Risk-neutral option pricing under GARCH intensity model
RRisk-neutral option pricing under GARCH intensitymodel
Kyungsub Lee ∗ Abstract
The risk-neutral option pricing method under GARCH intensitymodel is examined. The GARCH intensity model incorporates thecharacteristics of financial return series such as volatility clustering,leverage effect and conditional asymmetry. The GARCH intensity op-tion pricing model has flexibility in changing the volatility accordingto the probability measure change.
This paper develop the risk-neutral option pricing framework under theGARCH intensity model. The financial asset returns series have interestingcharacteristics. Large volatility tends to follow large volatility and smallvolatility tends to follow small volatility in the series of financial returnsand this is called volatility clustering. The leverage effect indicates thattoday’s volatility has a negative correlation with past returns. Conditionalasymmetry 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. Thevolatility clustering is well described by original GARCH model [4]. [13],[12], [8], and [14] incorporate the leverage effects and [1] capture the con-ditional asymmetry. The GARCH intensity model introduced by [5] alsodescribes the volatility clustering, leverage effect and conditional asymme-try in financial asset price dynamics and based on Poisson type intensityprocesses.The risk-neutral option pricing is a method to determine a no-arbitrageprices of financial options with underlying assets. The option pricing theory ∗ Department of Statistics, Yeungnam University, Gyeongsan, Gyeongbuk 38541, Korea a r X i v : . [ q -f i n . P R ] A ug f [3] and [11] is based on the risk-neutral pricing framework. The relationbetween risk-neutral pricing and the no-arbitrage principle was studied by[9] and [10]. [6] explained the risk-neutral option pricing under the GARCHmodel. This paper extends the risk-neutral pricing idea to the GARCHintensity model.The remainder of the paper is organized as follows: Section 2 reviews theGARCH intensity model. Section 3 examines the mathematical analysis onthe risk-neutral option pricing under the GARCH intensity model. Section4 extends the option pricing theory to the generalized GARCH intensitymodel. Section 5 concludes the paper. First, we review the GARCH intensity model introduced by [5]. In themodel, the asset price process movement is described by two Poisson-typeprocesses with time-varying intensity processes. A probability space (Ω , F = F ( T ) , P ) with a filtration F ( t ), 0 ≤ t ≤ T , is given. Assumption 2.1 ([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 t i = i ∆ t , 0 ≤ i ≤ N .(ii) (Conditional distribution) ( N ± ( t ) − N ± ( t i − )) |F ( t i − ) has Poisson dis-tribution with intensity λ ± ( t i − )( t − t i − ), t i − ≤ t ≤ t i . Hence P ( N ± ( t i ) − N ± ( t i − ) = k |F ( t i − )) = ( λ ± ( t i − )∆ t ) k k ! exp ( − λ ± ( t i − )∆ t ) . (iii) (Conditional independence) N + ( t ) − N + ( t i − ) and N − ( t ) − N − ( t i − ) areconditionally independent given F ( t i − ), t i − ≤ t ≤ t i .(iv) (Step process) λ + ( t ) = λ + ( t i − ) and λ − ( t ) = λ − ( t i − ), t i − ≤ t < t i .(v) (Predictability) λ + ( t i ) depends on N ± ( t i − k +1 ) and λ ± ( t i − k ), 1 ≤ k ≤ i −
1, and similarly for λ − ( t i ).(vi) (Asset price) With a constant δ >
0, the price process is S ( t ) = S (0) exp ( δ ( N + ( t ) − N − ( t ))) . With a price jump at time t , S ( t ) = e δ S ( t − ) or S ( t ) = e − δ S ( t − ) depend-ing on the direction of the jump. The asset price can also be represented bya stochastic differential equation given byd S ( t ) = ( e δ − S ( t − )d N + ( t ) + ( e − δ − S ( t − )d N − ( t ) . X ( t i ) = log S ( t i ) S ( t i − )be the log-return over the period [ t i − , t i ]. Then the integer-valued randomvariable M i defined by M i = X ( t i ) δ = N + ( t i ) − N − ( t i ) − ( N + ( t i − ) − N − ( t i − ))has the conditional Skellam distribution on F ( t i − ) f ( m | λ + ( t i − ) , λ − ( t i − ))= exp {− λ + ( t i − ) − λ − ( t i − ) } (cid:18) λ + ( t i − ) λ − ( t i − ) (cid:19) m/ I | m | (2 (cid:112) λ + ( t i − ) λ − ( t i − ))where I a is the modified Bessel function of the first kind defined by I a ( x ) = ∞ (cid:88) k =0 k ! Γ( k + a + 1) (cid:16) x (cid:17) k + a . Since the closed form of conditional probability density is exist, the maxi-mum likelihood estimation can be easily employed.
Definition 2.2 (Decomposition of Log-Return) . Define µ ( t i ), γ ( t i ), ε ( t i ) by µ ( t i ) = { ( e δ − λ + ( t i − ) + ( e − δ − λ − ( t i − ) } ∆ tγ ( t i ) = { ( e δ − − δ ) λ + ( t i − ) + ( e − δ − δ ) λ − ( t i − ) } ∆ tε ( t i ) = X ( t i ) − E [ X ( t i ) |F ( t i − )] . Recall that under Assumption 2.1, E [ X ( t i ) |F ( t i − )] = δ ( λ + ( t i − ) − λ − ( t i − ))∆ t Var( X ( t i ) |F ( t i − )) = δ ( λ + ( t i − ) + λ − ( t i − ))∆ t E [exp( X ( t i )) |F ( t i − )) = exp( µ ( t i ))and X ( t i ) = µ ( t i ) − γ ( t i ) + ε ( t i ) , where µ ( t i ) is a drift term, γ ( t i ) is an Itˆo correction factor, and ε ( t i ) is a F ( t i )-measurable shock occurred during time interval [ t i − , t i ].As in [5], to capture volatility clustering, GARCH[4]-type modeling isapplied: λ ± ( t i ) = ω ± + α ± ε ( t i ) + β ± λ ± ( t i − )3or some constants ω ± , α ± , and β ± . If β + = β − = β in the GARCH intensity model, then the GARCH-type time varying volatil-ity is obtained. To show this, let h ( t i ) be a one-step-ahead conditionalvariance of return at t i , then h ( t i ) = Var( X ( t i ) |F ( t i − )) / ∆ t = δ ( λ + ( t i − ) + λ − ( t i − )) , and h ( t i ) = δ ( ω + + ω − ) + βh ( t i − ) + δ ( α + + α − ) ε ( t i − ) . (1)This is consistent with conditional variance modeling in GARCH. In addi-tion, we also consider the GJR[8] GARCH-type intensity model: λ ± ( t i ) = ω ± + α ± ε ( t i ) + β ± λ ± ( t i − )where I ( t i ) = (cid:26) , ε ( t i ) < , ε ( t i ) ≥ . We propose an option pricing method for intensity models by constructingan equivalent measure under which the discounted stock price process is amartingale. We assume that the underlying asset pays no dividend and let r >
Definition 3.1.
Take a pair of positive r.c.l.l. adapted step processes (cid:101) λ + and (cid:101) λ − such that ( e δ − (cid:101) λ + ( t ) + ( e − δ − (cid:101) λ − ( t ) = r. (2)(We take the right hand side equal to r since the left hand side is regardedas drift under a risk-neutral measure.)(i) Let D ( t ) = λ + ( t ) + λ − ( t ) − (cid:101) λ + ( t ) − (cid:101) λ − ( t )for 0 ≤ t ≤ T , and let U (0) = 0 and for t i − < t ≤ t i define U ( t ) = ( N + ( t ) − N + ( t i − )) log (cid:101) λ + ( t ) λ + ( t ) + ( N − ( t ) − N − ( t i − )) log (cid:101) λ − ( t ) λ − ( t ) . (ii) Let Z (0) = 1 and for t i − < t ≤ t i define Z ( t ) = Z ( t i − ) exp { D ( t )( t − t i − ) + U ( t ) } . heorem 3.2. { Z ( t ) } ≤ t ≤ T is a P -martingale.Proof. For t i − < t ≤ t i , define Z + ,i ( t ) and Z − ,i ( t ) by Z ± ( t i − ) = 1 andd Z ± ,i ( t ) = Z ± ,i ( t − ) (cid:101) λ ± ( t i − ) − λ ± ( t i − ) λ ± ( t i − ) d( N ± ( t ) − λ ± ( t i − ) t ) . Then Z ± ,i ( t ) = exp (cid:26) ( λ ± ( t i − ) − (cid:101) λ ± ( t i − ))( t − t i − )+ ( N ± ( t ) − N ± ( t i − )) log (cid:101) λ ± ( t i − ) λ ± ( t i − ) (cid:27) . (For the details of the proof, see [7].) Since N ± ( t ) − λ ± ( t i − ) t are martingales, Z ± ,i ( t ) are martingales for t i − ≤ u < t and hence E [ Z ± ,i ( t ) |F ( t i − )] = 1 . Note that Z + ,i ( t ) Z − ,i ( t ) = Z ( t ) Z ( t i − ) . Since N + ( t ) − N + ( t i − ) and N − ( t ) − N − ( t i − ) are conditionally independentgiven F ( t i − ), we have E (cid:20) Z ( t ) Z ( t i − ) (cid:12)(cid:12)(cid:12)(cid:12) F ( t i − ) (cid:21) = E [ Z + ,i ( t ) |F ( t i − )] E [ Z − ,i ( t ) |F ( t i − )] = 1 . Take s, t such that t j − < s ≤ t j and s < t . Then E [ Z ( t j ) |F ( s )] = Z ( s )5nd E [ Z ( t ) |F ( s )]= E (cid:20) Z ( t ) Z ( t i − ) Z ( t i − ) Z ( t i − ) × · · · × Z ( t j +1 ) Z ( t j ) Z ( t j ) (cid:12)(cid:12)(cid:12)(cid:12) F ( s ) (cid:21) = E (cid:20) E (cid:20) Z ( t ) Z ( t i − ) Z ( t i − ) Z ( t i − ) × · · · × Z ( t j +1 ) Z ( t j ) Z ( t j ) (cid:12)(cid:12)(cid:12)(cid:12) F ( t i − ) (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) F ( s ) (cid:21) = E (cid:20) Z ( t i − ) Z ( t i − ) × · · · × Z ( t j +1 ) Z ( t j ) Z ( t j ) E (cid:20) Z ( t ) Z ( t i − ) (cid:12)(cid:12)(cid:12)(cid:12) F ( t i − ) (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) F ( s ) (cid:21) = E (cid:20) Z ( t i − ) Z ( t t − ) × · · · × Z ( t j +1 ) Z ( t j ) Z ( t j ) (cid:12)(cid:12)(cid:12)(cid:12) F ( s ) (cid:21) ...= E [ Z ( t j ) |F ( s )]= Z ( s ) . Definition 3.3.
Define an equivalent probability measure Q by Q ( A ) = (cid:90) A Z ( T )d P for A ∈ F . Now we change intensities.
Lemma 3.4.
The intensities of N + and N − under Q are given by (cid:101) λ + and (cid:101) λ − , respectively.Proof. Since { N + ( t ) − N + ( u ) }|F ( u ) has a Poisson distribution for t i − ≤ u < t ≤ t i , we have E P (cid:2) exp { η i ( N + ( t ) − N + ( u )) } (cid:12)(cid:12) F ( u ) (cid:3) = exp { λ + ( t i − )( t − u )( e η i − } for an F ( t i − )-measurable random variable η i . We will show that the samerelation holds for Q and (cid:101) λ + . Define Z ± ,i ( t ) as in the proof of Theorem 3.2.6or a constant ξ , we have E Q [exp { ξ ( N + ( t ) − N + ( t i − )) }|F ( t i − )]= E P (cid:20) exp { ξ ( N + ( t ) − N + ( t i − )) } Z ( t ) Z ( t i − ) (cid:12)(cid:12)(cid:12)(cid:12) F ( t i − ) (cid:21) = E P [exp { ξ ( N + ( t ) − N + ( t i − )) } Z + ,i ( t ) Z − ,i ( t ) |F ( t i − )]= E P [exp { ξ ( N + ( t ) − N + ( t i − )) } Z + ,i ( t ) |F ( t i − )] E P [ Z − ,i ( t ) |F ( t i − )]= E P [exp { ξ ( N + ( t ) − N + ( t i − )) } Z + ,i ( t ) |F ( t i − )]= E P (cid:20) exp { ξ ( N + ( t ) − N + ( t i − )) + ( λ + ( t i − ) − (cid:101) λ + ( t i − ))( t − t i − ) }× exp (cid:40) ( N + ( t ) − N + ( t i − )) log (cid:101) λ + ( t i − ) λ + ( t i − ) (cid:41) (cid:12)(cid:12)(cid:12)(cid:12) F ( t i − ) (cid:21) = exp { ( λ + ( t i − ) − (cid:101) λ + ( t i − ))( t − t i − ) }× E P (cid:34) exp (cid:40) ( N + ( t ) − N + ( t i − )) (cid:32) ξ + log (cid:101) λ + ( t i − ) λ + ( t i − ) (cid:33)(cid:41) (cid:12)(cid:12)(cid:12)(cid:12) F ( t i − ) (cid:35) = exp { ( λ + ( t i − ) − (cid:101) λ + ( t i − ))( t − t i − ) }× exp (cid:40) λ + ( t i − ) (cid:32) (cid:101) λ + ( t i − ) λ + ( t i − ) e ξ − (cid:33) ( t − t i − ) (cid:41) = exp { (cid:101) λ + ( t i − )( e ξ − t − t i − ) } where the last expression is the moment generating function of a Poissondistribution with intensity (cid:101) λ + ( t i − )( t − t i − ). For N − ( t ) − N − ( t i − ), theproof is identical.As in Definition 2.2 we define (cid:101) γ ( t i ) and (cid:101) ε ( t i ) in terms of (cid:101) λ + , (cid:101) λ − and Q . (cid:101) γ ( t i ) = { ( e δ − − δ ) (cid:101) λ + ( t i − ) + ( e − δ − δ ) (cid:101) λ − ( t i − ) } ∆ t (cid:101) ε ( t i ) = X ( t i ) − E Q [ X ( t i ) |F ( t i − )] . Then X ( t i ) = r ∆ t − (cid:101) γ ( t i ) + (cid:101) ε ( t i ) . Theorem 3.5.
The discounted stock price process is a Q -martingale, i.e.,for ≤ u ≤ t ≤ T E Q [ e − rt S ( t ) |F ( u )] = e − ru S ( u ) . roof. By the tower property it suffices to consider the case that t i − ≤ u Figure 1: GJR: Volatility smile: maturity 30 days and 60 days (from left toright) Table 1: Parameter setting for GJR intensity model δ . × − ω + . × − β + . × − α + . × γ + . × ω − . × − β − . × − α − . × γ − . × emark 3.6. Assume that λ ± ( t ) = λ ± and (cid:101) λ ± ( t ) = (cid:101) λ ± for some constants λ ± and (cid:101) λ ± for every t , and assume that the European call option price attime t is given by c ( t, S ( t )) = E Q [ e − r ( T − t ) ( S ( T ) − K ) + |F ( t )]where Q is the conditional variance preserving measure. By Iˆto’s formula,we have e − rt c ( t, S ( t )) = c (0 , S (0)) + A ( t )+ (cid:90) t (cid:104) c ( u, e δ S ( u )) − c ( u, S ( u )) (cid:105) d( N + ( u ) − (cid:101) λ + u )+ (cid:90) t (cid:104) c ( u, e − δ S ( u )) − c ( u, S ( u )) (cid:105) d( N − ( u ) − (cid:101) λ − u )where A ( t ) = (cid:90) t e − ru (cid:0) − rc ( u, S ( u )) + ∂c∂t ( u, S ( u )) + (cid:101) λ + (cid:104) c ( u, e δ S ( u )) − c ( u, S ( u )) (cid:105) + (cid:101) λ − (cid:104) c ( u, e − δ S ( u )) − c ( u, S ( u )) (cid:105) (cid:1) d u. Note that e − rt c ( t, S ( t )), N + ( t ) − (cid:101) λ + t and N − ( t ) − (cid:101) λ − t are Q -martingales.Hence A ( t ) is a martingale, and the integrand should be zero. Thus c satisfies − rc ( t, S ) + ∂c∂t ( t, S ) + (cid:101) λ + (cid:104) c ( t, e δ S ) − c ( t, S ) (cid:105) + (cid:101) λ − (cid:104) c ( t, e − δ S ) − c ( t, S ) (cid:105) = 0 . For small δ , consider the following approximations: c ( t, e ± δ S ) − c ( t, S ) ≈ ( e ± δ − S ∂c∂S ( t, S ) + 12 ( e ± δ − S ∂ c∂S ( t, S ) . Then − rc ( t, S ) + ∂c∂t ( t, S ) + (cid:101) λ + (cid:20) ( e δ − S ∂c∂S ( t, S ) + 12 ( e δ − S ∂ c∂S ( t, S ) (cid:21) + (cid:101) λ − (cid:20) ( e − δ − S ∂c∂S ( t, S ) + 12 ( e − δ − S ∂ c∂S ( t, S ) (cid:21) ≈ − rc ( t, S ) + ∂c∂t ( t, S ) + (cid:101) λ + (cid:20) ( e δ − S ∂c∂S ( t, S ) + 12 δ S ∂ c∂S ( t, S ) (cid:21) + (cid:101) λ − (cid:20) ( e − δ − S ∂c∂S ( t, S ) + 12 δ S ∂ c∂S ( t, S ) (cid:21) ≈ − rc ( t, S ) + ∂c∂t ( t, S ) + rS ∂c∂S ( t, S ) + 12 hS ∂ c∂S ( t, S ) = 0where h = δ ( λ + + λ − ). The goal of this section is to provide the generalized version for the previousmodel in which it is assumed to be that the sizes of stock price changesaffected by news are represented by independent and identically distributedrandom variables. Assumption 4.1. The assumptions (i)–(v) for asset price process are thesame as in Assumption 2.1(i)–(v) and we need an additional condition:(vi) (Asset price) The asset price S ( t ) satisfies S ( t ) = S (0) exp N + ( t ) (cid:88) j =1 δ + ,j − N − ( t ) (cid:88) j =1 δ − ,j for some i.i.d. random variables δ + ,j > δ − ,j > j ≥ Lemma 4.2. Under Assumption 4.1, we have E [ X ( t i ) |F ( t i − )] = (¯ δ + λ + ( t i − ) − ¯ δ − λ − ( t i − ))∆ t and Var( X ( t i ) |F ( t i − )) = ( δ λ + ( t i − ) + δ − λ − ( t i − ))∆ t where ¯ δ ± = E [ δ ± ] and δ ± = E [ δ ± ] . Proof. Recall that X ( t i ) = N + ( t i ) (cid:88) j = N + ( t i − )+1 δ + ,j − N − ( t i ) (cid:88) j = N − ( t i − )+1 δ − ,j . N ± ( t i ) (cid:88) j = N ± ( t i − )+1) δ ± ,j (cid:12)(cid:12) F ( t i − )is a compound Poisson distribution.Note that depending on distributions of δ + and δ − , the model allows theskewness in the conditional distribution of log-return X ( t ). For example, ifthe 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. Definition 4.3. We define µ ( t i ) = ( φ δ + − λ + ( t i − )∆ t + ( φ δ − − λ − ( t i − )∆ tγ ( t i ) = ( φ δ + − − ¯ δ + ) λ + ( t i − )∆ t + ( φ δ − − − ¯ δ − ) λ − ( t i − )∆ tε ( t i ) = X ( t i ) − E [ X ( t i ) |F ( t i − )]where φ δ + = E [ e δ + ] , and φ δ − = E [ e − δ − ]respectively.Note that, by direct computation, we have X ( t i ) = µ ( t i ) − γ ( t i ) + ε ( t i ) . (4)The mean correction factor γ ( t i ) appears because we model with log-return and this implies that E [exp( ε ( t i )) |F ( t i − )]= E [exp {− E [ X ( t i ) |F ( t i − )] + X ( t i ) } |F ( t i − )]= exp {− E [ X ( t i ) |F ( t i − )] }× E exp N + ( t i ) (cid:88) j = N + ( t i − )+1 δ + ,j (cid:12)(cid:12)(cid:12)(cid:12) F ( t i − ) × E exp N − ( t i ) (cid:88) j = N − ( t i − )+1 δ − ,j (cid:12)(cid:12)(cid:12)(cid:12) F ( t i − ) E [exp( ε ( t i )) |F ( t i − )]= exp { ( φ δ + − − ¯ δ + ) λ + ( t i − )∆ t + ( φ δ − − − ¯ δ − ) λ − ( t i − )∆ t } = exp( γ ( t i )) . The one step ahead expectation of future stock price can be represented asexponential of drift term µ ( t i ) multiplied by current stock price, that is E [ S ( t i ) |F ( t i − )]= S ( t i − ) exp( µ ( t i ) − γ ( t i )) E [exp( ε ( t i )) |F ( t i − )]= S ( t i − ) exp( µ ( t i )) . The derivation of equivalent martingale measure for extended version ofGRACH intensity model is similar to the previous version. Definition 4.4. Let f ± be probability density functions of δ ± , respectivelyand (cid:101) f ± are some probability density functions (which are desired probabilitydensity functions of δ ± under equivalent martingale measure).(i) Let (cid:101) φ δ + = E Q [ e δ + ] = (cid:90) ∞−∞ e x (cid:101) f + ( x )d x, (cid:101) φ δ − = E Q [ e − δ − ] = (cid:90) ∞−∞ e − x (cid:101) f − ( x )d x and (cid:101) λ + and (cid:101) λ − be two r.c.l.l. adapted step processes satisfying the equation( (cid:101) φ δ + − (cid:101) λ + ( t i ) + ( (cid:101) φ δ − − (cid:101) λ − ( t i ) = r ∆ t (5)for each i , and (cid:101) λ ± ( t ) = (cid:101) λ ± ( t i − )for t i − ≤ t < t i .(ii) Suppose that (cid:101) λ + ( t ) and (cid:101) λ − ( t ) are positive processes. Let κ i − = (cid:101) λ + ( t i − ) − λ + ( t i − ) + (cid:101) λ − ( t i − ) − λ − ( t i − ) ,Q i − ( t ) = N + ( t ) (cid:88) j = N + ( t i − )+1 log λ + ( t i − ) f ( δ + ,j ) (cid:101) λ + ( t i − ) (cid:101) f ( δ + ,j ) + N − ( t ) (cid:88) j = N − ( t i − )+1 log λ − ( t i − ) f ( δ − ,j ) (cid:101) λ − ( t i − ) (cid:101) f ( δ − ,j ) , ≤ i ≤ N . Define Z (0) = 1and Z ( t ) = Z ( t i − ) exp( − κ i − ( t − t i − ) − Q i − ( t ))for t i − < t ≤ t i , recursively. Remark 4.5. Note that F ( t i − )-measurable random variable κ i − is zerowhen the conditional variance of return distribution of the martingale mea-sure is equal to the variance under physical measure. Q i ( t ) depends on F ( t )random variables N ± and δ . Lemma 4.6. For ≤ t ≤ T , E [ Z ( t )] = 1 . Proof. For t i − < t ≤ t i , define Z + ,i ( t ) = exp ( λ + ( t i − ) − (cid:101) λ + ( t i − ))( t − t i − ) + N + ( t ) (cid:88) j = N + ( t i − )+1 log (cid:101) λ + ( t i − ) (cid:101) f ( δ + ,j ) λ + ( t i − ) f ( δ + ,j ) Z − ,i ( t ) = exp ( λ − ( t i − ) − (cid:101) λ − ( t i − ))( t − t i − ) + N − ( t ) (cid:88) j = N − ( t i − )+1 log (cid:101) λ − ( t i − ) (cid:101) f ( δ − ,j ) λ − ( t i − ) f ( δ − ,j ) . Then Z + ,i ( t ) and Z − ,i ( t ) are F ( t )-measurable and satisfy E [ Z + ,i ( t ) |F ( t i − )] = 1and E [ Z − ,i ( t ) | F ( t i − )] = 1 . Note that Z + ,i ( t ) Z − ,i ( t ) = Z ( t ) Z ( t i − ) . Since N + ( t ) − N + ( t i − ) and N − ( t ) − N − ( t i − ) are conditionally independentupon F ( t i − ), we have E (cid:20) Z ( t ) Z ( t i − ) (cid:12)(cid:12)(cid:12)(cid:12) F ( t i − ) (cid:21) = E [ Z + ,i ( t ) |F ( t i − ] E [ Z − ,i ( t ) |F i − ] = 1 . < t ≤ T , we have E [ Z ( t )] = E (cid:20) Z ( t ) Z ( t i − ) Z ( t i − ) Z ( t i − ) . . . Z ( t ) Z ( t ) Z ( t ) Z (0) (cid:21) = E (cid:20) E (cid:20) Z ( t ) Z ( t i − ) Z ( t i − ) Z ( t i − ) . . . Z ( t ) Z ( t ) Z ( t ) Z (0) (cid:12)(cid:12)(cid:12)(cid:12) F ( t i − ) (cid:21)(cid:21) = E (cid:20) Z ( t i − ) Z ( t t − ) . . . Z ( t ) Z ( t ) Z ( t ) Z (0) (cid:21) = ...= E (cid:20) Z ( t ) Z (0) (cid:21) = 1 . Since E [ Z ( T )] = 1, we use Z ( T ) in Definition 4.4 to construct a newprobability measure Q . We define Q ( A ) = (cid:90) A Z ( T )d P for A ∈ F . (6) Lemma 4.7. Under the measure Q in (6) , for every t i − < t ≤ t i , theconditional distributions ( N + ( t ) − N + ( t i − )) |F ( t i − ) and ( N − ( t ) − N − ( t i − )) |F ( t i − ) are Poisson distributions with new intensities (cid:101) λ + ( t i − ) and (cid:101) λ − ( t i − ) , respec-tively. Moreover, δ + ,j have probability density function (cid:101) f under Q .Proof. Define Z ± ,i ( t ) as in the proof of Lemma 4.6. For a constant u , wehave E Q [exp { u ( N + ( t ) − N + ( t i − )) }|F ( t i − )]= E P (cid:20) exp { u ( N + ( t ) − N + ( t i − )) } Z ( t ) Z ( t i − ) (cid:12)(cid:12)(cid:12)(cid:12) F ( t i − ) (cid:21) = E P [exp { u ( N + ( t ) − N + ( t i − )) } Z + ,i ( t ) Z − ,i ( t ) |F ( t i − )]= E P [exp { u ( N + ( t ) − N + ( t i − )) } Z + ,i ( t ) |F ( t i − )] E P [ Z − ,i ( t ) |F ( t i − )]= E P [exp { u ( N + ( t ) − N + ( t i − )) } Z + ,i ( t ) |F ( t i − )]15nd E P [exp { u ( N + ( t ) − N + ( t i − )) } Z + ,i ( t ) |F ( t i − )]= E P (cid:20) exp { u ( N + ( t ) − N + ( t i − )) + ( λ + ( t i − ) − (cid:101) λ + ( t i − ))( t − t i − ) }× exp N + ( t ) (cid:88) j = N + ( t i − )+1 log (cid:101) λ + ( t i − ) (cid:101) f ( δ + ,j ) λ + ( t i − ) f ( δ + ,j ) (cid:12)(cid:12)(cid:12)(cid:12) F ( t i − ) (cid:21) = exp { ( λ + ( t i − ) − (cid:101) λ + ( t i − ))( t − t i − ) }× E P exp N + ( t ) (cid:88) j = N + ( t i − )+1 (cid:32) u + log (cid:101) λ + ( t i − ) (cid:101) f ( δ + ,j ) λ + ( t i − ) f ( δ + ,j ) (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) F ( t i − ) . Put Y + ,j = u + log (cid:101) λ + ( t i − ) (cid:101) f ( δ + ,j ) λ + ( t i − ) f ( δ + ,j )and let Φ be the conditional moment generating function of Y + ,j , i.e.Φ( z ) = E P [exp( zY + ,j ) |F ( t i − )] . Note that Φ(1) = (cid:101) λ + ( t i − ) λ + ( t i − ) e u E P (cid:34) (cid:101) f ( δ + ,j ) f ( δ + ,j ) (cid:12)(cid:12)(cid:12)(cid:12) F ( t i − ) (cid:35) = (cid:101) λ + ( t i − ) λ + ( t i − ) e u (cid:90) ∞−∞ (cid:101) f ( x ) f ( x ) f ( x )d x = (cid:101) λ + ( t i − ) λ + ( t i − ) e u . Then E P exp N + ( t ) (cid:88) j = N + ( t i − )+1 Y + ,j (cid:12)(cid:12)(cid:12)(cid:12) F ( t i − ) = exp { λ + ( t i − ) (Φ(1) − 1) ( t − t i − ) } . Note that E P [exp { u ( N + ( t ) − N + ( t i − )) } Z + ,i ( t ) |F ( t i − )]= exp (cid:40)(cid:32) λ + ( t i − ) − (cid:101) λ + ( t i − ) + λ + ( t i − ) (cid:32) (cid:101) λ + ( t i − ) λ + ( t i − ) e u − (cid:33)(cid:33) ( t − t i − ) (cid:41) = exp { (cid:101) λ + ( t i − )( e u − t − t i − ) } , (cid:101) λ + ( t i − ). Hence N + ( t ) − N + ( t i − ) |F ( t i − ) is a Poissson distributionwith intensity (cid:101) λ + ( t i − ). For N − ( t ) − N − ( t i − ), 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 + , under Q . Withoutloss of generality, assume that first jump, i.e. the occurrence time of randomvariable δ + , is less than t . Thus, for a constant u , we have E Q [ e uδ + , ]= E P [ e uδ + , Z ( T )]= E P [ e uδ + , Z + , ( t ) Z − , ( t ) Z ( T ) /Z ( t )]= E P (cid:20) exp (cid:26) ( λ + ( t ) − (cid:101) λ − ( t ))∆ t + N + ( t ) log λ + ( t ) λ − ( t ) (cid:27) × e uδ + , (cid:101) f ( δ + , ) f ( δ + , ) N + ( t ) (cid:89) j =2 (cid:101) f ( δ + ,j ) f ( δ + ,j ) (cid:21) . The last equality holds since given random variables independent and since E [ Z − , ( t )] = E (cid:20) Z ( T ) Z ( t ) (cid:21) = 1 . Furthermore, because of independency and the fact that E P (cid:20) exp (cid:26) ( λ + ( t ) − (cid:101) λ − ( t ))∆ t + N + ( t ) log λ + ( t ) λ − ( t ) (cid:27)(cid:21) = 1(by Lemma 4.6 when δ is constant) and E P (cid:34) (cid:101) f ( δ + ,j ) f ( δ + ,j ) (cid:35) = 1for each j . Finally we have E Q [ e uδ + , ] = E P (cid:34) e uδ + , (cid:101) f ( δ + , ) f ( δ + , ) (cid:35) = (cid:90) ∞−∞ e ux (cid:101) f ( x ) f ( x ) f ( x )d x = (cid:90) ∞−∞ e ux (cid:101) f ( x )d x, which implies δ + , has a probability density function (cid:101) f under Q . For general j , the proof is the same. 17he choice for risk-neutral distributions (cid:101) f + and (cid:101) f − are related to theskewness in conditional distribution of log-return under risk-neutral mea-sure. This is similar to the fact that distributions f + and f − are relatedto the skewness under physical measure. Now we show that under theequivalent martingale measure Q , the discounted stock price process is amartingale. Theorem 4.8. Under the measure Q defined by (6) , we have E Q [ e − r ( t − u ) S ( t ) |F ( u )] = S ( u ) for < u < t < T .Proof. Take u, t such that t i − ≤ u < t ≤ t i . E Q [ S ( u ) |F ( t )]= E Q S ( t ) exp N + ( t ) (cid:88) j = N + ( u )+1 δ + ,j − N − ( t ) (cid:88) j = N − ( u )+1 δ − ,j (cid:12)(cid:12)(cid:12)(cid:12) F ( t ) = E Q S ( t ) exp N + ( t ) (cid:88) j = N + ( u )+1 δ + ,j − N − ( t ) (cid:88) j = N − ( u )+1 δ − ,j Z ( t ) Z ( u ) (cid:12)(cid:12)(cid:12)(cid:12) F ( t ) = S ( t ) exp { ( λ + ( t i − ) − (cid:101) λ + ( t i − ) + λ − ( t i − ) − λ − ( t i − ))( t − u ) }× E P exp N + ( t ) (cid:88) j = N + ( u )+1 (cid:32) δ + ,j + log (cid:101) λ + ( t i − ) (cid:101) f ( δ + ,j ) λ + ( t i − ) f ( δ + ,j ) (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) F ( t ) × E P exp − N − ( t ) (cid:88) j = N − ( u )+1 (cid:32) δ − ,j + log (cid:101) λ − ( t i − ) (cid:101) f ( δ − ,j ) λ − ( t i − ) f ( δ − ,j ) (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) F ( t ) . Let φ W + be the conditional moment generating function of W + ,j = δ + ,j + log (cid:101) λ + ( t i − ) (cid:101) f ( δ + ,j ) λ + ( t i − ) f ( δ + ,j )given filtration F ( t ). Then φ W + (1) = E P [ e W + ,j |F ( t )] = (cid:101) λ + ( t i − ) λ + ( t i − ) E P (cid:34) e δ + ,j (cid:101) f ( δ + ,j ) f ( δ + ,j ) (cid:12)(cid:12)(cid:12)(cid:12) F ( t ) (cid:35) = (cid:101) λ + ( t i − ) λ + ( t i − ) (cid:90) ∞−∞ e x (cid:101) f ( x ) f ( x ) f ( x )d x = (cid:101) λ + ( t i − ) λ + ( t i − ) (cid:101) φ δ + (cid:101) φ δ + is defined in Definition 4.4. Hence E P exp N + ( t ) (cid:88) j = N + ( u )+1 (cid:32) δ + ,j + log (cid:101) λ + ( t i − ) (cid:101) f ( δ + ,j ) λ + ( t i − ) f ( δ + ,j ) (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) F ( u ) = exp { λ + ( t i − )( t − u )( φ W + (1) − } = exp { ( (cid:101) λ + ( t i − ) (cid:101) φ δ + − λ ( t i − ))( t − u ) } and E Q [ S ( u ) |F ( t )]= S ( t ) exp (cid:110) ( λ + ( t i − ) − (cid:101) λ + ( t i − ) + λ + ( t i − ) − (cid:101) λ + ( t i − ))( t − u ) (cid:111) × exp { ( (cid:101) λ + ( t i − ) (cid:101) φ δ + − λ + ( t i − ) + (cid:101) λ − ( t i − ) (cid:101) φ δ + − λ − ( t i − ))( t − u ) } = S ( t ) exp { ( (cid:101) λ + ( t ) (cid:101) φ δ + − (cid:101) λ + ( t ) + (cid:101) λ − ( t ) (cid:101) φ δ + − (cid:101) λ − ( t ))( t − u ) } = S ( t ) e r ( t − u ) . The last equality is due to Definition 4.4(i). By applying the tower property,we obtain the desired result for arbitrary u and t . The risk-neutral option pricing framework for the GARCH intensity modelwas introduced. Equivalent martingale measures are provided and hence thethe risk-neutral option price is computed under the measure. The frameworkis consistent with the empirical characteristics such as volatility smile andspread. The theory is easily extended to the generalized version of theGARCH intensity model. References [1] Babsiri, M. E., and Jean-Michel Zakoian. :257 – 294.[2] Bakshi, G., and Dilip Madan. :1945–1956.[3] Black, F., and Myron S. Scholes. :637–654. 194] Bollerslev, T. :307–327.[5] Choe, G. H., and Kyungsub Lee. :197–224.[6] Duan, J.-C. :13–32.[7] Elliott, R. J., and P. Ekkehard Kopp. :157 – 167.[8] Glosten, L. R., Ravi Jagannathan, and David E. Runkle. :1779–1801.[9] Harrison, J. M., and David M. Kreps. :381–408.[10] Harrison, J. M., and Stanley R. Pliska. :215 – 260.[11] Merton, R. C. Nelson, D. B. :347–370.[13] Pagan, A. R., and G. William Schwert. :267–290.[14] Zakoian, J.-M.18