A Hull and White Formula for a General Stochastic Volatility Jump-Diffusion Model with Applications to the Study of the Short-Time Behavior of the Implied Volatility

Alòs, Elisa; León, Jorge A.; Pontier, Monique; Vives, Josep

doi:https://doi.org/10.1155/2008/359142

International Journal of Stochastic Analysis

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Research Article | Open Access

Volume 2008 | Article ID 359142 | https://doi.org/10.1155/2008/359142

A Hull and White Formula for a General Stochastic Volatility Jump-Diffusion Model with Applications to the Study of the Short-Time Behavior of the Implied Volatility

Elisa Alòs,¹Jorge A. León,²Monique Pontier,³and Josep Vives⁴

Academic Editor: Wenbo Li

Received01 Apr 2008

Revised03 Sept 2008

Accepted25 Nov 2008

Published10 Feb 2009

Abstract

We obtain a Hull and White type formula for a general jump-diffusion stochastic volatility model, where the involved stochastic volatility process is correlated not only with the Brownian motion driving the asset price but also with the asset price jumps. Towards this end, we establish an anticipative Itô's formula, using Malliavin calculus techniques for Lévy processes on the canonical space. As an application, we show that the dependence of the volatility process on the asset price jumps has no effect on the short-time behavior of the at-the-money implied volatility skew.

1. Introduction

It iswell known that classical stochastic volatility models, where the volatility is allowed to be a diffusion process, are able to capture the dependence of the implied volatility as a function of the strike (the smile or the skew). Nevertheless, they can not explain its dependence with respect to time to maturity (term structure). For example, empirical observations indicate that the at-the-money skew slope is approximately (here denotes the time to maturity), while the rate for these stochastic volatility models is (e.g., Lewis [1], Lee [2], or Medvedev and Scaillet [3]). The introduction of jumps in the asset price dynamics is a natural extension of classical stochastic volatility models proposed with the aim to capture this short-time behavior. Although the rate of the skew slope for models with jumps is still (see Medvedev and Scaillet [3]), they allow flexible modelling, and generate skews and smiles similar to those observed empirically (see Bates [4], Barndorff-Nielsen and Shephard [5, 6], or Carr and Wu [7]).

In Alòs et al. [8], the authors considered general jump-diffusion stochastic volatility models where the volatility is not necessarily a diffusion. They proved that for a volatility process independent of price jumps (as in the Bates [4] case) the at-the-money skew slope explodes if and only if the Malliavin derivative of the volatility process also does when . This allows us to consider new models—where the volatility is not required to be Markovian nor to be a diffusion process—that capture the short-time explosion of skew slopes. The basic idea in that work was to expand option prices around the classical Hull and White expression by means of Malliavin calculus, following similar ideas as in Alòs [9]. This gives us a decomposition of option prices that allows us to identify the effect of correlation (between the volatility and the Brownian motion driving stock prices) and the effect of jumps in the at-the-money skew slope when time to maturity tends to zero. This can be interpreted as an answer of a demand in Fouque et al. [10, end of page 54].

In this paper, we study jump-diffusion stochastic volatility models allowing the volatility process to be correlated also with the price jumps (see Bakshi et al. [11] and Duffie et al. [12], among others). Our main goal in this work is to describe analytically the effect of this extension in the at-the-money short-time behavior of the stochastic volatility. The idea will be again to decompose option prices around the Hull and White term, now by using Malliavin calculus for Lévy processes (see Solé et al. [13], Løkka [14] and Petrou [15]). In comparison with the formula obtained in Alòs et al. [8], here we obtain an extra term because the volatility depends now on the jump price. This representation allows us to show that the existence of correlation between the volatility process and the price jumps does not have any influence on the at-the-money skew of the implied volatility as time runs to expiry, confirming an heuristic idea explained in Gatheral [16, page 70].

The paper is organized as follows. In Section 2, we give the main hypotheses and notations. Section 3 is devoted to introduce the Malliavin calculus framework needed in the remaining of the paper. In Section 4, we obtain the Hull and White formula. In Section 5, we apply it to the problem of describing the at-the-money short time skew of the implied volatility. Finally, Section 6 is devoted to the conclusions.

2. Main Hypotheses and Notations

We consider a log-price process, under the market chosen risk-neutral probability measure, given bywhere, , is the current log-price, is the instantaneous interest rate, and are independent standard Brownian motions, , and is a compound Poisson process, independent of and , with intensity , finite Lévy measure , and with

We assume that the process is adapted to the filtration generated by and So, in this paper, generalizing Alòs et al. [8], we allow the volatility to have nonpredictable jump times as advocated by Bakshi et al. [11] and Duffie et al. [12], among others.

In the following, we denote by , and the filtrations generated by the independent processes , and , respectively. Moreover, we define

It is well known that if we price a European call with strike price by the formulawhere denotes the expectation with respect to a risk-neutral measure, there is no arbitrage opportunity. Thus is a possible price for this derivative.

In the sequel, we use the following notation: (i)The process with denotes the future average volatility.(ii)With we represent the classical Black-Scholes function with constant volatility , current log stock price , time to maturity strike price , and interest rate This function can be written aswhere is the future log-price at and is the cumulative probability function of the standard normal law.(iii)With we denote the Poisson random measure on such that Moreover, is the compensated Poisson random measure.(iv)We consider the operator which satisfies

3. Required Tools of Malliavin Calculus for Lévy Processes

3.1. Introduction

In this section, we introduce the tools of Malliavin calculus for Lévy processes that we need in the rest of the paper.

Consider a complete probability space and let be a càdlàg Lévy process with triplet See for example the book of Sato [17] for a general theory of Lévy processes.

It is well known that can be represented aswhere is a Brownian motion and is the Poisson random measure associated to It is also known that See for example, Solé et al. [13].

In general, the construction of a Malliavin calculus, based on a chaos expansion, for a certain process follows three main steps. First of all, to prove a chaotic representation property, secondly, to define formally the gradient and divergence operators, and finally, to give their probabilistic interpretations.

In the last years, several approaches to the Malliavin calculus for Lévy processes have been developed, with different probabilistic interpretations of gradient operators. Between them, we mention the approach of Nualart and Schoutens [18], the approach of Solé et al. [13], based on Itô [19], and the approach of Løkka [14] and Petrou [15]. In this paper, we follow the last one, because in it, the form of the gradient operator simplifies strongly our computations. As a tool for our results we develop in Sections 3.3 and 3.4 two transfer formulas between the second and the third methods. Finally, let us remark that we are under the conditions of Løkka's approach because our Lévy measure is finite.

3.2. The Chaotic Representation Property

Consider the space with its Borel family of sets We can introduce the centered independent random measure given bywhere and Its variance is given by

Note that can be seen as a centered independent Gaussian random measure on , and can be seen as a centered independent random measure on where Thus we can write

where is the Dirac's delta, that is, a unitary mass on the point

In this context, we can define stochastic multiple integrals with respect to with kernels in the Hilbert spaces

in the usual way, and to prove that if is the completed natural filtration of for any random variable we have the chaotic representationwhere the kernels are unique if we take them symmetric.

3.3. The Malliavin-Type Derivative

Let us denote by the set of random variables in , of the form (3.6), such that The Malliavin derivative of a random variable is the process defined by

In order to give the probabilistic interpretation of this operator, we assume, in the remaining, as in Solé et al. [13], that the underlying probability space is the canonical Lévy space That is, is the canonical Wiener space and is the canonical Lévy space of the compound Poisson process with Lévy measure Moreover, we assume that and are the canonical processes.

Let be an element of this space. So, is a continuous trajectory null at the origin and is a sequence of (jump instant, jump size) pairs

From Petrou [15], we havewhere denotes the classical Malliavin derivative with respect to the Brownian motion (e.g., Nualart [20]). Denote by its domain.

In order to obtain the probabilistic interpretation of for , we consider the following transformation.

Given , we can add to any a jump of size at instant , denote the new elementand write So, for any , we can define the operator As it is shown in Solé et al. [13, Proposition 4.8] this is a well-defined operator.

For all defineand denote by its domain.

This operator is related to Indeed, in Solé et al. [13] is considered the random measure

Observe thatwhere

Therefore, we obtain the transfer principlefor and Here is the annihilation operator used in Solé et al. [13].

Thus, combining results from Solé et al. [13] and Alòs et al. [21] it is easy to show that for ,and in this case

Observe that we have provedwhich follows from (3.8) and (3.10).

Observe also that it is immediate from (3.10), to see that

Finally, we have , due to Solé et al. [13, Section 2] and the equalities (3.12) and (3.14).

3.4. The Skorohod-Type Integral

Let . For almost all we have the chaotic decomposition (see Section 3.2)

where is symmetric in the last variables.

Let be the symmetrization in all variables. Then, we define the Skorohod integral of byin provided that means

Moreover, if and we have the duality relation

So, in this sense, is the dual of the operator Sometimes we will write instead of

Note, also, that the transfer principle (3.14) allows us to establish a transfer principle between and the divergence operator given in Solé et al. [13]. Namely,

The following lemma is useful for our purposes. A version of this lemma in the pure jump case is given in Di Nunno et al. [22, Theorem 3.13].Lemma 3.1. Let and such that Then,
and in this case Proof. This result follows using relations (3.16) and (3.21); Alòs et al. [21, Lemma 2.4 and Proposition 2.5] and the transfer formulas (3.14) and (3.22). Note that, in our case, does not need to be bounded. This is a consequence of the fact that if is a bounded random variable of and is finite, we have that and is also bounded.

In order to give the relation between and the pathwise integral with respect to we consider the following two sets.Definition 3.2. We define

Observe that if is a random field of we have, in particular, that and are in and , respectively. Moreover .Definition 3.3. We define as the subset of of random fields such that the following -a.s. left-limits exist and belong to Proposition 3.4. Assume that is a random field belonging to Assume where is the classical path-by-path integral defined by
Then, and in this case,
or equivalently, Proof. Assume as a first step that is bounded. Then, , and are also bounded on . In particular (3.26) is true.
We begin considering the following partition of
Then, we can define
Using Lemma 3.1, we have that for all and
First of all, observe that if and then and almost surely go to the same limit whatever and go to infinity or and By the theorem hypothesis this limit is
Observe now that being bounded, and having the same bound, and are also -limits. So, using that is a closed operator, the left-hand side in (3.31) goes to in if we prove that the terms on the right-hand side converge in to the limits defined by the proposition.
For the first term in the right-hand side, observe that coincides with the path-by-path integral because the integrand is deterministic. Then, using is bounded and the dominated convergence theorem we obtain the expected -limit. For the second term, we have also a direct application of dominated convergence theorem.
In order to prove the nonbounded case observe that we can assume that is positive, because the formula that we want to prove is linear. Then, for the general case, we simply have to apply the result separately to the positive and negative parts.
So, let and Of course, and converges increasingly to We have, as a consequence of the first step, that

Being and in we have that and go up to and in respectively. So, hypothesis (3.26), the monotone convergence theorem and the closeness of the operator yield the result.Remark 3.5. Notice from the proof of Proposition 3.4 that we can change the space by a similar space with left limits with respect to the time variable but with right limits with respect the space variable Remark 3.6. Observe that when is adapted to the filtration generated by . Therefore, in such caseThat is, in this case, the pathwise and Skorohod integrals with respect to are the same.

Observe that in the last two results there is no contribution of because on the operator coincides with the Skorohod-type integral with respect to , as the following result shows.Lemma 3.7. Let and be the adjoint operators of and , respectively, and Then also belongs to and Proof. This result is implied by (3.21) and (3.16).

3.5. The Anticipating Itô's Formula

The basic tool for our results is the following anticipative Itô formula. Recall that the process is introduced in (2.1) and is the future average volatility, which is an anticipative process, even is adapted.

Theorem 3.8. Let and be a bounded function in with bounded derivatives. Then, where is the Skorohod integral with respect to the Brownian motion and Proof. The proof is as in Alòs et al. [8] combined by Proposition 3.4 to treat the sum of jump terms.
We apply it to the random field Here, the independence between , and , the fact that is a continuous process and the fact that is a compound Poisson process with a finite number of jumps on every compact time interval play a key role.
Indeed, let denote these jump instants. Then,
The first term yields a standard Itô formula concerning continuous process, so Alòs et al. [8] results apply and we get the six first terms in the right-hand side of the Theorem 3.8 formula. On other hand, the sum of second terms is the path by path integralNote here that are left continuous so . Then, using Proposition 3.4 we get the last sum is equal to:

4. The Hull and White Formula

Now we have the following extension of the Hull and White formula.Theorem 4.1. Let and be as in Theorem 3.8. Then, where Remark 4.2. Note that in the case that only depends on the filtration generated by we have Consequently, in this case, we obtain the Hull and White formula given in Alòs et al. [8].Proof. This proof is similar to the one of the Theorem 4.2. in Alòs et al. [8]. Notice that Then, from (2.2) we have
Now, our idea is to apply the Itô formula (Theorem 3.8) to the process As the derivatives of are not bounded we use an approximating argument, changing byand by where for some such that for all and for all Now, applying Theorem 3.8 between and to functionand grouping terms according to the type of derivative we obtain
Notice that where
Also note that the classical relation between the Gamma, the Vega, and the Delta gives us that
Then we can write
Hence, taking into account that it follows that (using the fact that
Now, taking conditional expectations we obtain that
Let us remark that continuity of and left continuity of imply that
Finally, we obtain the result proceeding as in the proof of Theorem 3 in Alòs et al. [8]. That is, letting first , then and using the dominated convergence theorem.Remark 4.3. The additional term given by can be detailed as follows. Suppose that Then we can define
But for , ,where
For example, consider the following pure volatility jump case described in Álvarez [23]. See also Espinosa and Vives [24]. For , let be the jump instants and be the jump sizes of process with Assume that the dynamic of is given by with for a certain function In this case, we have and so, the explicit computation of gives

5. Short Time Behavior of the Implied Volatility

In this section, we show that the short-time behavior of the at-the-money implied volatility is the same as in the case where the volatility is independent of the filtration of even the Hull and White formula is different in the last case (see Remark 4.2). This is a fact that must be taken in account for pricing and hedging.

Let denote the implied volatility process. By definition it satisfies Assume that is as in model (2.1). Proceeding as in Alòs et al. [8], the derivative of the implied volatility with respect to the log-strike iswhere

Now, in order to study the limit of as we need to introduce the following hypotheses: (H1)(H2)There exists a constant such that, for all (H3)For every fixed as

Theorem 5.1. Under the Hypotheses (H1)–(H3) we have (1) assume that in (H2) is nonnegative and that there exists a -measurable random variable such that, for every a.s. as then(2)assume that in (H2) is negative and that there exists a -measurable random variable such that, for every a.s. as thenProof. We can write
The term is due to the fact that the following majoration is uniform on

Now the result follows as in Alòs et al. [8, Proposition 6 and Theorem 7].

6. Conclusion

The Malliavin calculus for Lévy processes appears to be a natural tool to deal with the future average volatility in jump-diffusion models where the volatility is correlated with both Brownian motion and compound Poisson process driving the stock price process. Proceeding as in Alòs et al. [8], this powerful calculus allows us to identify the effect of both correlations. In particular, we have seen that the correlation with the asset price jumps has no effect on the short-time behavior of the volatility skew.

Acknowledgments

The authors would like to thank two anonymous referees for their useful comments and suggestions. The first author is supported by Grants MEC FEDER MTM 2006 06427 and SEJ2006-13537, the second author is partially supported by the CONACyT Grant 45684-F, and the last two authors are supported by Grant MEC FEDER MTM 2006 06427.

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Copyright © 2008 Elisa Alòs et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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