A Clark-Ocone type formula via Ito calculus and its application to finance
aa r X i v : . [ q -f i n . M F ] J un A Clark-Ocone type formula via Itˆo calculus andits application to finance
Takuji Arai ∗ and Ryoichi Suzuki † June 18, 2019
Abstract
An explicit martingale representation for random variables describedas a functional of a L´evy process will be given. The Clark-Ocone theoremshows that integrands appeared in a martingale representation are givenby conditional expectations of Malliavin derivatives. Our goal is to ex-tend it to random variables which are not Malliavin differentiable. To thisend, we make use of Itˆo’s formula, instead of Malliavin calculus. As anapplication to mathematical finance, we shall give an explicit representa-tion of locally risk-minimizing strategy of digital options for exponentialL´evy models. Since the payoff of digital options is described by an indi-cator function, we also discuss the Malliavin differentiability of indicatorfunctions with respect to L´evy processes.
MSC codes: 60G51, 91G20, 60H07.Keywords: L´evy processes, Martingale representation theorem, Local risk-minimization, Digital options, Malliavin calculus.
An explicit martingale representation for random variables described as a func-tional of a L´evy process will be given by using Itˆo’s formula, instead of Malliavincalculus. As an application to mathematical finance, we provide a representa-tion of locally risk-minimizing (LRM) strategy of digital options for exponentialL´evy models.Consider a square integrable 1-dimensional L´evy process X expressed as X t = X + µt + σW t + Z R x e N ([0 , t ] , dx ) (1.1) ∗ Department of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo, 108-8345,Japan ([email protected]) † Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama,223-8522, Japan (r [email protected]) t ≥
0, where X ∈ R , µ ∈ R , σ ≥ R := R \ { } . Here, W is a1-dimensional standard Brownian motion, N is a Poisson random measure; and e N is the compensated measure of N , that is, it is represented as e N ( dt, dx ) = N ( dt, dx ) − ν ( dx ) dt, where ν is the L´evy measure of N satisfying R R x ν ( dx ) < ∞ . For a timehorizon T > f : R → R such that f ( X T ) is squareintegrable, the martingale representation theorem implies that f ( X T ) = E [ f ( X T )] + Z T u fs dW s + Z T Z R ϑ fs,x e N ( ds, dx ) (1.2)for some predictable processes u f and ϑ f . The Clark-Ocone theorem (see, e.g.,Theorem 3.5.2 of Delong [6]) says that u f and ϑ f are described as conditionalexpectations of Malliavin derivatives of f ( X T ) if f ( X T ) is Malliavin differen-tiable, that is, f ( X T ) belongs to the space D , defined in Section 2.2 of [6]. Onthe other hand, when f ( X T ) is not Malliavin differentiable, e.g., { X T ≥ } with σ >
0, there is no way to calculate u f and ϑ f explicitly. In this paper, we aimto give concrete representations of u f and ϑ f by using Itˆo’s formula, instead ofMalliavin calculus, under some conditions which have nothing to do with theMalliavin differentiability of f ( X T ). To this end, regarding the conditional ex-pectation E [ f ( X T ) | X t = x ] as a function on ( t, x ) ∈ [0 , T ] × R , denoted by F ,we apply Itˆo’s formula to F . As a result, we obtain a Clark-Ocone type formula(1.2) in which u f and ϑ f are given as a partial derivative and a difference of F ,respectively.Using the obtained Clark-Ocone type formula, we shall provide a represen-tation of LRM strategy of digital options for exponential L´evy models in thesecond part of this paper. Remark that LRM strategy is a well-known quadratichedging method, which has been studied very well for about three decades, forcontingent claims in incomplete markets. Consider a financial market composedof one risk-free asset with interest rate r ≥ S : S t := e rt + X t (1.3)for t ≥
0. Then, the payoff of digital options is expressed as { S T ≥ K } with K >
0. Note that we need to assume some conditions on X in order to useour Clark-Ocone type formula. Considering three L´evy processes: Merton jumpdiffusion, variance gamma (VG) and normal inverse Gaussian (NIG) processes,as examples of representative L´evy processes frequently appeared in mathemat-ical finance, Merton jump diffusion and NIG processes satisfy our conditions,but VG processes do not. However, it is known that { X T ≥ c } ∈ D , for c ∈ R if R R | x | ν ( dx ) < ∞ and σ = 0 such as VG processes. Thus, when X is a VGprocess, a representation of LRM strategy of digital options are given from Ex-ample 3.9 of Arai and Suzuki [3], which has provided a general expression ofLRM strategies for exponential L´evy models by means of Malliavin calculus.2n the other hand, as is well-known, { X T ≥ c } / ∈ D , whenever σ > { X T ≥ c } / ∈ D , holds if R R | x | ν ( dx ) = ∞ and σ = 0 such asNIG processes. In summary, our result in the second part provides the only wayto calculate LRM strategy of digital options for the case where X is a Mertonjump diffusion process or an NIG process.The remainder of this paper is organized as follows: A Clark-Ocone typeformula for f ( X T ) is shown in Section 2. In Section 2.4, explicit martingalerepresentations for various functions f will be introduced. Section 3 is devoted toLRM strategy of digital options. In the last subsection, we discuss the Malliavindifferentiability of indicator functions with respect to L´evy processes. For a L´evy process X described by (1.1) and a measurable function f : R → R ,we aim at providing a Clark-Ocone type formula for f ( X T ) using Itˆo’s formula. Before stating our main theorem, we need some preparations. Denoting thecharacteristic function of X T − X t by φ ( t, z ) for ( t, z ) ∈ [0 , T ] × C , we have anddenote φ ( t, z ) := E [ e iz ( X T − X t ) ] = E [ e iz ( X T − t − X ) ]= E (cid:20) exp (cid:26) iz (cid:18) µ ( T − t ) + σW T − t + Z R x e N ([0 , T − t ] , dx ) (cid:19)(cid:27)(cid:21) = exp (cid:26) ( T − t ) (cid:18) izµ − σ z Z R ( e izx − − izx ) ν ( dx ) (cid:19)(cid:27) =: exp { ( T − t ) ψ ( z ) } (2.1)by the L´evy-Khintchine formula (see, e.g., (8.8) in Sato [10]). Now, we giveassumptions on X as follows: Assumption 2.1. (1) There exists α > such that E (cid:2) e αX T (cid:3) < ∞ .(2) For any α > with E (cid:2) e αX T (cid:3) < ∞ and any t ∈ [0 , T ) , there exists anintegrable function h t ( v ) on R such that | φ ( t, iz v ) | (cid:18) | z v | + 1 | z v | (cid:12)(cid:12)(cid:12)(cid:12)Z R ( e − z v x − z v x ) ν ( dx ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ h t ( v ) for t ∈ [ t , T + t ] , where z v = iv − α . Remark 2.2.
By Proposition 3.14 of Cont and Tankov [5], the above condition(1) is equivalent to the following two conditions:(1) ′ There exists α > such that E (cid:2) e αX t (cid:3) < ∞ for any t ∈ [0 , T ] , ′′ There exists α > such that Z {| x |≥ } e αx ν ( dx ) < ∞ .On the other hand, under (2), X T has a bounded continuous density from Propo-sition 2.5(xii) of [10]. We introduce three examples of L´evy processes, which are frequently ap-peared in literature for mathematical finance, and discuss whether or not theysatisfy Assumption 2.1.
Example 2.3 (Merton jump diffusion processes) . A L´evy process X describedby (1.1) is called a Merton jump diffusion process, if σ > and ν ( dx ) = γ √ πδ exp (cid:26) − ( x − m ) δ (cid:27) dx, where m ∈ R , δ > and γ > . In this case, X consists of a Browniancomponent and compound Poisson jumps with intensity γ . Note that jump sizesare distributed normally with mean m and variance δ , and ν is finite, that is, ν ( R ) < ∞ . Obviously, X satisfies Assumption 2.1 (1) for any α > . For anyfixed α > , (cid:12)(cid:12)(cid:12)R R ( e − z v x − z v x ) ν ( dx ) (cid:12)(cid:12)(cid:12) is bounded on v , which implies | φ ( t, iz v ) | ≤ C exp (cid:26) − σ v ( T − t )2 (cid:27) for some constant C > . Thus, (2) is also satisfied. Example 2.4 (Variance gamma processes) . When σ = 0 and ν ( dx ) = C (cid:0) { x< } e Gx + { x> } e − Mx (cid:1) dx | x | with C, G, M > , X is called a variance gamma (VG) process. For any α ∈ (0 , M ) , X satisfies Assumption 2.1 (1), but (2) is not satisfied in general, sincewe have | φ ( t, iz v ) | ≤ C V G | v | − C ( T − t ) for some constant C V G > from the view of Proposition 4.7 in Arai et al. [2]. Example 2.5 (Normal inverse Gaussian processes) . X is called a normal in-verse Gaussian (NIG) process, if σ = 0 and ν ( dx ) = δaπ e bx K ( a | x | ) | x | dx, where a > , − a < b < a , δ > , and K is the modified Bessel function ofthe second kind with parameter . For more details on the function K , seeAppendix A of [5]. Since K ( x ) = e − x r π x (cid:0) O ( x − ) (cid:1) hen x → ∞ , Assumption 2.1 (1) is satisfied for α ∈ (0 , a − b ) . In addition,taking α ∈ (0 , a − b ) arbitrarily, we can find a constant C > such that (cid:12)(cid:12)(cid:12)(cid:12)Z R ( e − z v x − z v x ) ν ( dx ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (1 + | z v | ) and | φ ( t, iz v ) | ≤ Ce − ( T − t ) δ | v | for any v ∈ R from the view of Section 5.3.8 of Schoutens [11]. As a result, (2)also holds. Henceforth, we fix α > f additionally as follows: Assumption 2.6. (1) f ( X T ) ∈ L ( P ) .(2) f ( x ) e − αx is an L ( R ) function with finite variation on R . The following is a Clark-Ocone type formula for f ( X T ). Its proof is postponeduntil the next subsection. Theorem 2.7.
Under Assumptions 2.1 and 2.6, f ( X T ) is represented as f ( X T ) = E [ f ( X T )] + Z T ∂F∂x ( s, X s ) σdW s + Z T Z R (cid:16) F ( s, X s − + y ) − F ( s, X s − ) (cid:17) e N ( ds, dy ) , (2.2) where the function F is defined as F ( t, x ) := E [ f ( X T ) | X t = x ] = E [ f ( X T − X t + x )] (2.3) for ( t, x ) ∈ [0 , T ] × R . Remark 2.8.
As mentioned in Introduction, the Clark-Ocone theorem (see,e.g., Theorem 3.5.2 of [6]) gives the same type of representation as (2.2) : f ( X T ) = E [ f ( X T )] + Z T E [ D s, f ( X T ) |F s ] σdW s + Z T Z R E [ xD s,x f ( X T ) |F s − ] e N ( ds, dx ) , when f ( X T ) ∈ D , . Note that the Malliavin derivative operator D s,x for ( s, x ) ∈ [0 , T ] × R and the space D , are defined in Section 2.2 of [6]. For example,Proposition 2.6.4 of [6] implies that f ( X T ) ∈ D , if f is Lipschitz continuousand X T has a continuous density. Thus, taking the absolute value function as f , we have | X T | = E [ | X T | ] + Z T E [sgn( X ′ T − s + X s ) | X s ] σdW s Z T Z R E h | X ′ T − s + X s − + y | − | X ′ T − s + X s − | (cid:12)(cid:12)(cid:12) X s − i e N ( ds, dy ) , where X ′ T − s is an independent copy of X T − X s . This expression can be derivedfrom not only the Clark-Ocone theorem, but also Theorem 2.7 as far as Assump-tion 2.1 is satisfied. Remark that we need to decompose | X T | into X T { X T > } and − X T {− X T > } in order to get the above expression via Theorem 2.7. Onthe other hand, when f ( X T ) / ∈ D , , the Clark-Ocone theorem is not available,but Theorem 2.7 is still available as far as Assumptions 2.1 and 2.6 are satisfied.Some examples of such cases will be discussed in Section 2.4 below. From (2.8) and (2.10) appeared in Section 2.3 below, we can rewrite (2.2)as follows:
Corollary 2.9.
Under Assumptions 2.1 and 2.6, f ( X T ) is represented as f ( X T ) = E [ f ( X T )] + Z T π Z R ( − z v ) b g ( X s , − iz v ) φ ( s, iz v ) dvσdW s + Z T Z R π Z R b g ( X s − , − iz v ) φ ( s, iz v )( e − z v y − dv e N ( ds, dy ) , where the function b g ( x, z ) for ( x, z ) ∈ R × C is defined as b g ( x, z ) := Z R e izy f ( x + y ) dy = e − izx b g (0 , z ) . (2.4) Remark 2.10.
Theorem 14.9 of Di Nunno et al. [7] introduced the same resultas Corollary 2.9 for pure jump L´evy processes, that is, the case of σ = 0 , butit has not been generalized to the case of σ > as far as we know. (Probablythis generalization is possible by using Theorem 14.15 of [7].) Note that theirargument is based on the L´evy-Wick calculus, much different from our approach.The result of Corollary 2.9 is very useful to develop a numerical scheme basedon fast Fourier transform. First of all, we show F ∈ C , ((0 , T ) × R ). Fix t ∈ (0 , T ) and x ∈ R arbitrarily.Remark that F defined in (2.3) is represented as F ( t, x ) = 12 π Z R b g ( x, − iz v ) φ ( t, iz v ) dv by Proposition 2 in Tankov [16], where b g ( x, z ) is defined in (2.4). Assumption2.6 (2) ensures that there exists a constant b C > | z v b g (0 , − iz v ) | < b C, (2.5)6hich implies that, for any t ∈ (cid:2) t , T + t (cid:3) , (cid:12)(cid:12)(cid:12)(cid:12)b g ( x, − iz v ) ∂φ∂t ( t, iz v ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) e − z v x b g (0 , − iz v ) φ ( t, iz v )( − ψ ( iz v )) (cid:12)(cid:12) ≤ b Ce αx (cid:12)(cid:12) φ ( t, iz v ) (cid:12)(cid:12) (cid:18) | µ | + σ | z v | + 1 | z v | (cid:12)(cid:12)(cid:12)(cid:12)Z R ( e − z v y − z v y ) ν ( dy ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ b Ce αx h t ( v )for some integrable function h t by (2.1) and Assumption 2.1 (2). Hence, Theo-rem 2.27 b in Folland [8] provides that ∂F∂t ( t, x ) exists on (0 , T ) × R , and ∂F∂t ( t, x ) = 12 π Z R b g ( x, − iz v ) ∂φ∂t ( t, iz v ) dv = 12 π Z R b g ( x, − iz v ) φ ( t, iz v )( − ψ ( iz v )) dv = 12 π Z R b g ( x, − iz v ) φ ( t, iz v ) (cid:18) µz v − σ z v − Z R ( e − z v y − z v y ) ν ( dy ) (cid:19) dv (2.6)holds. Next, we focus on ∂F∂x and ∂ F∂x . Note that ∂ b g∂x ( x, − iz v ) = − z v e − z v x b g (0 , − iz v ) = − z v b g ( x, − iz v ) . Thus, for any x ≤ x , Assumption 2.1 (2), together with (2.5), implies that (cid:12)(cid:12)(cid:12)(cid:12) ∂ b g∂x ( x, − iz v ) φ ( t, iz v ) (cid:12)(cid:12)(cid:12)(cid:12) = e αx | ( − z v ) b g (0 , − iz v ) φ ( t, iz v ) | ≤ b Ce αx | φ ( t, iz v ) | and (cid:12)(cid:12)(cid:12)(cid:12) ∂ b g∂x ( x, − iz v ) φ ( t, iz v ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ e αx | z v b g (0 , − iz v ) φ ( t, iz v ) | ≤ b Ce αx | z v φ ( t, iz v ) | are integrable functions of v on R . Therefore, we obtain that F ∈ C , ((0 , T ) × R )by Theorem 2.27 in [8].Secondly we show that ∂F∂t ( t, X t ) + ∂F∂x ( t, X t ) µ + σ ∂ F∂x ( t, X t )+ Z R (cid:18) F ( t, X t + y ) − F ( t, X t ) − ∂F∂x ( t, X t ) y (cid:19) ν ( dy ) = 0 . (2.7)7e have ∂F∂x ( t, X t ) = 12 π Z R ∂ b g∂x ( X t , − iz v ) φ ( t, iz v ) dv = 12 π Z R ( − z v ) b g ( X t , − iz v ) φ ( t, iz v ) dv, (2.8)and ∂ F∂x ( t, X t ) = 12 π Z R z v b g ( X t , − iz v ) φ ( t, iz v ) dv. (2.9)Noting that F ( t, X t + y ) = 12 π Z R b g ( X t , − iz v ) φ ( t, iz v ) e − z v y dv holds for any y ∈ R , we have F ( t, X t + y ) − F ( t, X t ) − ∂F∂x ( t, X t ) y = 12 π Z R b g ( X t , − iz v ) φ ( t, iz v )( e − z v y − z v y ) dv. (2.10)Hence, (2.7) holds from (2.6) and (2.8)–(2.10).Finally, since F ∈ C , ((0 , T ) × R ), Itˆo’s formula (see, e.g., Theorem 9.4 in[7]) is available. Hence, (2.7) implies f ( X T ) = F ( T, X T )= F (0 , X ) + Z T ∂F∂t ( s, X s ) ds + Z T ∂F∂x ( s, X s ) µds + Z T ∂F∂x ( s, X s ) σdW s + 12 Z T ∂ F∂x ( s, X s ) σ ds + Z T Z R (cid:18) F ( s, X s + y ) − F ( s, X s ) − ∂F∂x ( s, X s ) y (cid:19) ν ( dy ) ds + Z T Z R (cid:16) F ( s, X s − + y ) − F ( s, X s − ) (cid:17) e N ( ds, dy )= F (0 , X ) + Z T ∂F∂x ( s, X s ) σdW s + Z T Z R (cid:16) F ( s, X s − + y ) − F ( s, X s − ) (cid:17) e N ( ds, dy ) , from which Theorem 2.7 follows. Here we illustrate martingale representations for various examples of f by usingTheorem 2.7 and Corollary 2.9. 8 xample 2.11 (Polynomial functions of X T ) . When f is a polynomial function,it does not have the Lipschitz continuity basically, but we can see that f ( X T ) ∈ D , under Assumption 2.1 by using Proposition 2.5 of Suzuki [15]. Thus, wecan obtain the following representation by not only Theorem 2.7 but also theClark-Ocone theorem: f ( X T ) = E [ f ( X T )] + Z T E [ f ′ ( X ′ T − s + X s ) | X s ] σdW s + Z T Z R E [ f ( X ′ T − s + X s − + y ) − f ( X ′ T − s + X s − ) | X s − ] e N ( ds, dy ) , where X ′ T − s is an independent copy of X T − X s . Example 2.12 ( p | X T | ) . We introduce a martingale representation of p | X T | by using Theorem 2.7 or Corollary 2.9. Note that the Clark-Ocone theorem isnot available in this case, since we cannot expect that p | X T | ∈ D , when σ > .Suppose that X satisfies Assumption 2.1. We have then E [ | X T | ] < ∞ , whichensures Assumption 2.6 (1). Since p | x | does not satisfy Assumption 2.6 (2) forany α > , we decompose it into √ x ∨ f + ( x )) and p ( − x ) ∨ f − ( x )) .For functions f + and f − , denoting Df ± ( x ) := (cid:26) f ′± ( x ) , if x = 0 , , if x = 0 , we have and denote − iz b g ± ( x, z ) := − iz Z R e izy f ± ( x + y ) dy = − ize − izx Z R e izy f ± ( y ) dy = e − izx Z R e izy Df ± ( y ) dy = Z R e izy Df ± ( x + y ) dy =: b g D ± ( x, z ) for ( x, z ) ∈ R × C . Thus, we have ∂∂x E [ f ± ( X ′ T − s + x )] (cid:12)(cid:12)(cid:12) x = X s = 12 π Z R ( − z v ) b g ± ( X s , − iz v ) φ ( s, iz v ) dv = 12 π Z R b g D ± ( X s , − iz v ) φ ( s, iz v ) dv, which implies f ± ( X T ) = E [ f ± ( X T )] + Z T π Z R b g D ± ( X s , − iz v ) φ ( s, iz v ) dvσdW s + Z T Z R E [ f ± ( X ′ T − s + X s − + y ) − f ± ( X ′ T − s + X s − ) | X s − ] e N ( ds, dy ) by Theorem 2.7 or Corollary 2.9. Therefore, since p | X T | = f + ( X T ) + f − ( X T ) ,we have p | X T | = E hp | X T | i + Z T π Z R b g D ( X s , − iz v ) φ ( s, iz v ) dvσdW s Z T Z R E hq | X ′ T − s + X s − + y | − q | X ′ T − s + X s − | (cid:12)(cid:12)(cid:12) X s − i e N ( ds, dy ) , (2.11) where b g D ( x, z ) := Z R e izy (cid:16) Df + ( x + y ) + Df − ( x + y ) (cid:17) dy . Remark that we can-not rewrite the second term of the above (2.11) into the conditional expectation Z T E [ Df + ( X ′ T − s + X s ) + Df − ( X ′ T − s + X s ) | X s ] σdW s , since Df ± ( ± x ) e − αx are not finite variation for any α > . Example 2.13 ( { X T ≥ c } ) . We take an indicator function as f , that is, f ( x ) = { x ≥ c } for c ∈ R . As seen in Section 3.4, f ( X T ) / ∈ D , when σ = 0 or R ∞ | x | ν ( dx ) = ∞ . Here, we illustrate a martingale representation of { X T ≥ c } by using Theorem 2.7. Suppose that X satisfies Assumption 2.1. On the otherhand, Assumption 2.6 is automatically satisfied. Denoting F ( t, x ) := E [ { X T ≥ c } | X t = x ] = E [ { X T − X t ≥ c − x } ] = P ( X T − X t ≥ c − x ) , we have ∂F∂x ( t, x ) = p t ( c − x ) = 12 π Z R ( − z v ) b g ( x, − iz v ) φ ( t, iz v ) dv, where p t is the density function of X T − X t , and b g ( x, z ) = − iz e iz ( c − x ) Note that Assumption 2.1 (2) ensures the existence of p t . Theorem 2.7 impliesthen the following martingale representation: { X T ≥ c } = P ( X T ≥ c ) + Z T p s ( c − X s ) σdW s + Z T Z R (cid:16) P ( X ′ T − s ≥ c − X s − − y | X s − ) − P ( X ′ T − s ≥ c − X s − | X s − ) (cid:17) e N ( ds, dy ) . Example 2.14 ( e X T { X T > } ) . We consider the case where f ( x ) = e x { x> } .Assume that Assumption 2.1 holds for some α ≥ . Assumption 2.6 is thenautomatically satisfied. Defining F ( t, x ) := E [ f ( X T ) | X t = x ] , we have F ( t, x ) = e x Z ∞− x e y p t ( y ) dy where p t is the density function of X T − X t . Thus, we obtain ∂F∂x ( t, x ) = F ( t, x ) + p t ( − x ) . s a result, Theorem 2.7 provides e X T { X T > } = E (cid:2) e X T { X T > } (cid:3) + Z T (cid:16) E h e X ′ T − s + X s { X ′ T − s + X s > } | X s i + p s ( − X s ) (cid:17) σdW s + Z T Z R E h e X ′ T − s + X s − (cid:16) e y { X ′ T − s + X s − + y> } − { X ′ T − s + X s − > } (cid:17) (cid:12)(cid:12)(cid:12) X s − i e N ( ds, dy ) . The main goal of this section is to provide a representation of LRM strategy ofdigital options for exponential L´evy models described as (1.3) by using Theorem2.7. Moreover, we discuss the Malliavin differentiability of { X T ≥ c } in the lastpart of this section. We consider a financial market with maturity
T >
0, which is composed of onerisk-free asset with interest rate r ≥ t ∈ [0 , T ] is described as S t := e rt + X t , where X is a L´evy process given by (1.1). Moreover, we denote by b S thediscounted asset price process, that is, b S t := e − rt S t , which is also given as asolution to the following stochastic differential equation: d b S t = b S t − (cid:18)b µdt + σdW t + Z R ( e x − e N ( dt, dx ) (cid:19) , (3.1)where b µ := µ + σ Z R ( e x − − x ) ν ( dx ) . Next, we give a definition of LRM strategy. The following definition is asimplified version based on Theorem 1.6 of Schweizer [13], since the originalone introduced by Schweizer [12] and [13] is rather complicated. Note that [13]treated the problem under the assumption that r = 0. For the case where r > Definition 3.1. (1) A strategy is defined as a pair ϕ = ( ξ, η ) , where ξ is apredictable process satisfying E "Z T ξ s d h b S i s < ∞ , (3.2) and η is an adapted process such that the discounted value of ϕ at time t ∈ [0 , T ] , defined as b V t ( ϕ ) := ξ t b S t + η t is a right continuous process with [ b V t ( ϕ )] < ∞ for every t ∈ [0 , T ] . Note that ξ t and η t represent theamount of units of the risky and the risk-free assets respectively which aninvestor holds at time t .(2) For a strategy ϕ , a process b C ( ϕ ) defined by b C t ( ϕ ) := b V t ( ϕ ) − Z t ξ s d b S s for t ∈ [0 , T ] is called the discounted cost process of ϕ . A strategy ϕ issaid to be self-financing if b C ( ϕ ) is a constant.(3) Let H be a square integrable random variable representing the payoff ofa contingent claim at the maturity T . A strategy ϕ is called locally risk-minimizing (LRM) strategy for H , if it replicates H , that is, it satisfies b V T ( ϕ H ) = b H , and [ b C ( ϕ H ) , c M ] is a uniformly integrable martingale, where c M is the martingale part of b S . Roughly speaking, a strategy ϕ H = ( ξ H , η H ), which is not necessarily self-financing, is called LRM strategy for H , if it is the replicating strategy mini-mizing a risk caused by b C ( ϕ H ) in the L -sense among all replicating strategies.Proposition 5.2 of [13] provides that, under the so-called structure condition(SC), an LRM strategy ϕ H = ( ξ H , η H ) for H ∈ L ( P ) exists if and only if b H (= e − rT H ) admits a F¨ollmer-Schweizer decomposition, that is, b H has thefollowing decomposition b H = b H + Z T ξ F Ss d b S s + L F ST , (3.3)where b H ∈ R , ξ F S is a predictable process satisfying (3.2) and L F S is a square-integrable martingale orthogonal to c M with L F S = 0. Moreover, ϕ H is givenby ξ Ht = ξ F St , η Ht = b H + Z t ξ Hs d b S s + L F St − ξ Ht b S t . As a result, it suffices to obtain a representation of ξ H or, equivalently, ξ F S inorder to get ϕ H . Thus, we identify ξ H with ϕ H in this paper.To discuss LRM strategy, we need to consider minimal martingale measure(MMM), denoted by P ∗ . It is defined as an equivalent martingale measureunder which any square-integrable P -martingale orthogonal to c M remains amartingale. Thus, L F S appeared in (3.3) is characterized as a martingale notonly under P but also under P ∗ , and orthogonal to c M , that is, h L F S , c M i = 0.The density of P ∗ is given as d P ∗ d P = exp ( − b µσσ + C W T − b µ σ σ + C ) T Z R log (cid:18) − b µ ( e x − σ + C (cid:19) e N ([0 , T ] , dx )+ T Z R (cid:18) log (cid:18) − b µ ( e x − σ + C (cid:19) + b µ ( e x − σ + C (cid:19) ν ( dx ) ) , where C := R R ( e x − ν ( dx ). Note that C is finite and P ∗ exists underAssumption 3.2 below. Moreover, by the Girsanov theorem, W ∗ t := W t + b µσσ + C t (3.4)and e N ∗ ([0 , t ] , dx ) := e N ([0 , t ] , dx ) + b µ ( e x − σ + C ν ( dx ) t (3.5)are a P ∗ -Brownian motion and the compensated Poisson random measure of N under P ∗ , respectively. We can then rewrite (3.1) as d b S t = b S t − (cid:18) σdW ∗ t + Z R ( e x − e N ∗ ( dt, dx ) (cid:19) . Remark that X is a L´evy process even under P ∗ , and the L´evy measure under P ∗ is given as ν ∗ ( dx ) := (cid:18) − b µ ( e x − σ + C (cid:19) ν ( dx ) . We shall show a representation of LRM strategy for digital options by usingTheorem 2.7 under P ∗ . Thus, we need to rewrite Assumption 2.1 into one under P ∗ . Note that, as mentioned in Example 2.13, Assumption 2.6 is automaticallysatisfied. Assumption 3.2. (1) R R ( e x − ν ( dx )(= C ) < ∞ , which implies that E P ∗ (cid:2) e αX T (cid:3) < ∞ holds for some α ≥ . Such an α is fixed throughoutthis section.(2) ≥ b µ > − σ − C .(3) For any t ∈ [0 , T ) , there exists an integrable function h ∗ t ( v ) on R such that | φ ∗ ( t, iz v ) | (cid:18) | z v | + 1 | z v | (cid:12)(cid:12)(cid:12)(cid:12)Z R ( e − z v x − z v x ) ν ∗ ( dx ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ h ∗ t ( v ) for t ∈ [ t , T + t ] , where φ ∗ ( t, z ) := E P ∗ [ e iz ( X T − X t ) ] for z ∈ C . Note that Assumption 3.2 (1) ensures the structure condition (SC); and MMM P ∗ exists as an equivalent probability measure to P by the above (2). Moreover,(3) is corresponding to Assumption 2.1 (2), and ensures that X T − X t has abounded continuous density under P ∗ , denoted by p ∗ t .13 emark 3.3. By a similar argument with Example 2.3, Merton jump diffu-sion processes satisfy Assumption 3.2 without any parameter restriction. As forNIG processes, taking α ∈ ( , , a > and − < b ≤ − , we can see thatAssumption 3.2 is satisfied from the view of Arai et al. [1]. On the other hand,VG processes violate Assumption 3.2 by a similar argument with Example 2.4.For more details on this matter, see Remark 3.5 below. Note that the formula-tions of ϕ ∗ and ν ∗ are given in [2] for Merton jump diffusion processes and VGprocesses, and in [1] for NIG processes, respectively. Theorem 3.4.
Under Assumption 3.2, the LRM strategy ξ H for the digitaloption { S T ≥ K } with K > is given by ξ Ht = e − rT b S t − ( σ + C ) (cid:18) κ t σ + Z R Ψ ∗ t − ( K, x )( e x − ν ( dx ) (cid:19) (3.6) for t ∈ [0 , T ] . Here κ t := p ∗ t (log K − rT − X t ) and Ψ ∗ t ( K, x ) := P ∗ ( X ′ T − t ≥ log K − rT − X t − x | X t ) − P ∗ ( X ′ T − t ≥ log K − rT − X t | X t ) , where X ′ T − t is an independent copy of X T − X t . Remark 3.5.
By Example 3.9 of [3], we can obtain the same result as Theorem3.4 by using Malliavin calculus for L´evy processes if { S T ≥ K } ∈ D , , where D , is defined in Section 2.2 of [6]. Indeed, as shown in Section 4.2 of Geiss et al.[9], if σ = 0 , R R | x | ν ( dx ) < ∞ and X t has a bounded density, then we have { X T ≥ c } ∈ D , , in other words, { S T ≥ K } ∈ D , . For example, VG processessatisfy all of these conditions, although they do not satisfy Assumption 3.2 asstated in Remark 3.3. In other words, when X is a VG process, Theorem 3.4is not available, but we can obtain the same result via Malliavin calculus. TheMalliavin differentiability of indicator functions will be discussed in Section 3.4below. Denoting by ξ t the right hand side of (3.6), and defining a P ∗ -martingale L H with L H = 0 as L Ht := E P ∗ " e − rT { S T ≥ K } − e − rT E P ∗ [ { S T ≥ K } ] − Z T ξ s d b S s (cid:12)(cid:12)(cid:12) F t , we have e − rT { S T ≥ K } = e − rT E P ∗ (cid:2) { S T ≥ K } (cid:3) + Z T ξ s d b S s + L HT . (3.7)14t is enough to show that (3.7) is the F¨ollmer-Schweizer decomposition of e − rT { S T ≥ K } , the discounted value of the payoff function. To this end, wesee that L H is a P -martingale orthogonal to c M .Defining a function F on [0 , T ] × R as F ( t, x ) := E P ∗ [ { S T ≥ K } | X t = x ] = P ∗ ( X T − X t ≥ log K − rT − x ) , we have F ( t, X t ) = F (0 , X ) + Z t ∂F∂x ( s, X s ) σdW ∗ s + Z t Z R (cid:16) F ( s, X s − + y ) − F ( s, X s − ) (cid:17) e N ∗ ( ds, dy )= F (0 , X ) + Z t κ s σdW ∗ s + Z t Z R Ψ ∗ s − ( K, y ) e N ∗ ( ds, dy )by Assumption 3.2 and Example 2.13. Thus, we have L Ht = e − rT F ( t, X t ) − e − rT F (0 , X ) − Z t ξ s d b S s = Z t e − rT κ s σdW ∗ s + Z t Z R e − rT Ψ ∗ s − ( K, x ) e N ∗ ( ds, dx ) − Z t ξ s b S s − (cid:18) σdW ∗ s + Z R ( e x − e N ∗ ( ds, dx ) (cid:19) . (3.8)To show that L H is a P -martingale, we calculate the following: (cid:16) e − rT κ s σ − ξ s b S s − σ (cid:17) b µσσ + C = e − rT (cid:18) κ s C − Z R Ψ ∗ s − ( K, x )( e x − ν ( dx ) (cid:19) b µσ ( σ + C ) and Z R (cid:16) e − rT Ψ ∗ s − ( K, x ) − ξ s b S s − ( e x − (cid:17) b µ ( e x − σ + C ν ( dx )= (cid:18)Z R e − rT Ψ ∗ s − ( K, x )( e x − ν ( dx ) − ξ s b S s − C (cid:19) b µσ + C = e − rT (cid:18)Z R Ψ ∗ s − ( K, x )( e x − ν ( dx ) − κ s C (cid:19) b µσ ( σ + C ) for s ∈ [0 , T ]. Therefore, (3.8), together with (3.4) and (3.5), implies that L Ht = Z t (cid:16) e − rT κ s − ξ s b S s − (cid:17) σdW s Z t Z R (cid:16) e − rT Ψ ∗ s − ( K, x ) − ξ s b S s − ( e x − (cid:17) e N ( ds, dx ) , from which L H is a P -martingale.Next, we see that L H is orthogonal to c M . To this end, we have only to see h L H , c M i = 0. Noting that c M is given as d c M t = b S t − (cid:18) σdW t + Z R ( e x − e N ( dt, dx ) (cid:19) , we have d h L H , c M i t = b S t − σ (cid:16) e − rT κ t − ξ t b S t − (cid:17) dt + b S t − Z R (cid:16) e − rT Ψ ∗ t − ( K, x ) − ξ t b S t − ( e x − (cid:17) ( e x − ν ( dx ) dt = b S t − e − rT (cid:18) κ t σ + Z R Ψ ∗ t − ( K, x )( e x − ν ( dx ) (cid:19) dt − ξ t b S t − ( σ + C ) dt = 0 . Consequently, (3.7) is the F¨ollmer-Schweizer decomposition of e − rT { S T ≥ K } ,which implies that ξ H = ξ . This complete the proof of Theorem 3.4. As seen in Remark 3.5, { X T ≥ c } ∈ D , holds true for any c ∈ R when X isa VG process. That is, we can obtain the same result as Theorem 3.4 for VGprocesses by using Example 3.9 of [3]. On the other hand, it is known that { X T ≥ c } / ∈ D , whenever σ >
0. In other words, if X includes a Browniancomponent such as Merton jump diffusion processes, we need to use Theorem3.4 to compute ξ H in (3.6). In addition, as seen in Proposition 3.6 below, evenif σ = 0, we have { X T ≥ c } / ∈ D , as long as R R | x | ν ( dx ) = ∞ such as NIGprocesses. As a result, we can say that Theorem 3.4 provides the only way tocalculate LRM strategy of digital options for Merton jump diffusion and NIGprocesses. Proposition 3.6.
Let X be a pure jump L´evy process with L´evy measure ν satisfying R [ − , | x | ν ( dx ) = ∞ . In addition, suppose that X T has a boundedcontinuous density function p . Then, we have { X T ≥ c } / ∈ D , for all c ∈ R with p ( c ) > .Proof. Fix c ∈ R with p ( c ) > ε > p ( x ) > p ( c )2 for any x ∈ ( c − ε, c + ε ). From the view of Proposition 5.4 ofSol´e et al. [14], it suffices to show that E "Z T Z R | Ψ s,x { X T ≥ c } | x ν ( dx ) ds = ∞ , Ψ s,x is the increment quotient operator defined in Section 5.1 of [14].Thus, we have E "Z T Z R | Ψ s,x { X T ≥ c } | x ν ( dx ) ds = Z T Z R E (cid:2) | Ψ s,x { X T ≥ c } | (cid:3) x ν ( dx ) ds = Z T Z R E (cid:20) | { X T + x ≥ c } − { X T ≥ c } | x (cid:21) x ν ( dx ) ds = Z T (cid:18)Z ∞ P ( c > X T ≥ c − x ) ν ( dx ) + Z −∞ P ( c − x > X T ≥ c ) ν ( dx ) (cid:19) ds ≥ T (cid:18)Z ε P ( c > X T ≥ c − x ) ν ( dx ) + Z − ε P ( c − x > X T ≥ c ) ν ( dx ) (cid:19) ≥ T p ( c )2 Z ( − ε,ε ) | x | ν ( dx ) = ∞ , since R I | x | ν ( dx ) = ∞ for any interval I ⊂ R including 0 as an interior point. (cid:3) Acknowledgments
Takuji Arai gratefully acknowledges the financial support of the MEXT Grantin Aid for Scientific Research (C) No.18K03422.
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