Chapter 43
JUMP DIFFUSION MODEL
SHIU-HUEI WANG, University of Southern California, USA
Abstract the jumps occur, and hence not adequate. In add- ition, Ahn and Thompson (1986) also examined Jump diffusion processes have been used in modern the effect of regulatory risks on the valuation of finance to capture discontinuous behavior in asset public utilities and found that those ‘‘jump risks’’ pricing. Various jump diffusion models are consid- were priced even though they were uncorrelated ered in this chapter. Also, the applications of jump with market factors. It shows that jump risks can- diffusion processes on stocks, bonds, and interest not be ignored in the pricing of assets. Thus, a rate are discussed. ‘‘jump’’ stochastic process defined in continuous time, and also called as ‘‘jump diffusion model’’ Keywords: Black–Scholes model; jump diffusion was rapidly developed. process; mixed-jump process; Bernoulli jump pro- The jump diffusion process is based on Poisson cess; Gauss–Hermite jump process; conditional process, which can be used for modeling systematic = jump dynamics; ARCH GARCH jump diffusion jumps caused by surprise effect. Suppose we observe model; affine jump diffusion model; autoregressive a stochastic process St, which satisfies the following jump process model; jump diffusion with condi- stochastic differential equation with jump: tional heteroskedasticity. dSt ¼ at dt þ st dWt þ dJt, t 0, (43:1) 43.1. Introduction where dWt is a standard Wiener process. The term In contrast to basic insights into continuous-time dJt represents possible unanticipated jumps, and asset-pricing models that have been driven by sto- which is a Poisson process. As defined in Gourier- þ chastic diffusion processes with continuous sample oux and Jasiak (2001), a jump process (Jt,t 2 R ) paths, jump diffusion processes have been used in is an increasing process such that finance to capture discontinuous behavior in asset pricing. As described in Merton (1976), the validity (i). J0 ¼ 0, of Black–Scholes formula depends on whether the (ii). PJ½ ¼tþdt Jt ¼ 1jJt lt dt þ o(dt), stock price dynamics can be described by a con- (iii). PJ½ tþdt Jt ¼ 0jJt ¼ 1 lt dt þ o(dt), tinuous-time diffusion process whose sample path is continuous with probability 1. Thus, if the stock where o(dt) tends to 0 when t tends to 0, and lt, price dynamics cannot be represented by stochastic called the intensity, is a function of the information process with a continuous sample path, the Black– available at time t. Furthermore, since the term dJt Scholes solution is not valid. In other words, as the is part of the unpredictable innovation terms we price processes feature big jumps, i.e. not continu- make E[DJt] ¼ 0, which has zero mean during a ous, continuous-time models cannot explain why finite interval h. Besides, as any predictable part of JUMP DIFFUSION MODEL 677 the jumps may be can be included in the drift 1. Normal vibrations caused by marginal infor- component at, jump times tj, j ¼ 1, 2, . . . . vary mation events satisfying a local Markov pro- by some discrete and random amount. Without perty and modeled by a standard geometric loss of generality, we assume that there are k pos- Brownian motion with a constant variance sible types of jumps, with size ai, i ¼ 1, 2, L, and per unit time. It has a continuous sample path. the jumps occur at rate lt that may depend on the 2. Abnormal vibrations caused by information latest observed St. As soon as a jump occurs, the shocks satisfying an antipathetical jump pro- jump type is selected randomly and independently. cess defined in continuous time, and modeled The probability of a jump of size ai, occuring is by a jump process, reflecting the nonmarginal given by pi. Particularly, for the case of the stand- impact of the information. ard Poisson process, all jumps have size 1. In short, Thus, there have been a variety of studies that the path of a jump process is an increasing stepwise explain too many outliers for a simple, constant- function with jumps equal to 1 at random rate variance log-normal distribution of stock price ser- D1, D2,..., Dt,...,. ies. Among them, Merton (1976) and Tucker and Related research on the earlier development of a Pond (1988) provide a more thorough discussion basic Poisson jump model in finance was by Press of mixed-jump processes. Mixed-jump processes (1967). His model can be motivated as the aggre- are formed by combining a continuous diffusion gation of a number of price changes within a fixed- process and a discrete-jump process and may cap- time interval. In his paper, the Poisson distribution ture local and nonlocal asset price dynamics. governs the number of events that result in price Merton (1976) pioneered the use of jump pro- movement, and the average number of events in a cesses in continuous-time finance. He derived an time interval is called intensity. In addition, he option pricing formula as the underlying stock assumes that all volatility dynamics is the result returns are generated by a mixture of both con- of discrete jumps in stock returns and the size of a jump is stochastic and normally distributed. tinuous and the jump processes. He posited stock Consequently, some empirical applications found returns as that a normal Poisson jump model provides a dS ¼ (a lk)dt þ sdZ þ d q (43:1) good statistical characterization of daily exchange S rate and stock returns. For instance, using Stand- where S is the stock price, a the instantaneous ard &Poor’s 500 futures options and assuming expected return on the stock, s2 the instantaneous an underlying jump diffusion, Bates (1991) variance of the stock return conditional on no found systematic behavior in expected jumps be- arrivals of ‘‘abnormal’’ information, dZ the stand- fore the 1987 stock market crash. In practice, by ardized Wiener process, q the Poisson process as- observing different paths of asset prices with re- sumed independent of dZ, l the intensity of the spect to different assets, distinct jump diffusion Poisson process, k ¼ «(Y 1), where Y˜ 1 is the models were introduced into literature by many random variable percentage change in stock price researchers. Therefore, in this chapter, we will if the Poisson event occurs; « is the expectation survey various jump diffusion models in current operator over the random variable Y. Actually, literature as well as estimation procedures for Equation (43.1) can be rewritten as these processes. dS ¼ (a lk)dt þ sdZ S 43.2. Mixed-Jump Processes if the Poisson does not occur (43:2) ¼ (a lk)dt þ sdZ þ (Y 1), The total change in asset prices may be comprised of two types of changes: if the Poisson occurs 678 ENCYCLOPEDIA OF FINANCE
Therefore, the option return dynamics can be re- to value options written on securities involving written as jump processes. Also, Tucker and Pond (1988) empirically inves- dW ¼ ðÞaw lkw dt þ swdZ þ dqw (43:3) tigated four candidate processes (the scaled-t distri- W bution, the general stable distribution, compound
Most likely, aw is the instantaneous expected re- normal distribution, and the mixed-jump model) 2 for characterizing daily exchange rate changes for turn on the option, sv is the instantaneous vari- ance of the stock return conditional on no arrivals six major trading currencies from the period 1980 to of ‘‘abnormal’’ information, qw is an Poisson pro- 1984. They found that the mixed-jump model cess with parameter l assumed independent of dZ, exhibited the best distributional fit for all six cur- kw ¼ «(Yw 1), (Yw 1) is the random variable rencies tested. Akgiray and Booth (1988) also found percentage change in option price if the Poisson that the mixed-diffusion jump process was superior event occurs, « is the expectation operator over the to the stable laws or mixture of normals as a model random variable Yw. The Poisson event for the of exchange rate changes for the British pound, option price occurs if and only if the Poisson French franc, and the West German mark relative event for the stock price occurs. Further, define to the U.S. dollar. Thus, both theoretical and em- the random variable, Xn, which has the same dis- pirical studies of exchange rate theories under un- tribution as the product of n independently and certainty should explicitly allow for the presence of identically distributed random variables. Each of discontinuities in exchange rate processes. In add- n independently and identically distributed ran- ition, the assumption of pure diffusion processes for dom variables has the identical distribution as the exchange rates could lead to misleading inferences random variable Y described in Equation (43.1). due to its crude approximation. As a consequence, by the original Black–Scholes option pricing formula for the no-jump case, 43.3. Bernoulli Jump Process W(S, t; E, r, s2), we can get the option price with jump component In the implementation of empirical works, Ball and Torous (1983) provide statistical evidence with X1 e lt(lt)n ÂÃ FsðÞ, t ¼ « {W(SX e lkt,t; E, s2, r)} the existence of log-normally distributed jumps in n! n n n¼1 a majority of the daily returns of a sample of (43:4) NYSE-listed common stocks. The expression of their Poisson jump diffusion model is as Equation Generally speaking, W satisfies the boundary con- (43.1), and jump size Y has posited distribution, ln ditions of partial differential equation (see Oksen- Y N(m, d2). dal, 2000), and can be rewritten as a twice Ball and Torous (1983) introduced the Bernoulli continuously differentiable function of the stock jump process as an appropriate model for stock price and time, W(t) ¼ F(S, t). Nevertheless, price jumps. Denote X as the number of events Equation (43.4) still not only holds most of the i that occur in subinterval i and independent distri- attractive features of the original Black–Scholes buted random variables. By stationary independent formula such as being regardless of the investor increment assumption, preferences or knowledge of the expected return on the underlying stock, but also satisfies the Xn Sharpe–Linter Capital Asset Pricing model as N ¼ Xi, long as the jump component of a security’s return i¼1 is uncorrelated with the market. In other words, where N is the number of events that occur in a time the mixed-jump model of Merton uses the CAPM interval of length t. Besides, define h ¼ t=n for any JUMP DIFFUSION MODEL 679 arbitrary integer n and divide (0, t) into n equal jump models by employing Gauss–Hermite quad- subintervals each of length h. Thus, Xi satisfies rature. Note that t ¼ 0 and t ¼ T are the current time Pr[X 0] 1 lh O(h) i ¼ ¼ þ and expiration date of the option, respectively, and Pr[Xi ¼ 1] ¼ lh þ O(h) for i ¼ 1, 2, ..., n Dt ¼ T=N. Consider a compound option that can only be exercised at the N interval boundaries Pr[Xi > 1] ¼ O(h) tk ¼ T kDt, k ¼ 0, ..., N. Let Ck(S) be the value For large n, Xi has approximately the Bernoulli of the compound option at time tk, the current value distribution with parameter lh ¼ lt=n. As a result, of the compound option is then CN (S) ; the value of N has the binomial distribution, approximately, an actual contingent claim with optimal exercise i.e. possible at any time is lim N ! CN (S). The com- pound option can be recursively valued by n lt k lt n k o Pr[N ¼ k] ffi 1 k ¼ 0, 1, 2, ..., n: È Â Ã r k n n Ckþ1(S) ¼ max EVk þ 1,e DtE Ck(Sk;S) , Now, assume that t is very small, they can approxi- where EVkþ1 is the immediate exercise value at mate N by the Bernoulli variate X defined by time tkþ1. Since S(t) is an unrestricted log-normal P[X ¼ 0] ¼ 1 lt, diffusion process from tk to tkþ1, ð 1 pffiffiffiffiffiffi P[X ¼ 1] ¼ lt: m0Dt þ zs (Dt) E[Ck] ¼ f(z)Ck Se dz, (43:5) The advantage of the Bernoulli jump process is that 1 more satisfactory empirical analyses are available. where z is an independent sample from a normal The maximum likelihood estimation can be prac- distribution with mean zero and variance one, f(z) 2 tically implemented and the unbiased, consistent, is its density function, and m0 ¼ r s =2 for a risk- and efficient estimators that attain the Cramer–Rao neutral valuation. A jump process approximation lower bound for the corresponding parameters. to the above with n jumps takes the form Moreover, the statistically most powerful test of Xn u the null hypothesis l ¼ 0 can be implemented. Ob- E[Ck] ffi pjCk(Se j), pj 0 for j ¼ 1, ..., n, viously, a Bernoulli jump process models informa- j¼1 tion arrivals and stock price jumps. This shows that Xn pffiffiffiffiffiffiffiffi 0 the presence of a jump component in common stock pj ¼ 1, uj ¼ m Dt þ zjs (Dt) returns can be possessed well. As a consequence, j¼1 Vlaar and Palm (1993) combined the GARCH (1,1) So, Omberg (1988) considers to use Gaussian in- and Bernoulli jump distribution to account for tegration to approximate an intergral of the skewness and leptokurtosis for weekly rates of the form as in Equation (43.5). For example, for the European Monetary System (EMS). Das (2002) intergral, ð considered the concept of Bernoulli approximation b to test the impact of Federal Reserve actions by I ¼ w(x)f (x)dx, Federal Funds’ rate as well. (See Section 43.9.2 a and Section 43.5, respectively.) we can approximate this equation by a weighted average of the function f(x)atn points 43.4. Gauss–Hermite Jump Process {x1, ..., xn}. Let { wi}and { xj}are selected to maximize the degree of precision m , which is a To ensure the efficiency properties in valuing com- integration rule, i.e. if the integration error is zero pound option, Omberg (1988) derived a family of for all polynomials f(x) of order m or less. {Pj(x)} 680 ENCYCLOPEDIA OF FINANCE is the set of polynomials with respect to the weight- investigated the effect of jump components of the ing function w(x), underlying processes on the term structure of inter- ð b est rates. They differ from the model of Cox et al. w(x)Pi(x)Pj(x)dx ¼ 0, for i 6¼ j, (1985) when they consider the state variables as a jump diffusion processes. Therefore, they sug- ð b gested that jump risks may have important impli- 2 w(x)Pj (x)dx ¼ gj 6¼ 0, for i ¼ j, cations for interest rate, and cannot be ignored for a the pricing of assets. In other words, they found Thus, the optimal evaluation points {Xj}are the n that Merton’s multi-beta CAPM does not hold in zeros of Pn(x) and the corresponding weights {wj} general due to the existence of jump component of are the underlying processes on the term structure of = interest rate. Also, Breeden’s single consumption (anþ1, nþ1 an, n)gn wj ¼ > 0: beta does not hold, because the discontinuous P 0 (x )P (x ) n j nþ1 j movements of the investment opportunities cannot The degree of precision is m ¼ 2n 1. If the be fully captured by a single consumption beta. weighting function w(x) is symmetric with regard Moreover, in contrast with the work of Cox et al. to the midpoint of the interval [a, b], then { xj}and (1981) providing that the traditional expectations {wj} are the Gaussian evaluation points { xj} theory is not consistent with the equilibrium and weights {wj}, respectively. Particularly, the models, they found that traditional expectations above procedure is called Gauss–Hermite quadra- theory is not consistent with the equilibrium ture to approximate the integration problem. What models as the term structure of interest rate is is shown in Omberg (1988) is the application of under the jump diffusion process, since the term Gauss–Hermite quadrature to the valuation of a premium is affected by the jump risk premiums. compound option, which is a natural way to Das (2002) tested the impact of Federal Reserve generate jump processes of any order n that are actions by examining the role of jump-enhanced efficient in option valuation. Thus, the Gauss–Her- stochastic processes in modeling the Federal Funds mite jump process arises as an efficient solution to rate. This research illustrated that compared to the the problem of replicating a contingent claim stochastic processes of equities and foreign ex- over a finite period of time with a portfolio of change rates, the analytics for interest rates are assets. With this result, he suggested the exten- more complicated. One source of analytical com- sion of these methods to option valuation prob- plexity considered in modeling interest rates with lems with multiple state variables, such as the jumps is mean reversion. Allowing for mean rever- valuation of bond options in which the state sion included in jump diffusion processes, the pro- variables are taken to be interest rates at various cess for interest rates employed in that paper is as terms. follows dr ¼ k(u r)dt þ ydz þ Jdp(h), (43:6) 43.5. Jumps in Interest Rates which shows interest rate has mean-reversing drift Cox et al. (1985a) proposed an influential paper and two random terms, a pure diffusion process that derived a general equilibrium asset pricing and a Poisson process with a random jump J. In model under the assumption of diffusion pro- addition, the variance of the diffusion is y2, and a cesses, and analyzed the term structure of interest Poisson process p represents the arrival of jumps rate by it. Ahn and Thompson (1988) applied Cox, with arrival frequency parameter h, which is de- Ingersoll, and Ross’s methodology to their model, fined as the number of jumps per year. Moreover, which is driven by jump diffusion processes, and denote J as jump size, which can be a constant or JUMP DIFFUSION MODEL 681 with a probability distribution. The diffusion and lower bound. Thus, they obtain the evidence that Poisson processes are independent of each other as jumps are an essential component of interest rate well as independent of J. models. Especially, the addition of a jump process The estimation method used here is the Ber- diminishes the extent of nonlinearity although some noulli approximation proposed in Ball and Torous research finds that the drift term in the stochastic (1983). Assuming that there exists no jump or only process for interest rates appears to be nonlinear. one jump in each time interval, approximate the Johannes (2003) suggested the estimated infini- likelihood function for the Poisson–Gauss model tesimal conditional moments to examine the statis- using a Bernoulli mixture of the normal distribu- tical and economic role of jumps in continuous-time tions governing the diffusion and jump shocks. interest rate models. Based on Johannes’s ap- In discrete time, Equation (43.6) can be ex- proach, Bandi and Nguyen (2003) provided a gen- pressed as follows: eral asymptotic theory for the full function estimates
2 of the infinitesimal moments of continuous-time Dr ¼ k(u r)Dt þ yDz þ J(m, g )Dp(h), models with discontinuous sample paths of the where y2 is the annualized variance of the Gaussian jump diffusion type. Their framework justifies con- shock, and Dz is a standard normal shock term. sistent nonparametric extraction of the parameters J(m, g2) is the jump shock with normal distribu- and functions that drive the dynamic evolution of tion. Dp(q) is the discrete-time Poisson increment, the process of interest. (i.e. the potentially nonaffine approximated by a Bernoulli distribution with par- and level dependent intensity of the jump arrival ameter q ¼ hDt þ O(Dt), allowing the jump inten- being an example). Particularly, Singleton (2001) sity q to depend on various state variables provided characteristic function approaches to conditionally. The transition probabilities for inter- deal with the Affine jump diffusion models of inter- est rates following a Poisson–Gaussian process are est rate. In the next section, we will introduce affine written as (for s > t): jump diffusion model. ! Â Ã (r(s) r(t) k(u r(t))Dt m)2 f r(s)jr(t) ¼ q exp 2 2 43.6. Affine Jump Diffusion model 2(yt Dt þ g )
pffiffi 1 For development in dynamic asset pricing models, (2p(y2Dt þ g2)) a particular assumption is that the state vector X t ! 2 follows an affine jump diffusion (AJD). An affine (r(s) r(t) k(u r(t))Dt) þ (1 q) exp 2 jump model is a jump diffusion process. In general, 2yt Dt as defined in Duffie and Kan (1996), we suppose 1 pffiffi , the diffusion for a Markov process Xis ‘affine’ if 2 (pyt Dt) m(y) ¼ u þ ky where q ¼ hDt þ O(Dt). This is an approximation XN for the true Poisson–Gaussian density with a mix- 0 ( j) s(y)s(y) ¼ h þ yjH , ture of normal distributions. As in Ball and Torous j¼1 (1983), by maximum-likelihood estimation, which : n : n n maximizes the following function L, where m D ! R and s D ! R , u is N 1, k is N N, h and H(j) are all N N and symmetric. YT  à The X’s may represent observed asset returns or L ¼ f r(t þ Dt)jr(t) , prices or unobserved state variables in a dynamic t¼1 pricing model, such as affine term structure we can obtain estimates that are consistent, un- models. Thus, extending the concept of ‘affine’ biased, and efficient and attain the Cramer-Rao to the case of affine jump diffusions, we can note 682 ENCYCLOPEDIA OF FINANCE that the properties for affine jump diffusions Thus, the ‘‘jump transform’’ u determines the are that the drift vector, ‘‘instantaneous’’ covar- jump size distribution. In other words, the ‘‘coeffi- iance matrix, and jump intensities all have affine cients’’ (K, H, l, u)ofX completely determine its dependence on the state vector. Vasicek (1977) and distribution. Their method suggests a real advan- Cox et al. (1985) proposed the Gaussian and tage of choosing a jump distribution v with an square root diffusion models which are among explicitly known or easily computed jump trans- the AJD models in term structure literature. Sup- form u. They also applied their transform analysis pose that X is a Markov process in some state to the pricing of options. See Duffle et al. (2000). space D Rn, the affine jump diffusion is Furthermore, Singleton (2001) developed several estimation strategies for affine asset pricing models dX ¼ m(X )dt þ s(X )dW þ dZt, t t t t based on the known functional form of the condi- where W is an standard Brownian motion tional characteristic function (CCF) of discretely in Rn, m : D ! Rn, s : D ! Rn n, and Z is a sampled observations from an affine jump diffu- pure jump process whose jumps have a fixed prob- sion model, such as LML-CCF (Limited-informa- ability distribution y on and arriving intensity tion estimation), ML-CCF (Maximum likelihood {l(Xt): t 0}, for some l: D ! [0, 1). estimation), and GMM-CCF estimation, etc. As Furthermore, in Duffie et al. (2000), they sup- shown in his paper, a method of moments estima- pose that X is Markov process whose transition tor based on the CCF is shown to approximate the semi-group has an infinitesimal generator of levy efficiency of maximum likelihood for affine diffu- type defined at a bounded C2 function f : D ! R sion models. with bounded first and second derives by  à 43.7. Geometric Jump Diffusion Model 1 T }f (x) ¼ fx(x)m(x) þ tr fxx(x)s(x)s(x) þ l(x): 2 Using Geometric Jump Diffusion with the instant- It means that conditional on the path of X, the aneous conditional variance, Vt, following a mean jump times of Z are the jump times of a reverting square root process, Bates (1996) showed Poisson process with time varying intensity that the exchange rate, S($=deutschemark(DM)) {l(Xs): 0 s t}, and that the size of the jump followed it: of Z at a jump time T is independent of pffiffiffiffi = X : s T dS S ¼ (m lk)dt þ VdZ þ kdq { s 0 }, and has the probability distri- pffiffiffiffi bution y. Consequently, they provide an analytical dV ¼ (a bV)dt þ sv VdZv treatment of a class of transforms, including Cov(dZ,dZv) ¼ pdt Laplace and Fourier transformations in the setting Pr (dq ¼ 1) ¼ ldt of affine jump diffusion state process. 1 The first step to their method is to show that the ln (1 þ k) N ln (1 þ k) d2, d2 , 2 Fourier transform of Xt and of certain related random variables are known in closed form. where m is the instantaneous expected rate of ap- Next, by inverting this transform, they show how preciation of the foreign currency, l is the numbers the distribution of Xt and the prices of options can of jumps in a year, k is the random percentage be recovered. Then, they fix an affine discount jump conditional on a jump occurring, and q is a rate function R: D ! R. Depending on coefficients Poisson counter with intensity l. (K, H, L, r), the affine dependence of m, ssT, l, R The main idea of this model illustrated that are determined, as shown in p.1350 of Duffie et al. skewed distribution can arise by considering non- n (2000). Moreover, for c 2 C , theÐ set of n-tuples of zero average jumps. Similarly, it also discusses that : complex numbers, let u(c) ¼ Rn exp (c z)dv(z). excess kurtosis can arise from a substantial jump JUMP DIFFUSION MODEL 683 component. In addition, this geometric jump where P stands for share price, dW is the increment diffusion model can see a direct relationship be- of a Wiener process with zero mean and unit vari- tween the magnitude of conditional skewness and ance, z is the percent change in share price resulting excess kurtosis and the length of the holding period from a jump, dp is a jump process (when dp ¼ 1, a as well. jump occurs; when dp ¼ 0, no jump occurs) and dp and dW are assumed to be independent. Jump amplitude is independent of dp and dW, but jumps 43.8. Autoregressive Jump Process Model may be serially correlated. s is the elapsed time A theory of the distribution of stock returns was between observed price Ptþs and Pt. The number derived by Bachelier (1900) and expanded using of jumps during the interval s is N, and Z(i) are the the idea of Brownian motion by Osborne (1959). jump size where Z(0) ¼ 1 and Z(i) 0 for However, the empirical works generally concluded i ¼ 1, ..., N. And the solution for Equation that the B-O model fits observed returns rather (43.7) is poorly. For example, a casual examination of P(t þ s) ¼ P(t) Z(0) Z(1) ...Z(N) exp transactions data shows that assumption of a con- pffiffiffiffiffiffi (43:8) 2= stant interval between transactions is not strictly {( a b 2)s þ b (s)W} valid. On the other hand, transactions for a given Divide Equation (43.8) by P(t) and take natural stock occur at random times throughout a day logarithms, then which gives nonuniform time intervals Also, the notion of independence between transaction re- ln [P(t þ s)=P(t)] ¼ (a b2=2)s turns is suspect. Niederhoffer and Osborne (1966) pffiffiffiffiffiffi XN (43:9) showed that the empirical tests of independence þ b (s)W þ ] log Z(i): using returns based on transaction data have gen- i¼1 erally found large and statistically significant nega- According to the Equation (43.9), we can see the tive correlation. Thus, it is reasonable to model third term of Equation (43.9) is the jump process. returns as a process with random time intervals If N ¼ 0 then ln [P(t þ s)=P(s)]is normally distrib- between transaction and serial correlation among uted with mean (a b2=2)s and variance b2s. If returns on individual trades. Accordingly, an auto- the ln Z(i) are assumed to be identically distrib- regressive jump process that models common stock uted with mean m and finite variance s2, a general returns through time was proposed by Oldfield et form of joint density for ln Z(i) can be represented al. (1977). This model consists of a diffusion pro- by: cess, which is continuous with probability 1 and ð 1 jump processes, which are continuous with prob- f (lnZ(1), ,lnZ(N)) ¼ f (lnZ(1), , ability 1. The jump process is assumed to operate 1 such that a jump occurs at each actual transaction, ln Z(N),W)dW, and allows the magnitudes of jumps to be auto- with : correlated. In addition, the model relies on the distribution of random time intervals between E[ln Z(i)] ¼ m, for i ¼ 1, , N, transactions. They suppose the dollar return of a Var[ln Z(i)] ¼ s2, for i ¼ 1, , N, common stock over a holding period of length s is 2 : the result of a process, which is a mixture process Cov[ln Z(i), ln Z(i j)] ¼ rjs , for j 0 composed of a continuous and jump process, where rj is the correlation between lnZ(i) and dP ln Z(i j). The index i represents the jump number ¼ adt þ bdW þ zdp, (43:7) P while the index j denotes the number of lags 684 ENCYCLOPEDIA OF FINANCE between jumps. The startling feature of this general Define the information set at time t to be the joint density is the autocorrelation among jumps. history of returns, Ft ¼ {Rt, , R1}The condi- Hence, some major conclusions are drawn from tional jump size Yt,k, given Ft 1, is presumed to be the data analysis: (1). independent and normally distributed with mean 2 A geometric Brownian motion process or a sub- ut and variance d . Denote nt as the discrete count- ordinated process does not alone describe the sam- ing process governing the number of jumps that ple data very well. (2). Stock returns seem to follow arrive between t 1 and t, which is distributed as a an autoregressive jump process based on the sam- Poisson random variable with the parameter ple means and variances of transaction returns. (3). lt > 0 and density
In contrast to the previous empirical work which is j not sufficiently detailed to determine the probabil- exp ( lt)lt : P(nt ¼ jjFt 1) ¼ ! , j ¼ 0, 1, 2, ity law for transaction returns, the probability j density for the time intervals between jumps is (43:11) gamma. The mean and variance for the Poisson random variable are both lt, which is often called the 43.9. Jump Diffusion Models with Conditional jump intensity. The jump intensity is allowed Heteroscedasticity time-varying. ht is measurable with respect to the information set F and follows a GARCH(p,q) 43.9.1. Conditional Jump Dynamics t 1 process,
The basic jump model has been extended in a Xq Xp 2 number of directions. A tractable alternative is to ht ¼ w þ ai«t i þ biht i, combine jumps with an ARCH=GARCH model in i¼1 i¼1 P discrete time. It seems likely that the jump prob- p where «t ¼ Rt m i¼1 fiRt i. «t contains the ability will change over time. Ho et al. (1996) expected jump component and it affects future formulate a continuous-time asset pricing model volatility through the GARCH variance factor. based on the work of Chamnerlain (1988), but Moreover, based on a parsimonious ARMA struc- include jumps. Their work strongly suggested that ture, let lt be endogenous. Denote the following both jump components and heteroscedastic ARJI(r,s) model: Brownian motions are needed to model the asset Xr Xs returns. As the jump components are omitted, the lt ¼ l0 þ rilt i þ gijt i estimated rate of convergence of volatility to its i¼1 i¼1 unconditional mean is significantly biased. More- l E[n F ] is the conditional expectation of over, Chan and Maheu (2002) developed a new t ¼ tj t 1 the counting process. j represents the innovation conditional jump model to study jump dynamics t i to l . The shock jump intensity residual is in stock market returns. They present a discrete- t i time jump model with time varying conditional jt i ¼ E[nt ijFt i] lt i jump intensity and jump size distribution. Besides, X1 : : (43 12) they combine the jump specification with a ¼ jP(nt i ¼ jjft i) lt i GARCH parameterization of volatility. Consider j¼0 the following jump model for stock returns: The first term of Equation (43.12) is average num-
Xt pffiffiffiffi Xnt ber of jumps at time t i based on time t i Rt ¼ m þ fRt i þ htzt þ Yt,k, information. Therefore, xi represents the unpre- (43:10) t i i¼1 k¼1 dictable component about the conditional mean 2 : zt NID(0,1), Yt,k N(ut, d ) of the counting process nt i. Moreover, having JUMP DIFFUSION MODEL 685
observed Rt, let f (Rtjnt ¼ j, Ft 1) denote the con- that the jump intensity is temporarily trending ditional density of returns given that j jumps occur away from its unconditional mean. On the other and the information set Ft 1, we can get the ex- hand, this model effectively captures systematic post probability of the occurrence of j jumps at changes in jump risk in the market. In addition, time t, with the filter defined as they find significant time variation in the condi- tional jump intensity and the jump size distribution f (R jn ¼ j, F )P(n ¼ jjF ) P(n ¼ jjF ) ¼ t t t 1 t t 1 , in their application for daily stock market returns. t t P(R F ) tj t 1 Accordingly, the ARJI model can capture system- j ¼ 0, 1, 2, , atic changes, and also forecast increases (de- (43:13) creases) in jump risk into the future. where, the definition of P(nt ¼ jjFt 1) is the same as Equation (43.11). The filter in Equation (43.13) 43.9.2. ARCH=GARCH Jump Diffusion Model is an important component of their model of time varying jump dynamics. Thus, the conditional As described in Drost et al. (1998), there exists density of return is a major drawback of Merton’s (1976) model which implies that returns are independent and X1 identically distributed at all frequencies that con- P(RtjFt 1) ¼ f (Rtjnt ¼ j, Ft 1)P(nt ¼ jjFt 1): j¼0 flict with the overwhelming evidence of conditional (43:14) heteroscedasticity in returns at high frequencies, because all deviations from log normality of ob- Equation (43.14) shows that this model is nothing served stock returns at any frequency can be attrib- more than a discrete mixture of distribution where uted to the jumps in his model. Thus, several papers the mixing is driven by a time varying Poisson consider the size of jumps within the models that distribution. Therefore, from the assumption of also involve the conditional heteroscedasticity. Equation (43.10), the distribution of returns con- Jorion (1988) considered a tractable specifica- ditional on the most recent information set and j tion combining both ARCH and jump processes jumps is normally distributed as for foreign exchange market: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 f (Rtjnt ¼ j, Ft 1) ¼ exp pffiffiffiffiffiffiffi Xnt 2 = 2p(ht þ jdt ) ln (Pt Pt 1)jt 1 ¼ m þ (bt)z þ ln (Yt), P ! i¼1 (R m l f R u j)2 b ¼ E (s2) ¼ a þ a (x m)2 t i¼1 i t i t : t t 1 t 0 1 t 1 2 2(ht þ jdt ) in which a1 is the autoregressive parameter induc- Equation (43.13) includes an infinite sum over the ing heteroskedasticity and the distribution of xt is possible number of jumps nt. However, practically, conditional on information at t 1 and define xt they consider truncating the maximum number of as the logarithm of price relative ln (Pt=Pt 1). jumps to a large value t, and then they set the A jump size Y is assumed independently log nor- 2 probability of t or more jumps to 0. Hence, the mally distributed, ln Y N(u, d ), nt is the actual first way to choose t is to check Equation (43.11) number of jumps during the interval. z is a stand- to be equal to 0 for j t. The second check on the ard normal deviate. Consequently, his results choice of t is to investigate t > t to make sure that reveal that exchange rate exhibit systematic discon- the parameter estimate does not change. tinuities even after allowing for conditional hetero- The ARJI model illustrates that conditional skedasticity in the diffusion process. In brief, in jump intensity is time varying. Suppose that we his work, the maximum likelihood estimation of > observe jt 0 for several periods. This suggests a mixed-jump diffusion process indicates that 686 ENCYCLOPEDIA OF FINANCE
< ignoring the jump component in exchange rates such that zh (with jjbh 1) is determined by can lead to serious mispricing errors for currency c ¼ hv{1 exp ( hu)}, options. The same findings also can be found in h bh ch exp ( hu) 1 Nieuwland et al. (1991) who allow for the model ah ¼ exp ( hu) bh, 2 ¼ , 1 þ b ch{1 þ exp ( 2hu)} 2 with conditional heteroscedasticity and jumps in h exchangerate market. Also, an application of a where u is the time unit and scale is denoted by v. GARCH jump mixture model has been given by f and y are slope parameters and f will denote : Vlaar and Palm (1993). They point out that the slopes in the (ah bh) plane, while v determines the GARCH specification cum normal innovation slope of the kurtosis at very high frequencies. cannot fully explain the leptokurtic behavior for Drost et al. (1998) employed the results of Drost high-frequency financial data. Both the GARCH and Werker (1996), which stated that for GARCH specification and the jump process can explain the diffusion at an arbitrary frequency h, the five dis- leptokurtic behavior. Hence, they permit autocor- crete-time GARCH parameters can be written in relation in the mean higher-order GARCH effect terms of only four continuous-time parameters, i.e and Bernoulli jumps. an over identifying restriction in GARCH diffu- A weak GARCH model can be defined as a sion, for proposing a test for the presence of jumps symmetric discrete-time process {y(h)t, t 2 h AA} with conditional heteroscedasticity, which is based with finite fourth moment and with parameter on the following Theorem 1. zh ¼ (fh, ah, bh, kh), if there exists a covariance- Theorem 1. Let {Yt: t 0}be a continuous- stationary process {s(h)t, t2ha}with time GARCH diffusion. Then u > 0 and l: 2 (0,1)
2 2 2 is defined by s(h)tþh ¼ fh þ ahy(h)t þ bhs(h)t, t 2 h A 4 u ¼ ln (a þ b) Ey(h)t ( and we denote kh ¼ 2 as the kurtosis of the 2 2 (Ey2 ) 2 {1 (a þ b) }(1 b) process. (h)t l ¼ 2ln (a þ b) a ab(a þ b) Roughly speaking, the class of continuous-time ) GARCH models can be divided into two groups. þ 6ln(a þ b) þ 2ln2 (a þ b) þ 4(1 a b) One is the GARCH diffusion in which the sample paths are smooth and the other, where the sample l exp ( u) 1 þ u k ¼ 3 þ 6 paths are erratic. Drost and Werker (1996) devel- 1 l u2 oped several properties of discrete-time data that are generated by underlying continuous-time pro- Thus, we set up the null and alternative hypo- cesses that accommodate both conditional hetero- theses: : : scedasticity and jumps. Their model is as follows. H0 {Yt t 0}is a GARCH diffusion model : : Let {Yt, t 0}be the GARCH jump diffusion and H1 {Yt t 0}is a GARCH jump diffusion model. with parameter vector zh ¼ (fh, ah, bh, kh) and > From Theorem 1, by simple calculation, we suppose ah0 for some h0 0. Then, there exists v 2 (0, 1), u 2 (0, 1), f 2 (0, 1), y 2 (0, 1) yield the relation between functions K and k: and c and k are given by h h K(a, b, k) ¼ k k(a, b) ¼ 0 y þ 2hu 4{exp ( hu) 1 þ hu} þ 2hu 1 þ for GARCH diffusion. Furthermore, in Drost and yf(2 þ f) c ¼ , Werker (1996), they showed that K(a, b, k) will h 1 exp ( 2hu) be strictly larger than 0 for any GARCH jump y exp ( hu) 1 þ hu k ¼ 3 þ þ 3yf(2 þ f) , diffusion model. As a result, H0 is equivalent h hu 2 (hu) to K(a, b, k) ¼ 0 and H1 is equivalent to JUMP DIFFUSION MODEL 687
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