# A Semigroup Expansion for Pricing Barrier Options

Authors: Takashi Kato; Akihiko Takahashi; Toshihiro Yamada
Journal: International Journal of Stochastic Analysis
 (2014)

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## Abstract

This paper presents a new asymptotic expansion method for pricing continuously monitoring barrier options. In particular, we develop a semigroup expansion scheme for the Cauchy-Dirichlet problem in the second-order parabolic partial differential equations (PDEs) arising in barrier option pricing. As an application, we propose a concrete approximation formula under a stochastic volatility model and demonstrate its validity by some numerical experiments.


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## Body

## 1. Introduction

Since the Merton’s seminal work [1] barrier options have been quite popular and important products in both academics and financial business for the last four decades. In particular, fast and accurate computation of their prices and Greeks is highly desirable in the risk management, which is a tough task under the finance models commonly used in practice. Thus, it has been one of the central issues in the mathematical finance community. Among various approaches to attacking the problem, this paper proposes a new semigroup expansion scheme under general diffusion setting.Firstly, let us note that the value of a continuously monitoring down-and-out barrier option is expressed as the following form under the so-called risk-neutral probability measure:(1)CBarrier(T,x)=E[e-∫0T‍c(Xrx)drf(XTx)1{τ>T}]=E[e-∫0T‍c(Xrx)drf(XTx)1{min⁡t∈[0,T]⁡Xt>L}].
Here, T>0 is a maturity of the option and f is an option payoff function at maturity. (Xtx)t denotes a vector process with initial value x under the risk-neutral probability measure, which includes a price process of the underlying asset (usually given as the solution of a certain stochastic differential equation (SDE)). Also, c stands for the risk-free interest rate process. Moreover, L stands for a constant lower barrier, that is L<x, and τ is the hitting time to L:
(2)τ=inf⁡{t∈[0,T]:Xtx≤L}.It is well-known that a possible approach in computation ofCBarrier(T,x) is the Euler-Maruyama scheme, which stores the sample paths of the process (Xtx)t through an n-time discretization with the step size T/n. When applying this scheme to pricing a continuously monitoring barrier option, one kills the simulated process, say (X¯tix)i if X¯tix exits from the domain (L,∞) until the maturity T. The usual Euler-Maruyama scheme issuboptimal since it does not control the diffusion paths between two successive dates ti and ti+1: the diffusion paths could have crossed the barriers and come back to the domain without being detected. It is also known that the error between CBarrier(T,x) and C¯Barrier(T,x), the barrier option price obtained by the Euler-Maruyama scheme is of order T/n, as opposed to the order T/n for standard plain-vanilla options (see [2]). Thus, various Monte-Carlo schemes have been proposed for improving the order of the error (see [3] for instance).One of the other tractable approaches for calculatingCBarrier(T,x) is to derive an analytical approximation. If we obtain an accurate approximation formula, it is a powerful tool for pricing continuously monitoring barrier options because we need not rely on Monte-Carlo simulations anymore. However, from a mathematical viewpoint, deriving an approximation formula by applying stochastic analysis is not an easy task since the Malliavin calculus cannot be directly applied. It is due to the nonexistence of the Malliavin derivative Dtτ (see [4]) and to the fact that the minimum (maximum) process of the Brownian motion has only the first-order differentiability in the Malliavin sense. Thus, neither approach in [5] nor in [6] can be applied directly to evaluation of continuously monitoring barrier options, while they are applicable to pricing discrete barrier options (see [7] for the detail).This paper proposes a new general method for the approximation of barrier option prices. Particularly, our objective is to price barrier options when the vector process of the underlying state variables is described by the following perturbed SDE under a filtered complete probability space with the risk-neutral probability measure, which will be concretely defined in the begging of the next section:(3)dXtɛ,x=b(Xtɛ,x,ɛ)dt+σ(Xtɛ,x,ɛ)dBt,X0ɛ,x=x,
where ɛ is a small parameter. In this case, the barrier option price (1) is characterized as a solution of the Cauchy-Dirichlet problem:
(4)∂∂tuɛ(t,x)+Lɛuɛ(t,x)=0,(t,x)∈[0,T)×(L,∞),uɛ(T,x)=f(x),x>L,uɛ(t,L)=0,t∈[0,T],
where the differential operator Lɛ is determined by the diffusion coefficients b and σ with the risk-free interest rate c, which will be explicitly defined in the next section.Next, we introduce an asymptotic expansion formula:(5)uɛ(t,x)=u0(t,x)+ɛv10(t,x)+⋯+ɛn-1vn-10(t,x)+O(ɛn),
where O denotes the Landau symbol. The function u0(t,x) is the solution of (4) with ɛ=0: if b(x,0) and σ(x,0) have some simple forms such as constants (as in the Black-Scholes model), we already know the closed form of u0(t,x) and hence obtain the price. Then, we are able to get the approximate value for uɛ(t,x) through evaluation of the coefficient functions v10(t,x),…,vn-10(t,x). In fact, they are also characterized as the solution of a certain PDE with the Dirichlet condition. By formal asymptotic expansions, (5) above and the following,
(6)Lɛ=L0+ɛL~10+⋯+ɛn-1L~n-10+⋯,
we can derive the following PDE which vk0(t,x) satisfies:
(7)∂∂tvk0(t,x)+L0vk0(t,x)+gk0(t,x)=0,vvvvvvvvvvvv(t,x)∈[0,T)×(L,∞),vk0(T,x)=0,x>L,vk0(t,L)=0,t∈[0,T],
where gk0(t,x) will be given explicitly in Section 2. Moreover, by applying the Feynman-Kac approach to the PDE (7), we obtain a semigroup representation of vk0. That is, for each k=1,…,n-1,
(8)vk0(T-t,x)=∑l=1k‍∑(βi)i=1l⊂Nl,∑iβi=k‍∫0t‍∫0t1‍⋯∫0tl-1‍Pt-t1DL~β10Pt1-t2Dhhhhhhhhhhhhhhhhhhhhhhhh×L~β20⋯Ptl-1-tlDL~βl0hhhhhhhhhhhhhhhhhhhhhhhh×PtlDf(x)dtl⋯dt1,
where (PtD)t is a semigroup defined in Section 2. We will justify the above argument in a mathematically rigorous manner in Section 2.The theory of the Cauchy-Dirichlet problem for this kind of the second order parabolic PDE is well understood for the case of bounded domains (see [8–10] e.g.). As for an unbounded domain case such as (4), [11] provides the existence and uniqueness results for a solution of the PDE and the Feynman-Kac type formula, the part of which will be cited as Theorem 1 in Section 2. However, some mathematical difficulty exists for applying the results of [11] to the PDE (7). More precisely, the function gk0(t,x) may be divergent at t=T. Hence, in order to obtain an asymptotic expansion (5), we generalize the result of [11] and the argument of the Feynman-Kac representation. Furthermore, we derive a new representation (8) for vk0(t,x) by using the semigroup (PtD)t. We notice that such a form is convenient for evaluation of vk0(t,x) in concrete examples.We also apply our method to pricing a barrier option in a stochastic volatility model. Then, as an example of (8) we obtain a new approximation formula of the barrier option price CBarrierSV,ɛ under a stochastic volatility model as follows: for the initial value of the logarithmic underlying price x, the maturity T and the lower barrier L,
(9)CBarrierSV,ɛ(T,ex)=E[e-cTf(STɛ,x)1{min⁡0≤t≤T⁡Stɛ>L}]≃PTDf¯(x)+ɛ∫0T‍PT-rDL~10PrDf¯(x)dr,
where (Stɛ,x)t is the underlying asset price process, f is a payoff function and f¯(x)=f(ex) and the expectation is taken under the risk-neutral probability measure. Here, PTDf¯(x) is regarded as the down-and-out barrier option price in the Black-Scholes model. Moreover, we confirm practical validity of our method through a numerical example given in Section 3.Finally, we remark that there exist the previous works on barrier option pricing such as [12–15], which start with some specific models (e.g., Black-Scholes model or some type of fast mean-reversion model), and derive approximation formulas for discretely or continuously monitoring barrier option prices. Our approach is to firstly develop a general semigroup expansion scheme for the Cauchy-Dirichlet problem under multidimensional diffusion setting; then as an application, we provide a new approximation formula under a certain class of stochastic volatility model.The organization of this paper is as follows: the next section firstly prepares the existence and uniqueness result for the Cauchy-Dirichlet problem in the second-order parabolic PDE associated with barrier option pricing. Then, we present our main result for an asymptotic expansion of barrier option prices. Section3 shows numerical examples under a stochastic volatility model. Section 4 concludes. Finally, Appendix provides the proofs of the results in the main text.


## 2. Asymptotic Expansion of Barrier Option Price

### 2.1. Main Results

This subsection states the setup and our main results. The relevant assumptions(A)–(H) will be explained in the next subsection.Suppose first that a filtered complete probability space(Ω,F,Q,{Ft}t∈[0,T]) is given, where Q denotes the risk-neutral probability measure, the filtration {Ft}t∈[0,T] satisfies the usual conditions, and T>0 is some fixed time horizon. Then, the d-dimensional underlying state variable is described by the following perturbed SDE:
(10)dXtɛ,x=b(Xtɛ,x,ɛ)dt+σ(Xtɛ,x,ɛ)dBt,X0ɛ,x=x,
where B is an m-dimensional Brownian motion and ɛ is a small parameter. Let b:Rd×I→Rd and σ:Rd×I→Rd⊗Rm be Borel measurable functions (d,m∈N) where I is an interval on R including the origin 0, for instance I=(-1,1).We consider the SDE (10) for any x∈Rd and ɛ∈I. We note that Assumption (A) introduced in the next subsection guarantees the existence and uniqueness of a solution of (10). We also remark that at least one element of Xɛ,x stands for the underlying asset price process of a barrier option.Next, we are interested in evaluation of the following barrier option price: for a smallɛ,
(11)uɛ(t,x)=E[exp⁡(-∫0T-t‍c(Xrɛ,x,ɛ)dr)vvvvvvv×f(XT-tɛ,x)1{τD(Xɛ,x)≥T-t}exp⁡(-∫0T-t‍c(Xrɛ,x,ɛ)dr)],(t,x)∈[0,T]×D¯
for Borel measurable functions f:Rd→R, c:Rd×I→R and a domain D⊂Rd.Also,D¯⊂Rd is the closure of D and τD(w),w∈C([0,T];Rd) stands for the first exit time from D; that is
(12)τD(w)=inf⁡{t∈[0,T];w(t)∉D}.We also remark that the expectation operatorE[·] is taken under the risk-neutral probability measure Q and that e-∫0T-t‍c(Xrɛ,x,ɛ)dr represents the discount factor with the risk-free interest rate process c.Let us define a second order differential operatorLɛ by
(13)Lɛ=12∑i,j=1d‍aij(x,ɛ)∂2∂xi∂xj+∑i=1d‍bi(x,ɛ)∂∂xi-c(x,ɛ),
where aij=∑k=1d‍σikσjk. We consider the following Cauchy-Dirichlet problem for a PDE of parabolic type:
(14)∂∂tuɛ(t,x)+Lɛuɛ(t,x)=0,(t,x)∈[0,T)×D,uɛ(T,x)=f(x),x∈D,uɛ(t,x)=0,(t,x)∈[0,T]×∂D.Under the assumptions(A)–(E) stated in the next subsection, we have the following existence and uniqueness result due to Theorem 3.1 in [11].Theorem 1.
Assume(A)–(E) which are given in Section 2.2. For each ɛ∈I, uɛ(t,x) defined with formula (11) is a (classical) solution of (14) and
(15)sup⁡(t,x)∈[0,T]×D¯⁡|uɛ(t,x)|1+|x|2m<∞.
Moreover, if wɛ(t,x) is also a solution of (14) satisfying the growth condition
(16)sup⁡(t,x)∈[0,T]×D¯⁡|wɛ(t,x)|1+|x|2m′<∞
for some m′∈N, then uɛ=wɛ.Our main purpose is to present an asymptotic expansion of the barrier option priceuɛ(t,x):
(17)uɛ(t,x)=u0(t,x)+ɛv10(t,x)+⋯+ɛn-1vn-10(t,x)+O(ɛn),ɛ⟶0.
Here, the coefficient functions vk0(t,x), k=1,…,n-1 are (formally) given as the solution of
(18)∂∂tvk0(t,x)+L0vk0(t,x)+gk0(t,x)=0,(t,x)∈[0,T)×D,vk0(T,x)=0,x∈D,vk0(t,x)=0,(t,x)∈[0,T]×∂D,
where gk0(t,x) is given inductively by
(19)gk0(t,x)=L~k0u0(t,x)+∑l=1k-1‍L~k-l0vl0(t,x).
Here, L~k0 is defined as follows:
(20)L~k0=1k!{12∑i,j=1d‍∂kaij∂ɛk(x,0)∂2∂xi∂xjvvvvvv+∑i=1d‍∂kbi∂ɛk(x,0)∂∂xi-∂kc∂ɛk(x,0)}.Then, we can show the next result, whose proof is given in AppendixA.Theorem 2.
Assume(A)–(H) which are stated in Section 2.2. Then, for each k=1,…,n-1, vk0 is the classical solution of (18) and satisfies
(21)|vk0(t,x)|≤Ck(1+|x|2mk),(t,x)∈[0,T]×Rd
for some Ck, mk>0.Note that the uniqueness of the solutions of (18) follows from the same arguments as in the proof of Theorem 5.7.6 in [16]. That is, we obtain the next proposition.Proposition 3.
For any functiong which has a polynomial growth rate in x uniformly in t, a classical solution of (18) is unique in the following sense: if v and w are classical solutions of (18) and |v(t,x)|+|w(t,x)|≤C(1+|x|2m) for some C,m>0, then v=w.Now, we are able to state our first main result on the asymptotic expansion. The proof is given in AppendixB.Theorem 4.
Assume(A)–(H) which are given in Section 2.2. There are positive constants Cn and m~n which are independent of ɛ such that
(22)|uɛ(t,x)-(u0(t,x)+∑k=1n-1‍ɛkvk0(t,x))|≤Cn(1+|x|2m~n)ɛn,(t,x)∈[0,T]×D¯.Next, we construct a semigroup corresponding to(Xt0,x)t (i.e., (Xtɛ,x)t with ɛ=0) and D. Then, based on this semigroup we can obtain more explicit representation for the coefficient function vk0(t,x) than the right hand side of (36) in Assumption (H), which will appear in Section 2.2.LetCb0(D¯) be the set of bounded continuous functions f:D¯→R such that f(x)=0 on ∂D. Obviously, Cb0(D¯) equipped with the supnorm becomes a Banach space.Fort∈[0,T] and f∈Cb0(D¯), we define PtDf:D¯→R by
(23)PtDf(x)=E[exp⁡(-∫0t‍c(Xv0,x,0)dv)×f(Xt0,x)1{τD(X0,x)≥t}(-∫0t‍c(Xv0,x,0)dv)],
where c(x,0) is nonnegative. We notice that PtDf(x) is equal to u0(T-t,x) with the payoff function f. Then, we have the following result.Proposition 5.
Under the assumptions(A)–(E) stated in Section 2.2, the mapping PtD:Cb0(D¯)→Cb0(D¯) is well-defined and (PtD)0≤t≤T is a contraction semigroup.Proof.
Letf∈Cb0(D¯). The relations P0Df=f, PtDf|∂D=0, and sup⁡D¯|PtDf|≤sup⁡D¯|f| are obvious. The continuity of PtDf is by Lemma 4.3 in [11]. The semigroup property is verified by a straightforward calculation.Remark 6.
Note that(PtD)t also has the semigroup property on the set Cp0(D¯) of continuous functions f, each of which has a polynomial growth rate and satisfies f(x)=0 on ∂D.Finally, we show our second main result on the semigroup representation of the coefficient functionvk0 in the expansion, whose proof is given in Appendix C.Theorem 7.
Under Assumptions(A)–(H) given in Section 2.2, for each k=1,…,n-1(24)vk0(T-t,x)=∑l=1k‍∑(βi)i=1l⊂Nl,∑iβi=k‍∫0t‍∫0t1‍⋯∫0tl-1‍Pt-t1DL~β10Pt1-t2Dhhhhhhhhhhhhhhhhhhhhhhhhhhh×L~β20⋯Ptl-1-tlDhhhhhhhhhhhhhhhhhhhhhhhhhhh×L~βl0PtlDf(x)dtl⋯dt1.


### 2.2. Assumptions

This subsection introduces a series of the assumptions necessary for our main results stated in the previous subsection. Particularly, the assumptions (A)–(E) are relevant for Theorem1, that is the existence and uniqueness result of the PDE (14), while (F)–(H) are additional assumptions necessary for the asymptotic expansion results, that is Theorems 2–7.(A)
There is a positive constantA1 such that
(25)|σij(x,ɛ)|2+|bi(x,ɛ)|2≤A1(1+|x|2),vvvvvvvx∈Rd,ɛ∈I,i,j=1,…,d.Moreover, for eachɛ∈I it holds that σij(·,ɛ),bi(·,ɛ)∈Lip⁡ for i,j=1,…,d, where Lip⁡ is the set of locally Lipschitz continuous functions defined on Rd:
(26)Lip⁡={f∈C(Rd;R);foranycompactsetK⊂Rd,∃CK>0suchthat|f(x)-f(y)|≤CK|x-y|,vvvvx,y∈KRd}.Remark 8.
Note that under (A), the existence and uniqueness of a solution of (10) are guaranteed on any filtered probability space equipped with a standard m-dimensional Brownian motion, and Corollary 2.5.12 in [17] and Lemma 3.2.6 in [18] imply
(27)E[sup⁡0≤r≤t|Xrɛ,x-x|2l]≤CltleClt(1+|x|2l),vvvvvv.vvv(t,x)∈[0,T]×Rd,l∈N
for some Cl>0, which depends only on l, A1, and CK. Moreover, (Xrx)r has the strong Markov property.(B)
The functionf(x) is continuous on D¯ and there are Cf>0 and m∈N such that |f(x)|≤Cf(1+|x|2m), x∈Rd. Moreover, f(x)=0 on Rd∖D.Remark 9.
The assumption(B) which corresponds toH2(2) in [11] guarantees the continuity of a solution of (14) (if it exists) on the so-called parabolic boundary Σ=∂D×[0,T)∪D¯×{T}. For the details, see page 8 in [11].(C)
c(x,ɛ) is nonnegative (i.e., c(x,ɛ)≥0). Moreover, for each ɛ∈I, it holds that c(·,ɛ)∈Lip⁡.(D)
The boundary∂D has the outside strong sphere property; that is, for each x∈∂D there is a closed ball E such that E∩D=∅ and E∩D¯={x}.Remark 10.
The assumption(D) provides the regularity of each point in ∂D. (compared to [8]) Also, [11] points out that (D) with the ellipticity of the matrix (aij(x,ɛ))ij in (E) below gives
(28)P(τD(Xϵ,x)=τD-(Xϵ,x))=1.(E)
The matrix(aij(x,ɛ))ij is locally elliptic in the sense that for each ɛ∈I and compact set K⊂Rd there is a positive number μɛ,K such that ∑i,j=1d‍aij(x,ɛ)ξiξj≥μɛ,K|ξ|2 for any x∈K and ξ∈Rd.Remark 11.
Note that although the condition (E) (local ellipticity) is necessary for the existence of classical solution of our PDE (see Remark 2.2 in [11]), the assumption can be removed through consideration of viscosity solutions rather than classical solutions by applying Theorem 8.2 in [19] and Theorems 4.4.3 and 7.7.2 in [18]. Note that we need the additional assumption such that I⊂[0,∞) by technical reason in this case.To study the asymptotic expansion, we put the following assumptions in addition to (A)–(E). Firstly, by the next condition we can properly defineL~k0, k∈N in (20) above.(F)
Letn∈N. The functions aij(x,ɛ), bi(x,ɛ), and c(x,ɛ) are n-times continuously differentiable in ɛ. Furthermore, each of derivatives ∂kaij/∂ɛk, ∂kbi/∂ɛk, ∂kc/∂ɛk, k=1,…,n-1, has a polynomial growth rate in x∈Rd uniformly in ɛ∈I.To state the existence of the functionsvk0(t,x) in the asymptotic expansion (17), we first prepare the following set.Definition 12.
The setHm,p of g∈C([0,T)×D¯) is defined to satisfy the following condition. There is a Mg∈C([0,T))∩Lp([0,T),dt) such that
(29)|g(t,x)|≤Mg(t)(1+|x|2m),t∈[0,T),x∈D¯.Given this definition of the setHm,p, we put the next condition on u0.(G)
u0∈Gm, where
(30)Gm={∂2u∂xi∂xju∈C1,2([0,T)×D)∩C([0,T]×D¯);∂u∂xi∈Hm,2,∂2u∂xi∂xj∈Hm,1,i,j=1,…,d}.Now we examine the conditions necessary for the classical solution to the PDE (18).Firstly, Let us start with the case ofk=1. By the assumption (G), we have g10∈Hm,1 for some m∈N by the definition of gk0 with k=1 in (19). Thus we can define
(31)v10(t,x)=E[∫0(T-t)∧τD(X0,x)‍exp⁡(-∫0r‍c(Xv0,x,0)dv)vvvvvvvvv×g10(t+r,Xr0,x)dr(-∫0r‍c(Xv0,x,0)dv)].
Therefore, if we assume that v10∈C1,2([0,T)×D) we can show that v10 is the solution of (18) with k=1; that is, we can confirm that
(32)∂∂tv10(t,x)+L0v10(t,x)+g10(t,x)=0.
Note that the relations v10(T,·)=0 and v10=0 on [0,T]×∂D are obvious.Next, let us give some comments on the smoothness ofv10. In many cases as in the Black-Scholes model (see (49) in Section 3) we can rewrite (31) as
(33)v10(t,x)=∫0T-t‍∫D‍g10(t+r,y)p(r,x,y)dydr
for some p(r,x,y). Thus, if p has a “good” smoothness property, the smoothness of v10 also holds such as
(34)∂∂tv10(t,x)=-lim⁡s→T∫D‍g10(s,y)p(s-t,x,y)dy+∫0t‍∫D‍∂∂tg10(t+r,y)p(r,x,y)dr,
if the limit in the right hand side exists, and
(35)∂∂xiv10(t,x)=∫0t‍∫D‍g10(t+r,y)∂∂xip(r,x,y)dr,∂2∂xi∂xjv10(t,x)=∫0t‍∫D‍g10(t+r,y)∂2∂xi∂xjp(r,x,y)dr.Moreover, ifv10 is in Gm1 for some m1∈N, we also have g20∈Hm~1,1 for some m~1∈N by the definition of gk0 with k=2 in (19). Then, we can define v20 similarly as v10. Furthermore, under some suitable smoothness conditions for v20, which may be given by the smoothness property of p(r,x,y), we are able to show that v20 is the classical solution of (18) with k=2.Thus, the observation above leads us to our final assumption.(H)
It holds thatvk0∈Gmn, k=1,…,n-1 for some mn∈N, where
(36)vk0(t,x)=E[∫0(T-t)∧τD(X0,x)‍exp⁡(-∫0r‍c(Xv0,x,0)dv)×gk0(t+r,Xr0,x)dr∫0(T-t)∧τD(X0,x)‍].


## 2.1. Main Results

This subsection states the setup and our main results. The relevant assumptions(A)–(H) will be explained in the next subsection.Suppose first that a filtered complete probability space(Ω,F,Q,{Ft}t∈[0,T]) is given, where Q denotes the risk-neutral probability measure, the filtration {Ft}t∈[0,T] satisfies the usual conditions, and T>0 is some fixed time horizon. Then, the d-dimensional underlying state variable is described by the following perturbed SDE:
(10)dXtɛ,x=b(Xtɛ,x,ɛ)dt+σ(Xtɛ,x,ɛ)dBt,X0ɛ,x=x,
where B is an m-dimensional Brownian motion and ɛ is a small parameter. Let b:Rd×I→Rd and σ:Rd×I→Rd⊗Rm be Borel measurable functions (d,m∈N) where I is an interval on R including the origin 0, for instance I=(-1,1).We consider the SDE (10) for any x∈Rd and ɛ∈I. We note that Assumption (A) introduced in the next subsection guarantees the existence and uniqueness of a solution of (10). We also remark that at least one element of Xɛ,x stands for the underlying asset price process of a barrier option.Next, we are interested in evaluation of the following barrier option price: for a smallɛ,
(11)uɛ(t,x)=E[exp⁡(-∫0T-t‍c(Xrɛ,x,ɛ)dr)vvvvvvv×f(XT-tɛ,x)1{τD(Xɛ,x)≥T-t}exp⁡(-∫0T-t‍c(Xrɛ,x,ɛ)dr)],(t,x)∈[0,T]×D¯
for Borel measurable functions f:Rd→R, c:Rd×I→R and a domain D⊂Rd.Also,D¯⊂Rd is the closure of D and τD(w),w∈C([0,T];Rd) stands for the first exit time from D; that is
(12)τD(w)=inf⁡{t∈[0,T];w(t)∉D}.We also remark that the expectation operatorE[·] is taken under the risk-neutral probability measure Q and that e-∫0T-t‍c(Xrɛ,x,ɛ)dr represents the discount factor with the risk-free interest rate process c.Let us define a second order differential operatorLɛ by
(13)Lɛ=12∑i,j=1d‍aij(x,ɛ)∂2∂xi∂xj+∑i=1d‍bi(x,ɛ)∂∂xi-c(x,ɛ),
where aij=∑k=1d‍σikσjk. We consider the following Cauchy-Dirichlet problem for a PDE of parabolic type:
(14)∂∂tuɛ(t,x)+Lɛuɛ(t,x)=0,(t,x)∈[0,T)×D,uɛ(T,x)=f(x),x∈D,uɛ(t,x)=0,(t,x)∈[0,T]×∂D.Under the assumptions(A)–(E) stated in the next subsection, we have the following existence and uniqueness result due to Theorem 3.1 in [11].Theorem 1.
Assume(A)–(E) which are given in Section 2.2. For each ɛ∈I, uɛ(t,x) defined with formula (11) is a (classical) solution of (14) and
(15)sup⁡(t,x)∈[0,T]×D¯⁡|uɛ(t,x)|1+|x|2m<∞.
Moreover, if wɛ(t,x) is also a solution of (14) satisfying the growth condition
(16)sup⁡(t,x)∈[0,T]×D¯⁡|wɛ(t,x)|1+|x|2m′<∞
for some m′∈N, then uɛ=wɛ.Our main purpose is to present an asymptotic expansion of the barrier option priceuɛ(t,x):
(17)uɛ(t,x)=u0(t,x)+ɛv10(t,x)+⋯+ɛn-1vn-10(t,x)+O(ɛn),ɛ⟶0.
Here, the coefficient functions vk0(t,x), k=1,…,n-1 are (formally) given as the solution of
(18)∂∂tvk0(t,x)+L0vk0(t,x)+gk0(t,x)=0,(t,x)∈[0,T)×D,vk0(T,x)=0,x∈D,vk0(t,x)=0,(t,x)∈[0,T]×∂D,
where gk0(t,x) is given inductively by
(19)gk0(t,x)=L~k0u0(t,x)+∑l=1k-1‍L~k-l0vl0(t,x).
Here, L~k0 is defined as follows:
(20)L~k0=1k!{12∑i,j=1d‍∂kaij∂ɛk(x,0)∂2∂xi∂xjvvvvvv+∑i=1d‍∂kbi∂ɛk(x,0)∂∂xi-∂kc∂ɛk(x,0)}.Then, we can show the next result, whose proof is given in AppendixA.Theorem 2.
Assume(A)–(H) which are stated in Section 2.2. Then, for each k=1,…,n-1, vk0 is the classical solution of (18) and satisfies
(21)|vk0(t,x)|≤Ck(1+|x|2mk),(t,x)∈[0,T]×Rd
for some Ck, mk>0.Note that the uniqueness of the solutions of (18) follows from the same arguments as in the proof of Theorem 5.7.6 in [16]. That is, we obtain the next proposition.Proposition 3.
For any functiong which has a polynomial growth rate in x uniformly in t, a classical solution of (18) is unique in the following sense: if v and w are classical solutions of (18) and |v(t,x)|+|w(t,x)|≤C(1+|x|2m) for some C,m>0, then v=w.Now, we are able to state our first main result on the asymptotic expansion. The proof is given in AppendixB.Theorem 4.
Assume(A)–(H) which are given in Section 2.2. There are positive constants Cn and m~n which are independent of ɛ such that
(22)|uɛ(t,x)-(u0(t,x)+∑k=1n-1‍ɛkvk0(t,x))|≤Cn(1+|x|2m~n)ɛn,(t,x)∈[0,T]×D¯.Next, we construct a semigroup corresponding to(Xt0,x)t (i.e., (Xtɛ,x)t with ɛ=0) and D. Then, based on this semigroup we can obtain more explicit representation for the coefficient function vk0(t,x) than the right hand side of (36) in Assumption (H), which will appear in Section 2.2.LetCb0(D¯) be the set of bounded continuous functions f:D¯→R such that f(x)=0 on ∂D. Obviously, Cb0(D¯) equipped with the supnorm becomes a Banach space.Fort∈[0,T] and f∈Cb0(D¯), we define PtDf:D¯→R by
(23)PtDf(x)=E[exp⁡(-∫0t‍c(Xv0,x,0)dv)×f(Xt0,x)1{τD(X0,x)≥t}(-∫0t‍c(Xv0,x,0)dv)],
where c(x,0) is nonnegative. We notice that PtDf(x) is equal to u0(T-t,x) with the payoff function f. Then, we have the following result.Proposition 5.
Under the assumptions(A)–(E) stated in Section 2.2, the mapping PtD:Cb0(D¯)→Cb0(D¯) is well-defined and (PtD)0≤t≤T is a contraction semigroup.Proof.
Letf∈Cb0(D¯). The relations P0Df=f, PtDf|∂D=0, and sup⁡D¯|PtDf|≤sup⁡D¯|f| are obvious. The continuity of PtDf is by Lemma 4.3 in [11]. The semigroup property is verified by a straightforward calculation.Remark 6.
Note that(PtD)t also has the semigroup property on the set Cp0(D¯) of continuous functions f, each of which has a polynomial growth rate and satisfies f(x)=0 on ∂D.Finally, we show our second main result on the semigroup representation of the coefficient functionvk0 in the expansion, whose proof is given in Appendix C.Theorem 7.
Under Assumptions(A)–(H) given in Section 2.2, for each k=1,…,n-1(24)vk0(T-t,x)=∑l=1k‍∑(βi)i=1l⊂Nl,∑iβi=k‍∫0t‍∫0t1‍⋯∫0tl-1‍Pt-t1DL~β10Pt1-t2Dhhhhhhhhhhhhhhhhhhhhhhhhhhh×L~β20⋯Ptl-1-tlDhhhhhhhhhhhhhhhhhhhhhhhhhhh×L~βl0PtlDf(x)dtl⋯dt1.


## 2.2. Assumptions

This subsection introduces a series of the assumptions necessary for our main results stated in the previous subsection. Particularly, the assumptions (A)–(E) are relevant for Theorem1, that is the existence and uniqueness result of the PDE (14), while (F)–(H) are additional assumptions necessary for the asymptotic expansion results, that is Theorems 2–7.(A)
There is a positive constantA1 such that
(25)|σij(x,ɛ)|2+|bi(x,ɛ)|2≤A1(1+|x|2),vvvvvvvx∈Rd,ɛ∈I,i,j=1,…,d.Moreover, for eachɛ∈I it holds that σij(·,ɛ),bi(·,ɛ)∈Lip⁡ for i,j=1,…,d, where Lip⁡ is the set of locally Lipschitz continuous functions defined on Rd:
(26)Lip⁡={f∈C(Rd;R);foranycompactsetK⊂Rd,∃CK>0suchthat|f(x)-f(y)|≤CK|x-y|,vvvvx,y∈KRd}.Remark 8.
Note that under (A), the existence and uniqueness of a solution of (10) are guaranteed on any filtered probability space equipped with a standard m-dimensional Brownian motion, and Corollary 2.5.12 in [17] and Lemma 3.2.6 in [18] imply
(27)E[sup⁡0≤r≤t|Xrɛ,x-x|2l]≤CltleClt(1+|x|2l),vvvvvv.vvv(t,x)∈[0,T]×Rd,l∈N
for some Cl>0, which depends only on l, A1, and CK. Moreover, (Xrx)r has the strong Markov property.(B)
The functionf(x) is continuous on D¯ and there are Cf>0 and m∈N such that |f(x)|≤Cf(1+|x|2m), x∈Rd. Moreover, f(x)=0 on Rd∖D.Remark 9.
The assumption(B) which corresponds toH2(2) in [11] guarantees the continuity of a solution of (14) (if it exists) on the so-called parabolic boundary Σ=∂D×[0,T)∪D¯×{T}. For the details, see page 8 in [11].(C)
c(x,ɛ) is nonnegative (i.e., c(x,ɛ)≥0). Moreover, for each ɛ∈I, it holds that c(·,ɛ)∈Lip⁡.(D)
The boundary∂D has the outside strong sphere property; that is, for each x∈∂D there is a closed ball E such that E∩D=∅ and E∩D¯={x}.Remark 10.
The assumption(D) provides the regularity of each point in ∂D. (compared to [8]) Also, [11] points out that (D) with the ellipticity of the matrix (aij(x,ɛ))ij in (E) below gives
(28)P(τD(Xϵ,x)=τD-(Xϵ,x))=1.(E)
The matrix(aij(x,ɛ))ij is locally elliptic in the sense that for each ɛ∈I and compact set K⊂Rd there is a positive number μɛ,K such that ∑i,j=1d‍aij(x,ɛ)ξiξj≥μɛ,K|ξ|2 for any x∈K and ξ∈Rd.Remark 11.
Note that although the condition (E) (local ellipticity) is necessary for the existence of classical solution of our PDE (see Remark 2.2 in [11]), the assumption can be removed through consideration of viscosity solutions rather than classical solutions by applying Theorem 8.2 in [19] and Theorems 4.4.3 and 7.7.2 in [18]. Note that we need the additional assumption such that I⊂[0,∞) by technical reason in this case.To study the asymptotic expansion, we put the following assumptions in addition to (A)–(E). Firstly, by the next condition we can properly defineL~k0, k∈N in (20) above.(F)
Letn∈N. The functions aij(x,ɛ), bi(x,ɛ), and c(x,ɛ) are n-times continuously differentiable in ɛ. Furthermore, each of derivatives ∂kaij/∂ɛk, ∂kbi/∂ɛk, ∂kc/∂ɛk, k=1,…,n-1, has a polynomial growth rate in x∈Rd uniformly in ɛ∈I.To state the existence of the functionsvk0(t,x) in the asymptotic expansion (17), we first prepare the following set.Definition 12.
The setHm,p of g∈C([0,T)×D¯) is defined to satisfy the following condition. There is a Mg∈C([0,T))∩Lp([0,T),dt) such that
(29)|g(t,x)|≤Mg(t)(1+|x|2m),t∈[0,T),x∈D¯.Given this definition of the setHm,p, we put the next condition on u0.(G)
u0∈Gm, where
(30)Gm={∂2u∂xi∂xju∈C1,2([0,T)×D)∩C([0,T]×D¯);∂u∂xi∈Hm,2,∂2u∂xi∂xj∈Hm,1,i,j=1,…,d}.Now we examine the conditions necessary for the classical solution to the PDE (18).Firstly, Let us start with the case ofk=1. By the assumption (G), we have g10∈Hm,1 for some m∈N by the definition of gk0 with k=1 in (19). Thus we can define
(31)v10(t,x)=E[∫0(T-t)∧τD(X0,x)‍exp⁡(-∫0r‍c(Xv0,x,0)dv)vvvvvvvvv×g10(t+r,Xr0,x)dr(-∫0r‍c(Xv0,x,0)dv)].
Therefore, if we assume that v10∈C1,2([0,T)×D) we can show that v10 is the solution of (18) with k=1; that is, we can confirm that
(32)∂∂tv10(t,x)+L0v10(t,x)+g10(t,x)=0.
Note that the relations v10(T,·)=0 and v10=0 on [0,T]×∂D are obvious.Next, let us give some comments on the smoothness ofv10. In many cases as in the Black-Scholes model (see (49) in Section 3) we can rewrite (31) as
(33)v10(t,x)=∫0T-t‍∫D‍g10(t+r,y)p(r,x,y)dydr
for some p(r,x,y). Thus, if p has a “good” smoothness property, the smoothness of v10 also holds such as
(34)∂∂tv10(t,x)=-lim⁡s→T∫D‍g10(s,y)p(s-t,x,y)dy+∫0t‍∫D‍∂∂tg10(t+r,y)p(r,x,y)dr,
if the limit in the right hand side exists, and
(35)∂∂xiv10(t,x)=∫0t‍∫D‍g10(t+r,y)∂∂xip(r,x,y)dr,∂2∂xi∂xjv10(t,x)=∫0t‍∫D‍g10(t+r,y)∂2∂xi∂xjp(r,x,y)dr.Moreover, ifv10 is in Gm1 for some m1∈N, we also have g20∈Hm~1,1 for some m~1∈N by the definition of gk0 with k=2 in (19). Then, we can define v20 similarly as v10. Furthermore, under some suitable smoothness conditions for v20, which may be given by the smoothness property of p(r,x,y), we are able to show that v20 is the classical solution of (18) with k=2.Thus, the observation above leads us to our final assumption.(H)
It holds thatvk0∈Gmn, k=1,…,n-1 for some mn∈N, where
(36)vk0(t,x)=E[∫0(T-t)∧τD(X0,x)‍exp⁡(-∫0r‍c(Xv0,x,0)dv)×gk0(t+r,Xr0,x)dr∫0(T-t)∧τD(X0,x)‍].


## 3. Application to Barrier Option Pricing in Stochastic Volatility Environment

This section demonstrates the effectiveness of our method in stochastic volatility environment. Section3.1 derives concrete approximation formulas, and Section 3.2 shows numerical examples.


### 3.1. Approximation of Barrier Option Prices in a Stochastic Volatility Model

We consider the following stochastic volatility model under the risk-neutral probability measure:(37)dStɛ=(c-q)Stɛdt+σtɛStɛdBt1,S0ɛ=S,dσtɛ=ɛλ(θ-σtɛ)dt+ɛνσtɛ(ρdBt1+1-ρ2dBt2),vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvσ0ɛ=σ,
where c,q>0, ɛ∈[0,1), λ,θ,ν>0, and ρ∈[-1,1] and B=(B1,B2) is a two dimensional Brownian motion. Here c and q represent a domestic interest rate and a foreign interest rate, respectively, when we consider the currency options.Clearly, applying Itô’s formula, we have its logarithmic process:(38)dXtɛ=(c-q-12(σtɛ)2)dt+σtɛdBt1,X0ɛ=x=log⁡S,dσtɛ=ɛλ(θ-σtɛ)dt+ɛνσtɛ(ρdBt1+1-ρ2dBt2),σ0ɛ=σ.
Also, its generator is expressed as
(39)Lɛ=(c-q-12σ2)∂∂x+12σ2∂2∂x2+ɛρνσ2∂2∂x∂σ+ɛλ(θ-σ)∂∂σ+ɛ212ν2σ2∂2∂σ2-c.
In this case, L~10 which is defined by (20) with k=1 is given as
(40)L~10=ρνσ2∂2∂x∂σ+λ(θ-σ)∂∂σ.
We will apply the asymptotic expansion in the previous section to (38) and give an approximation formula for a barrier option price, which is given under a risk-neutral probability measure as
(41)CBarrierSV,ɛ(T-t,ex)=E[e-c(T-t)f(ST-tɛ,ex)1{τ(L,∞)(Sɛ,ex)>T-t}],
where f stands for a payoff function, L(<S) is a barrier price and the expectation is taken with respect to the risk-neutral probability measure.Then,uɛ(t,x)=CBarrierSV,ɛ(T-t,ex) satisfies the following PDE:
(42)(∂∂t+Lɛ)uɛ(t,x)=0,(t,x)∈(0,T]×D,uɛ(T,x)=f¯(x),x∈D¯,uɛ(t,l)=0,t∈[0,T],
where f¯(x)=max⁡{ex-K,0}, D=(l,∞), and l=log⁡L. We obtain the 0th order u0 as
(43)u0(t,x)=PT-tDf¯(x)=E[e-c(T-t)f¯(XT-tx,0)1{τD(X0,x)>T-t}].Remark 13.
u
0 satisfies PDE (42) with ɛ=0. Although the condition (E) in Section 2 does not seem to be satisfied in this case, the volatility process (σtɛ)t becomes a constant σ>0, and so (38) is reduced to a one-dimensional SDE. Then, (42) with ɛ=0 becomes a nondegenerating PDE with fixed σ. Therefore, we need not take care of the lack of the condition (E) in this example.Settingα=c-q, we note that PT-tDf¯(x)=CBarrierBS(T-t,ex,α,σ,L) is the price of the down-and-out barrier call option under the Black-Scholes model:
(44)CBarrierBS(T-t,ex,α,σ,L)=CBS(T-t,ex,α,σ)-(exL)1-2α/σ2CBS(T-t,L2ex,α,σ).
Here, we recall that the price of the plain vanilla option under the Black-Scholes model is given as
(45)CBS(T-t,ex,α,σ)=e-q(T-t)exN(d1(T-t,x,α))-e-c(T-t)KN(d2(T-t,x,α)),
where
(46)d1(t,x,α)=x-log⁡K+αtσt+12σt,d2(t,x,α)=d1(t,x,α)-σtN(x)=∫-∞x‍n(y)dy,n(y)=12πe-y2/2.Note also that(47)P(τD(X0,x)≥t∣Xt0,x)=1-exp⁡(-2(x-l)(Xt0,x-l)σ2t)on⁡{Xt0,x>l}.
Therefore, for g∈Cp0(D¯) we have
(48)PtDg(x)=E[P(τD(X0,x)≥t∣Xtx)e-ctg(Xt0,x)1{Xt0,x>l}]=∫l∞‍e-ctg(y)p(t,x,y)dy,
where
(49)p(t,x,y)=12πσ2t(1-e-2(x-l)(y-l)/σ2t)×e-(y-x-μt)2/2σ2t,μ=α-σ22=(c-q-σ22).Then, we show the following main result in this section.Theorem 14.
We obtain an approximation formula for the down-and-out barrier call option under the stochastic volatility model (37):
(50)CBarrierSV,ɛ(T,ex)=CBarrierBS(T,ex,α,σ,L)+ɛv10(0,x)+O(ɛ2),
where
(51)v10(0,x)=e-cT∫0T‍∫l∞‍12πσ2s(1-e-2(x-l)(y-l)/σ2s)vvvvvvvvvvvv×e-(y-x-(α-(1/2)σ2)s)2/2σ2sϑ(s,y)dyds,(52)ϑ(t,x)=eα(T-t)ρνσexn(d1(T-t,x,α))×(-d2(T-t,x,α))+2eα(T-t)ρνα(exL)-2α/σ2×Ln(c1(T-t,x,α))T-t-eα(T-t)ρνσ(exL)-2α/σ2×Ln(c1(T-t,x,α))c1(T-t,x,α)-ec(T-t)ρν4ασ(exL)1-2α/σ2×{CBS(T-t,L2ex,α,σ)×{1+(x-log⁡⁡L)(1-2ασ2)}-(x-log⁡⁡L)e-q(T-t)L2exN(c1(T-t,x,α))}+λ(θ-σ)eα(T-t)exn(d1(T-t,x,α))T-t-λ(θ-σ)(exL)-2α/σ2eα(T-t)×Ln(c1(T-t,x,α))T-t-ec(T-t)λ(θ-σ)4ασ3(log⁡exL)(exL)1-2α/σ2×CBS(T-t,L2ex,α,σ),(53)c1(t,x,α)=2l-x-log⁡K+αtσt+12σt.Proof.
Firstly, note that whenk=1 in Theorem 7, we have
(54)v10(T-t,x)=∫0t‍Pt-rDL~10PrDf(x)dr.
Thus, we see the expansion
(55)CBarrierSV,ɛ(T-t,ex)=CBarrierBS(T-t,ex,α,σ,L)+ɛ∫0T-t‍PsDL~10PT-t-sDf¯(x)ds+O(ɛ2).
The first-order approximation term v10(t,x)=∫0T-t‍PsDL~10PT-t-sDf¯(x)ds is given by
(56)v10(t,x)=∫0T-t‍e-csP¯sDL~10e-c(T-t-s)P¯T-t-sDf¯(x)ds=e-c(T-t)∫0T-t‍P¯sDL~10P¯T-t-sDf¯(x)ds,
where P¯tD is defined by
(57)P¯tDf¯(x)=∫l∞‍12πσ2s(1-e-2(x-l)(y-l)/σ2s)×e-(y-x-(α-(1/2)σ2)s)2/2σ2sf¯(y)dy.
Define ϑ(t,x) as
(58)ϑ(t,x)=L~10P¯T-tDf(ex)=ec(T-t)ρνσ2∂2∂x∂σCBarrierBS(T-t,ex,α,σ,L)+ec(T-t)λ(θ-σ)∂∂σCBarrierBS(T-t,ex,α,σ,L).
A straightforward calculation shows that the above function agrees with the right-hand side of (52). Then we get the assertion.Remark that through numerical integrations with respect to times and space y in (51), we easily obtain the first order approximation of the down-and-out option prices.Next, as a special case of (37) we consider the following stochastic volatility model with no drifts:
(59)dStɛ=σtɛStɛdBt1,S0ɛ=S>0,dσtɛ=ɛνσtɛ(ρdBt1+1-ρ2dBt2),σ0ɛ=σ>0,
where ɛ∈[0,1), ρ∈[-1,1] and B=(B1,B2) is a two-dimensional Brownian motion. In this case, we can provide a simpler approximation formula than in Theorem 14.By Itô’s formula, the following logarithmic model is obtained:(60)dXtɛ=-12(σtɛ)2dt+σtɛdBt1,X0ɛ=x=log⁡S,dσtɛ=ɛνσtɛ(ρdBt1+1-ρ2dBt2),σ0ɛ=σ.
Again, the barrier option price is given by
(61)CBarrierSV,ɛ(T,ex)=E[f(STɛ)1{min⁡0≤u≤T⁡Suɛ>L}],
where f stands for a payoff function and L(<S) is a barrier price.The differential operatorsLɛ, L~10 and the PDE are same as (39)–(42) with c=q=0 and λ=0. Also, the barrier option price in the Black-Scholes model coincides with (44) with no drift; that is,
(62)CBarrierBS(T,S)=CBS(T,S)-(SL)CBS(T,L2S),
where CBS(T,S) is the driftless Black-Scholes formula of the European call option given by
(63)CBS(T,S)=SN(d1(T,log⁡S))-KN(d2(T,log⁡S))
with
(64)d1(t,x)=d1(t,x,0)=x-log⁡K+σ2t/2σt,d2(t,x)=d2(t,x,0)=d1(t,x)-σt.
Then, we reach the following expansion formula which only needs 1-dimensional numerical integration.Theorem 15.
C
B
a
r
r
i
e
r
S
V
,
ɛ
(
T
,
e
x
)
=
C
B
a
r
r
i
e
r
B
S
(
T
,
e
x
)
+
ɛ
v
1
0
(
0
,
x
)
+
O
(
ɛ
2
), where
(65)v10(0,x)=-12Tνρσ{exn(d1(T,x))d2(T,x)hhhhhhhhhh+Ln(c1(T,x))c1(T,x)}+νρL(x-l)log⁡⁡(L/K)2πσ×∫0T‍(T-s)1/2s3/2hhhhhh×exp⁡(-c2(T-s,L/K)+c2(s,L/ex)2)ds,c1(t,x)=log⁡(L2/exK)+σ2t/2σt,c2(t,y)=(log⁡y+σ2t/2σt)2.Proof.
See AppendixD.


### 3.2. Numerical Example

Finally, applying our approximation formulas in Theorems14 and 15, we present numerical experiments for European down-and-out barrier call prices. First, let us denote u0=CBarrierBS(T,S) and v10=v10(0,log⁡S). Then, we see
(66)CBarrierSV,ɛ(T,S)≃u0+ɛv10.In the following we report the results of the numerical experiments, where the numbers in the parentheses show the error rates (%) relative to the benchmark prices ofCBarrierSV,ɛ(T,S); they are computed by Monte-Carlo simulations with 100,000 time steps (n=105) and 1,000,000 trials. We note that in our experiments the standard deviations of the benchmark Monte-Carlo simulations are calculated as at most 0.006 with the order of discretization error being 0.002 (≈T/n=0.5/105 as stated in Introduction).We check the accuracy of our approximations by changing the model parameters. Cases 1–6 show the results for the stochastic volatility model with drifts of the underlying price process or/and the volatility process (37), while Case 7 shows the result for the stochastic volatility model with no drifts (59) (see Tables 1, 2, 3, 4, 5, 6, and 7). There, we apply the formula in Theorem 14 to Cases 1–6 and the formula in Theorem 15 to Case 7, respectively.Table 1
Case  1 (c=0.01, q=0.0, εν=0.2, ρ=-0.5, ελ=0.0, θ=0.0).
Strike
Benchmark
Our approximation (u0+εv10)
Barrier Black-Scholes (u0)
100
3.468
3.466 (−0.05%)
3.495 (0.80%)
102
2.822
2.822 (0.00%)
2.866 (1.57%)
105
1.986
1.986 (0.01%)
2.052 (3.36%)Table 2
Case  2 (c=0.01, q=0.0, εν=0.35, ρ=-0.7, ελ=0.0, θ=0.0).
Strike
Benchmark
Our approximation (u0+εv10)
Barrier Black-Scholes (u0)
100
3.421
3.423 (0.07%)
3.495 (2.18%)
102
2.753
2.757 (0.18%)
2.866 (4.13%)
105
1.885
1.890 (0.23%)
2.052 (8.88%)Table 3
Case  3 (c=0.05, q=0.0, εν=0.35, ρ=-0.7, ελ=0.0, θ=0.0).
Strike
Benchmark
Our approximation (u0+εv10)
Barrier Black-Scholes (u0)
100
4.352
4.349 (−0.07%)
4.399 (1.06%)
102
3.585
3.586 (0.02%)
3.665 (2.24%)
105
2.560
2.563 (0.11%)
2.696 (5.31%)Table 4
Case  4 (c=0.05, q=0.1, εν=0.2, ρ=-0.5, ελ=0.0, θ=0.0).
Strike
Benchmark
Our approximation (u0+εv10)
Barrier Black-Scholes (u0)
100
2.231
2.224 (−0.31%)
2.268 (1.64%)
102
1.758
1.754 (−0.27%)
1.812 (3.02%)
105
1.172
1.168 (−0.31%)
1.243 (6.05%)Table 5
Case  5 (c=0.01, q=0.0, εν=0.2, ρ=-0.5, ελ=0.2, θ=0.25).
Strike
Benchmark
Our approximation (u0+εv10)
Barrier Black-Scholes (u0)
100
3.523
3.517 (−0.16%)
3.495 (−0.77%)
102
2.891
2.888 (−0.09%)
2.866 (−0.85%)
105
2.066
2.065 (−0.06%)
2.052 (−0.64%)Table 6
Case  6 (c=0.01, q=0.0, εν=0.2, ρ=-0.5, ελ=0.5, θ=0.25).
Strike
Benchmark
Our approximation (u0+εv10)
Barrier Black-Scholes (u0)
100
3.587
3.594 (0.20%)
3.495 (−2.55%)
102
2.976
2.987 (0.39%)
2.866 (−3.68%)
105
2.170
2.183 (0.59%)
2.052 (−5.41%)Table 7
Case  7 (c=0.0, q=0.0, εν=0.2, ρ=-0.5, ελ=0.0, θ=0.0).
Strike
Benchmark
Our approximation (u0+εv10)
Barrier Black-Scholes (u0)
100
3.261
3.258 (−0.09%)
3.290 (0.90%)
102
2.640
2.639 (−0.02%)
2.686 (1.78%)
105
1.841
1.841 (0.01%)
1.911 (3.77%)In all the cases, we set the initial asset priceS=100, the initial volatility σ=0.15, the time to maturity T=0.5, the lower barrier L=95, and strike prices K=100,102,105. The other parameters (c, q, ɛν, ρ, ɛλ, θ) are listed in the caption of each table.Apparently, our approximation formulau0+ɛv10 improves the accuracy against CBarrierSV,ɛ(T,S), and it is observed that ɛv10 accurately compensates for the difference between CBarrierSV,ɛ(T,S) and CBarrierBS(T,S), which confirms the validity of our method.


## 3.1. Approximation of Barrier Option Prices in a Stochastic Volatility Model

We consider the following stochastic volatility model under the risk-neutral probability measure:(37)dStɛ=(c-q)Stɛdt+σtɛStɛdBt1,S0ɛ=S,dσtɛ=ɛλ(θ-σtɛ)dt+ɛνσtɛ(ρdBt1+1-ρ2dBt2),vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvσ0ɛ=σ,
where c,q>0, ɛ∈[0,1), λ,θ,ν>0, and ρ∈[-1,1] and B=(B1,B2) is a two dimensional Brownian motion. Here c and q represent a domestic interest rate and a foreign interest rate, respectively, when we consider the currency options.Clearly, applying Itô’s formula, we have its logarithmic process:(38)dXtɛ=(c-q-12(σtɛ)2)dt+σtɛdBt1,X0ɛ=x=log⁡S,dσtɛ=ɛλ(θ-σtɛ)dt+ɛνσtɛ(ρdBt1+1-ρ2dBt2),σ0ɛ=σ.
Also, its generator is expressed as
(39)Lɛ=(c-q-12σ2)∂∂x+12σ2∂2∂x2+ɛρνσ2∂2∂x∂σ+ɛλ(θ-σ)∂∂σ+ɛ212ν2σ2∂2∂σ2-c.
In this case, L~10 which is defined by (20) with k=1 is given as
(40)L~10=ρνσ2∂2∂x∂σ+λ(θ-σ)∂∂σ.
We will apply the asymptotic expansion in the previous section to (38) and give an approximation formula for a barrier option price, which is given under a risk-neutral probability measure as
(41)CBarrierSV,ɛ(T-t,ex)=E[e-c(T-t)f(ST-tɛ,ex)1{τ(L,∞)(Sɛ,ex)>T-t}],
where f stands for a payoff function, L(<S) is a barrier price and the expectation is taken with respect to the risk-neutral probability measure.Then,uɛ(t,x)=CBarrierSV,ɛ(T-t,ex) satisfies the following PDE:
(42)(∂∂t+Lɛ)uɛ(t,x)=0,(t,x)∈(0,T]×D,uɛ(T,x)=f¯(x),x∈D¯,uɛ(t,l)=0,t∈[0,T],
where f¯(x)=max⁡{ex-K,0}, D=(l,∞), and l=log⁡L. We obtain the 0th order u0 as
(43)u0(t,x)=PT-tDf¯(x)=E[e-c(T-t)f¯(XT-tx,0)1{τD(X0,x)>T-t}].Remark 13.
u
0 satisfies PDE (42) with ɛ=0. Although the condition (E) in Section 2 does not seem to be satisfied in this case, the volatility process (σtɛ)t becomes a constant σ>0, and so (38) is reduced to a one-dimensional SDE. Then, (42) with ɛ=0 becomes a nondegenerating PDE with fixed σ. Therefore, we need not take care of the lack of the condition (E) in this example.Settingα=c-q, we note that PT-tDf¯(x)=CBarrierBS(T-t,ex,α,σ,L) is the price of the down-and-out barrier call option under the Black-Scholes model:
(44)CBarrierBS(T-t,ex,α,σ,L)=CBS(T-t,ex,α,σ)-(exL)1-2α/σ2CBS(T-t,L2ex,α,σ).
Here, we recall that the price of the plain vanilla option under the Black-Scholes model is given as
(45)CBS(T-t,ex,α,σ)=e-q(T-t)exN(d1(T-t,x,α))-e-c(T-t)KN(d2(T-t,x,α)),
where
(46)d1(t,x,α)=x-log⁡K+αtσt+12σt,d2(t,x,α)=d1(t,x,α)-σtN(x)=∫-∞x‍n(y)dy,n(y)=12πe-y2/2.Note also that(47)P(τD(X0,x)≥t∣Xt0,x)=1-exp⁡(-2(x-l)(Xt0,x-l)σ2t)on⁡{Xt0,x>l}.
Therefore, for g∈Cp0(D¯) we have
(48)PtDg(x)=E[P(τD(X0,x)≥t∣Xtx)e-ctg(Xt0,x)1{Xt0,x>l}]=∫l∞‍e-ctg(y)p(t,x,y)dy,
where
(49)p(t,x,y)=12πσ2t(1-e-2(x-l)(y-l)/σ2t)×e-(y-x-μt)2/2σ2t,μ=α-σ22=(c-q-σ22).Then, we show the following main result in this section.Theorem 14.
We obtain an approximation formula for the down-and-out barrier call option under the stochastic volatility model (37):
(50)CBarrierSV,ɛ(T,ex)=CBarrierBS(T,ex,α,σ,L)+ɛv10(0,x)+O(ɛ2),
where
(51)v10(0,x)=e-cT∫0T‍∫l∞‍12πσ2s(1-e-2(x-l)(y-l)/σ2s)vvvvvvvvvvvv×e-(y-x-(α-(1/2)σ2)s)2/2σ2sϑ(s,y)dyds,(52)ϑ(t,x)=eα(T-t)ρνσexn(d1(T-t,x,α))×(-d2(T-t,x,α))+2eα(T-t)ρνα(exL)-2α/σ2×Ln(c1(T-t,x,α))T-t-eα(T-t)ρνσ(exL)-2α/σ2×Ln(c1(T-t,x,α))c1(T-t,x,α)-ec(T-t)ρν4ασ(exL)1-2α/σ2×{CBS(T-t,L2ex,α,σ)×{1+(x-log⁡⁡L)(1-2ασ2)}-(x-log⁡⁡L)e-q(T-t)L2exN(c1(T-t,x,α))}+λ(θ-σ)eα(T-t)exn(d1(T-t,x,α))T-t-λ(θ-σ)(exL)-2α/σ2eα(T-t)×Ln(c1(T-t,x,α))T-t-ec(T-t)λ(θ-σ)4ασ3(log⁡exL)(exL)1-2α/σ2×CBS(T-t,L2ex,α,σ),(53)c1(t,x,α)=2l-x-log⁡K+αtσt+12σt.Proof.
Firstly, note that whenk=1 in Theorem 7, we have
(54)v10(T-t,x)=∫0t‍Pt-rDL~10PrDf(x)dr.
Thus, we see the expansion
(55)CBarrierSV,ɛ(T-t,ex)=CBarrierBS(T-t,ex,α,σ,L)+ɛ∫0T-t‍PsDL~10PT-t-sDf¯(x)ds+O(ɛ2).
The first-order approximation term v10(t,x)=∫0T-t‍PsDL~10PT-t-sDf¯(x)ds is given by
(56)v10(t,x)=∫0T-t‍e-csP¯sDL~10e-c(T-t-s)P¯T-t-sDf¯(x)ds=e-c(T-t)∫0T-t‍P¯sDL~10P¯T-t-sDf¯(x)ds,
where P¯tD is defined by
(57)P¯tDf¯(x)=∫l∞‍12πσ2s(1-e-2(x-l)(y-l)/σ2s)×e-(y-x-(α-(1/2)σ2)s)2/2σ2sf¯(y)dy.
Define ϑ(t,x) as
(58)ϑ(t,x)=L~10P¯T-tDf(ex)=ec(T-t)ρνσ2∂2∂x∂σCBarrierBS(T-t,ex,α,σ,L)+ec(T-t)λ(θ-σ)∂∂σCBarrierBS(T-t,ex,α,σ,L).
A straightforward calculation shows that the above function agrees with the right-hand side of (52). Then we get the assertion.Remark that through numerical integrations with respect to times and space y in (51), we easily obtain the first order approximation of the down-and-out option prices.Next, as a special case of (37) we consider the following stochastic volatility model with no drifts:
(59)dStɛ=σtɛStɛdBt1,S0ɛ=S>0,dσtɛ=ɛνσtɛ(ρdBt1+1-ρ2dBt2),σ0ɛ=σ>0,
where ɛ∈[0,1), ρ∈[-1,1] and B=(B1,B2) is a two-dimensional Brownian motion. In this case, we can provide a simpler approximation formula than in Theorem 14.By Itô’s formula, the following logarithmic model is obtained:(60)dXtɛ=-12(σtɛ)2dt+σtɛdBt1,X0ɛ=x=log⁡S,dσtɛ=ɛνσtɛ(ρdBt1+1-ρ2dBt2),σ0ɛ=σ.
Again, the barrier option price is given by
(61)CBarrierSV,ɛ(T,ex)=E[f(STɛ)1{min⁡0≤u≤T⁡Suɛ>L}],
where f stands for a payoff function and L(<S) is a barrier price.The differential operatorsLɛ, L~10 and the PDE are same as (39)–(42) with c=q=0 and λ=0. Also, the barrier option price in the Black-Scholes model coincides with (44) with no drift; that is,
(62)CBarrierBS(T,S)=CBS(T,S)-(SL)CBS(T,L2S),
where CBS(T,S) is the driftless Black-Scholes formula of the European call option given by
(63)CBS(T,S)=SN(d1(T,log⁡S))-KN(d2(T,log⁡S))
with
(64)d1(t,x)=d1(t,x,0)=x-log⁡K+σ2t/2σt,d2(t,x)=d2(t,x,0)=d1(t,x)-σt.
Then, we reach the following expansion formula which only needs 1-dimensional numerical integration.Theorem 15.
C
B
a
r
r
i
e
r
S
V
,
ɛ
(
T
,
e
x
)
=
C
B
a
r
r
i
e
r
B
S
(
T
,
e
x
)
+
ɛ
v
1
0
(
0
,
x
)
+
O
(
ɛ
2
), where
(65)v10(0,x)=-12Tνρσ{exn(d1(T,x))d2(T,x)hhhhhhhhhh+Ln(c1(T,x))c1(T,x)}+νρL(x-l)log⁡⁡(L/K)2πσ×∫0T‍(T-s)1/2s3/2hhhhhh×exp⁡(-c2(T-s,L/K)+c2(s,L/ex)2)ds,c1(t,x)=log⁡(L2/exK)+σ2t/2σt,c2(t,y)=(log⁡y+σ2t/2σt)2.Proof.
See AppendixD.


## 3.2. Numerical Example

Finally, applying our approximation formulas in Theorems14 and 15, we present numerical experiments for European down-and-out barrier call prices. First, let us denote u0=CBarrierBS(T,S) and v10=v10(0,log⁡S). Then, we see
(66)CBarrierSV,ɛ(T,S)≃u0+ɛv10.In the following we report the results of the numerical experiments, where the numbers in the parentheses show the error rates (%) relative to the benchmark prices ofCBarrierSV,ɛ(T,S); they are computed by Monte-Carlo simulations with 100,000 time steps (n=105) and 1,000,000 trials. We note that in our experiments the standard deviations of the benchmark Monte-Carlo simulations are calculated as at most 0.006 with the order of discretization error being 0.002 (≈T/n=0.5/105 as stated in Introduction).We check the accuracy of our approximations by changing the model parameters. Cases 1–6 show the results for the stochastic volatility model with drifts of the underlying price process or/and the volatility process (37), while Case 7 shows the result for the stochastic volatility model with no drifts (59) (see Tables 1, 2, 3, 4, 5, 6, and 7). There, we apply the formula in Theorem 14 to Cases 1–6 and the formula in Theorem 15 to Case 7, respectively.Table 1
Case  1 (c=0.01, q=0.0, εν=0.2, ρ=-0.5, ελ=0.0, θ=0.0).
Strike
Benchmark
Our approximation (u0+εv10)
Barrier Black-Scholes (u0)
100
3.468
3.466 (−0.05%)
3.495 (0.80%)
102
2.822
2.822 (0.00%)
2.866 (1.57%)
105
1.986
1.986 (0.01%)
2.052 (3.36%)Table 2
Case  2 (c=0.01, q=0.0, εν=0.35, ρ=-0.7, ελ=0.0, θ=0.0).
Strike
Benchmark
Our approximation (u0+εv10)
Barrier Black-Scholes (u0)
100
3.421
3.423 (0.07%)
3.495 (2.18%)
102
2.753
2.757 (0.18%)
2.866 (4.13%)
105
1.885
1.890 (0.23%)
2.052 (8.88%)Table 3
Case  3 (c=0.05, q=0.0, εν=0.35, ρ=-0.7, ελ=0.0, θ=0.0).
Strike
Benchmark
Our approximation (u0+εv10)
Barrier Black-Scholes (u0)
100
4.352
4.349 (−0.07%)
4.399 (1.06%)
102
3.585
3.586 (0.02%)
3.665 (2.24%)
105
2.560
2.563 (0.11%)
2.696 (5.31%)Table 4
Case  4 (c=0.05, q=0.1, εν=0.2, ρ=-0.5, ελ=0.0, θ=0.0).
Strike
Benchmark
Our approximation (u0+εv10)
Barrier Black-Scholes (u0)
100
2.231
2.224 (−0.31%)
2.268 (1.64%)
102
1.758
1.754 (−0.27%)
1.812 (3.02%)
105
1.172
1.168 (−0.31%)
1.243 (6.05%)Table 5
Case  5 (c=0.01, q=0.0, εν=0.2, ρ=-0.5, ελ=0.2, θ=0.25).
Strike
Benchmark
Our approximation (u0+εv10)
Barrier Black-Scholes (u0)
100
3.523
3.517 (−0.16%)
3.495 (−0.77%)
102
2.891
2.888 (−0.09%)
2.866 (−0.85%)
105
2.066
2.065 (−0.06%)
2.052 (−0.64%)Table 6
Case  6 (c=0.01, q=0.0, εν=0.2, ρ=-0.5, ελ=0.5, θ=0.25).
Strike
Benchmark
Our approximation (u0+εv10)
Barrier Black-Scholes (u0)
100
3.587
3.594 (0.20%)
3.495 (−2.55%)
102
2.976
2.987 (0.39%)
2.866 (−3.68%)
105
2.170
2.183 (0.59%)
2.052 (−5.41%)Table 7
Case  7 (c=0.0, q=0.0, εν=0.2, ρ=-0.5, ελ=0.0, θ=0.0).
Strike
Benchmark
Our approximation (u0+εv10)
Barrier Black-Scholes (u0)
100
3.261
3.258 (−0.09%)
3.290 (0.90%)
102
2.640
2.639 (−0.02%)
2.686 (1.78%)
105
1.841
1.841 (0.01%)
1.911 (3.77%)In all the cases, we set the initial asset priceS=100, the initial volatility σ=0.15, the time to maturity T=0.5, the lower barrier L=95, and strike prices K=100,102,105. The other parameters (c, q, ɛν, ρ, ɛλ, θ) are listed in the caption of each table.Apparently, our approximation formulau0+ɛv10 improves the accuracy against CBarrierSV,ɛ(T,S), and it is observed that ɛv10 accurately compensates for the difference between CBarrierSV,ɛ(T,S) and CBarrierBS(T,S), which confirms the validity of our method.


## 4. Conclusion

This paper has proposed an approximation scheme for barrier option prices by applying a new semigroup expansion to the Cauchy-Dirichlet problem in the second order parabolic partial differential equations (PDEs). As an application, we have derived a semigroup expansion formula under a certain type of stochastic volatility model and confirmed the validity of our method through numerical examples. Developing concrete computational schemes under various models is our next research topic.



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