500. Subsequently, theimplied volatility surfaces depend on these values, but no other free parameters are involved(as we will see below, the drift and risk aversion function play only a minor role).

18

7 Expected return, risk premium and market price of risk

The estimation of the process parameters against the SP500 statistical properties is a straight-forward procedure, and the resulting parameter values corresponds to our intuition. Theexception is the drift µ which is notoriously difficult to estimate reliably because its value isstrongly sample dependent. But even for this parameter, the simple estimation given by themean sample return µ = 〈r〉 leads to very good results in reproducing the properties of theSP500.

Yet, using a constant value for the drift in the option pricing scheme leads to unintuitivevalues for the risk premium and market price of risk. This can be understood by consideringthe values for the risk-free rate rrf and the market volatility over the considered time period,as plotted on Fig. 11. The risk-free rate varies from a low of 0.25% early 2010 to a high of6.8% in the second half of 2000. The volatility ranges from a low 10% during the 2005/2006to a high of 60% in the Fall 2008 after the bankruptcy of Lehman Brothers, as it can be seenon the bottom plot of Figure 10. The dividend yield has a fairly innocuous dynamics, about0-3%, and not obviously related to the large macro-economic cycles. Inserting these timeseries into the definition of the risk premium (49) with a constant 6% drift leads to valuesranging from negative at the end of 2000 to above 4 early 2004. Clearly, at least a positivefloor has to be imposed, but the dynamics for the risk premium is too large.

An opposite solution is to fix the risk premium factor, for example to φ = 1.5, and to computethe drift from Equation (49), namely

µ = rrf − q + φσ2. (52)

Because the volatility can vary by a factor of 6 over the considered period, the risk premiumcan change by a factor 36. This is a too large range, leading to unrealistic drifts (or to toosmall risk premium factor). For example, taking φ = 1.5 results in an unrealistic drift for theSP500 of the order of 40% and above in periods of crisis.

Both propositions above are extreme, corresponding respectively to constant drift and con-stant risk premium factor. But there is no reason to choose the drift in this way. Our positionis that the drift should be estimated according to the prevailing market conditions at a giventime, so as to have a plausible forecast for the mean expected value for the index. A similarproposition was already put forward by [Merton, 1980], arguing that the risk premium shouldbe explicitly modeled. We have used the following form for the drift

µ = rrf − q + β ln

(1 +

σ

σrp

)(53)

where the coefficients β and σrp fix the risk premium associated with the volatility, and σ isgiven by a volatility forecast for the next 6 months. The description of the forecast was givenat the end of Section 5.2. On general grounds, the drift should be an increasing function ofthe volatility, leading to β > 0. It seems also reasonable to assume a concave dependence onthe volatility, so that µ increases less for high volatilities. The logarithm used in (53) is asimple increasing and concave function, but other choices are clearly possible. More generalforms can include for example a dependence on the risk-free rates (based on the ground thatthe risk premium should be higher when interest rates are high). The parameter σrp controlsthe concavity, and we have set σrp = 20%. Finally, the mean drift over the considered periodshould be compatible with the observed mean drift for the SP500, which is of the order of6%. This constraint restricts the value of β, and we have taken β = 0.075. These values lead

19

to risk premia ranging from 0.5 to 5 in the empirical sample (see Fig. 11 below), with a meanvalue around 1.8.

The market price of risk λ is also plotted on Fig. 11. Interestingly, the value is nearly constantwith λ ' 0.27. This can be understood by a first order Taylor expansion of (53) about σrp

λ ' β

σrp

(ln(2)− (ln(2)− 0.5)

σ − σrpσrp

+ · · ·). (54)

The coefficient of the linear term ln(2)− 0.5 ' 0.2 is small, and therefore the fairly constantvalue for the market price of risk. The market price of risk scales as

√∆T and can be used

to define a time horizon at which the drift starts to dominate the diffusion. At a given timehorizon ∆T , the market price of risk λ∆T is

λ∆T =µa∆T

σa√∆T

= λ√∆T (55)

where µa and σa are the annualized drift and volatility, and λ = λa is the annualized marketprice of risk. The drift of the process is comparable to the standard deviation at the timehorizon ∆T× such that λ∆T ' 1. This relation leads to the cross-over time

∆T× '(1

λ

)2

. (56)

The selected values for the parameters lead to ∆T× ' 13 years. This value roughly agrees withthe “buy-and-hold” strategy which advocates that investors engage in long-term investments.

Beside the risk aversion function f , the model for the drift is the only place where thecomputed implied volatility surfaces can be influenced. The previous discussion about thedependence of the drift µ on the volatility σ shows that the parameters are indeed fairlyrestricted. The results of these choices are discussed further in Sections 9 and 12.

Another path would be to break the consistence between µ and the risk premium φ, namelyto choose both parameters without enforcing relation (6). The rationale is that the driftis difficult to estimate, and the parameter µ in (6) could be estimated by the market to adifferent value compared to the underlying true drift. Such a choice essentially makes φ afree parameter that can be used to adjust the theoretical IV surface to the empirical one.Although convenient, such a path is odd in a theoretical computation. For the process, thedrift parameter µ is set by the user (and likely with a value different from an hypotheticaltrue market drift). In the subsequent option pricing, the risk premium is computed from thechosen process drift. As the option pricing scheme is derived theoretically, there is no reasonto break Equation (6). Essentially, the process and computation of the measure Q are underthe control of the user, and therefore the consistence given by (6) should be maintained. Asthe main role of the change of measure is to transform the drift into the risk-free rate, theimplied volatility surfaces are therefore mostly independent of the selected drift.

8 Numerical investigation of the pricing scheme

The salient properties of the option pricing scheme are investigated first in a simple set-up.The process is started with an history of constant price, and therefore no leverage dynamicsdue to the history is included in the initial conditions. All the internal states σ2k measuring

20

the variances at increasing time horizons are set to the mean variance, as if the volatility sincethe infinite past was at the preset level. The initial price is set to 1000 (e.g. $). Otherwise,the LM-ARCH process parameters are as given in Sec. 6, with a drift dependent on thevolatility given by (53). The risk premium is computed from the process drift and volatility.The risk-free rate and dividend yield correspond to the SP500 sample mean, respectively of3.5% and 1.5%. The mean volatility is set to 10%, 20% or 40%, leading to the followingmean drift µ and risk premium φ.

σ∞ µ φ

10 4.64 % 2.64

20 6.50 % 1.13

40 9.14 % 0.446

underlying

-1.5 -0.5 0.5 1.0 1.5

200400600800100012001400

-1.5

-1.0

-0.5

0.0R

realizedVol

0.2

0.4

0.6

0.8

1.0

200 800 1400

0.5

1.0

1.5

0.2 0.6 1.0

DQ_over_DP

Figure 3: Scatter plot of the main variables in a Monte Carlo simulation of size 1000. Thevolatility is 20%, the time-to-maturity is one year.

For a Monte Carlo simulation of 1000 paths and a time-to-maturity of one year, Figure 3presents a scatter plot of the main variables: final price Sn, path innovation R (22a), inte-

21

grated volatility (22c) and change of measure dQ/dP. The strong dependence between finalprices and path innovations is clear, with a good approximation given by Sn ' Fn exp(R). Adependence between final price and volatility is also observed, with large realized volatilityrelated to small final prices. This dependence is created by the leverage effect, that enhancescrashes. The large dispersion of the realized volatility is also noticeable, and is due to theheteroskedasticity included in the LM-ARCH process. There is also a dependence betweenchange of measures and final prices (or path innovations R). Yet, the dispersion shows thata simple approximation where the Radon-Nikodym derivative is a function of the path in-novation only is inappropriate. The same figure but for shorter time-to-maturity shows lessdispersion in the change of measures versus the final prices.

The distribution for the final prices is naturally obtained in the P measure. The mean changeof measure conditional on the final price dQ/dP(Sn) = E [dQ/dP | Sn] can be evaluated fromthe simulations. For short, we call this quantity the change of measure. Then, the distributionof final prices in the Q measure can be obtained by multiplying the two previous quantities.These numerical simulations of the distribution in P, the distribution in Q, and of the changeof measure are easy in the present framework.

The change of measure is known under various names in the finance literature: the as-set pricing kernel [Rosenberg and Engle, 2002], the state-price density per unit probabil-ity [Barone-Adesi et al., 2008], or the stochastic discount factor (see e.g. [Cochrane, 2001]for a review). It is generally accepted on economical ground that this should be a decreas-ing function. The probability distribution in the Q measure can also be evaluated empiri-cally using the second derivative with respect to the strike of the option prices, following aproposition by [Breeden and Litzenberger, 1978]. As the evaluation of the second derivativeincreases the noise, the empirical evaluation of such quantities is not easy, and many pa-pers have been devoted to this computation following different ideas to tame the instability(see e.g. [Aparicio and Hodges, 1998] for a review). These computations give access to theQ measure only, and the P measure or the change of measure should be inferred by othermeans.

Figure 4 shows these quantities for the volatilities σ = 10, 20 and 40% and for a time-to-maturity of one month and one year. The differences between the probability distributions inP and Q are fairly small, and roughly equivalent to a shift. This can be understood intuitivelyas the process is dominated by the diffusion at these time horizons, and the drift has only asmall contribution. Then, the whole construction of the Radon-Nikodym derivative has themain effect of changing the drift from µ to rrf.

As it can be seen on Figure 4, the change of measure is not necessarily decreasing for suf-ficiently large time-to-maturity on both tails of the distribution. [Barone-Adesi et al., 2008]already observed a similar shape in the lower tail, but using a different option pricing scheme(based on a GARCH(1,1) process expressed in the P and Q measures). The non-uniform be-havior can be understood intuitively as follows. The center of the distribution is dominatedby “typical” paths, and the two different drifts produce a small shift of the distributions.This small difference makes for the fairly linear change of measure. By contrast, for a suffi-ciently long time-to-maturity, the tails of the distribution correspond to persistent and largevolatility situations, as induced by the heteroskedasticity of the ARCH process. These pathsare dominated by the diffusion, and the drift plays a minor role. Hence, the probabilities inP and Q are close, and the change of measure tends to 1 for large negative R, as we can seefrom Figure 3.

22

The implied volatility surface can be computed easily for different processes in this scheme.A few reference processes are used in Fig. 5, all with a mean volatility of 20%. From topto bottom, the IV surface corresponds to a constant volatility with Student innovations(ν = 7), a GARCH(1,1) process (w∞ = 0.15, τ = 21 business day) with normal innovations,and a GARCH(1,1) process (same parameters) with Student innovations (ν = 7). Thissequence is interesting as the effect of the various ingredients on the smile can be clearlyobserved. With Student innovations, the short-term maturities have a clear smile. Thesmile is decreasing for increasing maturity, as the central limit theorem takes place and thereturns have a distribution approaching a normal distribution. Yet, an increasing differenceis observed for the put surface, negative moneyness, and increasing time-to-maturity. This isdue to using relative returns instead of logarithmic returns, leading to a difference for longtime-to-maturity in the lower tail. The same phenomenon is seen with normal innovationsand relative returns. Intuitively, there are no important differences when using logarithmic orrelative return for a short time interval. For a long time interval, the probability distributionsfor the returns differ due to the possibility of bankruptcy when using relative returns, andthe deep out-of-the money put options are sensitive to this part of the distribution. Forthe GARCH(1,1) volatility with normal innovations, there is no smile for short-term andthe curvature increases with increasing time-to-maturity. This is due only to the effect ofthe heteroskedastic volatility, which creates a non trivial distribution for the returns at anintermediate time-to-maturity. Finally, using both a GARCH(1,1) volatility and Studentinnovations leads to a smile at all time-to-maturities. Essentially, the short-term smile isdue to the fat-tail distribution for the innovations, while the long-term smile is due to thevolatility dynamics.

This set of surfaces is important as they allow to understand the various origins for the smile.For short time-to-maturities, it is the non Gaussian distribution for the returns that createsthe smile. As the difference between a normal distribution and a Student with 7 degreeof freedom is fairly large, this induces a significant smile. The graphs for the GARCH(1,1)volatility with normal innovations show that the short-term smile is of order δt. With increas-ing time-to-maturity, the volatility dynamics increases the difference compare to a constantvolatility, difference which becomes significant after one to three months. The combinationof a dynamical volatility and Student innovations create significant smiles for all time-to-maturities, due to the cross-over between aggregation and volatility dynamics. This order ofmagnitude for the size of the smile is used in Sec. 12 in order to understand the role of thedrift function, risk premium and risk aversion function.

Adding more features into the process, the long memory LM-ARCH process is used to com-pute the implied volatility surface in Fig. 6. The top graphs include no leverage, as e.g.appropriate for foreign exchange. The middle graphs include a medium-sized leverage, asappropriate for individual stocks. The bottom graphs include a large leverage effect, as ob-tained on the SP500 index. The increasing leverage (from top to bottom) increases the smirk,in particular for long-dated options. The short-term options are not affected by the leverage,because the chosen initial conditions include no directional moves. The similarity betweenthe GARCH(1,1) and LM-ARCH process (Fig. 5 bottom and 6 top) shows that the typicalvolatility smile is not very sensitive to the details of the volatility dynamics. The agreementbetween both surfaces is enhanced by the choice of a medium term characteristic time forthe GARCH(1,1) process, right in the middle of the computed term structure, and by the“static” initial conditions for the processes. Yet, the differences between the surfaces increasein a proper dynamical setting when the historical values for the underlying are used as initialconditions.

23

The implied volatility surfaces are displayed in Figure 7 for several levels of mean volatility:σ∞ = 10, 20 and 40%. The other parameters for the LM-ARCH process are as indicated inSec. 6. Notice that the moneyness is defined in units of the diffusion, namely the distancebetween strike and underlying price is measured with the volatility dependent unit

√n σ.

This natural choice leads to very similar surfaces, while the corresponding strike values areincreasing with increasing volatility. Another choice for the horizontal axis would show adependence on the volatility of the smirk and smile.

The put and call surfaces are very similar for increasing volatility, but not identical in partic-ular for negative moneyness. The similarity of the IV surfaces shows that using the impliedvolatility surface computed from vanilla options in order to price more complex derivatives islikely a good approximation, but it is not exact. For short maturities, the IV smile is fairlysymmetric, and the asymmetry grows with the time-to-maturity. This is due to the initial con-ditions of the process (no directional moves) coupled with the leverage effect. With increasingtime-to-maturity, the leverage induces more asymmetry in the path distribution, particularlywith the multi-scale leverage used in the present LM-ARCH process. Another interestingfeature is that the smile is very similar with increasing maturity, and is not obviously de-creasing. This fairly constant smile is due to the long-memory ARCH, with heteroskedasticitypresents at all time horizons up to one year. The multi-scale heteroskedasticity prevents thedistribution of the return to become Gaussian —as the central limit theorem for independentvariable would imply— leading to a clear smile even for long-dated options.

The risk premium (or the market price of risk) is the free parameter available in the optionpricing scheme. Figure 8 displays the implied volatility surface for β = 0.075 (i.e. the valuechosen for the SP500), and at half and double of this value. This corresponds to fairlylarge changes for the drift and for the risk premium, within the relation defined by (53).Nevertheless, the implied volatility surfaces are almost identical. The short-term smiles (greento blue curves) are essentially similar because the smile is due to the fat-tailed innovationsand the changes in the risk premium introduce only correction of order δt. For long-datedoptions (red curves), the cumulative effect in the change of measure introduces small butvisible changes. This stability shows that the present scheme is very weakly dependent on thedrift and risk premium. A similar independence is observed with the SP500 time series (seeSec. 12). This very weak dependence with respect to the drift and risk premium shows thatthe main driver of the option prices are the volatility process and the innovation distributionchosen to model the underlying.

24

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Figure 4: The probability density for the final price in the P (blue) and Q (red) measures (leftscale). The mean Radon-Nikodym derivative as function of the final price is plotted in black(right scale). The time-to-maturity is one month (left column) and one year (right column),the volatilities are σ∞ = 10% (top), 20% (middle) and 40% (bottom). The size of the MonteCarlo simulation is 100,000.

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Figure 5: The dependence on the processes of the implied volatility surface, for put (left)and call (right) options. The time-to-maturities are 1, 2, 5, 10, 21, 65, 130, 260, and 520days. The colors are according to the maturity, with a linear interpolation in the logarithmof the time-to-maturity through the colors green (1 day) to blue (2 weeks) to red (2 years).The simulation is done with 106 paths. The processes are constant volatility with Studentinnovations (top), GARCH(1,1) with normal innovations (middle) and GARCH(1,1) withStudent innovation (bottom). On the call side (right column) and with normal innovations(top and middle lines), the downward curve and missing points for moneyness above 2 is dueto numerical convergence issues, as very few paths end in this area.

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Figure 6: The dependence of the implied volatility on the leverage for the LM-ARCH process,for put (left) and call (right) options. The simulation is done with 106 paths and with a meanvolatility of 20%. The leverage parameters λlev are 0 (top), 0.5 (middle) and 0.95 (bottom).

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Figure 8: The dependence of the implied volatility on the risk premium, for put (left) andcall (right) options. The simulation is done with 106 paths and with a mean volatility of20%. The β coefficients for the drift in Eq. (53) are 0.0325 (top), 0.065 (middle) and 0.13(bottom), corresponding to the respective risk premia of 0.563, 1.13 and 2.253.

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9 Influence of the risk aversion function

The Radon-Nikodym derivative essentially depends on f(ν z)/E [f(ν z)] for each time step.Clearly the ratio of the risk aversion function f is invariant under a multiplicative constant.Similarly, f(a z) corresponds to a rescaling of ν. In the small δt expansion, these invariancesare reflected in a way that f1(ν) and f2(ν) are homogeneous of degree zero both in f andν. These invariances are used to simplify the space of risk aversion functions to explore. Forexample, all the functions f(z) = a exp(−b z) lead to the same results as f(z) = exp(−z).With a regularization around zero, a possible exponential function is 1/(1+exp(z−z0)), withone free constant z0. The computations in the previous section have been done with z0 = 0,but another choice for z0 leads to very similar results. Similarly, a simple exponential leadsto almost identical implied volatility values.

In order to alter the shape of the implied volatility surface, other functions can be used. Apower law such as f(z) = 1/(1 + zp) has a slower decay. Yet, Equation (18) for ν has asolution conditional on the value for the exponent p. Because the equation should be solvedat each time along each path, the existence of a solution requires p to be large, in practice ofthe order of 20. With such a large exponent, the behavior of the power law and exponentialare similar.

Another convenient class of functions is the stretched exponential f(z) = exp(−zp/p). Withthese functions, Equation (18) can be solved analytically, and there is no restriction on thevalue of p. Yet again, the implied volatility surfaces show only minor differences for p between0.1 to 4.

This exploration shows that the computed IV surfaces are robust with respect to the choice ofthe risk aversion function f . To phrase it otherwise, the risk aversion function cannot be usedto improve the match between computed and market implied volatility. This robustness canbe understood intuitively as follows. Equation (18) for ν fixes the value of f1 to be equal tothe market price of risk φi. Therefore, the leading term of order

√δt in the Radon-Nikodym

derivative cannot be adjusted by the choice of the risk aversion function f(z). At the nextorder in δt, the term σ2int (26c) depends on f1 and is therefore also not influenced by thechoice of f . Only the term in y (26b) introduces a real dependence on the choice of f(z)at order δt through f2. This contribution is small for a short time-to-maturity. For a longtime-to-maturity, this term depends on ε2i − 1 which averages to zero.

10 Extracting the main characteristics of the IV surfaces

In order to summarize with a few numbers the implied volatility surfaces, characteristic figuresare extracted using a kernel expectation over the surface. The kernel K(m, τ) is exponentialin the moneyness and in the logarithm of the time-to-maturity

K[m0, wm, τ0, wτ ](m, τ) = exp

(− |m−m0|

wm

)· exp

(− |ln(τ)− ln(τ0)|

log(wτ )

)(57)

with the centers located at m0 and τ0 and widths wm and wτ in the moneyness and time-to-maturity directions, respectively. For a given day, a kernel expectation over the impliedvolatility surface is given by

IV[m0, wm, τ0, wτ ] =

∑iK(mi, τi) IVi∑

iK(mi, τi)(58)

30

where the sums on the rhs are over the set of available put or call options on the chosen day,with the corresponding moneyness mi and time-to-maturity τi. The mean implied volatilityIV depends on the kernel parameters. Four main characteristics of the IV surface can bedefined with this kernel expectation: level IV (i.e. overall level of the surface), term structureof the backbone ∂τ IV (slope in the maturity direction and at-the-money), smirk ∂mIV (slopeof the smile at-the-money), and smile (curvature of the smile at-the-money). The first threeare respectively defined by

IV = IV[m = 0, wm = 1, τ0 = 2m, wτ = 2] (59a)

∂τ IV = IV[0, wm = 1, τ0 = 1y, wτ = 2]− IV[0, wm = 1, τ0 = 1m, wτ = 2] (59b)

∂mIV =IV[δmr, wm = 1, τ0 = 2m, wτ = 2]− IV[δml, wm = 1, τ0 = 2m, wτ = 2]

IV (δmr − δml)(59c)

The options are mostly traded at- and out-of-the money, but less options are available in-the-money. When computing the smirk with a symmetric definition δml = −1, δmr = 1 forthe put and the call options, this creates a significant noise due to the in-the-money side ofthe smirk definition. Therefore, an asymmetric smirk definition is used for the put and thecall options:

put options: δml = 0 δmr = 1

call options: δml = −1 δmr = 0

Because of the overall curvature of the smile, this asymmetry in the definition induces a smalldifference in the smirk measured on the put and on the call sides, namely the call smirksare smaller. The definition for the smirk contains a division by the volatility level in orderto remove the simple proportionality displayed in Fig. 7. In this figure, the vertical scaleis proportional to the volatility, and therefore the slope has the same proportionality. Thisdependence is removed in our smirk definition so as to extract only the non-trivial behavior.Let us emphasize that when other definitions are used, for example measuring the moneynessas ln(F/K) or omitting the 1/IV in the smirk definition, a very good agreement betweenmarket and computed slopes is obtained, with correlations above 90%. Yet, this is only dueto the dominant volatility dynamics on the slope that is not properly discounted.

The smile is defined as the discrete derivative of the smirk

∂2mIV =∂mIVput − ∂mIVcall

1(60)

where the denominator is the distance between mid-points at which the respective smirksare computed. Essentially, this measure the curvature at-the-money and around a time-to-maturity of 2 months. Notice that due to the strong smirk, it would be better to measurethe curvature around moneyness = -1, but we conform to the common practice.

11 Empirical investigation of European options on the SP500

In this section, the present option pricing scheme is applied to the SP500 underlying andcompared to the available options. The empirical analysis for the options is done on put andcall European options on the SP500. The option data is provided by OptionMetrics, andcovers the period from beginning 1996 to end of June 2010. For each date, two years of daily

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historical data for the SP500 is used as the build-up period for the process. The SP500 values,market implied volatilities, zero-coupon rates and dividend yields are taken as provided byOptionMetrics, without further processing. From the term-structure of the zero-curve, wecompute the corresponding risk-free rates for each required maturity by linear interpolationbetween adjacent rates in maturity. On a given day, all available options with a positivevolume are used, regardless of the moneyness and time-to-maturity: we emphasize here thatno data-cleaning procedure is applied. For the numerical computations, the same set ofoptions is used. The numerical option prices are computed using Monte Carlo simulationsfollowing the numerical implementation described previously. The computation is done foreach day in the sample, with a Monte Carlo simulation size of N = 10, 000. The resourcesrequired for such computations are fairly low, as no optimization needs to be done.

The risk-free rate rrf(τ) is time-to-maturity dependent. For the option replication, a bondof the same maturity τ should be used. As the risk premium depends on the risk-freerate rrf(τ), the Radon-Nikodym derivative becomes maturity dependent. This can be easilyaccommodated in the present scheme by computing the changes of measure dQ/dP (τ) foreach maturity τ , for a given path simulated up to the longest time-to-maturity. Essentially,for one path simulated in the P measure, the set of Radon-Nikodym derivatives dQ/dP (τ)is computed for the available time-to-maturities. These change of measures correspond tothe slightly different replication schemes that should be implemented for different maturities.Notice that the risk-free rate curve is naturally taken into account in this framework, whilesimulating the process in the Q measure requires to introduce different processes for eachmaturities.

The market and computed implied volatility surfaces are interesting to compare directly ona few selected dates, as shown in Fig. 9. Beware that the vertical scale is logarithmic onthese graphs in order to accommodate a large range of volatilities. The period around Fall2008 is unique because of the very large dynamics induced by the sub-prime crisis and thesubsequent worldwide instability. The plot on September 3 is before the start of the crisis andrepresentative of the low volatility periods, with a clear gap between the market and computedimplied volatility. On September 10, premises of the forthcoming crisis can be observed with araising volatility for short-term options. September 17 is shortly after Lehman Brothers filledfor bankruptcy (2008.09.15). The long-term options are well reproduced, but the differencesare increasing for shorter time-to-maturities. With such an abrupt raise in volatility, highfrequency data could improve the theoretical computations. Yet, the options with time-to-maturity shorter than 2 weeks are clearly too expensive, and the same is true in the wholesample. On October 8, AIG has been rescued (2008.09.23), the US senate has passed a bill torestore global financial stability (2008.10.01), Wachovia has been taken over by Wells Fargo(2008.10.03) and all the major banks in Iceland were in deep trouble. The volatility is higher,with a very good agreement between both surfaces. On December 3, the crisis has spreadworldwide, with an increased long-term volatility. Subsequently, the volatility decreased veryslowly, as seen on March 4, 2009. Overall, the agreement between market and computed IVare quite good, particularly for the smirks and smiles, while the levels show larger differences.

In order to summarize the amount of information contained in the IV surface, the results aregiven mainly as time traces for the main characteristics of the implied volatility. Figure 10(top) presents the value of the SP500 index, which is the single source of information in thisoption pricing scheme. The plot at the bottom gives the levels IV for the implied volatilitysurface, for put and call options, and for market and computed implied volatilities. The maindifference is between the empirical and the computed IV levels, while the put and call optionshave very similar values. Essentially, the market values are systematically higher, except in

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periods of crisis. This is particularly clear until 2001, but some systematic differences persistuntil 2010. The market implied volatility is computed by the mean of the bid and askimplied volatilities, and the difference with the computed values are much larger than whatcan be explained by the bid/ask spread. Similarly, the computed IV at-the-money are veryinsensitive with respect to the parameter values used in the computations, in particular withrespect to the drift and the risk premium, but also to the details of the process and theparameter values. Essentially, the computed ATM implied volatility is given by the historicalvolatility of the underlying, regardless of the details used in the option pricing scheme. Inother words, within this option pricing scheme, the computed IV cannot be adjusted to theempirical values.

The discrepancy between market and computed IV shows that options are too expensive inlow volatility periods. The difference is particularly large until 2001. A possible explanationis that the option market was dominated by the option writers, until the emergence of largeenough hedge funds that were able to arbitrage them. The arbitrage leads to very close valuesfor the market and computed IV during the period 2001-2002 and 2005-2006. Early 2007, themortgage crisis re-opened a gap between them. This difference is analyzed further in Fig. 16,in parallel with the variance risk premium.

Figure 11 displays the quantities relevant for the risk premium: risk-free interest rate, divi-dend yield, computed drift, and the total drift (the sum of the drift and the dividend yield).Interestingly, the total drift is fairly smooth as the drift model (53) creates larger drifts whenvolatility peaks, which compensate for the small dividend yields during the same period.Without the concavity induced by the logarithm (and with σrp ' σ∞), the peaks in thedrift are much higher and the total drift spikes markedly during the crises, which would becounter intuitive. The risk premium is displayed in the bottom graph (red line). The riskpremium ranges between 0.3 and 5, with the lowest values obtained during the high volatilityperiods. In short, there are less risk premia during crises, due to the higher volatility. Theaverage value for the risk premium is 1.8. The market price of risk is displayed on the samegraph (blue line). Its value is fairly constant, with a mean value of 0.25. The correspondingcross-over time —from diffusion dominated to drift dominated— defined in Equation (56) isof 17 year, which represents a reasonable value as argued earlier.

It is interesting to relate the observed differences between the market versus computed IVlevel with the risk premium. Essentially, high (low) risk premia occur with the large (small)difference in IV level, and with low (high) volatilities. This shows that the option marketfunctions more like an insurance business, with the option issuers bearing the risk of crisesand harvesting a profit during low volatility periods. Thus, options have a proper economicrole as crises insurances, and are not redundant contracts. This market pattern goes againstthe “short option” side of the replication argument used to derive the fair value of the option.

The time traces for the smirks (the relative slope in the moneyness direction) are plotted inFigure 12. The fairly high short-term noise is due to the day-to-day variability of the availableoptions, in conjunction with the the finite difference computation of the smirk. The scalesfor the put and call graphs are identical, making clear the overall level difference betweenthe put and call smirks. This is due to the asymmetry in the definition using at- and out-of-the money options. On each option side, the computed and market smirks are in the sameranges, but the market smirks exhibit a more dynamical behavior than the computed smirk,which looks somewhat flat. These differences in the level and dynamics can also be seen inthe cross-section on Fig. 14. It is not obvious to relate the behavior of the market smirkwith other IV stylized figures. For example, a scatter plot of the smirk and implied volatility

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level shows no obvious dependence. The difference between market and computed smirkscan have many origins. On the theoretical side, the leverage could include a dynamical part,with the leverage coefficient proportional to the volatility, for example. On the market side,the market might over-react in some situation and thus introduce an extra premium for out-of-the-money options. Clearly, more detailed studies are needed to explain this difference.Finally, let us emphasize that our definition for the smirk removes properly the dependence onthe volatility level, thanks to the definition of the moneyness and of using a relative derivative.Using another definition would keep a dependence on the volatility level, leading to a verygood agreement between market and computed smirks. Yet, this agreement between smirksis spurious as induced by the volatility dynamics. The same remark applies to the smile.

The time trace for the smile (the curvature in the moneyness direction) is plotted in Figure 13,top. Taking the second derivative is increasing further the day-to-day noise. Yet, the meanlevels are in agreement, and the dynamics during the 2001-2006 period and since 2009 arewell reproduced. This overall agreement is confirmed by the cross-section given on Fig. 14.But as for the smirks, the 2007-2008 crisis was a major disruption.

The time traces for the term structure of the backbone (the slope in the time-to-maturitydirection) are plotted in Figure 13, bottom. The market and computed term structures areessentially in agreement: the slopes are zero or small in normal times, but strongly negativeduring volatility spikes. This is due to the fast reaction of short-term volatilities to the recentevents, while long-term volatility components react more slowly. This behavior is correctlycaptured by the multi-scale LM-ARCH process, as already discussed in [Zumbach, 2011], andhence correctly incorporated in the computed backbones.

The variance risk premium is defined as

VRP0 = EP

[1

n

∑i

r2i

∣∣∣∣∣ F0

]− EQ

[1

n

∑i

r2i

∣∣∣∣∣ F0

]. (61)

It measures the difference between a variance forecast computed respectively in the P andQ measures. The second term corresponds to the undiscounted value of a variance swapwith zero variance strike. Using the present option pricing scheme, both terms in the rhscan be computed easily. These values depend on the time-to-maturity, and the same kernelexpectation is used to extract a single characteristic value per day from both variance termstructures. The time trace for the variance risk premium is reported in Fig. 15, in blackon the top graph. The value is mostly negative, with spikes during crisis and a small meanvalue, of the order of -0.2 %. For a volatility level of 20%, this corresponds to a correctionof the order of 5% on the variance. This is a small correction: to a good approximation, thetheoretical price of a variance swap is given by the expected variance in the P measure.

In order to have similar quantities appropriate for options, we define the variance gap andrelative variance gap as

σ2 = EP

[1

n

∑i

r2i

∣∣∣∣∣ Ft

](62)

VG = σ2 − IV2

(63)

RVG =σ2 − IV

2

σ2. (64)

where IV2is the market or computed ATM implied variance. In practice, the ATM variance

34

is extracted from the variance surface using the kernel expectation decribed above, and witha mean on the put and call sides. Essentially, all these definitions compare a variance forecastin P with different measures of variance in Q, corresponding either to options or to varianceswaps. The variance gaps are plotted on the same graph, in blue for the computed gap andin red for the market gap (multiplied by a factor 1/5 in order to be on the same scale). Thecorresponding relative variance gaps are displayed on Fig. 15, bottom.

The computed variance gap is small and positive, with a roughly constant relative variancegap of the order of 5%. This shows that the ATM implied volatility is smaller than a simplevolatility forecast by roughly 2.5%. This is due to the curvature of the smile and to the fairlyconstant smile with respect to the mean volatility, as can be observe on Fig. 7. As for avariance swap, this shows that the computed ATM implied variance is given essentially bythe expected variance forecast in the P measure.

Then, the market variance gap is roughly 5 times larger in magnitude than the computedgap, and without obvious relation to the computed values. Both quantities seem inverselyrelated, except during the crises during which both have positive peaks. Typically, the marketimplied variance is 10% larger than a variance forecast (and even larger before 2001). Onlyduring high volatility periods, the market implied variance moves closer to the computedimplied variance (but stays below). This is consistent with the explanation that the optionmarket works more as an insurance against high volatility, with expensive options duringquiet period and arbitrage prices during crisis. Part of the difference could be explained bythe cost of transactions incured during the replication and by the risk premium carried bythe discrete replication. Checking for this explanation requires to compute the Greeks and toimplement the option replication along each simulated trajectories. Such computations areclearly very expensive, and is the subject of a forthcoming paper.

The implied volatility is often described as a market forecast for the volatility. In this context,it is interesting to relate the market level for the IV, the computed level for the IV, andthe realized volatility. The market IV level is the mean between the put and call sides,and similarly for the computed level. The realized volatility is computed as

√1/∆T

∑r2

between the current time t and the option expiry t+∆T . This quantity depends on the time-to-maturity ∆T , and a single scalar is extracted using the same kernel weighting procedureas for the implied volatility. Let us emphasize that the realized volatility is computed ona larger information set, namely the prices up to t + ∆T . Figure 16 displays the threevolatilities. A close inspection shows that the market and computed IV tend to be closerthan the realized volatility. This is confirmed by the root-mean-square-error (RMSE) betweeneach pair of curves: RMSE(market, computed) = 4.8%, RMSE(market, realized) = 7.0% andRMSE(computed, realized) = 7.2%. Intuitively, the market and computed IV are build onthe same information set, while the realized volatility uses a larger information set. Thekey point is that the market IV and computed IV deliver a forecast of similar quality of therealized volatility. Thus, the larger market IV cannot be justified by a better view of thefuture.

Several distances between market and computed surfaces can be introduced, either for theprices, the time values or the implied volatilities. The distances based on the prices have theadvantage to measure monetary differences, but are sensitive to the option payoffs. Thesedistances therefore depend on the details of the definition (choice of moneyness, maturity,possible weighting), and on the available sample on a given day. The distances based on theimplied volatility are not directly related to the profit & loss, but yield values independentof the particular payoffs and of the available sample. All distances deliver roughly the same

35

overall message, but with less noise for the IV-based distances. Figure 16 shows the distancebetween the market and computed implied volatility surfaces. The distance is a L2 norm (i.e.RMSE) between implied volatilities, averaged over the surface with the exponential kerneldefined in Equation (58) and with parameters wm = 1.5 (width in the moneyness direction)and wτ = 4. The larger parameters are such that an overall distance over the whole surfaceis computed. The mean distances after 2001 are 3.5% for the put and 2.7% for the call.These are fairly impressive performances, particularly in view of the very small number ofparameters contained in our scheme.

12 Influence of the risk premium

In Sec. 7, the relations between the drift, risk premium and market price of risk are discussed.Using plausible arguments, the function (53) is introduced with a drift premium that growslogarithmically with the volatility forecast. This form is used for the computations, with theconstant of proportionality β = 0.075. Yet, the empirical statistics on the SP500 time seriesonly constrain the mean value of the drift, and offer no guidance for the analytical form of thedrift equation. This is clearly a place in the theory that can be changed in order to improvethe match between empirical and computed values.

The computations presented previously have been done for other choices of β (0.0375, 0.15),and with drift premia given by simple polynomials in σ (constant, linear, quadratic). Essen-tially, there are only tiny differences in the results, in particular all implied volatility levels,smirks, smiles and term structures are almost identical. The exception is the variance riskpremium, because the P expectation for the variance depends on the drift. This shows thatthe presented results are robust with respect to the choice of risk premium function and therelated parameters.

The intuitive explanation for the very weak influence of the risk premium is to be foundin the small δt expansion. At order

√δt, the term f1 in the Radon-Nikodym derivative is

proportional to the risk premium, and hence should influence the option pricing accordingly.Yet, the role for this term is to change the drift of the process from µ to rrf, namely toremove the dependence on the function used to model the drift. This cancels the influenceof the drift at order

√δt, leaving dependences at order δt. The small expansion parameter is

indeed σ√δt (namely the volatility at the daily level), which is of order of 1 to 2%. Therefore,

modifications in the risk aversion function f or in the model for the drift influence the optionprices and implied volatility at order δt, whereas the smile is of order O(1). This is indeedwhat is observed both with the simulations and with the historical SP500 options. Thus, onlythe volatility process and the distribution for the innovations change the implied volatilitysurface.

36

-2 -1 0 1 2

10

20

50

100

Moneyness

Implied Volatility [%]

2008.09.03

-2 -1 0 1 2

10

20

50

100

Moneyness

Implied Volatility [%]

2008.09.10

-2 -1 0 1 2

10

20

50

100

Moneyness

Implied Volatility [%]

2008.09.17

-2 -1 0 1 2

10

20

50

100

Moneyness

Implied Volatility [%]

2008.10.08

-2 -1 0 1 2

10

20

50

100

Moneyness

Implied Volatility [%]

2008.12.03

-2 -1 0 1 2

10

20

50

100

Moneyness

Implied Volatility [%]

2009.03.04

Figure 9: The implied volatility computed for the call options, on selected Wednesdays aroundthe September 2008 financial crisis. The empirical (computed) implied volatilities are givenby open (full) circles and joined by dotted (full) lines.

37

1996

1998

2000

2002

2004

2006

2008

2010

600

800

1000

1200

1400

1600

S&P500

market put

market call

computed put

computed call

1996

1998

2000

2002

2004

2006

2008

2010

0

10

20

30

40

50

60

70

IV level [%]

Figure

10:TheSP500level

(top)andthelevelsoftheim

plied

volatility

surfaces

(bottom

).

38

process drift

dividend yield

effective drift

risk free rate

1996

1998

2000

2002

2004

2006

2008

2010

02468

10

12

14

risk free rate and drift [%]

risk premium

market price of risk

1996

1998

2000

2002

2004

2006

2008

2010

012345

risk premium Figure

11:Therisk-freeinterest

rate

andthedrift

(top).

Therisk

premium

andmarket

price

ofrisk

(bottom

).

39

market put

computed put

1996

1998

2000

2002

2004

2006

2008

2010

05

10

15

20

smirk

market call

computed call

1996

1998

2000

2002

2004

2006

2008

2010

05

10

15

20

smirk

Figure

12:Thetimetraces

forthesm

irks,

fortheput(top)andcall(bottom)op

tion

s.

40

market smile

computed smile

1996

1998

2000

2002

2004

2006

2008

2010

-505

10

smile

market put

market call

computed put

computed call

1996

1998

2000

2002

2004

2006

2008

2010

-30

-20

-100

10

term structure

Figure

13:TheIV

smile(top)andtheterm

structure

(bottom).

41

0 5 10 15 200

5

10

15

20

market smirk

computed smirk

-5 0 5 10-5

0

5

10

market smile

computed smile

Figure 14: The cross sections for the smirk (left) and smile (right), with the market valuesgiven on the horizontal axis and the computed values on the vertical axis.

42

option variance gap: market (X 0.2)

option variance gap: computed

computed variance risk premium

1996

1998

2000

2002

2004

2006

2008

2010

-1012

variance risk premium [%]

relative variance gap: market (X 0.2)

relative variance gap: computed

1996

1998

2000

2002

2004

2006

2008

2010

-60

-40

-200

20

relative variance gap [%]

Figure

15:Thevarian

cerisk

premium

(top)andtherelativevariance

gap(bottom).

43

realized volatility

market IV

computed IV

1996

1998

2000

2002

2004

2006

2008

2010

0

10

20

30

40

50

60

70

volatilities [%]

1996

1998

2000

2002

2004

2006

2008

2010

02468

10

12

14

IV distance [%]

Figure

16:Themarket

IV,computedIV

andrealized

volatility

(top).

Thedistance

betweenthecomputedan

dmarket

values

fortheim

plied

volatility(bottom).

44

13 Conclusions

The present approach to option pricing is an ab initio scheme, and to a large extend is fairlystraighforward. An important ingredient is to have a good process for the underlying since theprices of the contingent claims derive from the future underlying values. In the present case,a multi-scale LM-ARCH process with leverage and fat-tailed innovations is used. The priceprocess is multiplicative and uses relative returns (instead of logarithmic returns) in order tohave well defined mathematical properties with fat-tailed innovations. As a by-product, noskew is required in the distribution for the innovations. The time increment of the processis discrete, and one day is used in our numerical computations. This process describes wellthe statistical properties of the SP500 from one day to one year, and has a minimal numberof parameters. These parameters are estimated as to heuristically reproduce at best theempirical statistics on the underlying. This simple procedure leads to parameter values thatare in adequation with most stylized facts. Unlike stochastic volatility processes, there are nohidden variables in an ARCH process and the estimation of the parameters is easily achieved(no log-likelihood is needed). Up to this point, only the underlying is involved.

For the option pricing, the key point is the analytical derivation of the Radon-Nikodym deriva-tive, following [Christoffersen et al., 2010] and [O’Neil and Zumbach, 2009]. The change ofmeasure is induced by the martingale condition and the discounting by a risk-free bond, anddepends on a risk aversion function that can be chosen by the user up to some restrictions.Then, the standard no arbitrage and replication arguments lead to the option price as adiscounted expectation. So far, the pricing is exact and does not require neither a continuumlimit, nor restrictions on the process equations, nor on the innovation distribution. In orderto turn the mathematical construction into a practical numerical scheme, a series expansionin

√δt is done, up to second order (where δt is the process time increment). This step is

“equivalent” to using Ito calculus in a continuum set-up. This derivation allows to obtain anexplicit equation for the change of measure dQ/dP where no quadratures are involved. Thechange of measure is path-dependent (contrarily to the B-S scheme) but is independent ofthe particular process equations or parameter values. Then, the option price is obtained asa discounted expectation in the P measure, where the integrand is the option payoff with aweight given by the (path-dependent) Radon-Nikodym derivative dQ/dP.

This scheme puts back in the central place the Pmeasure, and the process for the underlying isdefined only in P. The option replication leads to a martingale condition, but only the changeof measure is required in the pricing expectation. This procedure is simple and general, withthe soft technical condition on the process that the drift µi and volatility σi should be inthe filtration Fi. This simplicity contrasts with the approach found in the GARCH optionpricing literature, where the specific process for the underlying needs to be mapped to the Qmeasure (similarly to what can be done easily in the B-S approach). For realistic processes,such mappings require complex analytical computations, and/or an optimization against theempirical implied volatility surfaces.

The present ARCH scheme is also much simpler compared to processes using implied volatil-ity, or with more latent variables like skewness. First, the latent variables are a source ofnuisance for the parameter estimation. Then, they are a mixed blessing for the option pric-ing as they give additional degrees of freedom that can be optimized against the empiricalsurfaces (and so potentially yield good performances), but the optimization on each surfaceis an open trap for over-fitting. It is also conceptually unnatural to use the latent variablesin the underlying process to match better the implied volatility surface. Such a procedure

45

is better characterized as a complex fit or interpolation of the empirical IV surface, but thesurface is not computed from first principles.

Compared to the process used in [Zumbach et al., 2010], a significant change is the explicitmodel for the drift parameter µ(t). In order to model only the underlying time series, aconstant drift gives satisfactory enough results. The new option specific part is the riskpremium φ = (µ−rrf)/σ2 which appears in the Radon-Nikodym derivative. The broad rangeof historical values for the risk-free rate rrf and variance σ2 leads to constraints on µ in orderto have realistic values for both µ and φ. These constraints lead us to introduce a dynamicalmodel for the dependence of the drift on the volatility. On general grounds, the dependenceshould be an increasing function, i.e. an increasing reward is expected for an increasing risk.Furthermore, realistic values for the risk premium lead to a concave function, i.e. the rewardis lesser for taking even more risk. Given these plausible economic dependences, the driftdepends essentially on one parameter β that fixes both the mean drift and the mean riskpremium (the curvature parameter plays a lesser role). This double dependence constraintsstrongly the parameter value for β.

Overall, the present option pricing scheme is very constrained. The process for the underlyingis estimated on the underlying time series. The long memory ARCH process is very parsimo-nious, as essentially only six parameters should be adjusted. Then, essentially two specificchoices are left for the option pricing: the model for the drift and the function characterizingthe risk aversion. Both the analysis of the Radon-Nikodym formula and of the Monte Carlosimulations show that the dependence from these two functions are very weak, and essentiallybringing correction of order δt. It is only for long-dated options that the corrections accu-mulate enough to give a significant price change. As the final prices and implied volatilitysurfaces are essentially independent of these choices, the option prices are mainly given bythe process for the underlying. In short, there is not much space for “fudge factors” in thisscheme.

As the process can be changed easily in the present framework, the broad shape of the impliedvolatility surface can be related to the features included in the process. Inserting ingredientsone-by-one in the process shows that the short-term smile is due to the fat-tailed distributionfor the innovations, while the long-term smile is due to the heteroskedasticity. The smirk isdue to the leverage, and the dynamics of the term structure is related to the dynamics of theheteroskedasticity.

The freedom in the choice of the risk aversion function f(z) implies the non-unicity of theoption prices. Let us emphasize that this non-unicity occurs already for a process withconstant volatility and normal innovation, and is therefore not related to the choice of theprocess nor of the innovations distribution. Moreover, a replication strategy is available,regardless of f , the process, and the distribution for the innovations. Yet, the small

√δt

expansion shows that f influences the prices only at order δt. The Monte Carlo simulationsmake clear that the main determinant of the option prices is the process and its parameters,whereas f influences only weakly long-dated options. The implication is that option pricesare not unique, mainly because of the possible disagreement between option writers over theunderlying process and parameters, while disagreement over f can influence only long-datedoptions.

Using a long memory ARCH process in this pricing scheme, the IV surfaces for the SP500 arecorrectly captured, including the levels, smiles and smirks, and the terms structures of thebackbone. Given the simplicity of the present scheme and the small number of parameters,

46

this is quite an achievement. Compared to schemes based on implied volatility or on processesexpressed in the Q measure, the Occam’s razor favors clearly the present proposition.

Comparing the computed and market implied volatilities for the SP500 shows that the largestdifferences between implied volatility surfaces happen when the volatility is low, with a rel-ative difference of the order of 30%. During volatility peaks, both implied volatilities agreewell and possibly the market IV is smaller than the computed IV. This shows that an optionis not much of a redundant asset, but it acts more like an insurance against volatility crisesand large moves in the underlying. Essentially, options are used to hedge and to transfer risk,and the option issuers can charge a premium during low volatility periods. This differencesets a limit on the arbitrage argument used to derive the replicating price of a contingentclaim.

The very weak influence of the risk aversion and of the drift premium suggests ways tosimplify the full option pricing scheme. First, the risk aversion function can be taken as anexponential f(z) = exp(−z). With this choice, f1 = ν and f2 = ν2, leading to cancellationsin the Radon-Nikodym derivative. A stronger simplification is to add no risk premium inthe drift, namely to use µ = rrf − q, and to forget completely about the Radon-Nikodymderivative dQ/dP. Because the properties of the process used for the underlying dominatethe implied volatility surface, this is a very good approximation of the full scheme. Indeed,the computations using either approximation show barely any visual differences on the plots.

Finally, an even more drastic simplification is to dispose completely of the Monte Carlo sim-ulations. The analysis of the computed and empirical smirks and smiles shows that thesequantities are fairly static. On the one hand side, this is due to the definitions of the money-ness and the smirk that remove the dependence on the volatility. On the other hand side, theprocess has constant parameters for the leverage, heteroskedasticity and distribution for theinnovations, and therefore generates an almost constant smile shape. The empirical smilesare roughly in agreement, except for the crisis periods where they exhibit more dynamics.This static behavior contrasts with the level and term structure which show strong dynamics,related to the volatility dynamics of the underlying. These differences with respect to thedynamics suggest a simple model for the implied volatility surface given by IV surface =backbone × smile. The backbone can be obtained by a volatility forecast from an ARCHprocess, as used in this paper or in [Zumbach, 2011]. The smile can be taken as constantin time, and with smirk and smile parameters related to the statistical properties of theunderlying.

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