A Stochastic String with a Compound Poisson Process

Abstract

We investigate a compound Poisson infinite factor diffusion model which describes the relationship between the infinite-dimension random risk resource and the corresponding stochastic process. We derive the no-arbitrage condition on the drift of instantaneous forward rates in the compound model and study the impact of random jump on the price of the zero-coupon bond.

1. Introduction

In the study of the stochastic process, the expansion from single-factor model to multiple-factor model greatly improves its capability to describe the dynamic properties. For the term structure of interest rates, multiple-factor expansion enables itself to more flexibly describe the relationship between different maturities and their corresponding risk-free interest rates. The term structure curve contains rich information about the economy status and financial markets. The literatures have witted extensive researches on this topic for recent decades. For instance, the HJM model in [1, 2] provides insight about arbitrage-free condition of the shift and volatility of the dynamic process of forward interest rates under the risk-free condition. Kennedy [3, 4], Goldstein [5], and Santa-Clara and Didier [6] model each time-to-maturity forward rate driven by its own shock termed Brown Sheet, Random Field, or Stochastic String. In fact, the models in [3–6] expand the multiple-factor model to infinite stochastic factor with imperfect correlation among different maturities. At the same time, the correlations are not confined by the function form of volatilities. These results display greater variability in the correlation between different maturities.

In this paper, according to the stochastic string model presented in [6], we introduce a compound Poisson process with a constant jump intensity and random jump size to capture information burst and resulting discontinuous path. We derive the no-arbitrage condition in the compound Poisson infinite-dimensional diffusion process in which the drift and volatility of stochastic process under the risk-neutral measure are satisfied. Some of the results in [6] are extended.

The paper is organized as follows. Section 2 reviews the single-factor, multiple-factor model and stochastic string model, which are related to our study. In Section 3, we add a random jump process in the stochastic string model discussed in [6] and analyze its dynamic properties. An application of our main result is presented in Section 4.

2. Single-Factor, Multiple-Factor Model and Stochastic String Model

For a probability space (Ω,Φ,𝑃), the dynamics of a stochastic process 𝑓(𝑡,𝑥) is described by 𝑑𝑓(𝑡,𝑥)=𝛼(𝑡,𝑥)𝑑𝑡+𝜎(𝑡,𝑥)𝑑𝑊(𝑡), (1) where 𝑊(𝑡) is a standard Wiener process and 𝛼(𝑡,𝑥) and 𝜎(𝑡,𝑥) are drift and volatility terms, respectively. When 𝑓(𝑡,𝑥) is explained as the instantaneous forward rate at time 𝑡 for time-to-maturity 𝑥>0, (1) is the HJM model. The price at time 𝑡 of a bond with maturity 𝑠 is 𝐵(𝑡,𝑠)=exp−₀^𝑠−𝑡𝑓(𝑡,𝑥)𝑑𝑥. (2) The HJM model has no-arbitrage condition which is written by 𝛼()=()𝑡,𝑥𝜕𝑓𝑡,𝑥(𝜕𝑥+𝜎𝑡,𝑥)^𝑥₀𝜎𝑡,𝑦𝑑𝑦+𝜃(𝑡), (3) where the process 𝜃(𝑡) denotes the market price of risk at time 𝑡. Using Itô’s formula, we calculate the dynamics of bond prices as 𝑑𝐵(𝑡,𝑠)𝐵()=𝑓()−𝑡,𝑠𝑡,𝑠−𝑡₀^𝑠−𝑡𝛼+1𝑡,𝑦𝑑𝑦2₀^𝑠−𝑡𝜎𝑡,𝑦𝑑𝑦²−𝑑𝑡_0𝑠−𝑡𝜎𝑡,𝑦𝑑𝑦𝑑𝑊(𝑡). (4) Under the no-arbitrage condition, the instantaneous forward rate of time-to-maturity 𝑥 is 𝑓(𝑡,𝑥)=𝑓(0,𝑡+𝑥)+^𝑡₀𝜂()+𝑢,𝑡+𝑥−𝑢𝑑𝑢_𝑡0𝜎(𝑢,𝑡+𝑥−𝑢)𝑑𝑊(𝑢), (5) where 𝜂(𝑡,𝑥)≡𝜎(𝑡,𝑥)^𝑥₀𝜎(𝑡,𝑣)𝑑𝑣+𝜃(𝑡). (6) In fact, every term structure model driven by Brownian motion is a special case of the HJM model (e.g., the Vasicek model in [7], the Hull-White model in [8], etc.). However, the single-factor HJM model has a limitation that different time-to-maturities are completely correlated; that is, their correlation coefficient equals 1. This implies that(1)the term structure curve would parallel shift which limits its capability to generate richer class of dynamics and shapes of the term structure of interest rates,(2)it does not permit consistency with term structure innovation along with the time.

One way to extend the model is by introducing the 𝑁-dimensional Brownian motion to the term structure model written by 𝑑𝑓(𝑡,𝑥)=𝛼(𝑡,𝑥)𝑑𝑡+^𝑁_𝑖=1𝜎^𝑖(𝑡,𝑥)𝑑𝑊^𝑖. (7) One can obtain the correlation of the forward rate curve driven by the 𝑁-dimensional Brownian motion as 𝑐=∑𝑥,𝑦^𝑁𝑖=1𝜎^𝑖(𝑡,𝑥)𝜎^𝑖𝑡,𝑦∑^𝑁_𝑖=1𝜎^𝑖(𝑡,𝑥)²∑^𝑁_𝑖=1𝜎^𝑖𝑡,𝑦². (8) Different from the multiple-factor HJM model in (7), which implies all assets shocked by the common 𝑁 factors, Santa-Clara and Didier [6] model the forward rate as a stochastic string and allow each forward rate to be driven by its own random risk resource.

Assume that the dynamics of the instantaneous forward rates at 𝑡 with time-to-maturity 𝑥(𝑥>0) is modeled by 𝑑_𝑡𝑓(𝑡,𝑥)=𝛼(𝑡,𝑥)𝑑𝑡+𝜎(𝑡,𝑥)𝑑_𝑡𝑊(𝑡,𝑥), (9) where 𝑊(𝑡,𝑥) is a generalization of Brownian motion which depends on both time 𝑡 and time-to-maturity 𝑥. The notation 𝑑_𝑡() means that the increment is taken with respect to time 𝑡. Equation (9) is not the infinite-dimensional simple generalization of the multifactor HJM model, in which all forward rates are subject to the same (possibly infinite) set of Brownian motion shocks. In this model, there is one different stochastic shock for each time-to-maturity. Given time-to-maturity 𝑥, the stochastic perturbation 𝑊(𝑡,𝑥) is a standard Wiener process, which satisfies several requirements (see [6] for details). For example, the correlation coefficient matrix is positive semidefinite, which means that, for all functions ℎ(𝑥) in 𝐿² and for some large enough constant 𝑇, ^𝑇₀^𝑇₀ℎ(𝑥)𝑐𝑥,𝑦ℎ𝑦𝑑𝑦𝑑𝑥≥0. (10) The no-arbitrage condition of the drift of instantaneous forward rate is 𝛼()=()𝑡,𝑥𝜕𝑓𝑡,𝑥()×𝜕𝑥+𝜎𝑡,𝑥_𝑥0𝑐𝑥,𝑦𝜎𝑡,𝑦𝑑𝑦+^∞₀𝑐𝑥,𝑦𝜃𝑡,𝑦𝑑𝑦. (11) The solution of the stochastic differential equation for instantaneous forward interest rate is 𝑓(𝑡,𝑥)=𝑓(0,𝑡+𝑥)+^𝑡₀𝑎()+𝑢,𝑡+𝑥−𝑢𝑑𝑢_𝑡0𝜎(𝑢,𝑡+𝑥−𝑢)𝑑_𝑢𝑊(𝑢,𝑡+𝑥−𝑢), (12) where 𝑎()()×𝑡,𝑥≡𝜎𝑡,𝑥_𝑥0𝑐𝑥,𝑦𝜎𝑡,𝑦𝑑𝑦+^∞₀𝑐𝑥,𝑦𝜃𝑡,𝑦𝑑𝑦. (13) Under the arbitrage-free condition, we have 𝐵(𝑡,𝑠)=𝐵(0,𝑠)exp^𝑡₀𝑟(𝑢)𝑑𝑢−^𝑡₀𝑈()−𝑢,𝑠−𝑢𝑑𝑢_𝑡0𝑉(𝑢,𝑠−𝑢)𝑑_𝑢𝑊(𝑢,𝑠−𝑢), (14) where 𝑈(𝑡,𝑥)≡^∞₀𝜃𝑡,𝑦^𝑥₀𝑐𝜎()𝑉()≡𝑦,𝑧𝑡,𝑧𝑑𝑧𝑑𝑦,𝑡,𝑥^𝑥₀𝜎𝑡,𝑦𝑑𝑦. (15) The covariance of different time-to-maturity zero-coupon bond yields is ()cov𝑑𝐵𝑡,𝑠𝐵(),𝑡,𝑠𝑑𝐵𝑡,𝑠^′𝐵𝑡,𝑠^′=₀^𝑠−𝑡𝜎(𝑡,𝑥)^𝑠′₀^−𝑡𝑐𝑥,𝑦𝜎𝑡,𝑦𝑑𝑦𝑑𝑥𝑑𝑡. (16)

3. Compound Poisson Infinite-Dimensional Diffusion Process

Information burst, some emergency events, crises, and monetary target adjustment may cause the return and price of asset jump. There are more and more empirical lines of evidence which show that the interest rate models should incorporate the jump risk. The difference of the asset price of continuous path will disappear as the change of time converges to zero. We know that in the jump model, the difference does not converge to zero, although its probability of jump occurrence will converge to 0. Eberlein et al. [9] investigate the term structure models driven by Lévy processes.

Based on the continuous path of instantaneous forward rate used in Santa-Clara and Didier [6] and other works [3–5], we introduce a compound Poisson process to capture the yield curve jump. For each time-to-maturity 𝑥, assume that the dynamics of instantaneous forward rate is 𝑑_𝑡𝑓(𝑡,𝑥)=𝛼(𝑡,𝑥)𝑑𝑡+𝜎(𝑡,𝑥)𝑑_𝑡𝑊(𝑡,𝑥)+𝛾(𝑡,𝑥)𝑑𝐽(𝑡), (17) where 𝐽(𝑡) is a compound Poisson process and 𝐽(𝑡)=∑^𝑁(𝑡)_𝑚=1𝑌_𝑚. Poisson process 𝑁(𝑡) counts the number of jumps that occur at or before time 𝑡. 𝑌_𝑚(𝑚=1,2,…) are independent identically distributed discrete random variables with finite values 𝑎₁,𝑎₂,…𝑎_𝐾, with probability 𝑃(𝑦_𝑗=𝑎_𝑖)=𝑝_𝑖, ∑^𝐾_𝑖=1𝑝_𝑖=1. 𝛾(𝑡,𝑥) is a nonrandom function. The compound jump is independent of the stochastic string 𝑊(𝑡,𝑥).

Applying the decomposition theorem of a compound Poisson process (see [10]), 𝐽(𝑡)=∑^𝑁(𝑡)_𝑚=1𝑌_𝑚 may be rewritten as 𝐽(𝑡)=∑^𝐾_𝑖=1𝑎_𝑖𝑑𝑁_𝑖(𝑡). So, we can rewrite the dynamics of instantaneous forward rates as 𝑓(𝑡,𝑥)=𝑓(0,𝑥)+^𝑡₀𝛼(𝑢,𝑥)𝑑𝑢+^𝑡₀𝜎(𝑢,𝑥)𝑑_𝑢𝑊()+𝑢,𝑥_{𝐾𝑖=1𝑡0}𝛾(𝑢,𝑥)𝑎_𝑖𝑑𝑁_𝑖(𝑢), (18) where 𝑊(𝑡,𝑥) is the same random risk resource as defined in (9).

Assume that the market is complete, so that the existence and uniqueness of the risk market price are guaranteed. It can be proved that the continuous part of random risk factor is independent of the jump risk factor.

For the continuous risk resource part, define 𝑊(𝑡,𝑥)=𝑊(𝑡,𝑥)+^𝑡₀^∞₀𝜃𝑢,𝑦𝑐𝑥,𝑦𝑑𝑦𝑑𝑢, (19) or in differential form 𝑑_𝑡𝑊(𝑡,𝑥)=𝑑_𝑡𝑊(𝑡,𝑥)+^∞₀𝜃𝑡,𝑦𝑐𝑥,𝑦𝑑𝑦𝑑𝑡. (20) The correlation coefficient for different time-to-maturity random risk resource 𝑊(𝑡,𝑠) is the same as that of 𝑊(𝑡,𝑠). For fixed time-to-maturity 𝑥, let 𝑍₀()−𝑡,𝑥=exp^𝑡₀^∞₀𝜃𝑢,𝑦𝑐𝑥,𝑦𝑑𝑦𝑑_𝑢𝑊−1𝑢,𝑦2^𝑡₀^∞₀𝜃(𝑢,𝑧)𝑐(𝑥,𝑧)^∞₀𝜃𝑐,𝑍𝑢,𝑦𝑥,𝑦𝑑𝑦×𝑐𝑦,𝑧𝑑𝑧𝑑𝑢_𝑖(𝑡,𝑥)=exp𝜆_𝑖−𝜆_𝑖𝑡+𝑁_𝑖(𝑡)log𝜆_𝑖−𝜆_𝑖,𝑍()=𝑖=1,2,…,𝐾,𝑡,𝑥^𝐾_𝑖=0𝑍_𝑖(),𝑃(𝐴)=𝑡,𝑥_𝐴𝑍(𝑇,𝑥)𝑑𝑃∀𝐴∈Φ. (21) We obtain the risk-neutral measure 𝑃 under some conditions. Using the properties of risk-neutral measure 𝑃, for fixed time-to-maturity 𝑥, we know that 𝑊(𝑡,𝑥) is a martingale. At the same time, 𝐽(𝑡)=∑^𝐾_𝑖=1𝑎_𝑖𝑑𝑁_𝑖(𝑡) is a compound Poisson process with intensity 𝜆. The jump sizes 𝑌₁,𝑌₂⋯ are still I.I.D. random variables, while the jump intensity satisfies 𝑃(𝑌_𝑗=𝑎_𝑖)=𝑝(𝑎_𝑖)=(𝜆_𝑖/𝜆)(𝑖=1,2,…,𝐾) for arbitrary 𝑗, where ∑^𝐾_𝑖=1𝜆_𝑖=𝜆 and each 𝜆_𝑖(𝑖=1,2,…,𝐾) is a positive constant. Notice that, under the new measure 𝑃, 𝑊(𝑡,𝑥) is still independent of 𝑁_𝑖(𝑡), 𝑖=1,2,…,𝐾.

Theorem 1. If the market is complete, the arbitrage-free condition of the instantaneous forward rate satisfies 𝛼()=()𝑡,𝑥𝜕𝑓𝑡,𝑥()×𝜕𝑥+𝜎𝑡,𝑥_𝑥0𝜎𝑡,𝑦𝑐𝑥,𝑦𝑑𝑦+^∞₀𝜃𝑐−𝑡,𝑦𝑥,𝑦𝑑𝑦_𝐾𝑖=1𝑎_𝑖𝜆_𝑖𝛾(𝑡,𝑥)exp−𝑎_𝑖^𝑥₀𝛾𝑡,𝑦𝑑𝑦. (22)

Proof. The price at time 𝑡 of a zero-coupon bond with maturity 𝑠 is 𝐵(𝑡,𝑠)=exp[−∫₀^𝑠−𝑡𝑓(𝑡,𝑥)𝑑𝑥]. For fixed 𝑠, we have ln(𝐵(𝑡,𝑠))=−₀^𝑠−𝑡𝑓(𝑡,𝑥)𝑑𝑥. (23) Differentiating with respect to time 𝑡 in (23), we obtain 𝑑_𝑡ln𝐵(𝑡,𝑠)=𝑓(𝑡,𝑠−𝑡)𝑑𝑡−₀^𝑠−𝑡𝑑_𝑡𝑓()()𝑡,𝑥𝑑𝑥=𝑓𝑡,𝑠−𝑡𝑑𝑡−₀^𝑠−𝑡𝛼()−𝑡,𝑥𝑑𝑥𝑑𝑡_0𝑠−𝑡𝜎(𝑡,𝑥)𝑑_𝑡𝑊()−𝑡,𝑥𝑑𝑥_0𝑠−𝑡𝛾(𝑡,𝑥)^𝐾_𝑖=1𝑎_𝑖𝑑𝑁_𝑖(𝑡)=𝑓()−𝑑𝑥.𝑡,𝑠−𝑡₀^𝑠−𝑡𝛼()−𝑡,𝑥𝑑𝑥𝑑𝑡_0𝑠−𝑡𝜎(𝑡,𝑥)𝑑𝑥𝑑_𝑡𝑊()−𝑡,𝑥_{𝐾𝑖=10𝑠−𝑡}𝛾(𝑡,𝑥)𝑑𝑥𝑎_𝑖𝑑𝑁_𝑖(𝑡). (24) Applying jump process Itô’s formula (see [10]), we have 𝑑_𝑡𝐵()(𝑓()𝑡,𝑠=𝐵𝑡,𝑠)𝑡,𝑠−𝑡−₀^𝑠−𝑡𝛼()+1𝑡,𝑥𝑑𝑥2₀^𝑠−𝑡𝜎(𝑡,𝑥)₀^𝑠−𝑡𝜎𝑐()𝑡,𝑦𝑥,𝑦𝑑𝑦𝑑𝑥𝑑𝑡−𝐵𝑡,𝑠₀^𝑠−𝑡𝜎(𝑡,𝑥)𝑑𝑥𝑑_𝑡𝑊()(𝑡𝑡,𝑥+𝐵⁻,𝑠)^𝐾_𝑖=1exp−𝑎_𝑖0^𝑠−𝑡𝛾(𝑡,𝑥)𝑑𝑥−1𝑑𝑁_𝑖(𝑡). (25) Under the risk-neutral probability measure 𝑃, the price of zero-coupon bond satisfies 𝑑_𝑡𝐵()()𝑟(𝑡)()𝑡,𝑠=𝐵𝑡,𝑠𝑑𝑡−𝐵𝑡,𝑠₀^𝑠−𝑡𝜎(𝑡,𝑥)𝑑𝑥𝑑_𝑡𝑊()(𝑡𝑡,𝑥+𝐵⁻,𝑠)^𝐾_𝑖=1exp−𝑎_𝑖0^𝑠−𝑡𝛾()𝑁𝑡,𝑥𝑑𝑥−1×𝑑_𝑖(𝑡)−𝜆_𝑖𝑡. (26) Combining (25) with (26), we have 𝑟(𝑡)=𝑓(𝑡,𝑠−𝑡)−₀^𝑠−𝑡𝛼()+1𝑡,𝑥𝑑𝑥2₀^𝑠−𝑡𝜎(𝑡,𝑥)₀^𝑠−𝑡𝜎𝑐+𝑡,𝑦𝑥,𝑦𝑑𝑦𝑑𝑥_0𝑠−𝑡𝜎(𝑡,𝑥)^∞₀𝜃𝑐+𝑡,𝑦𝑥,𝑦𝑑𝑦𝑑𝑥_𝐾𝑖=1exp−𝑎_𝑖0^𝑠−𝑡𝛾(𝑡,𝑥)𝑑𝑥−1𝜆_𝑖, (27) which is the expression that the risk market prices satisfy. It includes market prices of 𝐾 jump risk factors and infinite random risk resource. Note that the compensated Poisson process 𝑁_𝑖(𝑡)−𝜆_𝑖𝑡 in (26) is a martingale. Let 𝑠−𝑡=𝑥, and applying the stochastic Fubini theorem, we rewrite (27) as 𝑟(𝑡)=𝑓(𝑡,𝑥)−^𝑥₀𝛼+1𝑡,𝑦𝑑𝑦2^𝑥₀𝜎(𝑡,𝑧)^𝑥₀𝜎𝑐+𝑡,𝑦𝑧,𝑦𝑑𝑦𝑑𝑧_𝑥0𝜎(𝑡,𝑧)^∞₀𝜃𝑐+𝑡,𝑦𝑧,𝑦𝑑𝑦𝑑𝑧_𝐾𝑖=1exp−𝑎_𝑖^𝑥₀𝛾𝑡,𝑦𝑑𝑦−1𝜆_𝑖. (28) Differentiating (28) with respect to 𝑥, we obtain 𝜕𝑓(𝑡,𝑥)()()×𝜕𝑥−𝛼𝑡,𝑥+𝜎𝑡,𝑥_𝑥0𝜎𝑡,𝑦𝑐𝑥,𝑦𝑑𝑦+^∞₀𝜃𝑐−𝑡,𝑦𝑥,𝑦𝑑𝑦_𝐾𝑖=1𝑎_𝑖𝜆_𝑖𝛾(𝑡,𝑥)exp−𝑎_𝑖^𝑥₀𝛾𝑡,𝑦𝑑𝑦=0. (29) Rearranging (29), we obtain the arbitrage-free condition of (22). It completes the proof.

Remark 2. When 𝑎_𝑖=0(𝑖=1,2,…𝐾), (22) reduces to 𝛼()=()𝑡,𝑥𝜕𝑓𝑡,𝑥()𝜕𝑥+𝜎𝑡,𝑥^𝑥₀𝜎𝑐()𝑡,𝑦𝑥,𝑦𝑑𝑦+𝜎𝑡,𝑥^∞₀𝜃𝑡,𝑦𝑐𝑥,𝑦𝑑𝑦, (30) which is the arbitrage-free condition presented in [6]. Moreover, for arbitrary time-to-maturity 𝑥 and 𝑦, if 𝑐(𝑥,𝑦)=1, and let ∫^∞₀𝜃(𝑡,𝑦)𝑑𝑦=Θ(𝑡), then the arbitrage-free condition becomes 𝛼()=()𝑡,𝑥𝜕𝑓𝑡,𝑥(𝜕𝑥+𝜎𝑡,𝑥)^𝑥₀𝜎𝑡,𝑦𝑑𝑦+Θ(𝑡), (31) which is the arbitrage-free condition of the one-dimensional HJM model.
Under the risk-neutral measure 𝑃, the dynamics of instantaneous forward rates which satisfy the arbitrage-free condition can be rewritten as 𝑑_𝑡𝑓(𝑡,𝑥)=𝐴(𝑡,𝑥)𝑑𝑡+𝜎(𝑡,𝑥)𝑑_𝑡𝑊()+𝑡,𝑥_𝐾𝑖=1𝑎_𝑖𝛾(𝑡,𝑥)𝑑𝑁_𝑖(𝑡), (32) where 𝐴()≡()𝑡,𝑥𝜕𝑓𝑡,𝑥()𝜕𝑥+𝜎𝑡,𝑥^𝑥₀𝜎𝑐−𝑡,𝑦𝑥,𝑦𝑑𝑦_𝐾𝑖=1𝑎_𝑖𝜆_𝑖𝛾(𝑡,𝑥)exp−𝑎_𝑖^𝑥₀𝛾𝑡,𝑦𝑑𝑦. (33)

Theorem 3. Under the risk-neutral measure 𝑃, the price at time 𝑡 of a zero-coupon bond with maturity 𝑠 is 𝐵()()𝑡,𝑠=𝐵0,𝑠×exp^𝑡₀𝑟(𝑢)𝑑𝑢−^𝑡₀₀^𝑠−𝑢𝜎(𝑢,𝑥)𝑑𝑥𝑑_𝑢𝑊()−1𝑢,𝑥2^𝑡₀𝑑𝑢₀^𝑠−𝑢𝜎(𝑢,𝑥)×₀^𝑠−𝑢𝜎𝑐−𝑢,𝑦𝑥,𝑦𝑑𝑦𝑑𝑥_{𝐾𝑖=1𝑡0}exp−𝑎_𝑖0^𝑠−𝑢𝛾(𝑢,𝑥)𝑑𝑥−1𝜆_𝑖−𝑑𝑢×exp^𝐾_𝑖=1^𝑡₀₀^𝑠−𝑢𝛾(𝑢,𝑥)𝑑𝑥𝑎_𝑖𝑑𝑁_𝑖(𝑢). (34)

Proof. We show that (34) is the solution of (26). Obviously, when 𝑡=0, the result is correct. Let 𝑌(𝑡)=𝐵(0,𝑠)exp[𝑋^𝑐(𝑡)], where 𝑋^𝑐(𝑡)≡^𝑡₀𝑟(𝑢)𝑑𝑢−^𝑡₀₀^𝑠−𝑢𝜎(𝑢,𝑥)𝑑𝑥𝑑_𝑢𝑊()−1𝑢,𝑥2^𝑡₀𝑑𝑢₀^𝑠−𝑢𝜎(𝑢,𝑥)₀^𝑠−𝑢𝜎𝑐−𝑢,𝑦𝑥,𝑦𝑑𝑦𝑑𝑥_{𝐾𝑖=1𝑡0}exp−𝑎_𝑖0^𝑠−𝑢𝛾(𝑢,𝑥)𝑑𝑥−1𝜆_𝑖𝑑𝑢. (35) Denote 𝐾(𝑡)≡exp−^𝐾_𝑖=1^𝑡₀₀^𝑠−𝑢𝛾(𝑢,𝑥)𝑑𝑥𝑎_𝑖𝑑𝑁_𝑖(𝑢). (36) Then, we have 𝑑𝑌(𝑡)=𝑌(𝑡)𝑟(𝑡)𝑑𝑡−₀^𝑠−𝑡𝜎(𝑡,𝑥)𝑑𝑥𝑑_𝑡𝑊()−𝑡,𝑥_𝐾𝑖=1exp−𝑎_𝑖0^𝑠−𝑡𝛾(𝑡,𝑥)𝑑𝑥−1𝜆_𝑖,(𝑡)(𝑡𝑑𝑡𝑑𝐾=𝐾⁻)×^𝐾_𝑖=1exp−𝑎_𝑖0^𝑠−𝑡𝛾(𝑡,𝑥)𝑑𝑥−1𝑑𝑁_𝑖(𝑡). (37)(38) Notice that as the length of time interval converges to 0, there is at most one jump in each time interval. Combining (37) with (38), we obtain 𝑑_𝑡𝐵()(𝑡)(𝑡)(𝑡)𝐾(𝑡)(𝑡)(𝑡)(𝑡)𝐾(𝑡𝑡,𝑠=𝑌𝑑𝐾+𝑑𝑌+𝑑𝑌𝑑𝐾=𝑌⁻)^𝐾_𝑖=1exp−𝑎_𝑖0^𝑠−𝑡𝛾(𝑡,𝑥)𝑑𝑥−1𝑑𝑁_𝑖(𝑡(𝑡)𝐾(𝑡𝑟(𝑡))+𝑌)𝑑𝑡−₀^𝑠−𝑡𝜎(𝑡,𝑥)𝑑𝑥𝑑_𝑡𝑊()−𝑡,𝑥_𝐾𝑖=1exp−𝑎_𝑖0^𝑠−𝑡𝛾(𝑡,𝑥)𝑑𝑥−1𝜆_𝑖(𝑡𝑑𝑡+0=𝐵⁻,𝑠)^𝐾_𝑖=1exp−𝑎_𝑖0^𝑠−𝑡𝛾()𝑁𝑡,𝑥𝑑𝑥−1×𝑑_𝑖(𝑡)−𝜆_𝑖𝑡+𝐵(𝑡,𝑠)𝑟(𝑡)𝑑𝑡−₀^𝑠−𝑡𝜎(𝑡,𝑥)𝑑𝑥𝑑_𝑡𝑊(𝑡,𝑥). (39) Therefore, (34) is the solution of (26). It completes the proof.

Remark 4. In (34), it shows that the price of zero-coupon bond is affected by two parts: continuous component and jump component. For the continuous part, besides the risk-free interest rate effect, the stochastic string risk resources till to the maturity would generate an accumulated effect on the price of zero-coupon bond. As the occurrence of jump, its long-term accumulative effects will be offset by part of −∑^𝐾_𝑖=1∫^𝑡₀[exp(−𝑎_𝑖∫₀^𝑠−𝑢𝛾(𝑢,𝑥)𝑑𝑥)−1]𝜆_𝑖𝑑𝑢. The correlation of different maturities from time 𝑡 to maturity 𝑠 has an explicit effect on the price. For the jump component, the effect is scaled by the function 𝛾(𝑡,𝑥), which captures the jump effect on the financial variables.

4. Application

The compound Poisson infinite diffusion model developed in Section 3 comprises the traditional jump-diffusion model. We consider that the jump sizes are discrete random variables. It is convenient to introduce the continuous jump size model whose size is, for instance, normally distributed random variable.

We consider the price of the short-term interest rate futures. Assume that the financial market is shocked by two kinds of random risk resource: one is continuous risk factor of 𝑊(𝑡,𝑥) and the other is a compound Poisson process. Assume that the market is complete. Consider a futures contract maturity at 𝑇_𝑓, 𝑡<𝑇_𝑓<𝑠, whose underlying asset is the zero-coupon bond maturity at 𝑠. Denote the futures price at time 𝑡 as 𝐹(𝑡,𝑇_𝑓,𝑠). Under the risk-neutral measure 𝑃, the futures price 𝐹(𝑡,𝑇_𝑓,𝑠) is a martingale. We have 𝐹𝑡,𝑇_𝑓,𝑠=𝐸_𝑡𝐹𝑇_𝑓,𝑇_𝑓,𝑠∣Φ_𝑡=𝐸𝑃_𝑡𝐵𝑇_𝑓,𝑠∣Φ_𝑡. (40)

Theorem 5. Under the risk-neutral measure 𝑃, the futures price with underlying asset as a zero-coupon bond is 𝐹𝑡,𝑇_𝑓,𝑠=𝐹0,𝑇_𝑓−,𝑠×exp^𝑡₀^𝑠_𝑇𝑓𝜎(𝑢,𝑥−𝑢)𝑑𝑥𝑑_𝑢𝑊()−𝑢,𝑥−𝑢_{𝐾𝑖=1𝑡0𝑠𝑇𝑓}𝛾(𝑢,𝑥−𝑢)𝑑𝑥𝑎_𝑖𝑑𝑁_𝑖(𝑢)−12^𝑡₀^𝑠_𝑇𝑓𝜎()×𝑢,𝑥−𝑢𝑑𝑥_𝑠𝑇𝑓𝜎𝑐+𝑢,𝑦−𝑢𝑥−𝑢,𝑦−𝑢𝑑𝑦𝑑𝑢_{𝐾𝑖=1𝑡0}𝜆_𝑖exp−𝑎_𝑖^𝑠_𝑇𝑓𝛾(𝑢,𝑥−𝑢)𝑑𝑥−1𝑑𝑢. (41)

Proof. Note that the zero-coupon bond price is a function of instantaneous forward rates 𝐵𝑇_𝑓,𝑠=exp−^{𝑠−𝑇𝑓}₀𝑓𝑇_𝑓,𝑥𝑑𝑥. (42) Therefore, we have ln𝐵𝑇_𝑓,𝑠=−^{𝑠−𝑇𝑓}₀𝑓𝑇_𝑓,𝑥𝑑𝑥=−^{𝑠−𝑇𝑓}₀𝑓0,𝑇_𝑓−+𝑥𝑑𝑥_{𝑠−𝑇𝑓0𝑇𝑓0}𝐴𝑢,𝑇_𝑓−+𝑥−𝑢𝑑𝑢𝑑𝑥_{𝑠−𝑇𝑓0𝑇𝑓0}𝜎𝑢,𝑇_𝑓+𝑥−𝑢𝑑_𝑢𝑊𝑢,𝑇_𝑓−+𝑥−𝑢𝑑𝑥_{𝐾𝑖=1𝑠−𝑇𝑓0𝑇𝑓0}𝛾𝑢,𝑇_𝑓+𝑥−𝑢𝑎_𝑖𝑑𝑁_𝑖(𝑢)𝑑𝑥=−^𝑠_𝑇𝑓𝑓(0,𝑥)𝑑𝑥−^𝑠_𝑇𝑓^𝑇𝑓₀𝐴(𝑢,𝑥−𝑢)𝑑𝑢𝑑𝑥−^𝑠_𝑇𝑓^𝑇𝑓₀𝜎(𝑢,𝑥−𝑢)𝑑_𝑢𝑊()−𝑢,𝑥−𝑢𝑑𝑥_{𝐾𝑖=1𝑠𝑇𝑓𝑇𝑓0}𝛾(𝑢,𝑥−𝑢)𝑎_𝑖𝑑𝑁_𝑖(𝑢)𝑑𝑥. (43) Applying the stochastic Fubini theorem and exchanging the integral order, we obtain ln𝐵𝑇_𝑓,𝑠=−^𝑠_𝑇𝑓𝑓()−0,𝑥𝑑𝑥_{𝑇𝑓0𝑠𝑇𝑓}𝐴()−𝑢,𝑥−𝑢𝑑𝑥𝑑𝑢_{𝑇𝑓0𝑠𝑇𝑓}𝜎(𝑢,𝑥−𝑢)𝑑𝑥𝑑_𝑢𝑊()−𝑢,𝑥−𝑢_{𝐾𝑖=1𝑇𝑓0𝑠𝑇𝑓}𝛾(𝑢,𝑥−𝑢)𝑑𝑥𝑎_𝑖𝑑𝑁_𝑖(𝑢). (44) Therefore, it has 𝐹𝑡,𝑇_𝑓𝑃,𝑠=𝐸_𝑡𝐵𝑇_𝑓,𝑠∣Φ_𝑡=𝐸𝑃_𝑡exp−^𝑠_𝑇𝑓𝑓(0,𝑥)𝑑𝑥−^𝑇𝑓₀^𝑠_𝑇𝑓𝐴(𝑢,𝑥−𝑢)𝑑𝑥𝑑𝑢−^𝑇𝑓₀^𝑠_𝑇𝑓𝜎(𝑢,𝑥−𝑢)𝑑𝑥𝑑_𝑢𝑊()−𝑢,𝑥−𝑢_{𝐾𝑖=1𝑇𝑓0𝑠𝑇𝑓}𝛾(𝑢,𝑥−𝑢)𝑑𝑥𝑎_𝑖𝑑𝑁_𝑖(𝑢)∣Φ_𝑡. (45) Taking out what is known, we obtain 𝐹𝑡,𝑇_𝑓−,𝑠=exp^𝑠_𝑇𝑓𝑓()−0,𝑥𝑑𝑥_{𝑇𝑓0𝑠𝑇𝑓}𝐴()−𝑢,𝑥−𝑢𝑑𝑥𝑑𝑢_{𝑡0𝑠𝑇𝑓}𝜎(𝑢,𝑥−𝑢)𝑑𝑥𝑑_𝑢𝑊()−𝑢,𝑥−𝑢_{𝐾𝑖=1𝑡0𝑠𝑇𝑓}𝛾(𝑢,𝑥−𝑢)𝑑𝑥𝑎_𝑖𝑑𝑁_𝑖(𝑢𝑃)×𝐸_𝑡exp−^𝑇𝑓_𝑡^𝑠_𝑇𝑓𝜎(𝑢,𝑥−𝑢)𝑑𝑥𝑑_𝑢𝑊(𝑢,𝑥−𝑢)∣Φ_𝑡×𝐸𝑃_𝑡exp−^𝐾_𝑖=1^𝑇𝑓_𝑡^𝑠_𝑇𝑓𝛾(𝑢,𝑥−𝑢)𝑑𝑥𝑎_𝑖𝑑𝑁_𝑖(𝑢)∣Φ_𝑡. (46) Applying independent theorem in [10], we have 𝐹𝑡,𝑇_𝑓,𝑠=exp−^𝑠_𝑇𝑓𝑓()−0,𝑥𝑑𝑥_{𝑇𝑓0𝑠𝑇𝑓}𝐴()−𝑢,𝑥−𝑢𝑑𝑥𝑑𝑢_{𝑡0𝑠𝑇𝑓}𝜎(𝑢,𝑥−𝑢)𝑑𝑥𝑑_𝑢𝑊()−𝑢,𝑥−𝑢_{𝐾𝑖=1𝑡0𝑠𝑇𝑓}𝛾(𝑢,𝑥−𝑢)𝑑𝑥𝑎_𝑖𝑑𝑁_𝑖(𝑢𝑃)×𝐸_𝑡exp−^𝑇𝑓_𝑡^𝑠_𝑇𝑓𝜎(𝑢,𝑥−𝑢)𝑑𝑥𝑑_𝑢𝑊(𝑃𝑢,𝑥−𝑢)×𝐸_𝑡exp−^𝐾_𝑖=1^𝑇𝑓_𝑡^𝑠_𝑇𝑓𝛾(𝑢,𝑥−𝑢)𝑑𝑥𝑎_𝑖𝑑𝑁_𝑖(𝑢), (47) where the first conditional expectation is 𝐸𝑃_𝑡exp−^𝑇𝑓_𝑡^𝑠_𝑇𝑓𝜎(𝑢,𝑥−𝑢)𝑑𝑥𝑑_𝑢𝑊()1𝑢,𝑥−𝑢=exp2^𝑇𝑓_𝑡^𝑠_𝑇𝑓𝜎()×𝑢,𝑥−𝑢𝑑𝑥_𝑠𝑇𝑓𝜎𝑢,𝑦−𝑢𝑐𝑥−𝑢,𝑦−𝑢𝑑𝑦𝑑𝑢. (48) The conditional expectation of the increments of jump process is 𝐸𝑃_𝑡exp−^𝐾_𝑖=1^𝑇𝑓_𝑡^𝑠_𝑇𝑓𝛾(𝑢,𝑥−𝑢)𝑑𝑥𝑎_𝑖𝑑𝑁_𝑖(𝑢)=exp^𝐾_𝑖=1^𝑇𝑓_𝑡𝜆_𝑖exp−𝑎_𝑖^𝑠_𝑇𝑓𝛾(𝑢,𝑥−𝑢)𝑑𝑥−1𝑑𝑢. (49) Thus, the price of futures is 𝐹𝑡,𝑇_𝑓,𝑠=exp−^𝑠_𝑇𝑓𝑓(0,𝑥)𝑑𝑥−^𝑇𝑓₀^𝑠_𝑇𝑓𝐴()−𝑢,𝑥−𝑢𝑑𝑥𝑑𝑢_{𝑡0𝑠𝑇𝑓}𝜎(𝑢,𝑥−𝑢)𝑑𝑥𝑑_𝑢𝑊()−𝑢,𝑥−𝑢_{𝐾𝑖=1𝑡0𝑠𝑇𝑓}𝛾(𝑢,𝑥−𝑢)𝑑𝑥𝑎_𝑖𝑑𝑁_𝑖(𝑢)+12^𝑇𝑓_𝑡^𝑠_𝑇𝑓𝜎()×𝑢,𝑥−𝑢_𝑠𝑇𝑓𝜎𝑢,𝑦−𝑢𝑐𝑥−𝑢,𝑦−𝑢𝑑𝑦𝑑𝑥𝑑𝑢×exp^𝐾_𝑖=1^𝑇𝑓_𝑡𝜆_𝑖exp−𝑎_𝑖^𝑠_𝑇𝑓𝛾(𝑢,𝑥−𝑢)𝑑𝑥−1𝑑𝑢. (50) Rearranging (50), we obtain the futures price expressed as (41). It completes the proof.

Acknowledgments

Thanks are due to the referees whose meaningful suggestions are very helpful to revise the paper. This work is supported by the Fundamental Research Funds for the Central Universities (JBK130401).