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.
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: Here, is a maturity of the option and is an option payoff function at maturity. denotes a vector process with initial value 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, stands for the risk-free interest rate process. Moreover, stands for a constant lower barrier, that is , and is the hitting time to :
It is well-known that a possible approach in computation of is the Euler-Maruyama scheme, which stores the sample paths of the process through an -time discretization with the step size . When applying this scheme to pricing a continuously monitoring barrier option, one kills the simulated process, say if exits from the domain until the maturity . The usual Euler-Maruyama scheme is suboptimal since it does not control the diffusion paths between two successive dates and : 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 and , the barrier option price obtained by the Euler-Maruyama scheme is of order , as opposed to the order 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 calculating 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 (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: where is a small parameter. In this case, the barrier option price (1) is characterized as a solution of the Cauchy-Dirichlet problem: where the differential operator is determined by the diffusion coefficients and with the risk-free interest rate , which will be explicitly defined in the next section.
Next, we introduce an asymptotic expansion formula: where denotes the Landau symbol. The function is the solution of (4) with : if and have some simple forms such as constants (as in the Black-Scholes model), we already know the closed form of and hence obtain the price. Then, we are able to get the approximate value for through evaluation of the coefficient functions . 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, we can derive the following PDE which satisfies: where will be given explicitly in Section 2. Moreover, by applying the Feynman-Kac approach to the PDE (7), we obtain a semigroup representation of . That is, for each , where 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 may be divergent at . 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 by using the semigroup . We notice that such a form is convenient for evaluation of 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 under a stochastic volatility model as follows: for the initial value of the logarithmic underlying price , the maturity and the lower barrier , where is the underlying asset price process, is a payoff function and and the expectation is taken under the risk-neutral probability measure. Here, 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. Section 3 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 will be explained in the next subsection.
Suppose first that a filtered complete probability space is given, where denotes the risk-neutral probability measure, the filtration satisfies the usual conditions, and is some fixed time horizon. Then, the -dimensional underlying state variable is described by the following perturbed SDE: where is an -dimensional Brownian motion and is a small parameter. Let and be Borel measurable functions () where is an interval on including the origin , for instance .
We consider the SDE (10) for any and . 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 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 small , for Borel measurable functions , and a domain .
Also, is the closure of and stands for the first exit time from ; that is
We also remark that the expectation operator is taken under the risk-neutral probability measure and that represents the discount factor with the risk-free interest rate process .
Let us define a second order differential operator by where . We consider the following Cauchy-Dirichlet problem for a PDE of parabolic type:
Under the assumptions stated in the next subsection, we have the following existence and uniqueness result due to Theorem 3.1 in [11].
Theorem 1. Assume which are given in Section 2.2. For each , defined with formula (11) is a (classical) solution of (14) and Moreover, if is also a solution of (14) satisfying the growth condition for some , then .
Our main purpose is to present an asymptotic expansion of the barrier option price : Here, the coefficient functions , are (formally) given as the solution of where is given inductively by Here, is defined as follows:
Then, we can show the next result, whose proof is given in Appendix A.
Theorem 2. Assume which are stated in Section 2.2. Then, for each , is the classical solution of (18) and satisfies for some , .
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 function which has a polynomial growth rate in uniformly in , a classical solution of (18) is unique in the following sense: if and are classical solutions of (18) and for some , then .
Now, we are able to state our first main result on the asymptotic expansion. The proof is given in Appendix B.
Theorem 4. Assume which are given in Section 2.2. There are positive constants and which are independent of such that
Next, we construct a semigroup corresponding to (i.e., with ) and . Then, based on this semigroup we can obtain more explicit representation for the coefficient function than the right hand side of (36) in Assumption , which will appear in Section 2.2.
Let be the set of bounded continuous functions such that on . Obviously, equipped with the supnorm becomes a Banach space.
For and , we define by where is nonnegative. We notice that is equal to with the payoff function . Then, we have the following result.
Proposition 5. Under the assumptions stated in Section 2.2, the mapping is well-defined and is a contraction semigroup.
Proof. Let . The relations , , and are obvious. The continuity of is by Lemma 4.3 in [11]. The semigroup property is verified by a straightforward calculation.
Remark 6. Note that also has the semigroup property on the set of continuous functions , each of which has a polynomial growth rate and satisfies on .
Finally, we show our second main result on the semigroup representation of the coefficient function in the expansion, whose proof is given in Appendix C.
Theorem 7. Under Assumptions given in Section 2.2, for each
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 Theorem 1, 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 constant such that  Moreover, for each it holds that for , where is the set of locally Lipschitz continuous functions defined on :
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 -dimensional Brownian motion, and Corollary 2.5.12 in [17] and Lemma 3.2.6 in [18] imply for some , which depends only on , , and . Moreover, has the strong Markov property.(B)The function is continuous on and there are and such that , . Moreover, on .
Remark 9. The assumption which corresponds to H2(2) in [11] guarantees the continuity of a solution of (14) (if it exists) on the so-called parabolic boundary . For the details, see page 8 in [11].(C) is nonnegative (i.e., ). Moreover, for each , it holds that .(D)The boundary has the outside strong sphere property; that is, for each there is a closed ball such that and .
Remark 10. The assumption provides the regularity of each point in . (compared to [8]) Also, [11] points out that with the ellipticity of the matrix in below gives (E)The matrix is locally elliptic in the sense that for each and compact set there is a positive number such that for any and .
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 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 define , in (20) above.(F)Let . The functions , , and are -times continuously differentiable in . Furthermore, each of derivatives , , , , has a polynomial growth rate in uniformly in .
To state the existence of the functions in the asymptotic expansion (17), we first prepare the following set.
Definition 12. The set of is defined to satisfy the following condition. There is a such that
Given this definition of the set , we put the next condition on .(G), where
Now we examine the conditions necessary for the classical solution to the PDE (18).
Firstly, Let us start with the case of . By the assumption (G), we have for some by the definition of with in (19). Thus we can define Therefore, if we assume that we can show that is the solution of (18) with ; that is, we can confirm that Note that the relations and on are obvious.
Next, let us give some comments on the smoothness of . In many cases as in the Black-Scholes model (see (49) in Section 3) we can rewrite (31) as for some . Thus, if has a “good” smoothness property, the smoothness of also holds such as if the limit in the right hand side exists, and
Moreover, if is in for some , we also have for some by the definition of with in (19). Then, we can define similarly as . Furthermore, under some suitable smoothness conditions for , which may be given by the smoothness property of , we are able to show that is the classical solution of (18) with .
Thus, the observation above leads us to our final assumption.(H)It holds that , for some , where
3. Application to Barrier Option Pricing in Stochastic Volatility Environment
This section demonstrates the effectiveness of our method in stochastic volatility environment. Section 3.1 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: where , , , and and is a two dimensional Brownian motion. Here and 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: Also, its generator is expressed as In this case, which is defined by (20) with is given as 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 where stands for a payoff function, is a barrier price and the expectation is taken with respect to the risk-neutral probability measure.
Then, satisfies the following PDE: where , , and . We obtain the th order as
Remark 13. satisfies PDE (42) with . Although the condition (E) in Section 2 does not seem to be satisfied in this case, the volatility process becomes a constant , and so (38) is reduced to a one-dimensional SDE. Then, (42) with becomes a nondegenerating PDE with fixed . Therefore, we need not take care of the lack of the condition (E) in this example.
Setting , we note that 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
Note also that Therefore, for we have where
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): where
Proof. Firstly, note that when in Theorem 7, we have Thus, we see the expansion The first-order approximation term is given by where is defined by Define as 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 time and space 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: where , and 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: Again, the barrier option price is given by where stands for a payoff function and is a barrier price.
The differential operators , and the PDE are same as (39)–(42) with and . Also, the barrier option price in the Black-Scholes model coincides with (44) with no drift; that is, where is the driftless Black-Scholes formula of the European call option given by with Then, we reach the following expansion formula which only needs 1-dimensional numerical integration.
Theorem 15. , where
Proof. See Appendix D.
3.2. Numerical Example
Finally, applying our approximation formulas in Theorems 14 and 15, we present numerical experiments for European down-and-out barrier call prices. First, let us denote and . Then, we see
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 of ; they are computed by Monte-Carlo simulations with 100,000 time steps () 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 ( 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.
In all the cases, we set the initial asset price , the initial volatility , the time to maturity , the lower barrier , and strike prices . The other parameters (, , , , , ) are listed in the caption of each table.
Apparently, our approximation formula improves the accuracy against , and it is observed that accurately compensates for the difference between and , 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.
Appendices
A. Proof of Theorem 2
First, by the definition of , we easily get for and for .
Next, fix any . By the Markov property, we have for each , where and is the Brownian filtration. This implies that is a local martingale. On the other hand, applying Ito’s formula, we have that for each . Thus, the uniqueness of decompositions of semimartingales gives us Therefore, for each fixed , holds for any small enough . Since , by letting , we obtain Finally we prove (21) by mathematical induction. When , the assertion is easily obtained by (27), (20), (F), and (G). Now we assume that (21) holds for . Then, by (20), (19), and (F), we have for some , where is a multi-index, and . By the induction hypothesis and (G)-(H), we see that for some and . Therefore, we get for some . Then we obtain by virtue of (27). Thus (21) also holds for . Now we complete the proof of Theorem 2.
B. Proof of Theorem 4
First, we generalize the definitions of , , and . For , we define where and are understood as and , respectively.
We consider the following Cauchy-Dirichlet problem:
For , we define . Obviously, we see
Proposition 16. The function is a solution of (B.2).
Proof. It is obvious that for and for .
We also recall in (14) that
Next, we apply Taylor’s theorem to to observe that
Since is the solution of (14) with , we get
Similarly, since is a solution of (18), we have
Then, replacing in the right hand side of (B.5) by the right hand side of (B.3) and taking (B.6) and (B.7) into account, we obtain
Thus, with the definition of above, we have for ,
This implies the assertion.
Set By (G)-(H), we find that there are , which are independent of and the function determined by such that The inequalities (27) and (B.11) imply for some which is also independent of .
Proposition 17. .
Proof. The assertion is easily obtained by the similar argument to the one in Theorem 5.1.9 in [20].
Proof of Theorem 4. By (B.3) and Proposition 17, we have . Our assertion is now immediately obtained by inequality (B.12).
C. Proof of Theorem 7
(1) Firstly, let us consider the case for . Let . Observe that and we obtain Thus, under the assumption , we see Thus, we have the assertion for .
(2) If the assertion holds for , then
Thus, our assertion is also true for . Then we complete the proof of Proposition 17 by mathematical induction.
D. Proof of Theorem 15
By the asymptotic expansion in Section 2 and Theorem 7 with , we see that the expansion holds with Then, we have the following proposition for an expression of . The proof is given in Appendix D.1.
Proposition 18. It holds that:
We remark that the expectation in the above equality can be represented as
where is the density function of the first hitting time to defined by
Now we evaluate
Note that
Then we have
Combining (D.7) and (D.9), we get
Substituting (D.10) into (D.4), we have
Thus we obtain
By Proposition 18, (D.4), (D.10), and (D.12), we reach the assertion.
D.1. Proof of Proposition 18
First, we notice the following relation: Then, using the relations and we get Also, we have Therefore, the function satisfies the following PDE Then Theorem 6.5.2 in [9] implies By (D.17) and (D.19), we get the assertion.
Disclosure
Mathematical Subject Classification (2010) 35B20, 35C20, 91G20.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are very grateful to Professor Seisho Sato in the University of Tokyo for his suggestions to our numerical computation. Also we would like to thank the anonymous referees for their valuable comments and suggestions.