Risk-neutral option pricing under GARCH intensity model

The risk-neutral option pricing method under GARCH intensity model is examined. The GARCH intensity model incorporates the characteristics of financial return series such as volatility clustering, leverage effect and conditional asymmetry. The GARCH intensity option pricing model has flexibility in changing the volatility according to the probability measure change.


Introduction
This paper develop the risk-neutral option pricing framework under the GARCH intensity model. The financial asset returns series have interesting characteristics. Large volatility tends to follow large volatility and small volatility tends to follow small volatility in the series of financial returns and this is called volatility clustering. The leverage effect indicates that today's volatility has a negative correlation with past returns. Conditional asymmetry is an asymmetric correlation between current and past volatility, depending on whether the current and past returns are positive or negative.
These characteristics are well captured by various GARCH models. The volatility clustering is well described by original GARCH model [4]. [13], [12], [8], and [14] incorporate the leverage effects and [1] capture the conditional asymmetry. The GARCH intensity model introduced by [5] also describes the volatility clustering, leverage effect and conditional asymmetry in financial asset price dynamics and based on Poisson type intensity processes.
The risk-neutral option pricing is a method to determine a no-arbitrage prices of financial options with underlying assets. The option pricing theory of [3] and [11] is based on the risk-neutral pricing framework. The relation between risk-neutral pricing and the no-arbitrage principle was studied by [9] and [10]. [6] explained the risk-neutral option pricing under the GARCH model. This paper extends the risk-neutral pricing idea to the GARCH intensity model.
The remainder of the paper is organized as follows: Section 2 reviews the GARCH intensity model. Section 3 examines the mathematical analysis on the risk-neutral option pricing under the GARCH intensity model. Section 4 extends the option pricing theory to the generalized GARCH intensity model. Section 5 concludes the paper.

GARCH intensity model
First, we review the GARCH intensity model introduced by [5]. In the model, the asset price process movement is described by two Poisson-type processes with time-varying intensity processes. A probability space (Ω, F = F(T ), P) with a filtration F(t), 0 ≤ t ≤ T , is given. 5]). We are given F(t)-adapted r.c.l.l. processes N + (t), N − (t) and positive F(t)-adapted r.c.l.l. processes λ + (t), λ − (t) for 0 ≤ t ≤ T satisfying the following conditions: (i) (Discrete observation time) ∆t = T /N and (vi) (Asset price) With a constant δ > 0, the price process is With a price jump at time t, S(t) = e δ S(t−) or S(t) = e −δ S(t−) depending on the direction of the jump. The asset price can also be represented by a stochastic differential equation given by be the log-return over the period [t i−1 , t i ]. Then the integer-valued random variable M i defined by has the conditional Skellam distribution on F(t i−1 ) where I a is the modified Bessel function of the first kind defined by Since the closed form of conditional probability density is exist, the maximum likelihood estimation can be easily employed.
Recall that under Assumption 2.1, and where µ(t i ) is a drift term, γ(t i ) is an Itô correction factor, and ε(t i ) is a F(t i )-measurable shock occurred during time interval [t i−1 , t i ]. As in [5], to capture volatility clustering, GARCH[4]-type modeling is applied: for some constants ω ± , α ± , and β ± . If in the GARCH intensity model, then the GARCH-type time varying volatility is obtained. To show this, let h(t i ) be a one-step-ahead conditional variance of return at t i , then and This is consistent with conditional variance modeling in GARCH. In addition, we also consider the GJR [8] GARCH-type intensity model: where

Risk-neutral option pricing
We propose an option pricing method for intensity models by constructing an equivalent measure under which the discounted stock price process is a martingale. We assume that the underlying asset pays no dividend and let r > 0 be the risk-free interest rate. In the following we choose new intensities for an equivalent martingale measure.
Definition 3.1. Take a pair of positive r.c.l.l. adapted step processes λ + and λ − such that (We take the right hand side equal to r since the left hand side is regarded as drift under a risk-neutral measure.) (ii) Let Z(0) = 1 and for t i−1 < t ≤ t i define Then (For the details of the proof, see [7]. .
Take s, t such that t j−1 < s ≤ t j and s < t. Then Definition 3.3. Define an equivalent probability measure Q by

Now we change intensities.
Lemma 3.4. The intensities of N + and N − under Q are given by λ + and λ − , respectively.
for an F(t i−1 )-measurable random variable η i . We will show that the same relation holds for Q and λ + . Define Z ±,i (t) as in the proof of Theorem 3.2.
For a constant ξ, we have where the last expression is the moment generating function of a Poisson distribution with intensity λ As in Definition 2.2 we define γ(t i ) and ε(t i ) in terms of λ + , λ − and Q. Then Theorem 3.5. The discounted stock price process is a Q-martingale, i.e., Proof. By the tower property it suffices to consider the case that Basically the GARCH intensity option pricing model has flexibility in changing the volatility according to the measure change. This allows us to construct a GARCH intensity option pricing model consistent with volatility spread [2].

A generalization of GARCH intensity model
The goal of this section is to provide the generalized version for the previous model in which it is assumed to be that the sizes of stock price changes affected by news are represented by independent and identically distributed random variables.

Lemma 4.2. Under Assumption 4.1, we have
and Proof. Recall that Now use the fact that each conditional distribution is a compound Poisson distribution.
Note that depending on distributions of δ + and δ − , the model allows the skewness in the conditional distribution of log-return X(t). For example, if the tail of the distribution of δ − is fatter than the tail of the distribution of δ + , then the conditional distribution of log-return X(t) is negatively skewed. respectively.
Note that, by direct computation, we have The mean correction factor γ(t i ) appears because we model with logreturn and this implies that and using the property of compound Poisson distributions, we have The one step ahead expectation of future stock price can be represented as exponential of drift term µ(t i ) multiplied by current stock price, that is The derivation of equivalent martingale measure for extended version of GRACH intensity model is similar to the previous version.
Definition 4.4. Let f ± be probability density functions of δ ± , respectively and f ± are some probability density functions (which are desired probability density functions of δ ± under equivalent martingale measure). (i) Let and λ + and λ − be two r.c.l.l. adapted step processes satisfying the equation for each i, and λ ± (t) = λ ± (t i−1 ) (ii) Suppose that λ + (t) and λ − (t) are positive processes. Let for t i−1 < t ≤ t i , recursively.
Remark 4.5. Note that F(t i−1 )-measurable random variable κ i−1 is zero when the conditional variance of return distribution of the martingale measure is equal to the variance under physical measure. Q i (t) depends on F(t) random variables N ± and δ.
Then Z +,i (t) and Z −,i (t) are F(t)-measurable and satisfy Note that .
Since E[Z(T )] = 1, we use Z(T ) in Definition 4.4 to construct a new probability measure Q. We define Lemma 4.7. Under the measure Q in (6), for every t i−1 < t ≤ t i , the conditional distributions are Poisson distributions with new intensities λ + (t i−1 ) and λ − (t i−1 ), respectively. Moreover, δ +,j have probability density function f under Q.
Proof. Define Z ±,i (t) as in the proof of Lemma 4.6. For a constant u, we have and let Φ be the conditional moment generating function of Y +,j , i.e.
Note that Then Note that which is the moment generating function of a Poissson distribution with intensity λ , the proof is the same. For the remaining part of lemma, to figure out the distribution of δ ±,j under Q, it is enough to check the distribution of δ +,1 under Q. Without loss of generality, assume that first jump, i.e. the occurrence time of random variable δ +,1 is less than t 1 . Thus, for a constant u, we have The last equality holds since given random variables independent and since Furthermore, because of independency and the fact that (by Lemma 4.6 when δ is constant) and for each j. Finally we have which implies δ +,1 has a probability density function f under Q. For general j, the proof is the same.
The choice for risk-neutral distributions f + and f − are related to the skewness in conditional distribution of log-return under risk-neutral measure. This is similar to the fact that distributions f + and f − are related to the skewness under physical measure. Now we show that under the equivalent martingale measure Q, the discounted stock price process is a martingale.

Concluding remark
The risk-neutral option pricing framework for the GARCH intensity model was introduced. Equivalent martingale measures are provided and hence the the risk-neutral option price is computed under the measure. The framework is consistent with the empirical characteristics such as volatility smile and spread. The theory is easily extended to the generalized version of the GARCH intensity model.