Stochastic Volatility Jump-Diffusion Model for Option Pricing

Journal of Mathematical Finance, 2011, 1, 90-97 doi:10.4236/jmf.2011.13012 Published Online November 2011 (http://www.SciRP.org/journal/jmf)

Nonthiya Makate, Pairote Sattayatham School of Mathematics, Suranaree University of Technology, Nakhon Ratchasima, Thailand E-mail: [email protected], [email protected] Received July 26, 2011; revised September 2, 2011; accepted September 15, 2011

Abstract

An alternative option pricing model is proposed, in which the asset prices follow the jump-diffusion model with square root stochastic volatility. The stochastic volatility follows the jump-diffusion with square root and mean reverting. We find a formulation for the European-style option in terms of characteristic functions of tail probabilities.

Keywords: Jump-Diffusion Model, Stochastic Volatility, Characteristic Function, Option Pricing

1. Introduction work by incorporating jumps in volatility and their model is given by Let ,, P be a probability space with filtration SS dddSStt tvWSYN tt  tt dt, (3)   t 0tT. All processes that we shall consider in this dddvvtvWZ   vv dN. section will be defined in this space. An asset price ttttv  t model with stochastic volatility has been defined by Eraker et al. [3] developed a likelihood-based estima- Heston [1] which has the following dynamics: tion strategy and provided estimates of parameters, spot dddSS tvWS , (1) volatility, jump times, and jump sizes using S&P 500 and tt tt Nasdaq 100 index returns. Moreover, they examined the v volatility structure of the S&P and Nasdaq indices and ddvvtv   dW, tttt indicated that models with jumps in volatility are pre- where S is the asset price, is the rate of return of t   ferred over those without jumps in volatility. But they the asset, vt is the volatility of asset returns,   0 is did not provide a closed-form formula for the price of a a mean-reverting rate,   is the long term variance, S v European call option.   0 is the volatility of volatility, Wt and Wt are In this paper, we would like to consider the problem of standard Brownian motions corresponding to the proc- finding a closed-form formula for a European call option esses St and vt , respectively, with constant correlation where the underlying asset and volatility follow the Model  . In 1996, Bate [2] introduced the jump-diffusion sto- (3). This formula will be useful for option pricing rather chastic volatility model by adding log normal jump Yt than an estimation of it as appeared in Eraker’s work. to the Heston stochastic volatility model. In the original The rest of the paper is organized as follows. In Sec- formulation of Bate, the model has the following form: tion 2, we briefly discuss the model descriptions for the ddSS tvdWSYNS  dS, (2) option pricing. The relationship between stochastic dif- tt tt ttt ferential equations and partial differential equations for ddvvtv   dWv , the jump-diffusion process with jump stochastic volatil- tttt ity is presented in Section 3. Finally, a closed-form for- S where Nt is the Poisson process which corresponds to the mula for a European call option in terms of characteristic underlying asset St , Yt is the jump size of asset price functions is presented. return with log normal distribution and St  means that there is a jump the value of the process before the jump 2. Model Descriptions is used on the left-hand side of the formula. Moreover, in 2003, Eraker Johannes and Polson [3] extended Bate’s It is assumed that a risk-neutral probability measure 

2 exists, the asset price St under this measure follows a jump- Nt Poisson random measure of the process R Z with diffusion process, and the volatility vt follows a pure mean t n n1 reverting and square root diffusion process with jump, i.e. 2 our models are governed by the following dynamics: intensity measure Z ddrt.

Let Uxx 12,  be a bounded real-valued function and dddSSr SS mtv W  SYNdS, (4) tttt ttt twice continuously differentiable with respect to x1 and x2 and ddvvtWZN   vddvv  , ttt ttt  12    1  2   uxx12,, t EUXTT , X X t xX1 , t x 2(6)  S v     where St , vt ,  ,  ,  , Wt and Wt are defined as S in Bate’s model, r is the risk-free interest rate, Nt and By the two dimensional Dynkin formula [5], u is a v Nt are independent Poisson processes with constant in- solution of the partial integro-differential equation (PIDE) S v tensities  and  respectively. Yt is the jump size uxx 12,, t of the asset price return with density Y y and 0,,uxx t t 12 EY t :m  and Zt is the jump size of the volatility with density  z . Moreover, we assume that the jump (1) Z  ux yxt,, uxx ,, t y d y S v  12 12Y  processes Nt and Nt are independent of standard  S v Brownian motions Wt and Wt . (2)  uxx12,, zt uxx 12 ,,d tZ z z 3. Partial Integro-Differential Equations 

subject to the final condition uxxT12,, Uxx 1 ,2  . Consider the process X  XX12,   where X 1 ttt t The notation  is defined by 2 and X t are processes in  and satisfy the following equations: uxx 12,, t uxx12 ,, t Au x12,, x t f 1  f 2 112    12   1 xx12 d,,,,X tttf11XXtd tg XXtW ttdt (5) 22 11  1 2 uxx12,, t uxx12,, t  XYtt dNt gg112  g2 2 x1 xx12 21212     2 d,,,,Xttttt f22XXtd tg XXtWdt 2 1 2  uxx12,, t 2  g2 2  ZNttd 2 x2 where f112,,gf and g2 are all continuously differenti- (7) 1 2 able, Wt and Wt are standard Brownian motions 12  1 2 with Corrd, WttdW   , Nt and Nt are 4. A Closed-Form Formula for the Price of a  European Call Option independent Poisson processes with constant intensities 1 2  and  respectively. Let C denote the price at time t of a European style call Since every compound Poisson process can be repre- option on the current price of the underlying asset St sented as an integral form of a Poisson random measure with strike price K and expiration time T . [4] then the last term on the right hand side of (5) can be The terminal payoff of a European call option on the written as follows: underlying stock St with strike price K is 1 ttNt 11 1 1     max SKT  ,0 . X ssYNdd s XY nnX s qJ Q sqd, 00n1  This means that the holder will exercise his right only 2 ttNt 2 if SKT  and then his gain is SKT  . Otherwise, if ZssddNZrJs n Rdr,  SKT  , then the holder will buy the underlying asset 00n1  from the market and the value of the option is zero. where Yn are i.i.d. random variables with density Y  y Assuming the risk-free interest rate r is constant 1 and NJtQ is a Poisson random measure of the process over the lifetime of the option, the price of the European 1 Qtn Ywith intensity measure Y ddqt, Zn are call at time t is equal to the discounted conditional n1 i.i.d. random variables with density Z z , and J R is a expected payoff

function of the stock log-return L denoted by CSvtKTtt,,;, t rT t CLvtkT ,,;,  CeLt ,, vteT;,k eE max SKSvTtt ,0 , tt t

 ln St ln K rT t  Ce,,; vtet , T e SPSSvST T tt,d T KPSSvS T tt ,d T KK  CStt,,v tKT;,, 1  i.e.,  SSPSSvS,d tTTttTrT t    rT t L e S ClvtkT ,,;,  e E maxeT K,0 L lv , v t K      tt  rT() t and satisfies the following PIDE:  Ke P (,)d STtt S v S T K C 0,ClvtkT ,;, 1  t   SStTP (,)dSTStvtST  S Cl yvtkT,,;, Cl ,,;, vtkT y d y ESSv[|,]TttK  Y     KerT() t PSSvS(,)d v    Ttt T  Clv,,;, ztkTCl ,,;, vtkTZ z d z K  rT() t SPSvtKTttt12(,,;,) Ke PSvtKT (,,;,)tt (10) Here the operator  as in (7) is defined by  (8)  S 1 CC  ClvtkT,,;,  r m v   v where E is the expectation with respect to the risk- 2 lv 22 neutral probability measure, P SSvTtt,  is the cor- 1 CC vv2  responding conditional density given Svtt,  and 2 l lv 2   1 2  C   vr2 C PSvtKT1 tt,,;,   SPT S T Sv tt, d S T E S T Sv tt, 2 v K In the current state variable, the last line of (8) becomes  l kr Tt  Note that P1 is the risk-neutral probability that C lvtkT,,;, eP12 lvtkT ,,;, e P lvtkT,,;, SKT  (since the integrand is nonnegative and the inte- (11) gral over 0, is one), and finally that  lk where PlvtkTjj ,,;, : PevteT ,,; , , j 1,2.   PSvtKT,,;,  P SSv, d S The following lemma shows the relationship between P1 2 tt   T tt T K  and P2 in the option value of (11). Pr ob(SKSvTt | ,t ) Lemma 1 The functions P and P in the option is the risk-neutral in-the-money probability. Moreover, 1 2 value of (11) satisfy the following PIDEs ESSv,e rT t S for t  0 .  Ttt t PP 11  0,PlvtkTv1 ,;,  Assume that the asset price St and the volatility vt tl satisfy (4), we would like to compute the price of a P European call option with strike price K and maturity vrmP1  S  v 1 T . To do this, we make a change of variable from St to L ln S, i.e. where S satisfies (4) and its inverse Sy tt t ePlyvtkTy1,,:,1 Y dy Lt   St  e. Denote k ln K the logarithm of the strike  price. By the jump-diffusion chain rule, ln S satisfies t and subject to the boundary condition at expiration time the SDE tT ; v SSt S  dSln ttt r m d t  vW d  ln 1  YNt d t(9) PlvTkT1  ,, ;,  1lk (12) 2 moreover, P satisfies the equation Applying the two-dimensional Dynkin formula [5] for 2  the price dynamics (9) and volatility v in system (4), P2 t 0,PlvtkTrP,;,, we obtain the value of a European-style option, as a t 22

and subject to the boundary condition at expiration time 2) The characteristic function f2 is given by tT ; ,  f22lvtxt,,;,  exp g  vh2  ixl r PlvTkT2 ,, ;, 1lk , (13) 22  The operator  is defined by  e 2 1  22  where h2   ,  flvtkT,,;, 2       e 2  22 22  2 S 11fff :rmv    v  v2 S 22lvl g2  rmix  r

2 2  f 1 2  f   vvrf   (14)  1 e 2 2  22  lv 2 v 2ln 1    2 2 22 S 2  f lyvtkTflvtkT,,;,  ,,;,Y yy d    v Sixyeyye1d   vzh2  1  zdz  f lv,,;, ztkT f lvTkT ,,;,Z z d z. YZ       ix  Note that 1lk 1 if lk and otherwise 10lk  . 2

The following lemma shows how to calculate the 22   and 22ix ix 1 . functions P1 and P2 as they appeared in Lemma 1.   In summary, we have just proved the following main Lemma 2 The functions P1 and P2 can be calculated theorem. by the inverse Fourier transforms of the characteristic Theorem 3 The value of a European call option of (4) is function, i.e. kr Tt ixk  l    eflvtkT,,;,  C lvtkT,,;, eP12 lvtkT ,,;, e P lvtkT,,;,  11 j  PlvtkTj ,,;,  Re  d x, 2 π 0  ix      where P1 and P2 are given in Lemma 2. for j 1, 2 with Re[ ] denoting the real component of a complex number. By letting  Tt. 5. Conclusions 1) The characteristic function f is given by 1 This paper has proposed asset price dynamics to accom- modate both jump-diffusion and jump stochastic vola- f11 lvtxt,,;,  exp g   vh1  ixl, tility. Under this proposed model, an analytical solution where is derived for a European call option via the characteris- 221 tic function. 11e 1 h1   2 1 11   11  e 6. Acknowledgements

SS This research is (partially) supported by The Centre of g1  rmixm   Excellence in Mathematics, the Commission on Higher  Education (CHE).  1 e 1  11   2ln 1   Address: 272 Rama VI Road, Ratchathewi District, 2 11  21 Bangkok, Thailand.   7. References S ix1 y  eyy1d Y  ,  [1] S. L. Heston, “A Close Form Solution for Options with  Stochastic Volatility with Applications to Bond and Cur- v ezzzh1 1d     Z  rency Options,” The Review of Financial Studies, Vol. 6,  No. 2, 1993, pp. 327-343. doi:10.1093/rfs/6.2.327  ix 1  [2] D. Bates, “Jump and Stochastic Volatility: Exchange Rate 1  Processes Implicit in Deutsche Mark in Options,” Review

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Appendix ePl  lvtkT , , ; , ekrTt P lvtkT,,;,   12 Proof of Lemma 1. We plan to substitute (11) into (10).  ly e eP11 l yvtkT,,;, P l yvtkT ,,;,  Firstly, we compute    ePll l yvtkT,,;,  eP lvtkT ,,;,  C PP  11 eel 12krTt  rekrTtP 2 krTt  tt  t e P11 l yvtkT,,;, P lvtkT ,,;,    ly C llP12krTtP ee1,,;, PlyvtkT  eePe 1  ll1 l  l  eP1  l yvtkT,,;, PlvtkT1 ,,;,   C l PP12krTt ee ekrTt P l yvtkT ,,;, P lvtkT ,,;,  vvv   21 and 2C 22PP P eeePelll112  krTt 2 Clv , ztkT ,;, ClvtkT ,,;,  ll22l 1 l2  l krTt  eP12 lv, ztkT ,;, e P lv,  ztkT ,;, 2  22     C llPP11krTt P2 eee ePl  lvtkT,,;, ekrTt P lvtkT,,;, lv  lv  v  lv 12  l 2  22 e P lv, ztkT ,;, P lvtkT ,,;,  C l PP12krTt 11 22ee 2, vv v krTt  e P22 lv, ztkT ,;, P lvtkT ,,;,  Cl yvtkT,,;, ClvtkT ,,;,  We substitute all terms above into (10) and separate it by assumed independent terms of P and P . This ePlyvtkTly   ,,;,  1 2  1 gives two PIDEs for the risk-neutralized probability for ePlyvtkTkrTt ,,;,  1  PlvtkTj (, ,; , ), j 1,2: (Equation (15))

 2  PPPP11111S 11P 0()2rmv  P11 v  v2   P tlvl22 l 

22PP1  P  vvr11 2 1PeSy1Ply ,v ,;tk ,TP ly ,v ,;tk ,TP  (,,;lvtk ,T ) y dy (15)  2 11 1 1Y  lv  v 2 v  v    Plv1 , ztkT,; , P1 lvtkT , ,; ,Z z d z  subject to the boundary condition at the expiration time 22 2  11PP222  P 2 tT according to (12). vv22   vrP 2 22lvlv By using the notation in (14), PIDE (15) becomes S   P22 l yvtkT,,;,  P lvtkT ,,;,Y y d y PP11   0, PlvtkTv ,;,   tl1  v PlvztkT, ,;, PlvtkT ,,;, z d z P1 S   22Z vrmP  1  v (16) SyePlyvtkTy1,,;, dy  1 Y subject to the boundary condition at the expiration time  tT according to (13). Again, by using the notation in P :,1  PlvtkT ,;, (14), PIDE (16) becomes t  1 P  2  For PlvtkT,,;, : 0[](,,;,) PlvtkTrP22  2  t  PPS 1 P  22  2 P2 0 rP2  r m v v :[](,,; PlvtkT ,). tl2 v t 22

The proof of Lemma 1 is now completed. and

For j 1, 2 the characteristic functions for yygvhix11  ly  eflyvtxtee1,,;,1 1     PlvtkTj ,,;, , yixy eeflvtxt1,,;,1   with respect to the variable k are defined by  Substituting all the above terms into (17) and after ixk  canceling the common factor of f , we get a simplified f jjlvtxT,,;, : e d P  lvtkT ,,;, , 1  form as follows: with a minus sign to account for the negativity of the 1 S  0 g11 vh   r   m  v ix measure dPj . 2

Note that f j also satisfies similar PIDEs 1 2 vvhvx 1    2 f j  flvtxT,,;, 0, (17) 1 22 S t jj vixh   vh   m 112 with the respective boundary conditions S eyyix1 y 1d     Y  ixk   f jjlvtxT,,;, e d P  lvtkT ,,;,   v zh1 ezz1d.Z        eklixk     dk  By separating the order v and ordering the remaining  eixl terms, we can reduce it to two ordinary differential equation (ODEs),  since d,,;,d1dPlvTkTjl  k Hlk    k ld k. 1122 12 hh11 1 ixhix  1 x(19) Proof of Lemma 2 222 1) To solve for the characteristic function explicitly, and letting  Tt be the time-to-go, we conjecture that SS g11 hrmix    m the function f1 is given by  ix 1 y S ey 1dy (20) f11lvtxt,,;,  exp g   vh1   ixl (18)    Y  and the boundary condition  v zh1 ez1d.Z z gh1100  0.   

This conjecture exploits the linearity of the coefficient Let 1  ix 1  and substitute it into (19). We get in PIDE (17). 11222 Note that the characteristic function f1 always exists.  1 hhhixix111221 In order to substitute (18) into (17), firstly we com- 2  pute 1 24422ix ix  1 2 11 f f  h1 2 1  1 2 2 g11 vh  f 1  ixf1  t l 2 24422ix ix  1 f1  f1 2 11   h f x f h1 11 2 1 2 2 v l  2 2  f1  f1 2 1 2 11 11   ixh1 f1 2  hf11  hh1122  lv v 2 

f1  l yvtxt,,;, f1 lvtxt ,,;,  22 where 11ix ix 1 . ixy eflvtxt1,,;,1   By the method of variable separation, we have

2dh1 2 f11 lv, ztxt ,;, f lvtxt ,,;,     d . 11 11  zh1 hh eflvtxt1,,;,  1122 1  

ixk Using partial fractions, we get 11 eflvtkT,,;, PlvtkT ,,;, Rej d x(21) j   2 π  ix 11 1 0  ddh  .    1 for j 1, 2 . 1 hh11 11 1122 To verify (21), firstly we note that Integrating both sides, we obtain Eeixln St  ln K ln SLvv ,  ttt 11 h   1 2 ix lk ln   C . Eeix Lt  k Llvv,d, e PlvtkT ,;,   1  tt j 11  h1  2   eePlvtkTeeixk ixld,,;, ixk ixk  lkk d Using boundary condition h1 (0)0 , we get j    

11  ixk C  ln .  eflvtkTj ,,;, . 11  Then Solving for h1 , we obtain  ixk 11 eflvtkTj ,,;,  22e1 1  Re dx 11 2 π  ix h1   . 0  2 1  11   11  e ixln St  ln K  Eeln SLvv ,  In order to solve g1  explicitly, we substitute h1   11  ttt  Re  dx into (20) and integrate with respect to  on both sides. 2 π   ix  0 Then we get  

SS  ixlk grmixm1   11 e   ExLlv  Re dtt  , v   2 π  ix 1 0   11 1 e  2ln 1    2 2 11 1 11 sin xl k ExLlvd,v    tt S ix1 y 2 π  x eyy1d  0    Y    11 sin x v zh1 Elksgn  d,xL lv v ezz1d Z    tt  2 π  x  0 Proof of 2). The details of the proof are similar to case 1). Hence, we have 11 ElkLlvv sgn tt , 22 f22lvtxt,,;, exp g  vh2  ixl r

ELlvv 1,lk t t where gh22 ,, 2 and 2 are as given in the Lemma.  We can thus evaluate the characteristic functions in  sin x where we have used the Dirichlet formula d1x  explicit form. However, we are interested in the risk-neu-  x  0 tral probabilities Pjj ,1,2 . These can be inverted from the characteristic functions by performing the following and the sgn function is defined as sgnx  1 if x  0 , integration 0 if x  0 and –1 if x  0 .