Exact and Approximated Option Pricing in a stochastic volatility jump ...

Exact and Approximated Option Pricing in a stochastic volatility jump-diffusion model Fernanda D’Ippolitia , Enrico Morettoa,b , Sara Pasqualia , and Barbara Trivellatoa,c a b c

CNR-IMATI, Via Bassini 15, IT - 20133 Milano

University of Parma, Via J.F. Kennedy 6, IT - 43100 Parma

Polytechnic University of Turin, Corso Duca degli Abruzzi 24, IT - 10129 Turin Abstract We extend the stochastic volatility model in Moretto et al. [MPT05] to a stochastic volatility jump-diffusion model. We provide a closed-form solution for the price of European-type options and develop a Monte Carlo method to approximate the price of more complex options. We apply the model to the DJ Euro Stoxx 50 market and test of goodness of fit on European options. Key words: Monte Carlo simulation, option pricing, stochastic volatility jump-diffusion models AMS: 91B28

1

Introduction

Heston [Hes93] introduced a path-breaking methodology to obtain semi-closed formulae of European-style option derivatives when the underlying is characterized, for instance, by stochastic volatility. Bakshi and Madan [BM00] proposed a general technique that encompasses Heston’s result and allows to evaluate a broad class of derivative assets under a general framework. These developments are due to the fact that there is a huge empirical evidence that volatility is far from being constant as in Black and Scholes [BS73]. For instance, in the last decade, there has been convincing evidence that stochastic volatility processes with jumps in returns are important to model index return and volatility (Bates [Bat96]) appropriately, but they are incapable of fully capturing the empirical features of equity index returns and option prices (Bakshi et al. [BCC97], Bates [Bat00] and Pan [Pan02]). As stated in Eraker et al. [EJP03], empirical evidence indicates that the stochastic volatility and jumps in the underlying are not sufficient 1

to reproduce real market behavior of returns such as, for example, Bates [Bat00], Duffie et al. [DPS00] and Pan [Pan02]. One way to tackle this issue is to introduce jumps into the volatility dynamics as well, as in Eraker et al. [EJP03]). We generalize the stochastic volatility model in Moretto et al. [MPT05], that, in the spirit of Heston [Hes93], admits a closed-form solution for European-style options. It is well known that, unfortunately, for more complex options, a closed-form solution does not exist anymore. Following Broadie and Kaya [BK06], we describe a Monte Carlo method to approximate their values. Broadie and Kaya propose an “exact simulation” of the jump-diffusion processes: loosely speaking, their technique consists in generating a sample from the final value of the variance, and then, using Fourier inversion methods, they get a sample from the integral of the variance. These two quantities allow to generate a sample for the stock price. Differently from Broadie and Kaya, we generate the integral of the variance through a rejection sampling algorithm. The method is applied to the DJ Euro Stoxx 50 market and we base the test of goodness of fit on European options. We plan to evaluate barrier and American options in a future work.

2

Stochastic Volatility Jump-diffusion Model

Let (Ω, F, P) a complete probability space where P is the historical measure and consider

t ∈ [0, T ]. We suppose that a bidimensional standard Wiener process W = (W1 , W2 ) and two

compound Poisson processes ZS and Zv are defined. We assume that W1 , W2 , ZS and Zv are

mutually independent. Under P, the dynamics of the stock price S(t) is

where

p

h i p dS(t) = S(t− ) µ dt + σS dW1 (t) + ξ v(t) dW2 (t) + dZS (t), ,

(1)

v(t) is the volatility process whose dynamics is specified later, and the parameters

µ, σS , ξ ∈ R are real constants. The process ZS (t) has constant intensity λ > 0 (annual

frequency of jumps) and log-normal distribution of jump sizes, that is, denoting with JS the
relative jump size, then log(1 + JS ) is distributed according to the N log(1 + jS ) − 12 δS2 , δS2 law, where jS is the unconditional mean.

Note that JS > 0 implies that the stock price remains positive for all t ∈ [0, T ].

The process v(t) evolves according to

dv(t) = k∗ (θ ∗ − v(t)) dt + σv

p

v(t) dW2 (t) + dZv (t),

(2)

where Zv (t) has the same intensity λ of ZS with exponentially distributed jump size Jv with mean jv , and the parameters k∗ , θ ∗ and σv are real constants. Variance v(t) is a mean reverting process where k∗ , θ ∗ and σv are respectively the speed of 2

adjustment, the long-run mean and the variation coefficient. If k∗ , θ ∗ , σv > 0, 2k∗ θ ∗ ≥ σv2 , v(0) ≥ 0 and Jv > 0, then the process v(t) is positive for all t ∈ [0, T ], with probability 1

(Lamberton and Lapeyre [LL97]). Jumps in both asset price and variance occur concurrently and the counting process will be denoted by Nt .

Note that the model (1)-(2) is a generalization of the stochastic volatility model presented in Moretto et al. [MPT05], obtained introducing jumps in the dynamics of both the underlying and the volatility. It is well known that the market is incomplete. Consequently, the principle of absence of arbitrage does not lead to a uniquely defined price. One obtains actually an entire range of prices and the structure of the investors choice has to come into play to determine the pricing measure or, equivalently, the market price of risk. As in Duffie et al. [DPS00], we make an ad-hoc choice of the market price of risk such that a risk-neutral measure Q equivalent to P exists. This choice involves the following form of the jump transform ζ(c1 , c2 ) of the bivariate jump-size distribution (JY , Jv ) 1 ζ(c1 , c2 ) = 2



    1 2 1 2 2 1 , exp log(1 + jS ) − δS c1 + δS c1 + 2 2 1 − jv c2

(3)

where c1 , c2 ∈ R. Under Q, the processes S and v evolve according to

h i p dS(t) = S(t− ) (r − λjS ) dt + σS dW1 (t) + ξ v(t) dW2 (t) + dZS (t) , dv(t) = k∗ (θ ∗ − v(t)) dt + σv

where r is the riskless rate.

3

p

v(t) dW2 (t) + dZv (t),

(4) (5)

A closed formula for European-style options

In this section we derive a closed-form solution for the price C(S, v, t) of a European call option with strike price K and maturity T written on the underlying asset S. The price of a European put option written on a non-paying dividend underlying asset is then obtained by applying the put-call parity P (S, v, t) = C(S, v, t) − S + Ke−r(T −t) , where r is the riskless rate. Assuming that, under Q, S and v evolve according to (4) and (5) respectively, the pricing equation for the value of any asset U = U (S, v, t) is 1 2


2 2 ∂2U σS2 + ξ 2 v S 2 ∂∂SU2 + ξσv vS ∂S∂v + 12 σv2 v ∂∂vU2 + (r − λjS ) S ∂U ∂S

+ [k∗ (θ ∗ − v)] ∂U ∂v − ru +

∂U ∂t

+ λU [ζ(S, v) − 1] = 0,

3

(6)

where the jump transform ζ is defined by (3). By analogy with the Black-Scholes and Heston formulæ, and following the scheme in Moretto et al. [MPT05], the European call option turns out to be of the form U (S, v, t) = SP1 (S, v, t) − Ke−r(T −t) P2 (S, v, t),

(7)

where the first term is the present value of the spot asset upon optimal exercise and the second term is the present value of the strike-price payment. Both terms U1 := SP1 and U2 := Ke−r(T −t) P2 must satisfy (6). Since ∂P1 ∂U1 ∂S = P1 + S ∂S , ∂ 2 U1 ∂ 2 P1 ∂P1 ∂S 2 = 2 ∂S + S ∂S 2 , ∂P1 ∂ 2 P1 ∂ 2 U1 ∂S∂v = ∂v + S ∂S∂v , ∂P1 ∂U1 ∂v = S ∂v , 2 ∂ 2 U1 = S ∂∂vP21 , ∂v2 ∂U1 ∂P1 ∂t = S ∂t ,

∂U2 −r(T −t) ∂P2 , ∂S = Ke ∂S ∂ 2 U2 ∂ 2 P2 −r(T −t) ∂S 2 = Ke ∂S 2 , ∂ 2 U2 ∂ 2 P2 −r(T −t) ∂S∂v = Ke ∂S∂v , ∂P2 ∂U2 −r(T −t) ∂v = Ke ∂v , 2 2 ∂ U2 = Ke−r(T −t) ∂∂vP22 , ∂v2   ∂U2 −r(T −t) rP + ∂P2 , = Ke 2 ∂t ∂t

(8)

replacing (8) into (6), it follows that P1 satisfies the following PDE   
 1 2 + ξ 2 v S 2 2 ∂P1 + S ∂ 2 P1 + ξσ vS ∂P1 + S ∂ 2 P1 σ v 2 S 2 ∂S ∂S∂v ∂S   ∂v ∂P1 ∂ 2 P1 ∂P1 1 2 + 2 σv vS ∂v2 + (r − λjS ) S P1 + S ∂S + S ∂t   1 − rSP1 + λSP1 [ζ(S, v) − 1] = 0, + [k∗ (θ ∗ − v)] S ∂P ∂v

(9)

and P2 satisfies

1 2


2 ∂ 2 P2 σS2 + ξ 2 v S 2 Ke−r(T −t) ∂∂SP22 + ξσv vSKe−r(T −t) ∂S∂v 2

2 + 12 σv2 vKe−r(T −t) ∂∂vP22 + (r − λjS ) SKe−r(T −t) ∂P  ∂S −r(T −t) rP + 2 + [k∗ (θ ∗ − v)] Ke−r(T −t) ∂P 2 ∂v + Ke

−Ke−r(T −t) rP2 + Ke−r(T −t) λP2 [ζ(S, v) − 1] = 0.

∂P2 ∂t



(10)

After some algebra, (9) and (10) become
2 2 ∂ 2 P1 σS2 + ξ 2 v S 2 ∂∂SP21 + ξσv vS ∂S∂v + 12 σv2 v ∂∂vP21 
 ∂P1 ∗ ∗ 1 + (r − λjS ) + σS2 + ξ 2 v S ∂P ∂S + [k (θ − v) + ξσv v] ∂v + 1 2

∂P1 ∂t

(11)

−λjS P1 + λP1 [ζ(S, v) − 1] = 0,

and 1 2


2 2 ∂ 2 P2 2 σS2 + ξ 2 v S 2 ∂∂SP22 + ξσv vS ∂S∂v + 12 σv2 v ∂∂vP22 + (r − λjS ) S ∂P ∂S

2 + [k∗ (θ ∗ − v)] ∂P ∂v +

∂P2 ∂t

+ λP2 [ζ(S, v) − 1] = 0.

(12)

Now, consider Pej (z) := Pj (ez ), 4

j = 1, 2

(13)

and note that ∂Pj 1 ∂ Pej = , ∂S S ∂Y

∂ 2 Pj 1 = 2 2 ∂S S

∂ Pej ∂ 2 Pej − 2 ∂Y ∂Y

!

,

∂ 2 Pj 1 ∂ 2 Pej = . ∂S∂v S ∂Y ∂v

Replacing (14) into (11) and (12), we obtain the following PDEs
 ∂ 2 Pe1 ∂ Pe1  ∂ 2 Pe1 1 1 2 ∂ 2 Pe1 2 + ξ2v σ + ξσv v ∂Y S 2 ∂v + 2 σv v ∂v2 ∂Y 2 − ∂Y 
 Pe1 Pe1 + (r − λjS ) + σS2 + ξ 2 v ∂∂Y + [k∗ (θ ∗ − v) + ξσv v] ∂∂v + +λPe1 [ζ(S, v) − 1 − jS ] = 0,

and

1 2

σS2 + ξ 2 v


 ∂ 2 Pe2

+ [k∗ (θ ∗ − v)]

∂Y 2 ∂ Pe2 ∂v +



∂ Pe2 ∂ 2 Pe2 1 2 ∂ 2 Pe2 + ξσv v ∂Y ∂Y ∂v + 2 σv v ∂v2 ∂ Pe2 e ∂t + λP2 [ζ(S, v) − 1] = 0.

−

(14)

∂ Pe1 ∂t

(15)

e

P2 + (r − λjS ) ∂∂Y

(16)

After some algebra, (15) and (16) become


 ∂ Pe1 ∂ 2 Pe1 1 1 2 ∂ 2 Pe1 2 2 + ξσv v ∂Y ∂v + 2 σv v ∂v2 + (r − λjS ) + 2 σS + ξ v ∂Y ∂ Pe1 ∂ Pe1 ∗ ∗ e + + λP1 [ζ(S, v) − 1 − jS ] = 0, + [k (θ − v) + ξσv v] 1 2

and

1 2

σS2 + ξ 2 v

σS2 + ξ 2 v

+ [k∗ (θ ∗

−


∂ 2 Pe1 ∂Y 2


∂ 2 Pe2

∂Y 2 Pe2 v)] ∂∂v

∂v

(17)

∂t

 ∂ 2 Pe2 1 2 ∂ 2 Pe2 + ξσv v ∂Y ∂v + 2 σv v ∂v2 + (r − λjS ) − e + ∂ P2 + λPe2 [ζ(S, v) − 1] = 0.

1 2

σS2 + ξ 2 v

∂t


 ∂ Pe2 ∂Y

(18)

The equations (17) and (18) can be written in one unique equation as follows
∂ 2 Pe 2e ∂ 2 Pe σS2 + ξ 2 v ∂Y 2j + ξσv v ∂Y ∂vj + 21 σv2 v ∂∂vP22
 Pe  2 + (r − λjS ) + 21 aj σS2 + ξ 2 v ∂∂Y ∂ Pe ∂ Pe + [k∗ (θ ∗ − v) + bj ξσv v] j + j + λPej [ζ(S, v) − 1 + cj jS ] = 0, 1 2

∂v

(19) j = 1, 2,

∂t

where a1 = 1, a2 = −1, b1 = 1, b2 = 0, c1 = −1 and c2 = 0. Note that Pej , j = 1, 2, are the conditional probabilities that the option expires in-the-money, that is

Pej (Y, v, t; log K) = Q{Y (T ) ≥ log K|Y (t) = Y, v(t) = v},

j = 1, 2,

where Y (t) := log S(t), and (Y, v) evolves according to   p 1 2 2 dY (t) = (r − λjS ) + aj (σS + ξ v(t)) dt + σS dW1 (t) + ξ v(t) dW2 (t) + dZY (t), 2 p dv(t) = k∗ (θ ∗ − v(t) + bj ξσv v(t)) dt + σv v(t) dW2 (t) + dZv (t), j = 1, 2.

Using a Fourier transform method one gets   −iu1 log K Z e ϕj (Y, v, t; u1 , 0) 1 1 ∞ e du1 , R Pj (Y, v, t; log K) = + 2 π 0 iu1 5

j = 1, 2,

(20)

(21) (22)

(23)

where the characteristic functions ϕj (Y, v, t; u1 , u2 ) also satisfy the PDE (19). The practice to solving this kind of equations is to guess the general form of the solution and set some boundary conditions. Following Heston [Hes93] and Duffie et al. [DPS00], we guess ϕj (Y, v, t; u1 , u2 ) = exp [Cj (τ ; u1 , u2 ) + Jj (τ ; u1 , u2 ) + Dj (τ ; u1 , u2 )v + iu1 Y ],

j = 1, 2, (24)

where τ = T −t. The explicit expressions of the characteristic functions are obtained as solutions

to (19) with terminal condition

ϕj (Y, v, T ; u1 , u2 ) = exp{iu1 Y (T )},

j = 1, 2.

(25)

In particular we have (see Appendix for details) Cj (τ ; u1 , u2 ) =




 − 12 u21 σS2 + iu1 r σv2 − k∗ θ ∗ Ej στ2  √ √ v p Ej +e ∆j τ (−Bj + ∆j ) √ ∆j τ − log ,

1 σ2 2 aj iu 1 S ∗ ∗

+ 2kσ2θ v

2

∆j

j = 1, 2,



log(1 + jS ) − 21 δS2 iu1 − 12 δS2 u21 p 2 1 √ + λσ2 v √ Ej ∆j τ 2 ∆j ([σv +jv Ej ]Ej −2jv ∆j Ej )  √ ) √ √ √ ∆j τ (σv2 +jv Ej )Ej −2jv ∆j Ej +(σv2 +jv Ej )( ∆j −Bj )e 2 ∆j j v Ej √ √ − σ2 +jv Ej log , 2 2

Jj (τ ; u1 , u2 ) = −λτ (iu1 jS + jS cj − 1) + 12 λτ exp ( (σv +jv Ej )Ej −2jv

v

(26)



∆j Ej +(σv +jv Ej )(

(27)

∆j −Bj )

j = 1, 2, p p −Bj − ∆j 2 ∆j (Bj + ∆j ) ,  √ Dj (τ ; u1 , u2 ) = + p p σv2 σv2 Bj + ∆j + e ∆j τ (−Bj + ∆j )

j = 1, 2,

(28)

with

Bj = iu1 σv ξ − k∗ + bj ξσv ,

∆j = Bj2 − σv2 iu1 ξ 2 (aj + iu1 ), p Ej = Bj + ∆j .

j = 1, 2

Finally, the densities pej (Y, v, t; log K) of the distribution functions Fej (Y, v, t; log K) = 1 − Pej (Y, v, t; log K) are then 1 pej (Y, v, t; log K) = − π

Z

∞

0

  R −e−iu1 log K ϕj (Y, v, t; u1 , 0) du1 ,

6

j = 1, 2.

(29)

4

Generating Sample Paths

Financial models usually specify the dynamics of the state variables as stochastic differential equations (SDEs). If these SDEs do not yield closed-form pricing formulæ, then numerical approximations (Monte Carlo simulations, among others) can be used. However, the approximation of continuous-time processes by discrete-time processes introduces bias into the simulated solutions and this bias causes several important problems when estimating the prices of derivative securities (Kamrad and Ritchken [KR91]). Following Broadie and Kaya

[BK06], we give a Monte Carlo simulation estimator to com-

pute option price derivatives without discretizing the processes S and v. The main idea is that by appropriately conditioning on the paths generated by the variance and jump processes, the evolution of the asset price can be represented as a series of lognormal random variables. The method is called Exact Simulation Algorithm for the Stochastic Volatility with Contemporaneous Jumps (SVCJ) Model. Let 0 = t0 < t1 < . . . < tM = T be a partition of the interval [0, T ] into M possibly unequal segments of length ∆ti = ti − ti−1 , for each i = 1, . . . , M . We are assuming that we eventu-

ally want to price a path-dependent option whose payoff is a function of the asset price vector

(S(t0 ), . . . , S(tM )) (note that we can take M = 1 for a path-independent option). To illustrate the algorithm, it will be useful to consider the integral form of the dynamics of S and v under Q, namely

and

n o R ti S(ti ) = S(ti−1 ) exp (r − λjS − 12 σS2 )(ti − ti−1 ) − 12 ξ 2 ti−1 v(q)dq o n R R ti p PN (t ) (k) ti v(q) dW2 (q) + k=Ni (ti−1 )+1 log(1 + JS ) , exp σS ti−1 dW1 (q) + ξ ti−1 ∗ ∗

v(ti ) = v(ti−1 )+ k θ (ti − ti−1 )− k

∗

Z

ti

v(q)dq + σv

ti−1

Z

ti ti−1

p

(30)

N (ti )

X

v(q) dW2 (q)+

Jv(k) . (31)

k=N (ti−1 )+1

where Nt is the counting process of the (contemporaneous) jumps . (k)

Remark. If N (ti ) − (N (ti−1 ) + 1) = ni , since log(1 + JS ) have the normal distribution
N log(1 + jS ) − 12 δS2 , δS2 for every k, then ni X

log(1 +

(k) JS )

k=1

that is,

Pni

k=1 log(1

∼N



ni



1 log(1 + jS ) − δS2 2



, nδS2



,

(32)


√ (k) + JS ) can be represented as ni log(1 + jS ) − 12 δS2 + ni δS R, where R is

a standard normal random variable.

7

Now, consider two consecutive time steps ti−1 and ti on the time grid and suppose to know v(ti−1 ). The algorithm can be summarized as follows. Step 1. Generate a Poisson random variable with mean λ(ti − ti−1 ) and simulate the number of

jumps ni . Determine the time of the next jump after ti−1 and denote this time as τi,1 . Set

u := ti−1 and t := τi,1 (u < t). If t > ti , skip Step 5 and Step 6. Step 2. Generate a sample from the distribution of v(t) given v(u). Step 3. Generate a sample from the distribution of Step 4. Recover

Rtp u

Rt u

v(q)dq given v(u) and v(t).

v(q)dW2 (q) from (31) given v(u), v(t), and

Rt u

v(q)dq.

Step 5. If t ≤ ti , generate Jv by sampling from an exponential distribution with mean jv . Update (1)

(1)

the variance value by setting ve(t) = v(t) + Jv , where Jv

is the first jump size of the

variance.

Step 6. If t < ti , determine the time of the next jump τi,2 after τi,1 . If τi,2 ≤ ti , set u := τi,1 and t := τi,2 . Repeat the iteration Step 2-Step 5 up to ti .

If τi,2 > ti , set u := τi,1 and t := ti . Repeat one time the iteration Step 2-Step 4. Step 7. Define the average variance between ti−1 and ti as σ 2i =

ni δS2 + σS2 (ti − ti−1 ) , ti − ti−1

and an auxiliary variable (

ξ2 βi = exp ni log (1 + jS ) − λjS (ti − ti−1 ) − 2

Z

ti

v(q)dq + ξ

ti−1

(33)

Z

ti ti−1

p

)

v(q)dW2 (q) .

Using the definitions (33) and (34), the value S(ti ) given S(ti−1 ) can be written as    p σ 2i (ti − ti−1 ) + σ i ti − ti−1 R , S(ti ) = S(ti−1 )βi exp r− 2

(34)

(35)

where R is a standard normal random variable. From (35), it follows that S(ti ) is a lognormal random variable.

4.1

Sampling from v(t) given v(u)

It is known (see [CIR85]) that the distribution of v(t) given v(u) for some u < t is, up to a scale factor, a non-central chi-squared distribution. Then !
∗ ∗ σv2 1 − e−k (t−u) 0 2 4k∗ e−k (t−u)
v(u) , v(t) = χd 4k∗ σv2 1 − e−k∗ (t−u) 8

u 1. Specifically:

Step 1. Sample x from g(x) and u from U ((0, 1)); Step 2. Check whether u
S(0) is given by e−rT (S(T ) − K)+ 1{max1≤i≤M S(ti )