Jump Models with delay -- option pricing and logarithmic Euler-Maruyama scheme
aa r X i v : . [ q -f i n . M F ] O c t JUMP MODELS WITH DELAY - OPTION PRICING ANDLOGARITHMIC EULER-MARUYAMA SCHEME
NISHANT AGRAWAL AND YAOZHONG HU
Abstract.
In this paper, we obtain the existence, uniqueness and positivity of thesolution to delayed stochastic differential equations with jumps. This equation is thenapplied to model the price movement of the risky asset in a financial market and theBlack-Scholes formula for the price of European option is obtained together withthe hedging portfolios. The option price is evaluated analytically at the last delayedperiod by using the Fourier transformation technique. But in general there is noanalytical expression for the option price. To evaluate the price numerically we thenuse the Monte-Carlo method. To this end we need to simulate the delayed stochasticdifferential equations with jumps. We propose a logarithmic Euler-Maruyama schemeto approximate the equation and prove that all the approximations remain positiveand the rate of convergence of the scheme is proved to be 0 .
1. Introduction
The risky asset in the classical Black-Scholes market is described by the geometricBrownian motion given by the stochastic differential equation driven by standard Brow-nian motion: dS ( t ) = S ( t ) [ rdt + σdW ( t )] , (1.1)where r and σ are two positive constants and W ( t ) is the standard Brownian motion.Ever since the seminal work of Black, Scholes and Merton there have been many researchworks to extend the Black-Scholes-Merton’s theory of option pricing from the originalBlack-Scholes market to more sophisticated models.One of these extensions is the delayed stochastic differential equation (SDDE) drivenby the standard Brownian motion (e.g. [3], see also [22, 24]). In these works the riskyasset is described by the following stochastic delay differential equation dS ( t ) = S ( t ) [ f ( t, S t ) dt + g ( t, S t ) dW ( t )] , where S t = { S ( s ) , t − b ≤ s ≤ t } or S t = S ( t − b ) for some constant b >
0. .On the other hand, there have been some recent discovery (see e.g. [18, 19, 6, 17])that to better fit some risky assets it is more desirable to use the hyper-exponential jumpprocess along with the classical Brownian motion: dS ( t ) = S ( t ) [ rdt + σdW ( t ) + βdZ ( t )] , where Z ( t ) is a hyper-exponential jump process (see the definition in the next section). Mathematics Subject Classification.
Key words and phrases.
L´evy process, hyper-exponential processes, Poisson random measure, sto-chastic delay differential equations, positivity, options pricing, Black-Scholes formula, logarithmic Euler-Maruyama method, convergence rate.supported by an NSERC discovery fund and a startup fund of University of Alberta.
Let N ( dt, dz ) be the Poisson random measure associated with a jump process whichincludes the hyper-exponential jump process as a special case and let ˜ N ( dt, dz ) denoteits compensated Poisson random measure. Then the above equation with σ = 0 is aspecial case of the following equation dS ( t ) = S ( t ) (cid:16) rdt + β Z [0 ,T ] × R z ˜ N ( dz, ds ) (cid:17) (1.2)and it has been argued in (eg. [4, 10, 8]) that the equation (1.2) is a better model forstock prices than (1.1).In this paper, we propose a new model to describe the risky asset by combining thehyper-exponential process with delay. More precisely, we propose the following stochasticdifferential equation as a model for the price movement of the risky asset: dS ( t ) = S ( t ) [ f ( t, S ( t − b )) dt + g ( t, S ( t − b )) dZ ( t )] , (1.3)where f and g are two given functions, and Z ( t ) is a L´evy process which include thehyper-exponential jump processes as a special case. The above model along with theBrownian motion component can be found in [14], where the coefficient of Brownianmotion cannot be allowed to be zero. In this work, we let the coefficient of the Brownianmotion to be zero and we use the Girsanov formula for the jump process to address theissue of completeness of the market and hedging portfolio missed in [14].With the introduction of this new market model, the first question is that whetherthe equation has a unique solution or not and if the unique solution exists whether thesolution is positive or not (since the price of an asset is always positive). We shallfirst answer these questions in Section 2, where we prove the existence, uniqueness andpositivity of the solutions to a larger class of equations than (1.3). To guarantee thatthe solution is positive, we need to assume that the jump part g ( t, S ( t − b )) dZ ( t ) of theequation is bounded from below by some constant (see the assumption (A3) in the nextsection for the precise meaning). The class of the equations our results can be appliedis larger in the following two aspects: The first one is that Z ( t ) can be replaced by amore general L´evy process or more general Poisson random measure and the second oneis that the equation can be multi-dimensional.Following the Black-Scholes-Merton’s principle we then obtain a formula for the fairprice for the European option and the corresponding replica hedging portfolio is alsogiven. To evaluate this formula during the last delay period, we propose a Fouriertransformation method. This method appears more explicit than the partial differentialequation method in the literature and is more closed to the original Black-Scholes formulain spirit. This is done in Section 4.Due to the involvement of f ( S ( t − b )) and g ( S ( t − b )) the above analytical expressionfor the fair option price formula is only valid in the last delay period. Then how do weperform the evaluation by using this option price formula? We propose to use Monte-Carlo method to get the numerical value approximately. For this reason we need tosimulate the equation (1.3) numerically. We observe that there have been a lot of works(eg. [20, 11, 25]) on Euler-Maruyama convergence scheme for SDDE models. There hasalready been study on the Euler-Maruyama scheme for SDDE models with jumps (e.g.[15]). However, in general the Euler-Maruyama scheme cannot preserve the positivityof the solution. Since the solution to the equation (1.3) is positive (when the initialcondition is positive), we wish all of our approximations of the solution is also positive.To this end and motivated by the similar work in the Brownian motion case (see e.g. [13])we introduce a logarithmic Euler-Maruyama scheme, a variant of the Euler-Maruyama UMP MODELS WITH DELAY 3 scheme for (1.3). With this scheme all the approximate solutions are positive and the rateof the convergence of this scheme is also 0 .
5. This rate is optimal even in the Brownianmotion case (e.g. [7]). Let us point out that the 0 . L sense. Not onlyour logarithmic Euler-Maruyama scheme preserves the positivity, its rate is 0 . L p forany p ≥
2. This is done in Section 3.Finally in Section 5 we present some numerical attempts and compared that withthe classical Black-Scholes price formula against the market price for some famous calloptions in the real financial market.
2. Delayed stochastic differential equations with jumps
Let (Ω , F , P ) be a probability space with a filtration ( F t ) { t ≥ } satisfying the usualconditions. On (Ω , F , P ) let Z ( t ) be a L´evy process adapted to the filtration F t . Weshall consider the following delayed stochastic differential equation driven by the L´evyprocess Z ( t ): ( dS ( t ) = f ( S ( t − b )) S ( t ) dt + g ( S ( t − b )) S ( t − ) dZ ( t ) , t ≥ ,S ( t ) = φ ( t ) , t ∈ [ − b, , (2.1)where(i) f, g : R → R are some given bounded measurable functions;(ii) b > φ : [ − b, → R is a (deterministic) measurable function.To study the above stochastic differential equation, it is common to introduce thePoisson random measure associated with this L´evy process Z ( t ) (see e.g. [2, 8, 9, 23] andreferences therein). First, we write the jump of the process Z at time t by∆ Z ( t ) := Z ( t ) − Z ( t − ) if ∆ Z ( t ) = 0 . Denote R := R \{ } and let B ( R ) be the Borel σ -algebra generated by the family ofall Borel subsets U ⊂ R , such that ¯ U ⊂ R . For any t > U ∈ B ( R ) wedefine the Poisson random measure , N : [0 , T ] × B ( R ) × Ω → R , associated with theL´evy process Z by N ( t, U ) := X ≤ s ≤ t, ∆ Z s =0 χ U (∆ Z ( s )) , (2.2)where χ U is the indicator function of U . The associated L´evy measure ν of the L´evyprocess Z is given by ν ( U ) := E [ N (1 , U )] (2.3)and the compensated Poisson random measure ˜ N associated with the L´evy process Z ( t )is defined by ˜ N ( dt, dz ) := N ( dt, dz ) − E [ N ( dt, dz )] = N ( dt, dz ) − ν ( dz ) dt . (2.4)For some technical reason, we shall assume that the process Z ( t ) has only boundednegative jumps to guarantee that the solution S ( t ) to (2.1) is positive. This means thatthere is an interval J = [ − R, ∞ ) bounded from the left such that ∆ Z ( t ) ∈ J for all t > Z ( t ) = Z [0 ,t ] × J zN ( ds, dz ) or dZ ( t ) = Z J zN ( dt, dz ) AGRAWAL AND HU and the equation (2.1) becomes dS ( t ) = (cid:20) f ( S ( t − b )) + g ( S ( t − b )) Z J zν ( dz ) (cid:21) S ( t ) dt + g ( S ( t − b )) S ( t − ) Z J z ˜ N ( dt, dz ) . It is a special case of the following equation: dS ( t ) = f ( S ( t − b )) S ( t ) dt + Z J g ( z, S ( t − b )) S ( t − ) ˜ N ( dt, dz ) . (2.5) Theorem 2.1.
Suppose that f : R → R and g : J × R → R are bounded measurablefunctions such that there is a constant α > satisfying g ( z, x ) ≥ α > − for all z ∈ J and for all x ∈ R , where J is the supporting set of the Poisson measure N ( t, dz ) . Then,the stochastic differential delay equation (2.5) admits a unique pathwise solution with theproperty that if φ (0) > , then for all t > , the random variable X ( t ) > almost surely.Proof First, let us consider the interval [0 , b ]. When t is in this interval f ( X ( t − b )) = f ( φ ( t − b )) and g ( z ; X ( t − b )) = g ( z ; φ ( t − b )) are known given functions of t (and z ).Thus, (2.5) is a linear equation driven by Poisson random measure. The standard theory(see e.g. [2, 23]) can be used to show that the equation has a unique solution. Moreover,it is also well-known (see the above mentioned books or [1]) that by Itˆo’s formula thesolution to (2.5) can be written as X ( t ) = φ (0) exp (cid:26) Z t f ( φ ( s − b )) ds + Z [0 ,t ] × J log [1 + g ( z, φ ( s − b ))] ˜ N ( ds, dz )+ Z [0 ,t ] × J (cid:16) log [1 + g ( z, φ ( s − b ))] − g ( z, φ ( s − b )) (cid:17) dsν ( dz ) (cid:27) . From this formula we see that if φ (0) >
0, then the random variable X ( t ) > t ∈ [0 , b ].In similar way, we can consider the equation (2.5) on t ∈ [ kb, ( k + 1) b ] recursively for k = 1 , , , · · · , and obtain the same statements on this interval from previous results onthe interval t ∈ [ − b, kb ].Since (2.1) is a special case of (2.5), we can write down a corresponding result of theabove theorem for (2.1). Corollary 2.2.
Let the L´evy process Z ( t ) have bounded negative jumps (e.g. ∆ Z ( t ) ∈ J ⊆ [ − R, ∞ ) ). Suppose that f, g : R → R are bounded measurable functions such thatthere is a constant α > satisfying g ( x ) ≤ α R for all x ∈ R . Then, the stochasticdifferential delay equation (2.1) admits a unique pathwise solution with the property thatif φ (0) > , then for all t > the random variable X ( t ) > almost surely.Proof Equation (2.1) is a special case of (2.5) with g ( z, x ) = zg ( x ). The condition g ( x ) ≤ α R implies g ( z, x ) ≥ α > − z ∈ J and for all x ∈ R . Thus, Theorem 2.1can be applied. Example 2.3.
One example of the L´evy process Z ( t ) we have in mind which is usedin finance is the hyper-exponential jump process, which we explain below. Let Y i , i =1 , , · · · be independent and identically distributed random variables with the probability UMP MODELS WITH DELAY 5 distribution given by f Y ( x ) = m X i =1 p i η i e − η i x I { x ≥ } + n X j =1 q j θ j e θ j x I { x< } , where η i > , p i ≥ , θ j > , q j ≥ , i = 1 , · · · , m, j = 1 , · · · , n with P mi =1 p i + P nj =1 q j = 1. Let N t be a Poisson process with intensity λ . Then Z ( t ) = N t X i =1 Y i is a L´evy process. If m = 1 , n = 1 then Z ( t ) is called a double exponential process. Theassumption on the boundedness of the negative jumps can be made possible by requiringthat q j = 0 for all j = 1 , · · · , n or by replacing the negative exponential distribution bytruncated negative exponential distributions, namely, f Y ( x ) = m X i =1 p i η i e − η i x I { x ≥ } + n X j =1 q j θ j − e − θ j R j e θ j x I {− R j
Suppose that f ij : R → R and g ij : J × R → R , ≤ i, j ≤ d are boundedmeasurable functions such that there is a constant α > satisfying g ij ( z, x ) ≥ α > − for all ≤ i, j ≤ d , for all z ∈ J and for all x ∈ R , where J is the common supporting setof the Poisson measures ˜ N j ( t, dz ) , j = 1 , · · · , d . If for all i = j , f ij ( x ) ≥ for all x ∈ R ,and φ i (0) ≥ , i = 1 , · · · , d , then, the stochastic differential delay equation (2.6) admitsa unique pathwise solution with the property that for all i = 1 , · · · , d and for all t > ,the random variable S i ( t ) ≥ almost surely. AGRAWAL AND HU
Proof
We can follow the argument as in the proof of Theorem 2.1 to show that thesystem of delayed stochastic differential equations (2.6) has a unique solution S ( t ) =( S ( t ) , · · · , S d ( t )) T . We shall modify slightly the method of [12] to show the positivity ofthe solution. Denote ˜ g ij ( t, z ) = g ij ( z, S ( t − b )). Let Y i ( t ) be the solution to the stochasticdifferential equation dY i ( t ) = Y i ( t − ) d X j =1 Z J ˜ g ij ( t, z ) ˜ N j ( dt, dz )with initial conditions Y i (0) = φ i (0). Since this is a scalar equation for Y i ( t ), its explicitsolution can be represented Y i ( t ) = φ i (0) exp (cid:26) d X j =1 log [1 + ˜ g ij ( s, z )] ˜ N j ( ds, dz )+ d X j =1 Z [0 ,t ] × J (cid:16) log [1 + ˜ g ij ( s, z )] − ˜ g ij ( s, z ) (cid:17) dsν j ( dz ) (cid:27) , where ν j is the associated L´evy measure for ˜ N j ( ds, dz ). Denote ˜ f ij ( t ) = f ij ( S ( t − b )) andlet p i ( t ) be the solution to the following system of equations dp i ( t ) = d X j =1 ˜ f ij ( t ) p j ( t ) dt , p i (0) = 1 , i = 1 , · · · , d . By the assumption on f we have that when i = j , ˜ f ij ( t ) ≥ p i ( t ) ≥ t ≥ S i ( t ) = p i ( t ) Y i ( t ) is the solution to (2.6) which satisfies that˜ S i ( t ) ≥ S i ( t ) = ˜ S i ( t ) for i = 1 , · · · , d . The theorem is then proved.
3. Logarithmic Euler-Maruyama scheme
The equation (2.1) or (2.5) is used in Section 4 to model the price of a risky asset ina financial market and its the solution is proved to be positive as in Theorem 2.1. Asit is well-known the usual Euler-Maruyama scheme cannot preserve the positivity of thesolution (e.g. [13] and references therein). Motivated by the work [13], we propose inthis section a variant of the Euler-Maruyama scheme (which we call logarithmic Euler-Maruyama scheme) to approximate the solution so that all approximations are alwaysnon-negative. For the convenience of the future simulation, we shall consider only theequation (2.1), which we rewrite here: dS ( t ) = f ( S ( t − b )) S ( t ) dt + g ( S ( t − b )) S ( t − ) dZ ( t ) , (3.1)where Z ( t ) = P N t i =1 Y i is a L´evy process. Here N t is a Poisson process with intensity λ and Y , Y , · · · , are iid random variables.The solution to the above equation can be written as S ( t ) = φ (0) exp (cid:16) Z t f ( X ( u − b )) du + X ≤ u ≤ t, ∆ Z ( u ) =0 ln(1 + g ( X ( u − b )) Y N ( u ) ) (cid:17) . (3.2)We shall consider a finite time interval [0 , T ] for some fixed T >
0. Let ∆ = Tn > n ∈ N . For any nonnegative integer k ≥ UMP MODELS WITH DELAY 7 denote t k = k ∆. We consider the partition π of the time interval [0 , T ]: π : 0 = t < t < · · · < t n = T .
On the subinterval [ t k , t k +1 ] the solution (3.2) can also be written as S ( t ) = S ( t k ) exp (cid:16) Z tt k f ( X ( u − b )) du + X t k ≤ u ≤ t, ∆ Z ( u ) =0 ln(1 + g ( X ( u − b )) Y N ( u ) ) (cid:17) , t ∈ [ t k , t k +1 ] . (3.3)Motivated by the formula (3.3), we propose a logarithmic Euler-Maruyama scheme toapproximate (2.1) as follows. S π ( t k +1 ) = S π ( t k ) exp (cid:16) f ( S π ( t k − b ))∆ (cid:17) · exp (cid:16) ln(1 + g ( S π ( t k − b ))∆ Z k ) (cid:17) , k = 0 , , , ..., n − S π ( t ) = φ ( t ) for all t ∈ [ − b, φ (0) >
0, then S π ( t k ) > k = 0 , , , ..., n . Then our approximations S π ( t k ) are always positive.Notice that the approximations from usual Euler-Maruyama scheme is always not positivepreserving (see e.g. [13] and references therein).We shall prove the convergence and find the rate of convergence for the above scheme.For the convergence of the usual Euler-Maruyama scheme of jump equation with delay,we refer to [15]. To study the convergence of the above logarithmic Euler-Maruyamascheme, we make the following assumptions. (A1) The initial data φ (0) > ρ > γ ∈ [1 / , t, s ∈ [ − b, | φ ( t ) − φ ( s ) | ≤ ρ | t − s | γ . (3.5) (A2) f is bounded. f and g are global Lipschitz. This means that there exists aconstant ρ > (cid:12)(cid:12)(cid:12) g ( x ) − g ( x ) (cid:12)(cid:12)(cid:12) ≤ ρ | x − x | ; (cid:12)(cid:12)(cid:12) f ( x ) − f ( x ) (cid:12)(cid:12)(cid:12) ≤ ρ | x − x | , ∀ x , x ∈ R ; (cid:12)(cid:12) f ( x ) (cid:12)(cid:12) ≤ ρ , ∀ x ∈ R (A3) The support J of the Poisson random measure N is contained in [ − R, ∞ ) forsome R > α > ρ > − ρ ≤ g ( x ) ≤ α R for all x ∈ R . (A4) For any q > ρ q > Z J (1 + | z | ) q ν ( dz ) ≤ ρ q , ∀ x ∈ R . (3.6)For notational simplicity we introduce two step processes ( v ( t ) = P ∞ k =0 I [ t k ,t k +1 ) ( t ) S π ( t k ) v ( t ) = P ∞ k =0 I [ t k ,t k +1 ) ( t ) S π ( t k − b ) . AGRAWAL AND HU
Define the continuous interpolation of the logarithmic Euler-Maruyama approximatesolution on the whole interval [ − b, T ] (not only on t k , k = 0 , · · · , n ) as follows: S π ( t ) = φ ( t ) t ∈ [ − b, φ (0) exp (cid:16) Z t f ( v ( u )) du + P ≤ u ≤ t, ∆ Z ( u ) =0 ln(1 + g ( v ( u )) Y N ( u ) ) (cid:17) t ∈ [0 , T ] . (3.7)With this interpolation, we see that S π ( t ) > t ≥ Lemma 3.1.
Let the assumptions (A1)-(A4) be satisfied. Then for any q ≥ thereexists K q , independent of the partition π , such that E h sup ≤ t ≤ T | S ( t ) | q i ∨ E h sup ≤ t ≤ T | S π ( t ) | q i ≤ K q . Proof
We can assume that q >
2. First, let us prove E h sup ≤ t ≤ T | S π ( t ) | q i ≤ K q . From(3.7) it follows E h sup ≤ t ≤ T | S π ( t ) | q i ≤ | φ (0) | q E h sup ≤ t ≤ T exp (cid:16) q Z t f ( v ( u )) du + q X ≤ u ≤ t, ∆ Z ( u ) =0 ln(1 + g ( v ( u )) Y N ( u ) ) (cid:17)i . Since | f ( t ) | ≤ ρ we have E h sup ≤ t ≤ T | S π ( t ) | q i ≤ φ (0) q e qρT E h sup ≤ t ≤ T exp (cid:16) q X ≤ u ≤ t, ∆ Z ( u ) =0 ln(1 + g ( v ( u )) Y N ( u ) ) (cid:17)i = φ (0) q e qρT E h sup ≤ t ≤ T exp (cid:16) q Z T ln(1 + zg ( v ( u ))) N ( du, dz ) (cid:17)i , (3.8)where and throughout the remaining part of this paper, we denote T = [0 , t ] × J . Nowwe are going to handle the factor I := E h sup ≤ t ≤ T exp (cid:16) q Z T ln(1 + zg ( v ( u ))) N ( du, dz ) (cid:17)i . Let h = ((1 + zg ( v ( u )) q − /z . Then I = E h sup ≤ t ≤ T exp (cid:16) Z T ln(1 + zh ) N ( du, dz ) (cid:17)i = E h sup ≤ t ≤ T exp (cid:16) Z T ln(1 + zh ) ˜ N ( du, dz ) + 12 Z T ln(1 + zh ) ν ( dz ) du (cid:17)i = E h sup ≤ t ≤ T exp (cid:16) Z T ln(1 + zh ) ˜ N ( du, dz ) + 12 Z T [ln(1 + zh ) − zh ] ν ( dz ) du (cid:17)i sup ≤ t ≤ T exp (cid:16) − Z T (1 + zg ( v ( u )) q − ν ( dz ) du (cid:17)i ≤ C q E h sup ≤ t ≤ T exp (cid:16) Z T ln(1 + zh ) ˜ N ( du, dz ) + 12 Z T [ln(1 + zh ) − zh ] ν ( dz ) du (cid:17)i , UMP MODELS WITH DELAY 9 where we used boundedness of g and the assumption (A4). Now an application of theCauchy-Schwartz inequality yields I ≤ C q (cid:26) E h sup ≤ t ≤ T M t i(cid:27) / , where M t := exp (cid:16) Z T ln(1 + zh ) ˜ N ( du, dz ) + Z T [ln(1 + zh ) − zh ] ν ( dz ) du (cid:17) . But ( M t , ≤ t ≤ T ) is an exponential martingale. Thus, E h sup ≤ t ≤ T M t i ≤ E h M T i = 2 . Inserting this estimate of I into (3.8) proves E h sup ≤ t ≤ T | S π ( t ) | q i ≤ K q < ∞ . In thesame way we can show E h sup ≤ t ≤ T | S ( t ) | q i ≤ K q < ∞ . This completes the proof of thelemma. Lemma 3.2.
Assume (A1)-(A4). Then there is a constant
K > , independent of π ,such that E Q (cid:12)(cid:12)(cid:12) S π ( t ) − v ( t ) (cid:12)(cid:12)(cid:12) p ≤ K ∆ p/ , ∀ t ∈ [0 , T ] . Proof
Let t ∈ [ t j , t j +1 ) for some j . Using | e x − e y | ≤ ( e x + e y ) | x − y | we can write (cid:12)(cid:12)(cid:12) S π ( t ) − v ( t ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) S π ( t ) − S π ( t j ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) S π ( t ) + S π ( t j ) (cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12) Z tt j f ( v ( s )) ds + X t j ≤ s ≤ t ln(1 + g ( v ( s )) Y N ( s ) ) (cid:12)(cid:12)(cid:12) . An application of the H¨older inequality yields that for any p > E h(cid:12)(cid:12)(cid:12) S π ( t ) − v ( t ) (cid:12)(cid:12)(cid:12) p i ≤ (cid:26) E h(cid:12)(cid:12)(cid:12) S π ( t ) + S π ( t j ) (cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)i p (cid:27) / E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z tt j f ( v ( s )) ds + X t j ≤ s ≤ t ln(1 + g ( v ( s )) Y N ( s ) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p / ≤ K p E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z tt j f ( v ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p + E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X t j ≤ s ≤ t ln(1 + g ( v ( s )) Y N ( s ) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p / ≤ K p ∆ p + E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X t j ≤ s ≤ t ln(1 + g ( v ( s )) Y N ( s ) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p / . (3.9)Now we want to bound I := E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X t j ≤ s ≤ t ln(1 + g ( v ( s )) Y N ( s ) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p . (we use the same notation I to denote different quantities in different occasions and thiswill not cause ambiguity). We write the above sum as an integral: I = E (cid:12)(cid:12)(cid:12) Z J Z tt j ln(1 + zg ( v ( s ))) N ( ds, dz ) (cid:12)(cid:12)(cid:12) p = E (cid:12)(cid:12)(cid:12) Z J Z tt j ln(1 + zg ( v ( s ))) ˜ N ( ds, dz )+ Z J Z tt j ln(1 + zg ( v ( s ))) ν ( dz ) ds (cid:12)(cid:12)(cid:12) p ≤ C p ∆ p + E (cid:12)(cid:12)(cid:12) Z J Z tt j ln(1 + zg ( v ( s ))) ˜ N ( ds, dz ) (cid:12)(cid:12)(cid:12) p ! . By the Burkholder-Davis-Gundy inequality, we have E (cid:12)(cid:12)(cid:12) Z J Z tt j ln(1 + zg ( v ( s ))) ˜ N ( ds, dz ) (cid:12)(cid:12)(cid:12) p ≤ E Z J Z tt j (cid:12)(cid:12)(cid:12) ln(1 + zg ( v ( s ))) (cid:12)(cid:12)(cid:12) ν ( dz ) ds ! p ≤ K p ∆ p . Thus, we have I ≤ K p,T ∆ p . Inserting this bound into (3.9) yields the lemma.Our next objective is to obtain the rate of convergence of our logarithmic Euler-Maruyama approximation S π ( t ) to the true solution S ( t ). Theorem 3.3.
Assume (A1)-(A4). Let S π ( t ) be the solution to (3.4) and let S ( t ) be thesolution to (3.1) . Then there is a constant K p,T , independent of π such that E Q h sup ≤ t ≤ T | S ( t ) − S π ( t ) | p i ≤ K p,T ∆ p/ . (3.10) Proof
We write S ( t ) = φ (0) exp ( X ( t )) and S π ( t ) = φ (0) exp ( p ( t )). Then (cid:12)(cid:12)(cid:12) S ( t ) − S π ( t ) (cid:12)(cid:12)(cid:12) p ≤ (cid:12)(cid:12)(cid:12) S ( t ) + S π ( t ) (cid:12)(cid:12)(cid:12) p (cid:12)(cid:12)(cid:12) X ( t ) − p ( t ) (cid:12)(cid:12)(cid:12) p . Hence by Lemma 3.1 we have for any r ∈ [0 , T ] E h sup ≤ t ≤ r | S ( t ) − S π ( t ) | p i ≤ E h sup ≤ t ≤ r (cid:12)(cid:12)(cid:12) S ( t ) + S π ( t ) (cid:12)(cid:12)(cid:12) p i / E h sup ≤ t ≤ r (cid:12)(cid:12)(cid:12) X ( t ) − p ( t ) (cid:12)(cid:12)(cid:12) p i / ≤ p − (cid:16) E h sup ≤ t ≤ r (cid:12)(cid:12)(cid:12) S ( t ) (cid:12)(cid:12)(cid:12) p i + E h sup ≤ t ≤ r (cid:12)(cid:12)(cid:12) S π ( t ) (cid:12)(cid:12)(cid:12) p i(cid:17) / h E sup ≤ t ≤ r (cid:12)(cid:12)(cid:12) X ( t ) − p ( t ) (cid:12)(cid:12)(cid:12) p i / ≤ K p h E sup ≤ t ≤ r (cid:12)(cid:12)(cid:12) X ( t ) − p ( t ) (cid:12)(cid:12)(cid:12) p i / = K p I / . (3.11) UMP MODELS WITH DELAY 11
Thus we need only to bound the above expectation I , which is given by the following. I = E h sup ≤ t ≤ r | X ( t ) − p ( t ) | p i ≤ E sup ≤ t ≤ r (cid:12)(cid:12)(cid:12) Z t ( f ( S ( u − b )) − f ( v ( u ))) du (3.12)+ X ≤ u ≤ t, ∆ Z ( u ) =0 ln(1 + g ( S ( u − b )) Y N ( u ) ) − ln(1 + g ( v ( u )) Y N ( u ) ) (cid:12)(cid:12)(cid:12) p . By the Lipschitz conditions we have I ≤ K p E Z r (cid:12)(cid:12)(cid:12) S ( u − b ) − v ( u ) (cid:12)(cid:12)(cid:12) p du + K p E sup ≤ t ≤ r (cid:12)(cid:12)(cid:12) X ≤ u ≤ t, ∆ Z ( u ) =0 ln(1 + g ( S ( u − b )) Y N ( u ) ) − ln(1 + g ( v ( u )) Y N ( u ) ) (cid:12)(cid:12)(cid:12) p ≤ K p h E Z r (cid:12)(cid:12)(cid:12) S ( u − b ) − S π ( u − b ) (cid:12)(cid:12)(cid:12) p du + E Z r (cid:12)(cid:12)(cid:12) S π ( u − b ) − v ( u ) (cid:12)(cid:12)(cid:12) p du i + K p E sup ≤ t ≤ r (cid:12)(cid:12)(cid:12) X ≤ u ≤ t, ∆ Z ( u ) =0 ln(1 + g ( S ( u − b )) Y N ( u ) ) − ln(1 + g ( v ( u )) Y N ( u ) ) (cid:12)(cid:12)(cid:12) p = I + I + I . (3.13)By Lemma 3.2 and by the assumption (A1) about the H¨older continuity of the initialdata φ we have I ≤ K p,T ∆ p . (3.14)We write the above sum I with jumps as a stochastic integral: I = E sup ≤ t ≤ r (cid:12)(cid:12)(cid:12) X ≤ u ≤ t, ∆ Z ( u ) =0 ln(1 + g ( S ( u − b )) Y N ( u ) ) − ln(1 + g ( v ( u )) Y N ( u ) ) (cid:12)(cid:12)(cid:12) p = E sup ≤ t ≤ r (cid:12)(cid:12)(cid:12) Z J Z t [ln(1 + zg ( S ( u − b ))) − ln(1 + zg ( v ( u )))] ˜ N ( du, dz )+ Z J Z t [ln(1 + zg ( S ( u − b ))) − ln(1 + zg ( v ( u )))] ν ( dz ) du (cid:12)(cid:12)(cid:12) p = 4 p E sup ≤ t ≤ r (cid:12)(cid:12)(cid:12) Z J Z t [ln(1 + zg ( S ( u − b ))) − ln(1 + zg ( v ( u )))] ˜ N ( du, dz ) (cid:12)(cid:12)(cid:12) p +4 p E sup ≤ t ≤ r (cid:12)(cid:12)(cid:12) Z J Z t [ln(1 + zg ( S ( u − b ))) − ln(1 + zg ( v ( u )))] ν ( dz ) du (cid:12)(cid:12)(cid:12) p =: I + I . Using the Lipschitz condition on g and (A3), we have I ≤ K p E (cid:16) Z r (cid:12)(cid:12)(cid:12) g ( S ( u − b )) − g ( v ( u )) (cid:12)(cid:12)(cid:12) du (cid:17) p ≤ K p,T E sup ≤ t ≤ r | S ( t − b )) − S π ( t − b ) | p . Using the Burkholder-Davis-Gundy inequality we have I ≤ K p E (cid:16) Z J Z r (cid:12)(cid:12)(cid:12) ln(1 + zg ( S ( u − b ))) − ln(1 + zg ( v ( u ))) (cid:12)(cid:12)(cid:12) ν ( dz ) du (cid:17) p . Similar to the bound for I , we have I ≤ K p,T E sup ≤ t ≤ r | S ( t − b )) − S π ( t − b ) | p . Combining the estimates for I and , we see I ≤ K p,T E sup ≤ t ≤ r | S ( t − b )) − S π ( t − b ) | p . (3.15)It is easy to verify I ≤ K p,T E sup ≤ t ≤ r | S ( t − b )) − S π ( t − b ) | p . (3.16)Inserting the bounds obtained in (3.14)-(3.17) into (3.13), we see that I ≤ K p,T E sup ≤ t ≤ r | S ( t − b )) − S π ( t − b ) | p + K P,T ∆ p . (3.17)Combining this estimate with (3.11), we see E h sup ≤ t ≤ r | S ( t ) − S π ( t ) | p i ≤ K p,T (cid:20) E sup ≤ t ≤ r | S ( t − b ) − S π ( t − b ) | p (cid:21) / + K P,T ∆ p/ (3.18)for any p ≥ r ∈ [0 , T ]. Now we shall use (3.18) to prove the theorem onthe interval [0 , kb ] recursively for k = 1 , , · · · , [ Tb ] + 1. Since S π ( t ) = S ( t ) = φ ( t ) for t ∈ [ − b, r = b , we have E h sup ≤ t ≤ b | S ( t ) − S π ( t ) | p i ≤ K p,T ∆ p/ (3.19)for any p ≥
2. Now taking r = 2 b in (3.18), we have E h sup ≤ t ≤ b | S ( t ) − S π ( t ) | p i ≤ K p,T (cid:20) E sup − b ≤ t ≤ b | S ( t )) − S π ( t ) | p (cid:21) / + K P,T ∆ p/ ≤ K p,T [ K p,T ∆ p ] / + K P,T ∆ p/ ≤ K p,T ∆ p/ . (3.20)Continuing this way we obtain for any positive integer k ∈ N , E h sup ≤ t ≤ kb | S ( t ) − S π ( t ) | p i ≤ K k,p,T ∆ p/ . (3.21)Now since T is finite, we can choose a k such that ( k − b < T ≤ kb . This completesthe proof of the theorem. UMP MODELS WITH DELAY 13
4. Option Pricing in Delayed Black-Scholes market with jumps
In this section we consider the problem of option pricing in a delayed Black-Scholesmarket which consists of two assets. One is risk free, whose price is described by dB ( t ) = rB ( t ) dt , or B ( t ) = e rt , t ≥ . (4.1)Another asset is a risky one, whose price is described by the delayed equation (2.1) or(3.1), namely, dS ( t ) = f ( S ( t − b )) S ( t ) dt + g ( S ( t − b )) S ( t − ) dZ ( t ) , (4.2)where Z ( t ) = P N t i =1 Y i is a L´evy process, N t is a Poisson process with intensity λ , and Y , Y , · · · , are iid random variables. As in Section 2, we introduce the Poisson randommeasure N ( dt, dz ) and its compensator ˜ N ( dt, dz ). The above delayed equation can bewritten as dS ( t ) = (cid:20) f ( S ( t − b )) + g ( S ( t − b )) Z J zν ( dz ) (cid:21) S ( t ) dt + g ( S ( t − b )) S ( t − ) Z J z ˜ N ( dt, dz ) . Denote L = Z J zf Y ( z ) dz , (4.3)where f Y is the probability density of Y i (whose support is J ). Then Z J zν ( dz ) = λL . Set ˜ S ( t ) = S ( t ) B ( t ) . Then by Itˆo’s formula we have d ˜ S ( t ) = ˜ S ( t − ) g ( S ( t − b )) (cid:16) Z J z (cid:2) θ ( t ) ν ( dz ) dt + ˜ N ( dt, dz ) (cid:3)(cid:17) , (4.4)where θ ( t ) = f ( S ( t − b ))+ g ( S ( t − b )) − rλLg ( S ( t − b )) . We shall keep the assumptions (A1)-(A4) made inprevious section and we need to make an additional assumption: (A5) There is a constant α ∈ (1 , ∞ ) such that Z J ν ( dz ) ≥ α (cid:12)(cid:12)(cid:12) f ( s ) + g ( s ) − rg ( t ) (cid:12)(cid:12)(cid:12) ∀ s, t ∈ [0 , ∞ )To find the risk neutral probability measure we apply Girsanov theorem for L´evy process(see [9, Theorem 12.21]). The θ ( t ) is predictable for t ∈ [0 , T ]. From the assumptionsabove we also have that 0 < θ ( s ) ≤ α . Thus, Z [0 ,T ] × J (cid:16) | log(1 + θ ( s )) | + θ ( s ) (cid:17) ν ( dz ) ds ≤ K < ∞ . Now define S θ ( t ) := exp (cid:16) Z [0 ,t ] { log (cid:0) − θ ( s ) (cid:1) + θ ( s ) } ν ( dx ) ds + Z [0 ,t ] log (cid:0) − θ ( s ) (cid:1) ˜ N ( dx, ds ) (cid:17) . In order for us to obtain an equivalent martingale measure we need to verify the followingNovikov condition: E h exp (cid:16) Z [0 ,T ] × J { (1 − θ ( s )) log(1 − θ ( s )) + θ ( s ) } ν ( dz ) ds (cid:17)i < ∞ (4.5)This is a consequence of our assumption (A5). In fact, we have first | θ ( s ) | = | f ( S ( t − b )) − r | λLg ( S ( t − b )) ≤ α < . Hence we have Z [0 ,T ] { (1 − θ ( s )) log(1 − θ ( s )) + θ ( s ) } ds < ∞ . But ν ( dz ) = λf Y ( z ) dz , we have Z J ν ( dz ) = Z J λf Y ( z ) dz < ∞ . Thus, we have (4.5).Now since we have verified the Novikov condition (4.5) we have then E [ S θ ( T )] = 1.Define an equivalent probability measure Q on F T by d Q := S θ ( T ) d P . (4.6)On the new probability space (Ω , F T , Q ) (new probability Q ) the random measure˜ N Q ( dz, ds ) = θ ( t ) ν ( dz ) ds + ˜ N ( dz, ds ) , (4.7)is a compensated Poisson random measure. The corresponding L´evy measure is denotedby ν Q . With this new Poisson random measure we can write (4.4) as d ˜ S ( t ) = ˜ S ( t − ) Z J zg ( S ( t − b )) ˜ N Q ( dt, dz ) . (4.8)The following result gives the fair price formula for the European call option as well asthe corresponding hedging portfolio. Theorem 4.1.
Let the market be given by (4.1) and (4.2) , where the coefficients f and g satisfy the assumptions (A1)-(A5). Then the market is complete. Let T be the maturitytime of the European call option on the stock with payoff function given by X = ( S T − K ) + .Then at any time t ∈ [0 , T ] , the fair price V(t) of the option is given by the formula V ( t ) = e − r ( T − t ) E Q (cid:16) ( S T − K ) + |F t (cid:17) (4.9) where Q is the martingale measure on (Ω , F T ) given by (4.6) .Moreover, if Z J z j ν Q ( dz ) < ∞ , Z R + g ( t ) j dt < ∞ for j = 1 , , , , there is an adapted andsquare integrable process ψ ( z, t ) ∈ L ( J × [0 , T ]) such that E Q (cid:16) e − rT ( S T − K ) + |F t (cid:17) = E Q (cid:16) e − rT ( S T − K ) + (cid:17) + Z [0 ,t ] × J ψ ( z, s ) ˜ N Q (( dz, ds ) and the hedging strategy is given by π S ( t ) := Z J ψ ( z, t ) ˜ N Q ( dz, t )˜ S ( t ) g ( S ( t − b )) , π B ( t ) := U ( t ) − π S ( t ) ˜ S ( t ) , t ∈ [0 , T ] , (4.10) where U ( t ) = E Q ( e − rT ( S T − K ) + |F t ) . UMP MODELS WITH DELAY 15
Proof
Applying the Itˆo formula to (4.8) we get˜ S ( T ) = exp (cid:16) Z [0 ,T ] × J { ln(1 + zg ( S ( t − b ))) − zg ( S ( t − b ) } ν Q ( dz ) dt + Z [0 ,T ] × J ln(1 + zg ( S ( t − b ))) ˜ N Q ( dt, dz ) (cid:17) (4.11)Denote X = ( S T − K ) + and consider U ( t ) := E Q ( e − rT X |F t ) . In order to apply martingale representation theorem for L´evy process (see e.g. [2,Theorem 5.3.5]) we shall first show that U t ∈ L , which is implied by E Q [ S T ] < ∞ .Write h = g ( S ( t − b )). Then we can write˜ S T = exp (cid:16) Z [0 ,T ] × J { ln(1 + zh ) − zh } ν Q ( dz ) dt + Z [0 ,T ] × J ln(1 + zh ) ˜ N Q ( dt, dz ) (cid:17) . (4.12)Denoting T = [0 , T ] × J and taking ˜ h = (1+ zh ) − z we have˜ S T = exp (cid:16) Z T { ln(1 + z ˜ h ) − z ˜ h } ν Q ( dz ) dt + 12 Z T ln(1 + z ˜ h ) ˜ N Q ( dt, dz ) (cid:17) . exp (cid:16) Z T (cid:16) z ˜ h − zh (cid:17) ν Q ( dz ) dt (cid:17) . Applying the H¨older inequality we have E Q (cid:2) ˜ S T (cid:3) ≤ h E Q exp (cid:16) Z T { ln(1 + z ˜ h ) − z ˜ h } ν Q ( dz ) dt + Z T ln(1 + z ˜ h ) ˜ N Q ( dt, dz ) (cid:17)i / · h E Q exp (cid:16) Z T (cid:16) z ˜ h − zh (cid:17) ν Q ( dz ) dt (cid:17)i / = h E Q exp (cid:16) Z T (cid:16) z ˜ h − zh (cid:17) ν Q ( dz ) dt (cid:17)i / . From the definition of ˜ h , we have z ˜ h = (1 + zh ) −
1. Then z ˜ h − zh = (1 + zh ) − − zh = z h + 4 z h + 6 z h + 2 zh . Thus, E Q (cid:2) ˜ S T (cid:3) ≤ exp (cid:16) Z T (cid:16) z h + 4 z h + 6 z h + 2 zh (cid:17) ν Q ( dz ) dt (cid:17) which is finite by the assumptions of the theorem.From the martingale representation theorem (see e.g. [2, theorem 5.3.5]) there existsa square integrable predictable mapping ψ : T × Ω → R such that U ( t ) = E Q ( e − rT ( S T − K ) + ) + Z t Z J ψ ( s, z ) ˜ N ( ds, dz ) . Define π S ( t ) := Z J ψ ( z, t ) ˜ N Q ( dz, t )˜ S ( t ) g ( S ( t − b ))= Z J ψ ( z, t ) ˜ S ( t ) g ( S ( t − b )) d ˜ S ( t )˜ S ( t ) g ( S ( t − b )) ,π B ( t ) := U ( t ) − π S ( t ) ˜ S ( t ) , t ∈ [0 , T ] . Consider the strategy { ( π B ( t ) , π S ( t )) : t ∈ [0 , T ] } to invest π B ( t ) units in the riskylessasset B ( t ) and π S ( t ) units in the risky asset S ( t ) at time t . Then the value of the portfolioat time t is given by V ( t ) := π B ( t ) e rt + π S ( t ) S ( t ) = e rt U ( t )By the definition of the strategy we see that dV ( t ) = π B ( t ) de rt + π S ( t ) dS ( t ) = e rt dU ( t ) + U ( t ) de rt . Hence the strategy is self-financing. Moreover, we have V ( T ) = e rT U ( T ) = ( S T − K ) + . Hence the claim (referring to the European call option) is attainable stand therefore themarket { S ( t ) , B ( t ) : t ∈ [0 , T ] } is complete.The pricing formula (4.9) is hard to evaluate analytically and we shall use a generalMonte-Carlo method to find the approximate values. But when the time fall in the lastdelay period, namely, when t ∈ [ T − b, T ] we have the following analytic expression forthe price. Theorem 4.2.
Assume the conditions of Theorem 4.1. When t ∈ [ T − b, T ] , then pricefor the European Call option is given by V ( t ) = e rt lim v →∞ π Z ∞−∞ iξ ( e ivξ − e iwξ ) A ( t ) · ˜ S ( t ) exp (Z Tt Z J (cid:16) (1 + zg ( S ( u − b ))) (1 − iξ ) − (1 − iξ ) ln(1 + zg ( S ( u − b ))) − (cid:17) ν Q ( dz ) du o − Ke rt lim v →∞ π Z ∞−∞ iξ ( e ivξ − e iwξ ) A ( t ) · ˜ S ( t ) exp (Z Tt Z J (cid:16) (1 + zg ( S ( u − b ))) − iξ + iξ ln(1 + zg ( S ( u − b ))) − (cid:17) ν Q ( dz ) du o , (4.13) where w = ln( K/A ) − rT and A ( t ) = exp (cid:16) Z Tt Z J { ln (1 + zg ( S ( u − b ))) − zg ( S ( u − b )) ν Q ( dz ) du (cid:17) . (4.14) UMP MODELS WITH DELAY 17
Proof
By (4.9) for any time t ∈ [0 , T ] we have V ( t ) = e − r ( T − t ) E Q (cid:16) ( S ( T ) − K ) + | F t (cid:17) = e rt E Q (cid:16) ( ˜ S ( T ) − Ke − rT ) + | F t (cid:17) = e rt E Q (cid:16) ˜ S ( T ) I { ˜ S ( T ) ≥ Ke − rT } | F t (cid:17) − Ke rt Q ( ˜ S ( T ) ≥ Ke − rT )=: V ( t ) − V ( t ) . (4.15)First, let us compute V ( t ) and V ( t ) can be computed similarly. The solution ˜ S ( t ) isgiven by (4.11), which we rewrite here:˜ S ( T ) = ˜ S ( t ) exp n Z Tt Z J { ln (1 + zg ( S ( u − b ))) − zg ( S ( u − b )) } ν Q ( dz ) du + Z Tt Z J ln (1 + zg ( S ( u − b ))) ˜ N Q ( dz, du ) o . (4.16)When u ∈ [ t, T ] and t ∈ [ T − b, T ], we see that S ( u − b ) is F t -measurable. Hence whilecomputing the conditional expectation of h ( ˜ S ( T )) with respect to F t , we can consider theintegrands ln(1 + zg ( S ( u − b ))) and ln(1 + zg ( S ( u − b ))) − zg ( S ( u − b )) as “deterministic”functions. Thus, the analytic expression for the conditional expectation is possible. Butit is still complicated. To find the exact expression and to simplify the presentation, letus use the notation (4.14) and introduce Y = Z Tt Z J ln (1 + zg ( S ( u − b ))) ˜ N Q ( dz, du ) . With these notation we have ˜ S ( T ) = ˜ S ( t ) A exp Y .
To calculate E Q (cid:16) e Y I { v ≥ Y ≥ w } (cid:17) we first express I [ w,v ] as the (inverse) Fourier transformof exponential function because E ( e iξY ) is computable. Since the Fourier transform of I { w,v } is Z ∞−∞ e ixξ I [ w,v ] dx = 1 iξ ( e ivξ − e iwξ )we can write I [ w,v ] ( x ) = 12 π Z ∞−∞ iξ ( e i [ v − x ] ξ − e i [ w − x ] ξ ) dξ . Therefore we have E Q ( e Y I { v ≥ Y ≥ w } | F t ) = 12 π Z ∞−∞ E Q (cid:16) iξ ( e i [ v − Y ] ξ + Y − e i [ w − Y ] ξ + Y ) | F t (cid:17) dξ = 12 π Z ∞−∞ iξ ( e ivξ − e iwξ ) E Q ( e Y (1 − iξ ) | F t ) dξ . Denote T t = [ t, T ] × J . Then we have E Q ( e Y − iY ξ ) = E Q (cid:16) exp Z T t (1 − iξ ) ln (1 + zg ( S ( u − b ))) ˜ N ( dz, du ) | F t (cid:17) = E Q (cid:16) exp Z T t (1 − iξ ) ln (1 + zg ( S ( u − b ))) ˜ N ( dz, du ) (cid:17) = exp (cid:16) Z T t { e (1 − iξ ) ln(1+ zg ( S ( u − b ))) − (1 − iξ ) ln(1 + zg ( S ( u − b ))) − } ν Q ( dz ) du (cid:17) = exp (cid:16) Z T t { (1 + zg ( S ( u − b ))) (1 − iξ ) − ln(1 + zg ( S ( u − b ))) (1 − iξ ) − } ν Q ( dz ) du (cid:17) . Hence E Q ( e Y I { v ≥ Y ≥ w } | F t ) = 12 π Z ∞−∞ iξ ( e ivξ − e iwξ ) exp (cid:16) Z T t { (1 + zg ( S ( u − b ))) (1 − iξ ) − ln(1 + zg ( S ( u − b ))) (1 − iξ ) − } ν Q ( dz ) du (cid:17) dξ . Taking w = ln( K/A ) − rT , v → ∞ in the above formula we can evaluate (4.15) as follows. V ( t ) = e rt E Q (cid:16) ˜ S ( T ) I { ˜ S ( T ) ≥ Ke − rT } | F t (cid:17) = e rt lim v →∞ π Z ∞−∞ iξ ( e ivξ − e iwξ ) A · ˜ S ( t ) · exp (cid:16) Z T t { (1 + zg ( S ( u − b ))) (1 − iξ ) − ln(1 + zg ( S ( u − b ))) (1 − iξ ) − } ν Q ( dz ) du (cid:17) dξ = e rt lim v →∞ π Z ∞−∞ iξ ( e ivξ − e iwξ ) A · ˜ S ( t ) . exp (cid:16) Z T t { (1 + zg ( S ( u − b ))) (1 − iξ ) − ln(1 + zg ( S ( u − b ))) (1 − iξ ) − } ν Q ( dz ) du (cid:17) dξ . Exactly in the same way (and now without the factor e Y ), we have V ( t ) = Ke rt lim v →∞ π Z ∞−∞ iξ ( e ivξ − e iwξ ) A · ˜ S ( t ) . exp (cid:16) Z T t { (1 + zg ( S ( u − b ))) − iξ ) − ln(1 + zg ( S ( u − b ))) − iξ − } ν Q ( dz ) du (cid:17) dξ . This gives (4.13).
5. Numerical attempt
In this section we make an attempt to carry out some numerical computations of ourformula (3.39) against the American call options Microsoft stock traded in Questradeplatform. To apply our model in the financial market, we need to estimate all theparameters including the delay factor b from the real data. To the best of our knowledgethe theory on the parameter estimation is still unavailable even in the case of the classicalmodel of [3]. Motivated by the work of [19], we try our best guess of the parameters inthe model (3.31)-(3.32). UMP MODELS WITH DELAY 19
The real market option prices we consider is for the American call option on Microsoftstock. The data we use is from Questrade trading/investment platform on October 5,2020 at 12:25 PM (EDT). We take T to be one, three and six months active tradingperiod respectively. The real prices of the options of different strike prices are listed inthe last column of the three tables below.The readers may wonder that since the option pricing formulas for both our modeland the classical Black-Scholes model are for the European call option, why we use themarket price for the American option. The reason is that we can only find the marketprice for the American option. On the other hand, as stated in [21, p.251] “There is noadvantage to exercise an American call prematurely when the asset received upon earlyexercise does not pay dividends. The early exercise right is rendered worthless whenthe underlying asset does not pay dividends, so in this case the American call has thesame value as that of its European counterpart”. See also [16, p.61, Theorem 6.1]. Thisjustifies our use of the market price for the American option.Using Monte-Carlo simulation we calculate the prices of European option given by (4.9)and the analogous Black-Scholes formula obtained from the model: dS ( t ) = S ( t )[ αdt + σdW ( t )]. We simulate 2000 paths of the solutions to both equations using the logarith-mic Euler-Maruyama scheme [for Black-Scholes model the logarithmic Euler-Maruyamascheme is the same by replacing the jump process by Brownian motion]. In the simula-tions we take the time step ∆ to be the trading unit minute. So when T = 1 month,there are n = trading hours × × trading days = 6 . × ×
22 = 8580minutes. So ∆ = . We do the same for T = 3 and T = 6.In our calculation for the delayed jump model we use the double exponential jumpprocess as our Y i ’s with parameters p = . , q = 1 − p = . , η = 12 . , θ = 8 .
40 withthe intensity λ = .
03. The interest rate r = .
01 is the risk free rate. The delay factorwas taken to be one day which is b = . × because there are trading 6 . f ( x ) was taken to be a fixed constant f ( x ) = . g ( x ) = . ∗ sin( x/ .
11) and φ ( x ) = exp( αx/n ) with α = .
11. We choose α = .
11 since theinitial price we have taken is 209 .
11 and the predicted average price target of Microsoftstock for next one year (around 12 months from October 5, 2020) is 230 which is 11%.For the simulation of the Black-Scholes model, based on stock prices for the year 2019we take volatility of the Microsoft stock as σ = 15% to calculate Black-Scholes price. Wehave taken r = 1% since in the last one year the range of 10 year treasury rate has beenbetween .52% to 1.92%.The computations are summarized in the following tables. Notice an interesting phe-nomenon that the price we obtain by using our formula is comparable to the Black-Scholesprice for shorter maturities and is more closer to the real market price for longer maturity.This may be because of our choice of the parameters by guessing. Call Option price comparison for T = 1 month for Microsoft stockStrike Price Black-Scholesoption price(European)with 1 monthexpiration (nodelay) Option priceof jump model(European)with 1 monthexpiration Market Priceof Americanoption withexpiration 1month195 16.27 16.08 18.3200 11.41 11.05 15.15205 7.65 6.91 12210 4.54 3.62 9.43215 2.05 1.48 7220 .83 .61 5.15Call Option price comparison for T = 3 month for Microsoft stockStrike Price Black-Scholesoption price(European)with 3 monthexpiration (nodelay) Option priceof jump model(European)with 3 monthexpiration Market Priceof Americanoption withexpiration 3months195 21.37 21.27 24.40200 16.72 16.99 21.35205 13.08 14.50 18.55210 9.65 11.43 15.95215 6.35 8.58 13.65220 4.31 7.51 11.55Call Option price comparison for T = 6 month for Microsoft stockStrike Price Black-Scholesoption price(European)with 6 monthexpiration (nodelay) Option priceof jump model(European)with 6 monthexpiration Market Priceof Americanoption withexpiration 6months195 28.41 29.53 29.00200 23.85 26.11 26.15205 19.49 24.44 23.50210 16.24 21.15 21.05215 12.83 18.39 18.80220 10.58 17.97 16.70
6. Conclusion
In this paper we introduce and study a stochastic delay equation with jump and de-rive a formula for the fair price of the European call option. We assume that the jumpis dictated by a compensated L´evy process, which includes the process like asymmetricdouble exponential, hyper-exponential jump process. In the numerical execution we con-sider the asymmetric double exponential process. Furthermore, we propose a logarithmic
UMP MODELS WITH DELAY 21
Euler-Maruyama scheme (a variant of Euler-Maruyama scheme) which preserve the pos-itivity of the approximate solutions and show that the convergence rate of this schemeis 0 . L p norm, the optimal rate for the classical Euler-Maruyama scheme for thestochastic differential equations driven by standard Brownian motion (see e.g. [7]). Fromthe above tables we see that the parameters guessed here may not be the best possiblevalues but our formula still gives a good fit to the real market prices compared to theBlack-Scholes formula. We note further that potential research problem of parameterestimation is still open before we can come up with the best possible simulated results. This research was funded by an NSERC discovery fund and a startup fundof University of Alberta.References References [1] Nishant Agrawal, Yaozhong Hu, and Neha Sharma. General product formula of multiple integralsof l´evy process.
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Department of Mathematical and Statistical Sciences, University of Alberta at Edmonton,Edmonton, Canada, T6G 2G1
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