THE MINIMAL MARTINGALE MEASURE FOR THE PRICE PROCESS WITH POISSON SHOT NOISE JUMPS

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1 Communications on Stochastic Analysis Vol. 5, No. 4 11) Serials Publications TH MINIMAL MARTINGAL MASUR FOR TH PRIC PROCSS WITH POISSON SHOT NOIS JUMPS JUN YAN* Abstract. In this article, we consider the problem of the minimal martingale measures for the price process with Poisson shot noise jumps. We give out the minimal martingale measure for this price process by the exponential martingale method. 1. Introduction Let St), t T be a price process which is defined on probability space Ω,F,F,P), where F t denotes the completion of σsu), u t), F F t, t T, and St) evolves as St) : exp bt+σbt) 1 Nt) σ t 1+aHt T k,y k )) 1.1) b, σ are positive constants, a is a constant, Bt), t T is a standard Brown motion, Y k,k 1 are a sequence of i.i.d random variables, their common distribution is F ), T k,k 1 are the jump times of the Poissonprocess Nt), t T with non random intensity process λt), and we assume T λu)du < +, the bivariate function Ht,y) is nonnegative, and Ht,y) for t <, and we make an assumption that for any y, H,y) does not increase. We also assume that aht T k,y k ) > 1, otherwise St) may take negative value, which conflicts with the reality. By employing the random measure, we can rewrite St) by St) exp bt+σbt) 1 σ t+ exp bt+σbt) 1 σ t+ Nt) log1+aht T k,y k )) 1.) log1+aht u,y))ndy,du) where Ndy,du) is a Poisson random measure with intensity λu)fdy)du, and we denote Ñdy,du) Ndy,du) λu)fdy)du, Ñdy,du) is usually called martingale measure. In financial theory, we usually need to consider the discounted Received ; Communicated by J. Xiong. Mathematics Subject Classification. Primary 6H3; Secondary 6J75, 91B4. Key words and phrases. Minimal martingale measure, Poisson shot noise process, exponential martingale. * Research supported by Youth Foundation of Tianyuan MathematicsNo ). 671

2 67 JUN YAN price process which is defined by Ŝt) St)e rt with respect to price process S, where r denotes the constant interest rate. If a, the price process 1.1) decreases into the following form S B t) : exp bt+σbt) 1 σ t 1.3) whichisusuallycalledgeometricbrownmotion, andweknows B t)isacontinuous stochastic process, however, in financial market, the price process may take jumps, so we can extend S B t) into S J t)[1]) which evolves as S J t) : exp bt+σbt) 1 Nt) σ t 1+Y k ) 1.4) where Y j, j 1 denote the size of jumps, and Nt) denotes the time of jump before t. In reality, the influence of the jumps should fade away as time goes, in order to reflect this effect, we use Ht T k,y k ) to replace Y k, indeed, due to the fact that Ht,y) does not increase with respect to the first variable, the influence of Y k fades away as time goes. The part Nt) log1+aht T k,y k )) is usually called Poisson shot noise process. Recall that if any probability measure Q P, and satisfies that Ŝ is a martingale under Q, then we usually call Q the equivalent martingale measure. It is obvious that S is a price process with jumps, according to the theory of mathematical finance, the market should be incomplete, and the contingent claim is not always attainable, while the equivalent martingale measure is not unique, how to select an equivalent martingale measure as the pricing measure is a problem. Many principles such as minimal entropy martingale measure[4], [5], [3]), minimal Hellinger martingale measure[18]), q-optimal martingale measure[8], [11]) are employed to choose a equivalent martingale measure. Recently, many literature is interested in minimal martingale measures of all kinds of processes. Schweizer [17]) studies the minimal martingale measure for the semimartingale, he provides several characterizations of the minimal martingale measure. Takuji[19]) studies the minimal martingale measure for the jump diffusion process and Paola[15]) discusses the pure jump case. Altmann, Schmidt and Winfried[1]) study the minimal martingale measure of shot noise model, for detail, they discuss the following price process S h t) : exp bt+σbt) 1 Nt) σ t 1+Y k ht T k )) 1.5) which is a particular case of our model 1.1) by taking Ht,y) yht), where the function h ) is nonnegative and non increasing on the positive real line. Motivated by above studies, in this article, we establish the minimal martingale measure for the price process S with Poisson shot noise jumps by constructing an exponential martingale, and the definition of the minimal martingale measure is as follows.

3 MMM FOR PRIC PROCSS WITH POISSON SHOT NOIS JUMPS 673 Definition 1.1. An equivalent martingale measure Q is called minimal if any square integrable P martingale which is orthogonal to the martingale part of Ŝ under P remains a martingale under Q, and Q is usually called minimal martingale measure. Now let us define the probability measure Q by dq dp Ft Zt) exp σgu)dbu)+ c t u,y)ht u,y)ndy,du) : exp σgu)dbu)+ c t u,y)ht u,y)ndy,du)) where G ) which is nonnegative and c t, ) satisfy the following condition: H) : T G u)du+expc T Y 1 )H,Y 1 )) < + 1.6) where c T y) sup u [,T] c T u,y). As follows, we introduce a lemma in which we construct an exponential martingale, the lemma plays a key role in this paper, and the proof will be postponed for conciseness. The process Zt) in 1.6) is not explicit, with the help of the following key lemma, we can give a clear expression for Zt). Lemma 1.. If there exists a δ such that e δgtu,y)h,y) λu)fdy)du < +, for all t [,T], where g t u,y) is a non random function. Then for any Borel measurable function θ ) satisfying sup t T θt) < δ, we have e θu)dlu)) exp where Lt) Nt) g tt k,y k )Ht T k,y k ) ) e θu)gtu,y)ht u,y) 1 λu)fdy)du 1.7) g t u,y)ht u,y)ndy,du). Take θu) 1, g t, ) c t, ) in above lemma, then we can have ) exp σ Gu)dBu) + c t u,y)ht u,y)ndy,du) σ exp G u)du+ by noting the fact that exp σ ) 1.8) e ctu,y)ht u,y) 1 λu)fdy)du ) σ Gu)dBu) exp G u)du so we can give a explicit expression for Zt), that is, Zt) exp σ Gu)dBu) + c t u,y)ht u,y)ndy,du) σ G u)du 1.9) ) e ctu,y)ht u,y) 1 λu)fdy)du 1.1)

4 674 JUN YAN by Itô formula [16]), we can achieve that Zt) 1+σ Zu )Gu)dBu)+ Zu ) e ctu,y)ht u,y) 1)Ñdy,du) 1.11) Recall that, in the case of Brown motion, that is, for the price process S B t), we can define the equivalent probability measure by dq B dp Ft Z B t) 1.1) with Z B t) satisfying the stochastic differential equation dz B t) σat)z B t)dbt) 1.13) where at) makes the corresponding discounted price process Ŝ B t) : e rt S B t) exp bt+σbt) 1 σ t rt a martingale under Q B, for detail, at) satisfies indeed, by Itô formula, we have by Girsanov-Meyer theorem[16]) 1.14) b r)t+σ au)du, t T 1.15) dŝbt) b r)ŝbt)dt+σŝbt)dbt) 1.16) dl B t) :σŝbt)dbt) Z B t)) 1 d σŝbt)dbt) σ at)ŝbt)dt [ Z B,σ ] Ŝ B u)dbu) t 1.17) is a Q B local martingale, where [, ] denotes the quadratic variation[16]), and we can rewrite dŝbt) by dŝbt) b r)ŝbt)dt+σ at)ŝbt)dt+dl B t) 1.18) so under the condition 1.15), Ŝ B t) is a Q B local martingale, and under some slight condition, we can get ŜBt) is truly a Q B martingale, motivated by above discussion, we assume, in our model 1.1), Zt) satisfies the follow stochastic differential equation dzt) γt)zt )σdbt) + dht)) 1.19) where γ ) is an undetermined function, and Ht) Ht u,y)ñdy,du). Compare 1.11) and 1.19), we know that Gu) γu) e ctu,y)ht u,y) 1.) 1 γu)ht u,y) that is Gu) γu) c t u,y) log1+γu)ht u,y)) Ht u,y) 1.1)

5 MMM FOR PRIC PROCSS WITH POISSON SHOT NOIS JUMPS 675 and we can claim that γ ) satisfies the follow condition which is usually called martingale condition: b r)t+σ γu)du+a Ht u,y)+γu)h t u,y))λu)fdy)du 1.) with t T, we denote Q in 1.6) which is determined by γu) satisfying 1.) by Q, that is dq dp Ft exp σ σ γu)dbu) + γ u)du log1+γu)ht u,y))ndy,du) γu)ht u, y)λu)fdy)du where γ ) is determined by 1.). For Q, we have the following lemma. 1.3) Lemma 1.3. Under the conditions H) and 1.), the discounted price process Ŝt) is a martingale under the measure Q. The proof of Lemma 1.3 is postponed to the next section.. Main Result In this section, we give out our main result, and the proof of Lemma 1. and Lemma 1.3 are also presented. Here is our main result. Theorem.1. Under the condition H), the equivalent martingale measure Q is the minimal martingale measure with respect to the price process S. Proof. By Itô formula Ŝt) 1+σ +b r) 1+σ +b r) Ŝu )dbu) + a Ŝu )du Ŝu )dbu) + a Ŝu )du+a Ŝu )Ht u,y)ndy,du) Ŝu )Ht u,y)ñdy,du) Ŝu )Ht u, y)λu)fdy)du.1) obviously, the part σ Ŝu )dbu)+a Ŝu )Ht u,y)ñdy,du) is a P martingale. Let L be any square integrable P martingale satisfying )) Lt) σ Ŝu )dbu) + a Ŝu )Ht u,y)ñdy,du).) i.e., L is orthogonal to σ Ŝu )dbu)+a Ŝu )Ht u,y)ñdy,du), then we have Lt)Zt)).3) that is Q Lt)).4)

6 676 JUN YAN i.e., L is a Q local martingale, and furthermore L is truly a martingale, indeed we know sup Z t)) t T exp σ G u)du+ exp σ G u)+λu))du+ ) λu)fdy)du e ctu,y)ht u,y) 1 e ctu,y)h,y) λu)fdy)du.5) therefore, by the condition H) furthermore, L is a square integrable martingale, i.e. by Hölder inequality, we can conclude that sup Lt)Zt)) t T sup Z t)) < +.6) t T sup L t)) < +.7) t T sup t T L t)) sup t T Z t)) < +.8) i.e., Lt)Zt) is bounded under P, equivalently, Lt) is bounded under Q, so we have proved that L is truly a martingale under Q. The proof of Lemma 1.. Proof. Firstly, we can infer Lt) g t u,y)ht u,y)λu)fdy)du,g t,p is a martingale, where G t σns),s t σy k,k Nt). Indeed Lt) G s ) Ls)+ Nt) Ns) and by total probability formula Nt) Ns) m m1 g t T k+ns),y k+ns) )H ) t T k+ns),y k+ns) G s.9) g t T k+ns),y k+ns) )H ) t T k+ns),y k+ns) G s I Nt) Ns)m g t T k+ns),y k+ns) )H ) ) t T k+ns),y k+ns) G s.1)

7 MMM FOR PRIC PROCSS WITH POISSON SHOT NOIS JUMPS 677 then by Fubini theorem m I Nt) Ns)m g t T k+ns),y k+ns) )H ) ) t T k+ns),y k+ns) G s m1 mk I Nt) Ns)m g t T k+ns),y k+ns) )H ) ) t T k+ns),y k+ns) G s I Nt) Ns) k g t T k+ns),y k+ns) )H ) ) t T k+ns),y k+ns) G s I Tk t s I,t s] u) s g t s T k,y)h t s T ) k,y Fdy) G s ) g t s u,y)ht s u,y)fdy)λu+s)du G s g t u,y)ht u,y)λu)fdy)du.11) which implies Lt) g t u,y)ht u,y)fdy)du,g t,p is a martingale, where T k infu : Ns + u) Ns) k. Let Dt) θu)dlu), then by Itô formula and martingale properties of Lt) g t u,y)ht u,y)λu)fdy)du,g t,p, we can get that e Dt)) 1+ e Du )) θu)g t u,y)ht u,y)λu)fdy)du ) + e DTk 1) pθt k )g t T k,y k )Ht T k,y k )) 1+ where px) e x 1 x, so we have e θu)dlu)) exp ) I Tk t e Du )) ) e θu)gtu,y)ht u,y) 1 λu)fdy)du Remark.. From the proof of Lemma 1., we know ) e θu)gtu,y)ht u,y) 1 λu)fdy)du. g t u,y)ht u,y)ndy,du) λu)fdy)du).1).13) g t u,y)ht u,y)ñdy,du).14) is a P martingale, that is why we call Ñdy,du) martingale measure. The proof of Lemma 1.3.

8 678 JUN YAN Proof. We know from the proof of Theorem.1 where M H t) a a Ŝt) 1+M B t)+m H t)+ct).15) M B t) σ Ŝu )Ht u,y)ñdy,du) Ct) b r) +a Ŝu )dbu) σgu)du).16) ) Ŝu )Ht u,y) e ctu,y)ht u,y) 1 λu)fdy)du Ŝu )du+σ Ŝu )Gu)du Ŝu )Ht u,y)e ctu,y)ht u,y) λu)fdy)du.17).18) by Girsanov-Meyer theorem[16]), we know M B t) is a Q local martingales, for M H t), according to the formula of integration by parts[16]), we have Zt)M H t) σ +a +a Zu )dm H u)+ M H u )Zu )Gu)dBu) M H u )dzu)+[z ),M H )] t Zu )Ŝu )Ht u,y)ectu,y)ht u,y) Ñdy,du) M H u )Zu ) e ctu,y)ht u,y) 1)Ñdy,du).19) which implies that Zt)M H t) is a P local martingale, equivalently, M H t) is a Q local martingale, for detail see [1], so if we need to make Ŝ a local martingale under Q, we should insure that Ct), t T.) take 1.1) in mind, we know the above equation is equivalent to 1.). In order to confirm that Ŝt) is a martingale, we need to check ) Q sup Ŝt) < +, t T.1) t T according to Theorem 51 in Chapter I of [16], indeed ) Q sup Ŝt) t T e b r T Q sup t T ) NT) e σbt) Q exp log1+ a H,Y k )).)

9 MMM FOR PRIC PROCSS WITH POISSON SHOT NOIS JUMPS 679 by maximal inequality for martingales and noting that σbt) σ G u)du is a Q martingale, we have ) Q sup exp σbt) σ G u)du t T e ))) 1+ sup q σbt) Q σ G u)du e 1 t T e )).3) 1+ sup exp σbt) σ Q G u)du e 1 t T e ) T 1+exp σ G u)du < + e 1 where qx) x + e x and x + : maxx,, so we can obtain ) Q sup e σbt) exp t T T σ e e 1 exp G u)du σ T Q G u)du sup exp σbt) σ 3σ T +exp G u)du t T ) G u)du ) furthermore, by Hölder inequality NT) Q exp log1+ a H,Y k )) NT) ZT)exp log1+ a H,Y k )) NT) Z T)) exp log1+ a H,Y k )) < +.4).5) by Lemma 1. and noting the condition H), we know Z T)) exp σ T T G u)du+ T exp σ G u)+λu))du+ ) λu)fdy)du e ctu,y)ht u,y) 1 T e ctu,y)h,y) λu)fdy)du < +.6)

10 68 JUN YAN and by Lemma 1. again, we achieve that NT) exp log1+ a H,Y k )) T.7) exp λu)du a H,y)+ a H,y))Fdy) < + so we have NT) Q exp log1+ a H,Y k )) < +.8) finally,.4) combining with.8) implies.1), and we complete the proof. Acknowledgment. The author would like to thank the referees for their helpful suggestions and comments. Further, we thank the support of the Youth Foundation of Tianyuan Mathematics of ChinaNo ). References 1. Altmann, T., Schmidt, T., and Winfried, S.: A Shot Noise Model for Financial Assets, International Journal of Theoretical and Applied Finance 1).. mbrechts, P., Kluppelberg, C., and Mikosch, T.: Modeling xtremal vents for Insurance And Finance, Springer Verlag, sche, F., and Schweizer, M.: Minimal entropy preserves the Lévy property: how and why, Stoch. Proc. Appl ) Frittelli, M.: The minimal entropy martingale measure and the valuation problem in incomplete markets, Math. Finance 1 ) Fujiwara, T. and Miyahara, Y.: The minimal entropy martingale measure for geometric Lévy processes, Finance Stoch. 7 3) Fuqing G. and Jun, Y.: Functional large deviations and moderate deviations for Markovmodulated risk models with reinsurance, J. Appl. Probab. 45 8) Goll, T. and Rüschendorf, L.: Minimax and minimal distance martingale measures and their relationship to portfolio optimization, Finance Stoch. 5 1) Grandits, P.: The p-optimal martingale measure and its asymptotic relation with the minimal ertropy martingale measure, Bernoulli ) He, S. W., Wang, J. G., and Yan, J. A.: Semimartingale Theory and Stochastic Calculus, Science Press & CRC Press Inc., Beijing & Boca Raton, Jacod, J. and Shiryaev, A. N.: Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, Jeanblanc, M., Klöppel, S., and Miyahara, Y.: Minimal f q martingale measures for exponential Lévy process, Ann. Appl. Prob. 17 7) Merton, R. C.: Option pricing when underlying stock return are discontinuous, J. Financial. con ) Miyahara, Y.: Pricing model and related estimation problems, Asia-Pacific Financial Markets 8 1) Möller, M.: Pricing PCS options with use of sscher transforms, in: Afir Colloquium Nürnberg, Oct ). 15. Paola, T.: Minimal martingale measure: Pricing and hedging in a pure jump model under restriced information, Nonlinear Anal. 71 9) e1771 e Protter, P.., Stochastic Integration and Differential quation, Springer-Verlag, Berlin, 5.

11 MMM FOR PRIC PROCSS WITH POISSON SHOT NOIS JUMPS Schweizer, M.: On the minimal martingale measure and the Föllmer-Schweizer decomposition, Stoch. Ana. Appl ) Tahir, C. and Christophe, S.: Comparing the minimal Hellinger martingale measure of order q to the q-optimal martingale measure, Stoch. Ana. Appl ) Takuji, A.: Minimal martingale measures for jump diffusion processes, J. Appl. Prob. 41 4) Terence, C.: Pricing contingent claims on stocks driven by Lévy processes, Ann. Appl. Prob ) Thorsten, S. and Winfried, S.: Shot-noise processes and the minimal martingale measure, Stat. Proba. Lett. 9 7) Jun Yan: School of Mathematical Sciences, Yangzhou University, Yangzhou 5, P.R.China -mail address: yanjunwh@gmail.com