A Semi-group Expansion for Pricing Barrier Options

This paper presents a new asymptotic expansion method for pricing continuously monitoring barrier options. In particular, we develops a semi-group 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.


Introduction
Numerical methods for the Cauchy-Dirichlet problem have been a topic of great interest in stochastic analysis and its applications. For example, in mathematical finance the Cauchy-Dirichlet problem naturally arises in valuation of continuously monitoring barrier options: (1.1) Here, T > 0 is a maturity of the option, and (X x t ) t denotes a price process of the underlying asset starting from x (usually given as the solution of a certain stochastic differential equation (SDE)). Also, L stands for a constant lower barrier, that is L < x, and τ is the hitting time to L: It is well-known that a possible approach in computation of C Barrier (T, x) is the Euler scheme, which stores the sample paths of the process (X x t ) t through an n-time discretization with the step size T /n. In applying this scheme to pricing a continuously monitoring barrier option, one kills the simulated process (say, (X x t i ) i ) ifX x t i exits from the domain (L, ∞) until the maturity T . The usual Eular scheme is suboptimal since it does not control the diffusion paths between two successive dates t i and t i+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 C Barrier (T, x) andC Barrier (T, x) (the barrier option price obtained by the Euler scheme) is of order T /n, as opposed to the order T /n for standard plain-vanilla options. (See [7]) Therefore, to improve the order of the error, many schemes for the Monte-Carlo method have been proposed. (See [16] for instance.) One of the other tractable approaches for calculating C Barrier (T, x) is to derive an analytical approximation. If we obtain a closed form approximation formula, then it is a powerful tool for evaluation of continuously monitoring barrier options because we do not have to 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, which is due to the non-existence of the Malliavin derivative D t τ (see [4]) and to the fact that the minimum (maximum) process of the Brownian motion has only first-order differentiability in the Malliavin sense. Thus, neither approach in [11] nor in [19] can be applied directly to valuation of continuously monitoring barrier options while they are applicable to pricing discrete barrier options. (See [18] for the detail.) In this paper, we propose a new general method for approximating the solution to the Cauchy-Dirichlet problem. Roughly speaking, our objective is to pricing barrier options when the dynamics of the underlying asset price is described by the following perturbed SDE: dX ε,x t = b(X ε,x t , ε)dt + σ(X ε,x t , ε)dB t , X ε,x 0 = x, where ε is a small parameter, which will be defined precisely later in the paper. In this case, the barrier option price (1.1) is characterized as a solution of the Cauchy-Dirichlet problem: (1.4) where the differential operator L ε is determined by the diffusion coefficients b and σ. Next, we introduce an asymptotic expansion formula: where O denotes the Landau symbol. The function u 0 (t, x) is the solution of (1.4) with ε = 0: if b(t, x, 0) and σ(t, x, 0) have some simple forms such as constants (as in the Black-Scholes model), we already know the closed form of u 0 (t, x) and hence obtain the price. Then, we are able to get the approximate value for u ε (t, x) through evaluation of v 0 1 (t, x), . . . , v 0 n−1 (t, x). In fact, they are also characterized as the solution of a certain PDE with the Dirichlet condition. By formal asymptotic expansions, (1.5) as well as L ε = L 0 + εL 0 1 + · · · + ε n−1L 0 n−1 + · · · , 2 we can derive the PDEs corresponding to v 0 k (t, x) of the form: where g 0 k (t, x) will be given explicitly later in this paper. Moreover, by applying the Feynman-Kac approach, we are able to obtain their stochastic representations. We will justify the above argument in a mathematically rigorous way with necessary assumptions in Section 2.
The theory of the Cauchy-Dirichlet problem for this kind of second order parabolic PDE is well understood in the case of bounded domains (see [5], [6] and [14] for instance). As for an unbounded domain case such as (1.4), [17] provides the existence and uniqueness results for a solution of the PDE and the Feynman-Kac type formula (cited as Theorem 1 below). However, some mathematical difficulty exists for applying the results of [17] to the PDE (1.6). More precisely, the function g 0 , the existence and uniqueness of (1.6) are guaranteed: see [5].) To overcome this difficulty, we generalize the Levi's parametrix method (which is used to construct a classical solution of the PDE) in Theorem 2. Furthermore, we show another representation of v 0 k (t, x) by using the corresponding semi-group in Section 3. We notice that such a form is convenient for evaluation of v 0 k (t, x) in concrete examples. In Section 4, we apply our method to pricing a barrier option in a stochastic volatility model: is a two dimensional Brownian motion. Then, we obtain a new approximation formula: for x = log S, where S ε,x t = exp(X ε,x t ), (P D t ) t is a semi-group defined in Section 3, f is a payoff function andf (x) = f (e x ). Here, P D Tf (x) is regarded as the down-and-out barrier option price, C BS Barrier (T, e x ) in the Black-Scholes model. Moreover, we confirm practical validity of our method through a numerical example given in Section 4. Notice also that our example does not satisfy the assumptions introduced in Section 2. Thus, we generalize our main result and present weaker (but a little bit complicated) version of the assumptions in Section 5.1. Furthermore, Section 5.2-5.4 list the proofs of our results.
Finally, we remark that in the contrast to the previous works ( [2], [3], [8], [9] for example), which start with some specific models (the Black-Scholes model or some type of fast mean-reversion model) and derive approximation formulas for (discretely or continuously monitoring) barrier option prices, we firstly develop a general asymptotic expansion scheme for the Cauchy-Dirichlet problem under multi-dimensional diffusion setting; then, as an application, we provide a new approximation formula under a certain class of stochastic volatility model that can be widely applied in practice (e.g. in currency option markets).

Main Results
where I is an interval on R including the origin 0 (for instance I = (−1, 1).) We consider the SDE (1.3) for any x ∈ R d and ε ∈ I; we will introduce the assumptions for existence and uniqueness of a weak solution of (1.3) later.
We are interested in evaluation of the following: for a small ε, As mentioned in Section 1, the right-hand side of (2.1) corresponds to a barrier option price of knock-out type with maturity T in finance. We regard (X ε,x t ) t as the underlying aseet prices and the expectation E [·] is taken under a risk-neutral probability measure. The boundary ∂D of the domain means the trigger points of the option and f represents a payoff at maturity. The function c represents a short-term interest rate. Our setting includes the case of D = R d , which corresponds to a price of an European option: For applications to option pricing, see Section 4 for the details. Now we introduce our assumptions.
[A] There is a positive constant A 1 such that Moreover, for each ε ∈ I it holds that σ ij (·, ε), b i (·, ε) ∈ L for i, j = 1, . . . , d, where L is the set of locally Lipschitz continuous functions defined on R d .
[B] The function f (x) is continuous onD and there are C f > 0 and m ∈ N such that Note that under [A], the existence and uniqueness of a solution of (1.3) are guaranteed on any filtered probability space equipped with a standard d-dimensional Brownian motion, and Corollary 2.5.12 in [10] and Lemma 3.2.6 in [15] imply for some C l > 0 which depends only on l and A 1 . Moreover, (X x r ) r has the strong Markov property. Although the assumptions [A]- [B] are not always satisfied in our example in Section 4, we can weaken them, and will introduce more general conditions in Section 5.1.
We continue to state our assumptions. 4 [C] There is a positive constant A 2 such that c(x, ε) ≥ −A 2 for x ∈D, ε ∈ I. Moreover, for each ε ∈ I, it holds that c(·, ε) ∈ L.
[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 = φ and E ∩D = {x}.
[E] The matrix (a ij (x, ε)) ij is elliptic in the sense that for each ε ∈ I and compact set K ⊂ R d there is a positive number µ ε,K such that In the case of ε = 0, we further assume Let us define a second order differential operator L ε by σ ik σ jk . We consider the following Cauchy-Dirichlet problem for a PDE of parabolic type The following is obtained by Theorem 3.1 in [17].
is also a solution of (2.3) satisfying the growth condition To study an asymptotic expansion of u ε (t, x), we assume [G] Let n ∈ N. The functions a ij (x, ε), b i (x, ε) and c(x, ε) are n-times continuously differentiable in ε. Furthermore, each of derivatives ∂ k a ij /∂ε k , ∂ k b i /∂ε k , ∂ k c/∂ε k , k = 1, . . . , n − 1, has a polynomial growth rate in x ∈ R d uniformly in ε ∈ I.
Our purpose is to present an asymptotic expansion such that To state the existence of such a function v 0 k (t, x), we prepare the set H m,α,p of g ∈ C([0, T )× D) satisfying the following conditions: Then, we have the next theorem of which proof is given in Section 5.2.
Let g ∈ H m,α,p for some p > 1/α. Then, the following PDE has a classical solution v such that for some C > 0 which depends only on a(·, 0), b(·, 0), c(·, 0), D and M g . Moreover, if w is another classical solution of (2.10) which satisfies |w(t, We also put the next assumption: It is easy to see that the assumptions . . , v 0 k exist and are subject to G m k ,α,p for some m k ∈ N, then the unique classical solution v 0 k+1 of (2.7) exists. We introduce our final assumption.
We remark that v 0 k (t, x) has the stochastic representation: The proof is almost the same as Theorem 5.1.9 in [13]. Now we are prepared to state our main result whose proof is given in Section 5.3.
There are positive constants C n andm n which are independent of ε such that

Semi-Group Representation
In this section we construct a semi-group corresponding to (X 0,x t ) t and D, and give another form of (2.12). We always assume [A]-[I] (or the generalized assumptions in Section 5.1.) We only consider the case where c(t, x, 0) is non-negative and independent of t; we simply denote obvious. The continuity of P D t f is by Lemma 4.3 in [17]. The semi-group property is verified by a straightforward calculation.
Note that (P D t ) t also becomes a semi-group on the set C 0 p (D) of continuous functions f , each of which has a polynomial growth rate and satisfies f (x) = 0 on ∂D.
Let g ∈ H m,α,p . Observe that and we obtain Thus, under the assumption [H], we see Similarly we get the following.
Proof. By (3.1), we have the assertion for k = 1. If the assertion holds for 1, . . . , k − 1, then Thus, our assertion is also true for k. Then we complete the proof of Theorem 5 by mathematical induction.
In particular, when d = 1, D = (l, ∞), b(x, 0) ≡ µ, σ(x, 0) ≡ σ and c(x) ≡ 0 with constants l, µ ∈ R and σ > 0, the process X 0,x t is explicitly represented as X 0,x t = x + µt + σB t , and it is well-known that Therefore, for g ∈ C 0 p (D) we have where We remark that (3.3) is useful for explicit evaluation of (3.1), which is demonstrated in the next section.
respectively when we consider the currency options. Clearly, applying Itô's formula, we have its logarithmic process: Also, its generator is expressed as In this case,L 0 1 defined by (2.5) is given as We will apply Theorem 12 to (4.1) with d = 2 and d ′ = 1 and give an approximation formula for a barrier option of which value is given under a risk-neutral probability measure as where f stands for a payoff function and L(< S) is a barrier price. u ε (t, x) = C SV,ε Barrier (T − t, e x ) satisfies the following PDE: (4.5) wheref (x) = max{e x − K, 0}, D = (l, ∞) and l = log L. We obtain the 0-th order u 0 as is the price of the down-and-out barrier call option under the Black-Scholes model: Here, we recall that the price of the plain vanilla option under the Black-Scholes model is given as where We show the following main result in this section.
Theorem 6. We obtain an approximation formula for the down-and-out barrier call option under the stochastic volatility model (4.1): and Proof. By Theorem 12 and the equality (3.1), we see the expansion The first-order approximation term v 0 whereP D t is defined by (3.3) with the density (3.4), that is, Define ϑ(t, x) as

α, σ, L).
A straightforward calculation shows that the above fucntion agrees with the right-hand side of (4.11). Then we get the assertion.
Remark that through numerical integrations with respect to time s and space y in (4.10), we easily obtain the first order approximation of the down-and-out option prices.
Next, as a special case of (4.1) we consider the following stochastic volatility model with no drifts: where ε ∈ [0, 1), ρ ∈ [−1, 1] and B = (B 1 , B 2 ) is a two dimensional Brownian motion. In this case, we can give a slightly simple approximation formula compared with Theorem 6. By Itô's formula, the following logarithmic model is obtained.
This model is regarded as a SABR model with β = 1 and known as the log-normal SABR (see [12]). Again, the barrier option price is given by where f stands for a payoff function and L(< S) is a barrier price.
The differentiation operators L ε ,L 0 1 and the PDE are same as (4.3)-(4.5) with c = q = 0 and λ = 0. Also, the barrier option price in the Black-Scholes model coincides with (4.7) with no drift, that is, where C BS (T, S) is the driftless Black-Scholes formula of the European call option given by Then, we reach the following expansion formula which only needs 1-dimensional numerical integration.
Proof. By Theorem 12 and the equality (3.1), we see that the expansion Then, we have the following proposition for an expression of v 0 1 (0, x). The proof is given in Section 5.5.
We remark that the expectation in the above equality can be represented as where h(s, x − l) is the density function of the first hitting time to l defined by Now we evaluate Note that Then we have and Combining (4.18), (4.20) and (4.21), we get Substituting (4.22) into (4.16), we have Thus we obtain ds. Apparently, our approximation formula u 0 + εv 0 1 improves the accuracy for C SV,ε Barrier (T, S), and it is observed that εv 0 1 accurately compensates for the difference between C SV,ε Barrier (T, S) and C BS Barrier (T, S), which confirms the validity of our method.

Generalization of Main Results
There are several cases in practice that our assumptions [A]-[B] are not satisfied. Hence, in this section we weaken the assumptions. Let d ′ ∈ {1, . . . , d}, and we regard X ε,x,i for a stochastic volatility and a stochastic interest rate for i > d ′ . For a technical reason introduced later, we assume I ⊂ [0, ∞) in this section.
Then, u ε (t, x) is a viscosity solution of (2.3). Moreover, u ε (t, y) is a viscosity solution of Proof. The latter assertion is by the similar argument to the proof of Proposition 6. Then, the simple calculation gives the former assertion.
Here we give another version of generalized assumptions.
[D'] The domain D is given as where U is a domain in R d ′ whose boundary ∂U satisfies the outside strong shpere property.
[I"] The condition [I] holds replacing G m,α,p with G m,α,p .
Theorem 2 implies the next theorem.
Using the above theorem instead of Theorem 2 itself, we can prove the following theorem similarly to Theorem 10.

Proof of Theorem 2
We consider the following PDE which is equivalent to (2.10) with changing variable t to We defineH m,α,p as the same as H m,α,p replacing [0, T ) in the definition with (0, T ]. We divide the proof of Theorem 2 into the following two propositions. Proposition 2. For any g, a classical solution of (5.4) is unique in the following sense: if v and w are classical solutions of (5.4) and |v(t, x)| + |w(t, x)| ≤ C exp(β|x| 2 ) for some C, β > 0, Proposition 2 is obtained by the same argument as the proof of Theorem 2.4.9 in [5].
Hence,ṽ n is a viscosity subsolution of (5.7). By the similar argument, we also find thatṽ n is a viscosity supersolution. By the definition ofṽ n , we easily getṽ n (T, x) = 0 for x ∈ D and v n (t, x) = 0 for (t, x) ∈ [0, T ] × ∂D.
To see the equivalence v ε n =ṽ ε n , we need to give a new proof of Proposition 5 under the assumptions of Theorem 10.
Proof of Proposition 5. Setū ε n (t, x) = u 0 (t, x) + n−1 k=1 ε k v 0 k (t, x) + ε nṽε n (t, x). The analogous argument of the proof of Proposition 4 implies thatū ε n is a viscosity solutions of (2.3). We easily see thatū ε n has a polynomial growth rate in x uniformly in t. Then, Theorem 9 leads us toū ε n = u ε . This equality and (5.8) imply the assertion. Now, we obtain the assertion of Theorem 10 by the same way as that of Theorem 3.

Proof of Proposition 1
First, we notice the following relation: