Stochastic jump intensity models

Abstract

Stochastic volatility models have proven relevant for various purposes, e.g. explaining the term structure of implied volatility or correlations in absolute returns. When used in conjunction with jump models, they provide powerful tools that have been applied successfully in recent years. Volatility is however but one aspect of risk: in the frame of jump processes, the (local) intensity of jumps is another one, which is equally important. It thus seems natural to supplement stochastic volatility models with “stochastic jump intensity” ones, where the local intensity of jumps evolves according to a random dynamics. The aim of this preliminary work is merely to set up such models as extensions of the celebrated CGMY process, leaving their exploration both for risk management and for option pricing to further studies.

Keywords

Tempered stable processes CGMY model stochastic jump intensity risk management VaR

1. Background and motivation

Increment-stationary stochastic processes are typically unable to account for certain important features of financial markets, such as reproducing empirical Values at Risk (VaR) or implied volatility patterns. Introducing non stationarity requires to specify how the characteristic parameters of the models evolve in time in a way that is simple enough so as to lead to tractable subsequent treatment. Arguably the most popular ways to do so are through the use of a time-varying volatility. Two major classes of such models are stochastic volatility [13] and local volatility [6] ones. In the Gaussian and jump-diffusion frames, this amounts to replacing the instantaneous variance either by a stochastic process or by a function of the asset. In pure jump processes such as the CGMY model [4], this translates into letting the scale parameter C evolve. Quoting [4]: “For example, if one were to construct a model with a stochastic aggregate activity rate, then one could model C as an independent positive process, possibly following a square root law of its own”.

However, since the main feature of CGMY and similar processes is a fine modelling of jumps, and since it is precisely by taking into account jumps that these models improve upon diffusion ones, one may wonder why the parameter responsible for the jump intensity should be considered constant, while C would be allowed to vary. Indeed, contrarily to volatility, no local or stochastic jump intensity models have been put to use in financial engineering. A desirable extension of, say, CGMY should let all parameters evolve in time, and in particular the Y parameter, that finely tunes the jumps. Analogously to stochastic volatility and local volatility models, one should then design “stochastic jump exponent” and “local jump exponent” models.

A technical reason why models with varying jump intensity have not been widely considered so far is that their very definition is more challenging that, say, stochastic volatility ones. An operational reason is that, when it comes to risk control, the central notion considered in the literature is volatility, typically understood as a measure of variance, and this is why a huge number of models have been proposed that allow practitioners to finely tune it.

While volatility is obviously a central source of risk, it is not the only one. It has long been recognized that jumps occur in most financial logs [2,5] and that they also contribute to risk. Models with increasing precisions taking this aspect into account have been studied.

Our long-term program is to develop and study stochastic processes allowing one to model all dimensions of risk in a non-stationary way. The present article is a modest first step in this direction. We first recall in Section 2 some empirical studies from [7] showing (a) that the parameters $C, G, M, Y$ calibrated on the S&P 500 do evolve significantly in time, and (b) that these evolutions, in particular the one of Y, have a notable impact on risk as measured by VaR, which has become, for better or for worse, a standard tool for risk management [14], in particular in the frame of Solvency II [1]. In practice, VaR is computed either in a non-parametric way from historical logs or based on a model which is calibrated on market data [15]. In most studies, and in particular in the frame of Gaussian models, VaR is essentially governed by volatility, i.e. instantaneous variance. Our point is that instantaneous variance is but one dimension of risk. In the frame of CGMY and similar models, it condenses into one value influences of the four parameters $C, G, M$ and Y. A fine study of VaR should rather isolate the contributions of each of these parameters. In this work, we study the impact on VaR of variations of the overall level of activity, as measured by C, as well as of the intensity of jumps, as controlled by Y (studying the effects of G and M would of course also be interesting, but falls outside the scope of this paper). Our main motivation is that the presence of jumps on financial records is a well recognized fact [2], that have a strong impact on VaR.

We then present some preliminary steps towards the design of operational models with varying jump intensity in Section 3, in the hope that they will foster subsequent studies. We will base our developments on tempered versions of so-called multistable motions and consider various extensions. Multistable motions [8,11,19,20] are generalization of Lévy motions where the jump intensity is allowed to vary in time in an exogenous manner. Tempered multistable motions were introduced in [7,12] and further studied in [17] where they are shown to be semimartingales and where the series representations of their field-based and independent increments asymmetrical forms are derived. See also [21] for a related process. In order to provide useful models in finance and other fields, one needs to specify the time evolution of jump intensity. In that view, we consider analogues of various stochastic volatility models in Section 4.

2. Numerical evidences

2.1. Numerical evidences of non-stationarity

We analyse in this section a log of the S&P 500, and estimate the functional parameters $C (t), G (t), M (t)$ and $Y (t)$ of a “locally” CGMY model: that is, we assume that, on any sufficiently small time region, the parameters $C, G, M$ , and Y may be considered constant. Calibration is made using maximum likelihood. In details, to obtain the parameters at time $t_{i}$ , we use a sliding window that incorporates the sample points $t_{i} - 2000, \dots, t_{i}$ (which corresponds roughly to eight years of historical data), and perform maximum likelihood estimation on this set of data. Although rather crude, this procedure is still more refined that estimating the parameters on the whole log assuming a priori increments stationarity.

S&P 500 data between 01/03/2002 and 01/02/2012 are shown on Fig. 1. Figure 2 displays the instantaneous $C, G, M$ and Y parameters.

Fig. 1.

S&P 500 index between 01/03/2002 and 01/02/2012.

Fig. 2.

Evolution of C (top left), G (top right), M (bottom left) and Y (bottom right) for the S&P 500 index between 01/01/2002 and 31/12/2012.

As one can see it from the graphs, all parameters do vary significantly through time. Several interesting information may be drawn from those figures:

Except for the 2008–2009 crisis, movements on the parameters cannot be easily deduced from ones of the index log: the time evolution of the parameters reveal information that is concealed on the price variation.

Relative variations of the parameters are very large, with C ranging in $[0.02, 0.7]$ , G in $[28, 134]$ , M in $[26, 134]$ and Y in $[0.1, 1.15]$ .

Consistent patterns appear: most notably, an increase in C always corresponds to a decrease in Y, while $C, G$ and M tend to be positively correlated. In other words, an increase of overall activity coincides with a larger intensity of jumps, which at the same time undergoes stronger tempering.

The graphs may be roughly split into three zones (this is specially apparent from the evolution of C): from 01/03/2002 to 01/09/2006, characterized by small values of C, moderate values of G and M, and large values of Y; from 02/09/2006 to 01/12/2008, where C reaches large values, as do G and M, while Y drops to small ones; and finally from 02/12/2008 to 01/02/2012, where $C, G$ and M all assume their lowest values as Y increases notably. We do not have, at this time, an explanation of the empirical observation.

Since the three zones mentioned in item 4 are notably different and each one is more or less homogeneous, it seems relevant to analyse them separately. Global estimates for the three zones are as follows:

	C	G	M	Y
Zone 1	0.03	84	99	0.67
Zone 2	0.16	68	87	0.30
Zone 3	0.03	50	63	0.58

To assess the goodness of fit, comparisons between historical data and globally fitted CGMY models for each of the three zones are displayed through densities and QQ plots on Figs 3 to Fig. 5.

Fig. 3.

Densities (left) and QQ plots for zone 1; curve with lowest peak: historical data; curve with highest peak: CGMY model.

Fig. 4.

Densities (left) and QQ plots for zone 2; curve with lowest peak: historical data; curve with highest peak: CGMY model.

Fig. 5.

Densities (left) and QQ plots for zone 3; curve with lowest peak: historical data; curve with highest peak: CGMY model.

Fig. 6.

VaR as a function of Y (bottom curve) and C (top curve), when all other parameters are kept constant, region 1.

2.2. Impact on VaR of varying C and Y

Characterizing the impact on VaR of the various parameters in the CGMY model is a complex issue: it depends on the confidence threshold and on a complex interplay between the values of the parameters. For instance, it would seem intuitive that decreasing Y should increase VaR. However this is only the case when the threshold is in a certain range, which in turn depends on the values of the other parameters. Since VaR measures extreme events, one might think of using the asymptotic formulas (7.6) and (7.7) of [3] (see also [18]). However, for typical parameters values observed on markets, and at the threshold of, say, $1 %$ , it appears that we are not yet in the domain where these formulas provide a reasonable approximation. We must therefore resort to numerical estimations.

The aim of this section is to provide graphical illustrations of how changes of C and Y impact VaR, and to compare the influences of these two parameters. Indeed, Section 2.1 suggests that both C and Y do vary in time. A natural question is then whether such variations have a significant impact on VaR, or if they may be safely ignored. In order to elucidate this point, we shall make use of the empirical results of the previous section to set the ranges of variation of the parameters. It occurs that using the whole ranges between 01/01/2002 and 31/12/2012 results in unrealistically large variations of VaR. For this reason, we will rather work with the three roughly homogeneous zones identified above.

We draw two kinds of graphs;

VaR as a function of C when Y is kept constant, and as a function of Y when C is kept constant. The value of the parameter which is kept constant is set to the one estimated globally on the considered region. The varying parameter spans the range of the local estimates in the region. This is on Figs 6 to 8.

VaR either as a function of C when the variance is kept constant or as a function of Y when the variance is kept constant: recall that the variance of a CGMY process is equal to $\begin{matrix} (1) & σ^{2} = C Γ (2 - Y) (G^{Y - 2} + M^{Y - 2}) . \end{matrix}$ Varying only C or Y thus induces a variation of instantaneous variance. As a consequence, the impact on VaR in the previous experiment is a “mixed effect”. This is why we perform this second experiment: the variance is first computed using the values of the parameters that have been estimated globally on the region. Then, when, e.g., C is varied in the range of the locally estimated $C (t)$ in the region, Y is adjusted so that σ, as computed with (1), remains constant. See Figs 9 to 11.

We display two more graphs: Fig. 12 plots the evolution of C as a function of Y that yields a constant VaR (the bounds on Y are the ones of region 2): this is intended to stress the fact that a given fixed VaR may be obtained with significantly different levels of global activity C when Y varies in a realistic range. Figure 13 shows how the variance evolves when C spans its range of local estimates in region 1 and Y is adjusted accordingly so as to keep the VaR constant (equal to the one estimated with the global parameters in region 1). This is to emphasize that the instantaneous variance alone is not sufficient to characterize VaR: here, the same VaR is obtained with a variance undergoing a variation of 40%.

Fig. 7.

VaR as a function of Y (bottom curve) and C (top curve), when all other parameters are kept constant, region 2.

Fig. 8.

VaR as a function of Y (bottom curve) and C (top curve), when all other parameters are kept constant, region 3.

Fig. 9.

VaR as a function of Y (decreasing curve) and C (increasing curve), when the variance is kept constant, region 1.

Fig. 10.

VaR as a function of Y (decreasing curve) and C (increasing curve), when the variance is kept constant, region 2.

Fig. 11.

VaR as a function of Y (decreasing curve) and C (increasing curve), when the variance is kept constant, region 3.

Fig. 12.

C as a function of Y that yields a constant VaR (bounds of region 2).

Fig. 13.

Variance as a function of C, where Y is adjusted so that VaR remains constant (bounds of region 1).

3. Tempered multistable processes with stochastic jump intensity

In this section, we give two constructions of tempered multistable processes and their generalization to the case of a random jump intensity.

3.1. Construction through characteristic function

Fix parameters $C, G, M$ , and let $κ (Y, v)$ denote the Lévy–Khintchine–Laplace exponent of the tempered stable process with parameters $C, G, M, Y$ at time 1 (we single out Y because we will let it vary in the sequel): $\begin{array}{l} κ (Y, v) \\ = log E_{C, G, M, Y} [exp (v X_{1})] \\ = C Γ (- Y) [{(M - v)}^{Y} - M^{Y} \\ + {(G + v)}^{Y} - G^{Y}] \\ if Y \neq 1 \\ = C log [(1 - \frac{v}{M}) (1 + \frac{v}{G})] \\ if Y = 1 \end{array}$ which is defined for $- G < Re (v) < M$ .

The density of the associated Lévy measure reads: $\begin{matrix} π (Y, x) = \frac{C}{| x |^{1 + Y}} exp (- G x_{-} - M x_{+}) . \end{matrix}$ Assume now that Y is a continuous deterministic function of t ranging in $(- \infty, 2)$ . Denote $ℓ_{Y} (t, x) = π (Y (t), x)$ , that is $\begin{matrix} ℓ_{Y} (t, x) = \frac{C}{| x |^{1 + Y (t)}} exp (- G x_{-} - M x_{+}) \end{matrix}$ and let $λ_{Y}$ be the measure on $R_{+} \times R$ whose density is $ℓ_{Y}$ . Since it is σ-finite and atomless, it has an associated random Poisson measure, say $Λ_{Y}$ , whose law is characterized by the following equality: $\begin{array}{l} E exp (- \int h (t, x) d Λ_{Y} (t, x)) \\ = exp [- \int (e^{h (t, x)} - 1) ℓ_{Y} (t, x) d t d x] \end{array}$ where h is an arbitrary measurable positive function. Finally, we define the process $(L_{t})$ through the equality: $\begin{matrix} L_{t} = \int_{] 0, t] \times R} x d Λ_{Y} (s, x) \end{matrix}$ General results on Poisson point processes [16] ensure that this integral is well-defined as soon as: $\begin{matrix} \int_{] 0, t] \times R} | x \land 1 | ℓ_{Y} (t, s) d s d x < \infty \end{matrix}$ or: $\begin{array}{l} \int_{] 0, t] \times R} \frac{C}{| x |^{Y (s)}} exp (- G x_{-} - M x_{+}) d s d x \\ < \infty . \end{array}$ This inequality is verified when $Y < 1$ on $[0, t]$ , a condition that we assume for a moment. The general case will be delt with below.

Campbell formula allows us to compute the characteristic function of the random variable $L_{t}$ : $\begin{array}{l} E e^{i u L_{t}} \\ = exp [\int_{] 0, t] \times R} (e^{i u x} - 1) ℓ_{Y} (s, x) d s d x] \\ = exp [\int_{] 0, t] \times R} (e^{i u x} - 1) \\ \times C exp (- G x_{-} - M x_{+}) \frac{d x}{| x |^{1 + Y (s)}} d s] \\ = exp [\int_{] 0, t]} κ (Y (s), i u) d s] \end{array}$ as expected. In particular, the above expression confirms that L has indeed independent increments and that $(L_{a + t} - L_{a})$ follows the law of a $CGM Y_{0}$ on any interval $[a, b]$ on which Y assumes the constant value $Y_{0}$ .

In the general case, we need to compensate the Poisson measure. The compensated measure $\tilde{Λ_{Y}}$ is defined by: $\begin{matrix} d \tilde{Λ_{Y}} (t, x) = d Λ_{Y} (t, x) - ℓ_{Y} (s, x) d s d x . \end{matrix}$ Note that $\tilde{Λ_{Y}} (A)$ exists for all A compactly included in $R_{+} \times (R ∖ {0})$ .

Let φ be a measurable function on $R_{+} \times R$ verifying: $\begin{array}{l} \int_{(0, t] \times R} (φ {(s, x)}^{2} \land | φ (s, x) |) ℓ_{Y} (s, x) d s d x \\ (2) & < \infty . \end{array}$ Then the integral $\begin{matrix} I_{t} (φ) = \int_{(0, t] \times R} φ (s, x) d \tilde{Λ_{Y}} (s, x) \end{matrix}$ is well defined and its characteristic function reads: $\begin{array}{l} E e^{i u I_{t} (φ)} \\ = exp [\int_{(0, t] \times R} (e^{i u φ (s, x)} - 1 - i u φ (s, x)) \\ \times ℓ_{Y} (s, x) d s d x] . \end{array}$

Note that $φ (s, x) = | x | 1_{| x | < 1}$ verifies (2) when Y is continuous and ranges in $(- \infty, 2)$ . As a consequence, the process $\begin{matrix} J_{t} = \int_{(0, t] \times (- 1, 1)} x d \tilde{Λ_{Y}} (t, x) \end{matrix}$ is well defined. Denoting $\begin{matrix} B_{t} = \int_{(0, t] \times (- 1, 1)} x ℓ_{Y} (s, x) d x d s, \end{matrix}$ the compensated process $\tilde{L_{t}}$ is equal to $L_{t} + J_{t} - B_{t}$ , and its characteristic function is equal to $\begin{array}{l} exp [- u B_{t} + \int_{(0, t] \times R} (e^{i u x} - 1 - i u x 1_{| x | < 1})) \\ \times ℓ_{Y} (s, x) d s d x] . \end{array}$

The process $\tilde{L_{t}}$ is defined on a probability space $Ω_{Y}$ that depends on Y. Alternatively, considering the canonical space of càdlàg function $D (R_{+})$ , the construction above may be seen as a measure $P_{Y}$ on $D (R_{+})$ .

Let us now move to the case of real interest to us, that is, assume now that Y is an element of $C (R_{+})$ , that is a continuous stochastic process ranging in $(- \infty, 2)$ , and let $Q$ denote its law.

We construct the CGMY process with stochastic jump exponent as the mixture characterized by the following measure on $D (R_{+})$ : $\begin{matrix} P (d X) = \int_{C (R_{+})} P_{Y} (d X) Q (d Y) . \end{matrix}$

For this definition to make sense, we need to verify that $P_{Y}$ is a measurable function of Y, which follows from the fact that the increments are independent.

3.2. Series representation

A CGMY process is a tempered stable process in the sense of [22] with (we use the notations of [22]): $\begin{array}{l} σ (d u) = C (δ_{1} (u) + δ_{- 1} (u)), \\ q (r, u) = exp (- M r) δ_{1} (u) + exp (- G r) δ_{- 1} (u), \\ Q (d s | u) = δ_{M} (s) δ_{1} (u) + δ_{G} (s) δ_{- 1} (u), \\ Q (A) = C (δ_{M} (A) + δ_{- G} (A)), \\ R = C (M^{Y} δ_{\frac{1}{M}} + G^{Y} δ_{- \frac{1}{G}}) . \end{array}$ as straightforward computations show. A series representation on $[0, T], T > 0$ may be obtained by applying Theorem 5.3 in [22]. We will need five independent sequences of random variables:

${(γ_{j})}_{j}$ is a sequence of arrival times of a Poisson process with unit rate,

${(τ_{j})}_{j}$ is a sequence of i.i.d. r.v. uniformly distributed on $[0, T]$ ,

${(u_{j})}_{j}$ is a sequence of i.i.d. r.v. uniformly distributed on $[0, 1]$ ,

${(e_{j})}_{j}$ is a sequence of i.i.d. r.v. which are exponentially distributed with parameter 1,

${(ε_{j})}_{j}$ is a sequence of i.i.d. r.v. with $P (ε_{j} = 1) = P (ε_{j} = - 1) = \frac{1}{2}$ .

The first four series are the ones considered in [22], pages 696 and 701. As for the last one, it is obtained as follows: [22] on page 696 introduces a sequence of i.i.d. r.v.

{(v_{j})}_{j}

with distribution

\frac{Q}{σ ({- 1, 1})}

. In our case, this just means that

v_{j}

takes values M and

- G

with equal probabilities

\frac{1}{2}

. As a consequence,

ε_{j} : = \frac{v_{j}}{| v_{j} |}

is a Bernoulli random variable as stated above. In addition, one may write

| v_{j} | = \frac{M + G}{2} + ε_{j} \frac{M - G}{2}

. Theorem 5.3 in [22] then yields

\begin{array}{l} X_{t} = & \sum_{j = 1}^{\infty} ε_{j} ({(\frac{γ_{j} Y}{2 C T})}^{- \frac{1}{Y}} \land \frac{e_{j} u_{j}^{\frac{1}{Y}}}{\frac{M + G}{2} + ε_{j} \frac{M - G}{2}}) \\ (3) & \times 1_{(0, t]} (τ_{j}) + t b_{Y}, \end{array}

where

b_{Y} = 0

for

Y < 1

and

\begin{matrix} b_{Y} = - C Γ (1 - Y) (M^{Y - 1} - G^{Y - 1}) \end{matrix}

when

Y ⩾ 1

The value of $b_{Y}$ is computed from Formula (5.3) in [22], remarking that, in the notations of [22], $\begin{matrix} x_{0} : = E (\frac{v_{j}}{| v_{j} |}) = 0 \end{matrix}$ and $\begin{matrix} x_{1} : = \int_{R} x R (d x) = C (M^{Y - 1} - G^{Y - 1}), \end{matrix}$ which is valid for $Y \neq 1$ .

Representation (3) allows us to define two versions of multistable tempered processes as follows:

Theorem and Definition 1.
Let $Z (t, Y)$ be the random field defined over $[0, T] \times [a, b]$ , $a < b < 2$ , by: $\begin{array}{l} Z (t, Y) \\ = \sum_{j = 1}^{\infty} ε_{j} ({(\frac{γ_{j} Y}{2 C T})}^{- \frac{1}{Y}} \land \frac{e_{j} u_{j}^{\frac{1}{Y}}}{\frac{M + G}{2} + ε_{j} \frac{M - G}{2}}) \\ (4) & \times 1_{(0, t]} (τ_{j}) + t b_{Y}, \end{array}$ and let Y be a continuous function ranging in $[a, b]$ .

The field-based tempered multistable process parametrized by Y is the random process $\begin{array}{l} X_{t}^{(F)} = & Z (t, Y_{t}) \\ = & \sum_{j = 1}^{\infty} ε_{j} ({(\frac{γ_{j} Y_{t}}{2 C T})}^{- \frac{1}{Y_{t}}} \\ \land \frac{e_{j} u_{j}^{\frac{1}{Y_{t}}}}{\frac{M + G}{2} + ε_{j} \frac{M - G}{2}}) \\ (5) & \times 1_{(0, t]} (τ_{j}) + t b_{Y_{t}} . \end{array}$

The independent-increment tempered multistable process parametrized by Y is the random process $\begin{array}{l} X_{t}^{(I)} = & \sum_{j = 1}^{\infty} [ε_{j} ({(\frac{γ_{j} Y_{τ_{j}}}{2 C T})}^{- \frac{1}{Y_{τ_{j}}}} \\ \land \frac{e_{j} u_{j}^{\frac{1}{Y_{τ_{j}}}}}{\frac{M + G}{2} + ε_{j} \frac{M - G}{2}}) \\ (6) & \times 1_{(0, t]} (τ_{j}) + t b_{Y_{τ_{j}}}] . \end{array}$

Remark that we do not make any assumption of regularity on Y other than continuity at this stage.
Proof.
The proof that the series above converge almost surely is a straightforward extension of the ones in [22]. □

The generalization to the case of random jump exponent is then straightforward:
Theorem and Definition 2.
Let $Z (t, Y)$ be the random field defined by ( 4 ) and let Υ be a continuous random process ranging in $[a, b]$ .

The field-based tempered multistable process with random jump exponent parametrized by Υ is the random process $\begin{array}{l} X_{t}^{(F)} = & Z (t, Υ_{t}) \\ = & \sum_{j = 1}^{\infty} ε_{j} ({(\frac{γ_{j} Υ_{t}}{2 C T})}^{- \frac{1}{Υ_{t}}} \\ \land \frac{e_{j} u_{j}^{\frac{1}{Υ_{t}}}}{\frac{M + G}{2} + ε_{j} \frac{M - G}{2}}) \\ (7) & \times 1_{(0, t]} (τ_{j}) + t b_{Υ_{t}} . \end{array}$

The independent-increment tempered multistable process with random jump exponent parametrized by Υ is the random process $\begin{array}{l} X_{t}^{(I)} = & \sum_{j = 1}^{\infty} [ε_{j} ({(\frac{γ_{j} Υ_{τ_{j}}}{2 C T})}^{- \frac{1}{Υ_{τ_{j}}}} \\ \land \frac{e_{j} u_{j}^{\frac{1}{Υ_{τ_{j}}}}}{\frac{M + G}{2} + ε_{j} \frac{M - G}{2}}) \\ (8) & \times 1_{(0, t]} (τ_{j}) + t b_{Υ_{τ_{j}}}] . \end{array}$

Remark that we do not require independence between Υ and Z in these definitions. This will be however needed in applications to avoid technical complications.
4. Tempered multistable model with stochastic evolution of Y

In order to explore the benefit of using models with a time-varying local jump exponent in financial applications, the basic idea is to consider stochastic time evolutions of the form: $\begin{matrix} (9) & S_{t} = S_{0} exp (μ t + σ L_{t} - κ_{t}) \end{matrix}$ where $L_{t}$ is one version of a CGMY process with varying Y and $κ_{t}$ is the Lévy–Khintchine–Laplace exponent of $L_{t}$ . Depending on the version, $exp (- μ t) S_{t}$ will be a process with constant mean or a martingale.

For such models to be practical and amenable e.g. to option pricing or risk management, it is necessary to specify the evolution of Y. In a roughly similar way to local and stochastic volatility models, there are two ways to do so. The first one consists in defining “self-stabilizing” processes, where the evolution of Y depends on the value of the process. This path is explored in [9,10]. In this work we deal with the second possibility: we specify the evolution of Y through a separate stochastic differential equation, defining “stochastic local jump exponent” models, much in the spirit of stochastic volatility models. Our aim here is merely to propose reasonable models, leaving their theoretical as well as experimental study for further work.

4.1. General frame

The general idea is to let Y be a continuous ergodic Markov process ranging in a real interval I, independent of the variables $(γ_{i}, Γ_{i})$ , and $f : I \mapsto (\infty, 2)$ be a continuous function. The first examples that come to mind is to choose diffusion models for Y of Ornstein–Uhlenbeck or CIR type. We then study (9) where $L_{t}$ is a multistable process with random jump exponent given by $f (Y_{t})$ .

Let us start with a property related to the martingale character in the independent increments case:

Proposition 1.
The process $\begin{matrix} Z_{t} = exp (X_{t}^{(I)} - \int_{0}^{t} κ (Υ) d s) \end{matrix}$ is an $(F_{t} \lor F_{t}^{Υ})$ martingale.

The proof of this result is standard if we assume Υ deterministic. In the random case, it follows from the independence between the random variables that govern the jumps of $X^{(I)}$ and Υ.
4.2. A tempered multistable motion with random local jump exponent, Hull and White variant

Our first model mimics the Hull and White model where we have substituted Y for the volatility, and replaced the Wiener driving noise of the spot price by a tempered multistable one. It reads as follows: $\begin{matrix} (10) & \{\begin{matrix} d S_{t} = μ (t, S_{t}) d t + S_{t} d Z_{t}^{Y_{t}}, \\ d ln (2 - Y_{t}) \\ = θ (μ - ln (2 - Y_{t})) d t + γ d W_{t}, \\ Y_{0} < 2, \end{matrix} \end{matrix}$ where $Z^{Y}$ is an independent increment multistable CGMY process as in (8) and W is an independent Wiener process.

4.3. A tempered multistable motion with random local jump exponent, Heston variant

The second model mimics the Heston model where we have substituted Y for the volatility, and replaced the Wiener driving noise of the spot price by a multistable one. $\begin{matrix} (11) & \{\begin{matrix} d S_{t} = μ S_{t} d t + σ S_{t} d Z_{t}^{Y_{t}}, \\ d (2 - Y_{t}) = κ (θ - (2 - Y_{t})) d t \\ + ζ \sqrt{2 - Y_{t}} d W_{t}, \\ Y_{0} < 2, \end{matrix} \end{matrix}$ where again $Z^{Y}$ is an independent increment multistable CGMY process as in (8), W is an independent Wiener process and $2 κ θ ⩾ ζ^{2}$ .

Since Y appears as an exponent in the definition of multistable processes, it may seem more relevant to model the evolution of its logarithm, leading to $\begin{matrix} (12) & d β_{t} = κ (θ - β_{t}) d t + ζ \sqrt{β_{t}} d W_{t}, β_{0} > 0, \end{matrix}$ where $β = ln (\frac{2}{Y})$ .

Figures 14 and 15 display realizations of these two variants sampled on one thousand points, with $S_{0} = 2, ν_{0} = 1.4, μ = 0.02, σ = 0.5, κ = 1, θ = 1.4, ζ = 1$ and $ρ = 0$ . One can check that regions where the local jump intensity process is “small” correspond to times with larger jumps.

Fig. 14.

Spot price and local jump intensity process for model (11).

Fig. 15.

Spot price and local jump intensity process for model (12).

Footnotes

Acknowledgements

J. Lévy Véhel gratefully acknowledges support from SMA Group as well as many fruitful discussions with P. Desurmont, O. Le Courtois, M. Majri, H. Rodarie and C. Walter.

References

http://ec.europa.eu/internal_market/insurance/solvency/future/index_en.htm.

Ait-Sahalia

and Jacod

, Testing wether jumps have finite or infinite activity, Ann. Statist.37(1) (2009), 184–222. doi:10.1214/07-AOS568.

Albin

J.M.P.

and Sundén

, On the asymptotic behaviour of Lévy processes, part I: Subexponential and exponential processes, Stoch. Proc. Appl.119(1) (2009), 281–304. doi:10.1016/j.spa.2008.02.004.

Carr

Geman

Madan

and Yor

, The fine structure of asset returns: An empirical investigation, J. Business75 (2002), 305–332. doi:10.1086/338705.

Cont

and Tankov

, Financial Modelling with Jump Processes, Chapman & Hall / CRC Press, 2003.

Dupire

, Pricing with a smile, Risk7(1) (1994), 18–20.

El Mekeddem

and Lévy Véhel

, Value at risk with tempered multistable motions, in: 30th International French Finance Association Conference, Lyon, France, May 2013, 2013, available at hal-00868634.

Falconer

K.J.

and Lévy Véhel

, Multifractional, multistable, and other process with prescribed local form, J. Theoret. Probab. (2008). doi:10.1007/s10959-008-0147-9.

Falconer

K.J.

and Lévy Véhel

, Self-stabilizing processes based on random signs, 2018, available at https://arxiv.org/pdf/1802.03231.pdf.

10.

Falconer

K.J.

and Lévy Véhel

, 2018, Self-stabilizing processes, available at https://arxiv.org/pdf/1802.02543.pdf.

11.

Falconer

K.J.

and Liu

, Multistable processes and localisability, Stochastic Models28 (2012), 503–526. doi:10.1080/15326349.2012.699766.

12.

Fan

and Lévy Véhel

, Tempered Fractional Multistable Motion and Tempered Multifractional Stable Motion ESAIM PS, 2016, to appear.

13.

Heston

S.L.

, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies6(2) (1993), 327–343. doi:10.1093/rfs/6.2.327.

14.

Jorion

, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd edn, McGraw-Hill, 2006.

15.

Kelani

and Quittard-Pinon

, A General Approach to Compute Standard Risk Measures International Conference of the French Finance Association (AFFI), 2011.

16.

Kingman

, Poisson Processes, Oxford Studies in Probability, 1993.

17.

Le Courtois

, Some further results on the tempered multistable approach, Asia – Pacific Financial Markets (2018). doi:10.1007/s10690-018-9240-y.

18.

Le Courtois

and Walter

, The computation of risk budgets under the Lévy process assumption, Finance35 (2014), 2.

19.

Le Guével

and Lévy Véhel

, A Ferguson–Klass–LePage series representation of multistable multifractional motions and related process, Bernoulli18(4) (2012), 1099–1127. doi:10.3150/11-BEJ372.

20.

Le Guével

Véhel

J.L.

and Liu

, On two multistable extensions of stable Lévy motion and their semimartingale representation, Journal of Theoretical Probability28(3) (2012), 1125–1144.

21.

Meerschaert

M.M.

and Sabzikar

, Tempered fractional stable motion, Journal of Theoretical Probability29 (2016), 681–706. doi:10.1007/s10959-014-0585-5.

22.

Rosinski

, Tempering stable processes, Stochastic Processes and Their Applications117(6) (2007), 677–707. doi:10.1016/j.spa.2006.10.003.