Analytical pricing of single barrier options under local volatility models

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1 Quantitative Finance ISSN: (Print (Online Journal homepage: Analytical pricing of single barrier options under local volatility models Hideharu Funahashi & Masaaki Kijima To cite this article: Hideharu Funahashi & Masaaki Kijima (6 Analytical pricing of single barrier options under local volatility models, Quantitative Finance, 6:6, , DOI:.8/ To link to this article: Published online: Nov 5. Submit your article to this journal Article views: 5 View related articles View Crossmark data Citing articles: View citing articles Full Terms & Conditions of access and use can be found at Download by: [Wagner College] Date: 6 September 7, At: :8

2 Quantitative Finance, 6 Vol. 6, No. 6, , Analytical pricing of single barrier options under local volatility models HIDEHARU FUNAHASHI and MASAAKI KIJIMA Mizuho Securities Co. Ltd. and Tokyo Metropolitan University, Otemachi First Square -5-, Otemachi, Chiyoda-ku, Tokyo -4, Japan Graduate School of Social Sciences, Tokyo Metropolitan University, - Minami-Ohsawa, Hachiohji, Tokyo 9-397, Japan (Received 9 February 5; accepted 4 September 5; published online November 5 Downloaded by [Wagner College] at :8 6 September 7. Introduction This paper considers a single barrier option under a local volatility model and shows that any downand-in option can be priced by a combination of three standard European options whose volatility functions are connected through symmetrization. The symmetrized volatility function is approximated by a sequence of smooth functions that converges to the original one. An approximation formula is developed to price the standard European options with the approximated volatility functions. Finally, we apply the Aitken convergence accelerator to obtain an approximate price of the down-and-in option. Other single barrier options are priced in a similar fashion. Keywords: Barrier option; Put-call symmetry; Local volatility; Symmetrization A barrier option is a path-dependent option that resembles standard European options except that the payoff depends on whether or not the underlying asset price crosses a certain barrier level during the option s life. Barrier options are very popular in exchanges worldwide and also in over-the-counter markets. This paper focuses on single barrier options which include eight possible types: up (down-and-in (out call (put options. For example, a down-and-in call option gives the option holder the payoff of a European call option if the underlying asset price does reach a lower barrier level before the expiration date. Rubinstein and Reiner (99 derive closed-form pricing formulas for all the eight types of single barrier options under the geometric Brownian motion (GBM assumption on the asset price process. However, it is well known that the Black and Scholes model (973 cannot consistently price European options in the market, since implied volatility surfaces are usually skew- or smile-shaped. This tendency holds for the case of barrier options as well, and so these formulas may not be suitable for practical use. When the underlying asset price process is a constant elasticity of variance (CEV model, Davydov and Linetsky ( obtain analytical formulas for the prices of path-dependent Corresponding author. kijima@tmu.ac.jp More complicated barrier options include double barrier options, two-dimensional barrier options, barrier options with cash rebates, and so on. The approximation method developed in this paper cannot be applied to these barrier options directly. 5 Informa UK Limited, trading as Taylor & Francis Group (barrier and look-back options. The CEV model is capable of reproducing the volatility skew observed in the actual market and is thus very popular in practice. However, these formulas are extremely complicated for practical use and require much effort for the computation of their option prices. Also, for the general diffusion case, no explicit solution has been obtained. As a result, practitioners must use Monte Carlo methods for the pricing of such path-dependent options. In the standard Monte Carlo simulation, however, there is a problem that arises from their path-dependence nature. Gobet ( points out that the weak convergence rate against the time-discretization gets worse when compared with the standard path-independent case due to the failure to observe hitting between two time steps. He showed that the convergence order of Euler Maruyama approximation is one-half, which is much slower than the standard case whose order is unity. This means that we must build finer time steps and collect more simulation paths in order to obtain reliable results for the pricing of barrier options than for the standard European options. Therefore, a closed-form approximation formula might be a better solution for pricing such path-dependence options. Static (or semi-static hedging techniques have been developed for path-dependent options. In particular, Carr and Other numerical methods for the pricing of barrier options include the tree method (see, e.g. Boyle and Tian 998 and the PDE (partial differential equation method (see, e.g. Zvan et al.,. For the case of multidimensional barrier options, we refer to Bernis et al. (3. Carr and Chou (997 are the first to bring attention to this remarkable property. Since then, several papers such as (Carr et al. 998, Fink 3, Nalholm and Poulsen 6, and Rheinländer and Schmutz

3 868 H. Funahashi and M. Kijima Downloaded by [Wagner College] at :8 6 September 7 Lee (9 have fully investigated put-call symmetry and applied it to develop a semi-static hedge of barrier-type options. The symmetry holds true when the volatility function satisfies a certain type of symmetric property, but does not hold when the underlying process is a general diffusion, including the CEV case. In this paper, we consider a single barrier option under a local volatility (LV model. First, we apply put-call symmetry, after symmetrization to price a symmetrized down-and-in option. Then, using the basic relationship between down-andin and down-and-out options, we show that any down-andin option can be priced by a combination of three standard European options whose volatility functions are connected through the symmetrization. The symmetrized volatility function is approximated by a sequence of smooth functions that converges to the original one. Employing the chaos expansion technique developed by Funahashi and Kijima (5, an approximation formula is obtained to price the European options with the approximate volatility functions. Finally, we apply the Aitken convergence accelerator to obtain an approximate price of the down-and-in option. Other single barrier options can be priced similarly. This paper is organized as follows. In the next section, we outline the model and define the key parity relationship between down-and-in and down-and-out options. It is shown that any down-and-in option can be priced by a combination of three standard European options. Based on this result, we then derive an approximation formula for down-and-in option prices by applying the chaos expansion technique in section 3. Some numerical examples are given to demonstrate the effectiveness and accuracy of our approximation method in section 4. Section 5 is devoted to some discussions about the Akahori Imamura symmetrization (4 and some applications. Finally, section 6 concludes this paper. Proofs are provided in appendix for the interested reader.. The setup Throughout this paper, the current time is fixed to be t = and the option maturity is given by T >. Also, (, F, Q, {F t } t T denotes a filtered probability space where the filtration {F t } t T satisfies the usual conditions. The probability measure Q is a risk-neutral measure and the expectation operator under Q is denoted by E. Let us denote the time-t price of a risky asset by S t, and consider a single barrier B with S > B. The first hitting time of S t to the barrier B is defined by 3 investigate semi-static hedging in financial markets. For more general duality properties in option pricing, we refer to, e.g. Eberlein et al. (8. See, e.g. Carr and Mayob (7 for the case that the underlying asset process is a jump diffusion. When the barrier crossing is monitored only in the discrete time, the problem of option pricing becomes more complicated. See, e.g. Kou (3 and Fusai and Recchioni (7 for details. The idea of symmetrization was first introduced in a series of papers by Akahori and Imamura (4, Imamura et al. (, and Imamura et al. (4. This paper considers only the case of a down barrier. The up-barrier case can be treated similarly and is omitted. τ = inf {t : S t B}, S > B, (. where τ = if the event is empty. Let G(S be the payoff function of an option at the maturity T. A down-and-in option is defined as G(S T χ {τ T }, where χ denotes the indicator function, i.e. χ A = ifa is true and χ A = otherwise. Similarly, a down-and-out option is defined as G(S T χ {τ>t}. It follows that G(S T = G(S T χ {τ>t} + G(S T χ {τ T }, and we have the following basic relationship for the three options: E(S; G = DO(S; G + DI(S; G, (. where DO(S; G and DI(S; G denote the prices of down-andout and down-and-in options, respectively, when the underlying asset is S t and the payoff function is G, and where E(S; G is the price of the corresponding standard European option with the same underlying and same payoff function. Under some symmetry assumption, Carr and Lee (9 show that the following conditional expectations are identical for any bounded Borel function G(S: [ ( ST B ] E τ [G(S T ]=E τ B G. (.3 It can be shown that this observation yields a semi-static hedge for the down-and-in option G(S T χ {τ T }. Namely, at time, purchase and hold a European claim Ɣ(S T = G(S T χ {ST B} + S T B G ( B S T S T χ {ST B}. (.4 If and when the barrier knocks in, exchange the Ɣ(S T claim for a claim on G(S T at zero cost, because [ ] E τ [G(S T ]=E τ G(ST χ {ST B} + G(S T χ {ST >B} [ ( = E τ G(S T χ {ST B} + S T B G B ] χ S {B /S T >B} T [ ( = E τ G(S T χ {ST B} + S T B G B ] χ {ST <B} = E τ [Ɣ(S T ]. If the barrier never knocks in, we have S T > B and the claim on Ɣ(S T expires worthless. Therefore, the price DI(S; G of down-and-in option can be obtained as the price E(S; Ɣ of the standard European claim Ɣ(S T. In particular, for a down-and-in call option G(S T = (S T K + with K > B, where (x + = max{x, }, we have Ɣ(S T = S ( T B + K, B S T since K > B and S T > B together imply Ɣ(S T = ifτ>t. It follows that Ɣ(S T = K ( B B K S T + = K B S T (S T B K + + B K B S T, (.5 where the second equality is due to the put-call parity relation. Now, suppose that the underlying asset price S t follows a general diffusion. That is, we consider the stochastic differential equation (SDE for short ds t S t = rdt + σ(s t dw t, t T, (.6

4 Analytical pricing of single barrier options under local volatility models 869 Downloaded by [Wagner College] at :8 6 September 7 under the risk-neutral probability measure Q, where the volatility function σ(s is smooth enough and W t denotes the standard Brownian motion under Q. In particular, when σ(s = σ S β, S >, (.7 for some constants σ>and β, <β, the LVmodel (.6 is called the CEV model that is frequently used in practice. Of course, when β =, the CEV model is reduced to the Black Scholes model (973. Let v(t, T be the time-t price of the discount bond maturing at time T. Under the constant interest-rate environment, we have v(t, T = e r(t t.letm t = S t /v(t, T = e r(t t S t be the forward price of S t. It is readily seen that dm t = σ M (t, M t dw t, t T, (.8 M t so that M t is a martingale under Q subject to the regularity condition on σ M (t, M, where σ M (t, M = σ(e r(t t M. Let X t = log(m t /M τ, t τ, where M τ = e r(t τ B. According to theorem 5.5 in Carr and Lee (9, a sufficient condition for the symmetry (.3 is that f (t, x = σ(e r(t τ e x+log B, t τ, (.9 is an even function in x. In particular, for the CEV case (.7, we have f (t, x = σ e r(β (t τ e (β (x+log B, t τ, which cannot be even in x, except the trivial case that β =, i.e. the Black Scholes model. Let us define { σ(e f (t, x = r(t τ e x+log B, x, σ(e r(t τ e x+log B, x <. Evidently, the function f (t, x is even in x. Associated with f (t, x is the symmetrized volatility function given by { σ(s, S B, σ(t, S = σ(e r(t τ B (. /S, S < B, where τ t T. However, it will become apparent below that we need to extend the volatility structure (. to the whole time domain t [, T ] in order to price down-and-in options. One way for doing this is to assume that r =. In this case, we define the symmetrized volatility function by { σ(s, S B, σ(t, S := σ(s = σ(b (. /S, S < B, and consider the SDE ds t = σ(s t dw t, t T, (. S t under the risk-neutral probability measure Q. Here and hereafter, we always assume that the strong solution to the SDE (. exists, which we denote by S t. However, it should be It is straightforward to extend our method to the case that the volatility function σ(t, S depends on time t. Justification of this assumption for practical use is found on p. 67 of Carr et al. (998. Following their idea in section 5, bounds on the option values can be obtained for the case r >. Alternatively, we can define the barrier in terms of the martingale process M t.in this case, the barrier is given as a curved boundary B t = Be r(t t, t T. See section 5. for the discussion of this case. noted that the volatility function σ(s may not be smooth, because it may not be differentiable at S = B even if σ(s is differentiable with respect to S. When the underlying asset S t follows the SDE (., we can enjoy the semi-static hedge for the down-and-in option whose price, DI( S; G, is given by the price, E( S,Ɣ, of European claim Ɣ( S T defined in (.4.Also, due to the basic relationship (., we have E( S; G = DO( S; G + E( S; Ɣ, (.3 because DI( S; G = E( S; Ɣ. However, on the event {τ >T }, the volatility σ(s t is the same as the volatility σ(s t of the SDE (.6, so that the price DO( S; G of down-and-out option with the underlying S t and the payoff G(S is the same as the price DO(S; G with the underlying S t and the payoff G(S. Therefore, we obtain the following fundamental result for down-and-in options by using (. and (.3. Theorem. Under the given assumptions, the price DI (S; G of down-and-in option with the underlying S t and the payoff G(S is given by a combination of three standard European options, i.e. DI(S; G = E( S; Ɣ +[E(S; G E( S; G], (.4 where S t is the symmetrized price process and the payoff function Ɣ(S is defined in (.4. In particular, for a down-and-in call option G(S T = (S T K + with K = B, i.e. the strike and the barrier are the same level, we have from (.5 that Ɣ(S T G(S T = (S T K + +K S T (S T K + = K S T. Hence, the next result is immediate. Corollary. Under the given assumptions, the price DI (S; G of down-and-in call option with strike K = Bisthe same as the put option price with strike K, i.e. DI(S; G = E(S; G (S K. Also, when β =, i.e. the Black Scholes case, the volatility function is constant and hence it is symmetric. In this case, we have S t = S t and so, from (.4, we obtain DI(S; G = E(S; Ɣ. It follows from (.5 that the price of the down-and-in call option is given by DI(S; G = K B E [ ( S T B K + ] + B K B S, because S t is a martingale under Q. Therefore, we have DI(S; G = K (S (d B B K (d σ T + B K B S, (.5 where d = log(s K/B σ + σ T T. See Hull (9, p. 55 for another derivation of this result. 3. Approximation formula From theorem., all we need is to evaluate the three Europeanstyle options for the pricing of down-and-in option when the

5 87 H. Funahashi and M. Kijima Downloaded by [Wagner College] at :8 6 September 7 underlying is a general diffusion. At the first glance, this can be done by standard Monte Carlo simulation easily, because we do not need to identify whether the barrier knocks in or not for each simulation path. However, if the symmetrized volatility function σ(s defined in (. has a kink at S = B, there may exist a failure in detecting the change of the shape in the volatility function between two time steps, i.e. the problem raised by Gobet ( can be inherited. In this section, in order to avoid this drawback, we propose an analytical approximation method based on the chaos expansion technique developed by Funahashi and Kijima (5 to evaluate the three European-style options. Note, however, that one of the key assumption in this technique is that the volatility function is smooth enough. To overcome this deficit, we utilize the following well-known result. Lemma 3. Let h(x = x for x > and h(x = x for < x <. Then, we have exp ( ɛ + (log x h(x = O(ɛ. Consider a down-and-in option with payoff G(S, and let us define x = S/B. Letη(x = σ(bx, and consider the SDE (. in terms of x t, i.e. dx t = η(x t dw t, t T, (3. x t where x = S /B. Under our assumption, the strong solution to (3. exists and is given by [ x t = x exp η(x u dw u ] η (x u du, (3. although the volatility function η(x is not smooth at x =. Now, for a sufficiently small ɛ>, the symmetrized volatility function σ(s is approximated as ( σ(s η(ɛ, x; η(ɛ, x = σ B exp ( ɛ + (log x, (3.3 and consider the SDE (3. with the approximated volatility function η(ɛ, x for each fixed ɛ>. In this setting, the strong solution always exists and is given by [ xt ɛ = x exp η(ɛ, xu ɛ dw u ] η (ɛ, xu ɛ du for each ɛ>. Note that η(ɛ, x is differentiable with respect to x and η(x = lim ɛ η(ɛ, x due to lemma 3.. In the following, in order to simplify the notation, we omit the argument ɛ and simply write η(ɛ, x ɛ as η(x. Hence, we assume that, for each ɛ>, the solution is given by [ x t = x exp η(x u dw u ] η (x u du (3.4 and it converges to the solution (3. asɛ. In the rest of this section, we intend to approximate the solution x T at the maturity T for each fixed ɛ>, to derive the probability density function of the approximated x T, and to obtain the European option prices in (.4. Although we assume the convergence of (3.4 to the solution (3. with non-smooth coefficient as ɛ, the convergence can be proved for a certain class of volatility functions σ(s by the uniformity condition given in lemma 3., as pointed out by an anonymous referee. See, e.g. theorem 5. on p. 8 of Friedman ( Wiener Itô chaos expansion Let x t ( = and, using (3.4, define x t (m successively by [ x t (m+ = x exp η(x u (m dw u ] η (x u (m du. Denoting g t thus have x (m+ t = g (udu and J t (g = g(udw u,we [ = x exp J t ( η m ] η m t, (3.5 where η m (t = η(x t (m. The significant difference between the expansion method in this paper and the one proposed in Funahashi and Kijima (5 is the choice of the starting point x t ( =. Because of the form of the symmetrized volatility function η(x = σ(bx, we must expand it around x =. Suppose that x t (m converges to the solution x t in (3.4 as m. It then follows that { } x t = x t ( + x t (m+ x t (m. (3.6 m= On the other hand, by applying the Hermite polynomial expansion to (3.5, we have x t (m+ η m n ( t Jt ( η m = + h n, x n! η m t where h n (x is the Hermite polynomial of order n. Hence, defining I m,n (t = n! we obtain from (3.6 that { η m n t h n η m n t h n x t = x ( t + x m, ( Jt ( η m η m ( t Jt ( η m η m t }, (3.7 I m,n (t. (3.8 Our approximation strategy in this paper is to approximate the random quantity x t by x t = x t ( + x Ĩ m,n (t, (3.9 m, where x t ( and Ĩ m,n (t are approximations of x t ( and I m,n (t, respectively. The approximated random quantities must be of easy form. Namely, as in Funahashi and Kijima (5, each of these quantities is given as a sum of iterated stochastic integrals with deterministic integrands. The next result justifies a truncation of the infinite sum in (3.9 at a certain level. See Funahashi and Kijima (5 for the proof. Proposition 3. Consider the iterated integral I n = n σ (t σ (t σ n (t n dw t dw tn. For example, we have h (x = x, h (x = x, h 3 (x = x 3 3x, etc. See Funahashi and Kijima (5 for the superiority of this expansion over the Maclaurin counterpart.

6 Analytical pricing of single barrier options under local volatility models 87 Downloaded by [Wagner College] at :8 6 September 7 If the volatilities σ k (t are deterministic functions and ˆσ(t = max k σ k (t L ([, t] for all t, then we have E[In ] σ n t /n!. To be more specific, from proposition 3., the sum of iterated integrals of greater than the nth order can be considered to be zero in the L sense for large n. In this paper, we ignore the terms of iterated integrals of higher than the second order. Now, from the Wiener Itô chaos expansion of log-normal process (3.5 with m =, we first have x ( t x = + n η (t η (t η (t n dw t dw tn, where η (t = η(x t (. Since x t ( = by definition, we obtain η (t = η(ɛ, = σ(be ɛ, which is a deterministic function (in fact, it is a constant. We approximate x t ( by truncating the sum at n =, i.e. by x ( t = x [ + η (t dw t + (3. In order to approximate the term I m,n (t, we employ Taylor s expansion around x t (m. Recall that J t ( η m = η m(udw u = η(x(m u dw u and that η(x is differentiable infinitely many times. It follows that J t ( η m J t ( η m t [ ] + η (n n n! m (u x u (m x u (m dwu (3. η (t η (t dw t dw t ]. for sufficiently large N, where η m (n (t = x (n η(x (m x=x.also, t by definition, we have x t (m+ x t (m = x I m,n (t x n 3 m I m,n (t. (3. Recall that our strategy is to neglect the terms of iterated integrals of higher than the second order. Now, in order to approximate I, (t, recall that I, (t = J t ( η J t ( η by definition and so, from (3., we have I, (t n! + nx x η (n [ ] η (n n (u x u ( x u ( dwu. Since x t ( = and x t ( is approximated by (3., by ignoring the higher terms, we obtain [ t I, (t Ĩ, (t := n! (x n η (n (udw u ] η (udw u dw s. (s (3.3 An extension to a higher order approximation is straightforward, although tedious, and omitted. See Funahashi and Kijima (5 for details. According to our numerical experiences, the second-order approximation seems enough for practical use. We note that η ( (t =, η ( (t = σ ( (Be ɛ Beɛ, ɛ η (3 (t σ ( (Be ɛ = 3Beɛ, ɛ and so on, where σ ( (x = x σ(x and η (n (t = x (n η (x x= since x t ( = ; hence, η (n (t are all constant. In particular, when β =, we have η (n (t = for all n. Similarly, we obtain I, (t Ĩ, (t ( t := n! (x n + η (n (s + and m, m!n! (x m+n η (s η (n (udw u dw s η (udw u dw s (3.4 η (m (s η (n (udw u dw s I, (t Ĩ, (t := x (x n η ( n! (s η (n (udw u t + x m!n! (x m+n m, dw s η (m+ (s η (n (udw u dw s. (3.5 Proofs of (3.4 and (3.5 are given in appendix. On the other hand, we can show that I m,n (t for all m, n. To see this, note that I, (t = { } (Jt ( η Jt ( η ( η t η t = η (s η (udw u dw s η (s η (udw u dw s. (3.6 But, since η(x is differentiable infinitely many times, we have η (s = η (s + K s, K s η (n (s [ ] n x s ( x s (. n! Also, from (3., we have x s ( x s ( = x I, (s. It follows from (3.6 that I, (t = η (s K u dw u dw s + K s ( η (u + K u dw u dw s, which can be neglected due to our approximation strategy, because these terms involve only iterated integrals of higher than the second order from (3.3. Other higher terms can be proved to be neglected in a similar manner.

7 87 H. Funahashi and M. Kijima Downloaded by [Wagner College] at :8 6 September Approximation formulas The proposed approximations developed so far are put together to conclude the following. Theorem 3. Let X t := x t x. Then, X t p (sdw s + p (s η (udw u dw s where and + p (s = η (s + p 3,n (s p (s = η (s + x + N η (n (udw u dw s, n! (x (n n η (s, (x n (n! n! (x (n n η (s p 3,n (s = n! (x n [ η (s + x η ( (s + N ( η (m (s + x η (m+ ] (s m= η (n (s m! (x m for n =,,... Note that p (s, p (s and p 3,n (s are all deterministic functions. In order to calculate the probability density function of X t, we intend to derive an approximated characteristic function of X t, which can then be inverted back to derive an approximation of the probability density function of X t. This idea has been widely used for deriving approximated distributions. Before proceeding, for the sake of notational simplicity, we define a (t = p (sdw s and a (t = + p (s η (udw u dw s p 3,n (s η (n (udw u dw s. Then, from theorem 3., we have X t a (t + a (t. Note that a (t follows a normal distribution with zero mean and variance t = p (sds. The moments of a (t conditional on the normal variate a (t can be obtained explicitly. Let the characteristic function of X t be (ξ = E[e iξ X t ]. We approximate it as [ (ξ E e {iξ(a (t+a (t} ] [ ] = E e iξa (t ( + iξa (t + R 3, where R 3 consists of the third or higher order multiple stochastic integrals. Note that, since [ ] [ ] E e iξa(t R 3 E e iξa(t R 3 ( [ E e iξa(t ] ( [ E R 3 ] ( [ = E R 3 ], we regard E [ e iξa(t ] R 3 as for the previous case. Taking the conditional expectation on a (t, we then have [ ] (ξ E[e iξa(t ]+iξe e iξa(t E[a (t a (t]. (3.7 The conditional expectation E[a (t a (t] can be evaluated explicitly. Namely, we have ( x E[a (t a (t = x] =q(t, t t where t = p (sds and q(t = p (sp (s + p (sp 3,n (s η (up (udu ds η (n (up (udu ds. Recall that a (t follows a normal distribution with zero mean and variance t. Let us denote the density function of X t by f t (x. By applying the Fourier inversion formula to each term of the characteristic function (3.7, we obtain an approximation of the density function as f t (x n (x;, t x {E[a (t a (t = x]n (x;, t }, which leas to the following result. Here, n(x; a, b denotes the normal density function with mean a and variance b. Theorem 3. The probability density function of X t is approximated as { ( } q(t x f t (x n(x;, t 3 h 3 +, t t where n(x; a, b denotes the normal density function with mean a and variance b. Recall that X t = x t x and x t = S t /B. Also, the price of the European option with payoff G(S and maturity T written on the symmetrized process S t is given by E( S; G = E [ G( S T ] = E [G(Bx ( + X T ] = G(Bx ( + x f T (xdx, where f T (x is the density function of X T. In particular, when G(S = (S K +, i.e. a call option with strike K,we have E( S; G = Bx (x K + f T (xdx, K where K = + K S. Calculating the integral by using the approximated density function given in theorem 3., we conclude the following. The proof is straightforward and omitted.

8 Analytical pricing of single barrier options under local volatility models 873 Downloaded by [Wagner College] at :8 6 September 7 Theorem 3.3 The value of a European call option with maturity T and strike K written on the symmetrized process S t is approximated as E( S; G S K + S T exp + π + ( K S { } q(t (K S exp (K S S T 3 π T } { (K S S T ( K S, S T where T = T p (sds and (x is the cumulative distribution function of the standard normal distribution. Next, we need to calculate the value of the European option with payoff Ɣ(S given in (.5. The next result is readily obtained from theorem 3.3. Corollary 3. The value of down-and-in call option with maturity T and strike K written on the symmetrized process S t is approximated as E( S; Ɣ B Kx + Kx T exp + π + ( B Kx { } q(t (B Kx exp (B Kx K x T 3 π T { } (B Kx K x T ( B Kx, Kx T where x = S /B. Note that, when K = B, we have Kx = S so that we obtain E( S; Ɣ E( S; G = K S from theorem 3.3 and corollary 3., which agrees with the result in corollary.. Finally, the value of the European call option with the original underlying S t can be evaluated by the method proposed by Funahashi and Kijima (5. Proposition 3. The value of a European call option with maturity T and strike K written on the original process S t is approximated as E(S; G S [ ] n( K ;, T T v(t K ( T + S K ( K / T, where K = S K, T = σ (S T and v(t = σ 4 (S T /. Using these results, we can calculate the three terms in the right-hand side of (.4, which leads to the price of the downand-in call option with maturity T and strike K. Recall that the option pricing formula given so far is an approximation formula for the down-and-in call option with the volatility function η(ɛ, S/B. When ɛ = m, we denote it by DI m (S; G in order to emphasize the degree of approximation of the volatility function. The important issue at this point is how to determine appropriate number for m (see remark 3. below. In order to overcome the difficulty, we employ a convergence accelerator technique. Namely, for each volatility function η(m, S/B, the European-style options E m ( S; G and E m ( S; Ɣ are evaluated by theorem 3.3 and corollary 3., respectively. It follows from theorem. that DI m (S; G = E m ( S; Ɣ+[E(S; G E m ( S; G] DI(S; G (3.8 as m under some regularity condition. In actual computation, we employ the Aitken convergence accelerator, i.e. DI(S; G DI m+ (S; G (DI m+ (S; G DI m+ (S; G DI m+ (S; G DI m+ (S; G + DI m (S; G, (3.9 for some appropriate m. Remark 3. (Choice of ɛ and the order n of Taylor s Expansion We note that the parameter ɛ and the order n of Taylor s expansion are closely related. Recall that the centre of Taylor s expansion, x ( =, is exactly where there is a kink in the symmetrized volatility function η(x = σ(bx. Hence, as noted above, there appear terms involving ɛ k in the derivatives of η (x. For example, η ( (t ɛ, η (3 (t ɛ and η (4 (t ɛ 3. This means that, if we choose very small ɛ > to recover the symmetry, one may not safely neglect these higher terms. Hence, the use of smaller ɛ requires higher order terms of Taylor s series (i.e. larger n. As an illustrative example, suppose that the original volatility function is specified as σ(s = σ/ S with S =, σ =.5 and B = 95. In figure, weplotη(x (red line, η(ɛ,x (green line and the third-order Taylor s approximation (blue line of η(ɛ, x with respect to ɛ = /m, m =,,...,9. It is explicitly observed that, as ɛ becomes smaller, η(ɛ, x converges to η(x. However, at the same time, the discrepancy between η(ɛ, x and its Taylor s expansion becomes significant for x outside the kink. In the following numerical examples, we truncate the Taylor series at n = 3 in order to avoid using too small ɛ. In this case, as figure reveals, the parameter ɛ = /m between m = 3 and 5 is typically selected. In actual computation, in order to accelerate convergence, we employ the Aitken convergence accelerator (3.9 with m = Numerical examples In this section, the accuracy of our approximation method is studied by numerical examples. To do this, we consider the CEV model first and then a non-linear LV model to compare our approximated option prices with either the formula (.5 or Monte Carlo simulation results that are used as the benchmark. As the base case parameters, we set S =, B = 95 and T =.5. See, e.g. p. of Press et al. (7 for details. Alternatively, we can use the three-point Richardson extrapolation as is often used in the finance literature; see, e.g. Geske and Johnson (984 and Carr et al. (998. For the benchmark Monte Carlo simulation, we adopt the simple Euler approximation with the Brownian bridge technique (see, e.g. section 6.4 in Glasserman (4. The time interval [, ] is divided into subintervals with equal space, and sample paths are generated for one simulation experiment.

9 874 H. Funahashi and M. Kijima.5 ε=/ η (x η ~ (x TE.5 ε=/3 η (x η ~ (x TE η ~ (x. η ~ (x x x.5 ε=/4 η (x η ~ (x TE.5 ε=/5 η (x η ~ (x TE.4.4 Downloaded by [Wagner College] at :8 6 September 7 η ~ (x η ~ (x x ε=/6 η (x η ~ (x TE η ~ (x η ~ (x x ε=/7 η (x η ~ (x TE x x.5 ε=/8 η (x η ~ (x TE.5 ε=/9 η (x η ~ (x TE η ~ (x. η ~ (x x x Figure. Convergence and separation of the functions η(x (red line, η(ɛ, x (green line and the third-order Taylor s approximation of η(ɛ, x (blue line with respect to ɛ = /m, m =,...,9.

10 Analytical pricing of single barrier options under local volatility models 875 Downloaded by [Wagner College] at :8 6 September CEV model Suppose that the volatility function is specified as σ(s = σ S β, where σ>and β, β, are constants. If β =, the asset price S t follows a GBM as in the Black Scholes model (973. In this subsection, the volatility parameter is set to satisfy either σ(s =.5 (low volatility case or σ(s =.3 (high volatility case. Hence, if β =, then σ =.5 for low volatility case and σ =.3 for high volatility case. If β =.5, then σ =.5 for low volatility case and σ = 3. for high volatility case. Other cases are similarly defined. Also, we consider the three cases for maturity; T =.5 year, T = year and T = 3 years. In this model, the symmetrized volatility function σ(s defined in (. is given by { σ x σ(s = η(x; η(x = β, x, σ x (β (4., x <, where x = S/B and σ = σ B β. Further, form lemma 3., the approximated (smooth volatility function is given by ( η(ɛ, x = σ exp (β ɛ + log S (4. B for each fixed ɛ>. Note that we have η ( (t =, and η (3 η ( (t = σ(β e(β ɛ ɛ (t = 3 σ(β e(β ɛ, ɛ respectively. Also, in this case, the biggest difference between η(x and η(ɛ, x occurs at S = B (i.e. x = and max{ η(ɛ, x η(x} = σ(e (β ɛ = O(ɛ. x Figure shows the convergence of the approximated volatility function (upper panel and the approximated down-andin call option price (lower panel for the CEV model with β =.5. For each m, the approximated volatility function η(m, x, where x = S/B, is given by (4. with β =.5, while the option price DI m (S; G is calculated by (3.8. As noted in remark 3., if we choose small ɛ > (largem in this case to recover the original volatility function, we have to employ higher order Taylor s expansion. However, in our numerical examples, because we truncate Taylor s expansion at the third order, we cannot make m too large. In fact, as the lower panel reveals, our approximation certainly deteriorates after some point m. Remark 4. Since the payoff function Ɣ(S given in (.5 is convex in S and the volatility function η(m, x increases as m gets large, the option price E m ( S; Ɣin (3.8 is increasing in m. See, e.g. corollary 3. in Kijima ( for the monotonicity If β =, then the asset price S t is normally distributed and the model contradicts the ordinary finance assumption. Also, if <β<.5, then the volatility function is not smooth enough that violates our assumption, although these cases are often used in practice. In our numerical examples, we use these cases in order to check the accuracy of our approximation formula. of option prices. On the other hand, the second term [E(S; G E m ( S; G] in (3.8 is decreasing in m by the same reason. The total effect of these two terms in (3.8 seems indeterminate. However, this observation can help how to choose m, because the convergence is expected to be monotonic. Namely, we employ the Aitken convergence accelerator (3.9 with some appropriate m. By looking at figure, an appropriate choice of m would be m = 3. On the other hand, table presents the numerical results for down-and-in call option prices for the GBM case (i.e. β =. Note that the analytical solution is given by (.5 for this case. Also, because of (4. with β =, we have η (n (t = for all n. Hence, the problem pointed out in remark 3. does not exist in this case. In table, we examine three cases: (A low volatility and short maturity, i.e. σ =.5, T =.5, (B low volatility and long maturity, i.e. σ =.5, T =. and (C high volatility and short maturity, i.e. σ =.3, T =.5. In the table, Exact means the option prices calculated by (.5, while Approx indicates the approximated prices calculated by our approximation method. As a comparison purpose, we also list the results obtained by the Monte Carlo simulation with Brownian-bridge technique, the standard practice for the evaluation of barrier-type products. Here, MC(daily, MC(weekly and MC(monthly mean the option prices obtained by using daily, weekly and monthly step Monte Carlo simulation with Brownian-bridge technique, respectively. The number in the parenthesis represents the relative error (% from the benchmark (the exact solution in this case in order to check the accuracy of these approximations. We observe that, in the entire range of strikes, our approximation method outperforms the Brownian-bridge Monte Carlo simulation significantly. The relative error becomes larger for the out-of-the-money (OTM strikes, higher volatility and longer maturity. However, even in these cases, the errors of our approximation method are smaller than the bid-ask spread observed in the actual market. In tables 5, we consider the three cases under the CEV model with β =.75, β =.5, β =.5 and β =, respectively. While MC means the benchmark Monte Carlo simulation results, (3.9 indicates the prices calculated by our approximation method. Also, in order to check the convergence, the columns Approx(/m are appended to show the approximated prices when approximated volatility functions (5. are used with ɛ = /m, m =, 3,...,9. Recall that we fix n = 3, the third-order Taylor s expansion, in order to avoid using too small ɛ (see remark 3.. As a result, because of the discrepancy between η(ɛ, x and its Taylor s expansion (see figure, the sequence of approximated prices, Approx(/m, often overshoots the true price from above. This is the reason why we employ the Aitken convergence accelerator (3.9 to estimate the limit point. As a comparison purpose, we also list the results obtained by the Brownian-bridge Monte Carlo simulation as in table. The number in the parenthesis represents the relative error (% from the benchmark in order to check the accuracy of these approximations. As the base case parameters, For the Brownian-bridge Monte Carlo simulation, we adopt the simple Euler approximation and sample paths are generated for one simulation experiment as for the benchmark Monte Carlo simulation.

11 876 H. Funahashi and M. Kijima σ (x η (x η ~ (,x η ~ (/,x η ~ (/3,x η ~ (/4,x η ~ (/5,x η(x. Downloaded by [Wagner College] at :8 6 September 7 Option Price S MC WIC(ε= WIC(ε=/ WIC(ε=/3 WIC(ε=/4 WIC(ε=/5 WIC(ε=/6 WIC(ε=/7 WIC(ε=/8 WIC(ε=/9 WIC(ε=/ Strike Figure. Convergence of volatility function (upper panel and option price (lower panel for the CEV model with β =.5. Upper Panel: σ(x denotes the original CEV volatility (.7, η(x = σ(bx the symmetrized volatility function (.and η(/m, x the approximated volatility function (4. form =,...,5. Lower Panel: MC denotes the benchmark Monte Carlo simulation results, whereas WIC(ɛ = /m indicates the down-and-in call option prices calculated by (3.8. Parameters are set as S =, B = 95, T =.5andσ =.5.

12 Analytical pricing of single barrier options under local volatility models 877 Downloaded by [Wagner College] at :8 6 September 7 Table. Down-and-in call option prices for the GBM case. Exact means the option prices calculated by the exact solution (.5. While Approx indicates the approximated prices calculated by our approximation method, MC (daily, MC (weekly and MC (monthly represent the option prices calculated by the daily, weekly and monthly time steps Monte Carlo simulation with Brownian-bridge technique, respectively. The number in the parenthesis represents the relative error (% from the exact solution. Other parameters are set as S = and B = 95. K Exact Approx MC (daily MC (weekly MC (monthly (A σ =.5, T = (..96 ( ( ( (..69 ( (..6 ( (.3.44 ( (.5.37 ( (.3. ( (.9.7 ( (..4 ( ( ( (.6.87 ( ( ( (.4.73 ( (.73.7 ( (..6 ( ( ( (.45.5 ( (..49 ( (.8.4 ( (.4.4 ( (.3.35 ( (.5.33 (.7 (B σ =.5, T = ( ( ( ( ( ( ( ( (..86 ( ( ( (.4.58 ( ( ( (.6.3 ( ( (.76.. (.7.8 ( 5.3. ( ( (.7.87 ( ( ( (.6.67 ( ( ( (..49 ( ( ( (.6.33 ( ( ( (.6.8 ( ( (.7 (C σ =.3, T = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (.56.8 ( ( (.83

13 878 H. Funahashi and M. Kijima Downloaded by [Wagner College] at :8 6 September 7 we set S = and B = 95. The volatility parameter σ is set to satisfy σ S β =.5 for the low volatility case and σ S β =.3 for the high volatility case. Also, we consider a longer maturity T = 3. as case (B instead of (B with T =. in tables and 4 with β =.75 and β =.5, respectively. From tables 5, we conclude that, in the entire range of strikes, our approximation method calculated by (3.9 isbetter than, or at least comparable with, the daily step Brownianbridge Monte Carlo simulation, the market standard method to evaluate barrier-type options. Our method provides very good approximation results for in-the- and at-the-money options; however, it deteriorates for the OTM options. Also, we find that our approximation gets worse for high volatility, long maturity and small value of β. Recall that, as is shown in Funahashi and Kijima (5, the chaos expansion approximation works well even for the cases of long maturity and high volatility. Hence, the deterioration of our approximation seems the result from the discrepancy between η(ɛ, x and its Taylor s expansion. In fact, the deterioration is not observed for the GBM case, because the volatility function is constant and so Taylor s expansion is unnecessary in this case. Finally, when K = B, the results in Approx(/m are the same for all m, because the option price does not depend on the symmetrized process S t in this case (see corollary Non-linear volatility model In this subsection, we consider a general volatility function. Namely, we assume that the volatility function is specified as σ(s = (α + β SS e μ S S (4.3 for some positive constants α, β and μ. The symmetrized volatility function defined in (. is then given by ( α + β S B σ(s = η(x; η(x = x e μ S B x, x, (α + β B S x e μ S B x, x <, where x = S/B. Furthermore, form (3.3, the approximate (smooth volatility function is obtained as ( η(ɛ, x = α + β B e ɛ +(log x ɛ +(log x. (4.4 S e μ B S e It follows that its st, nd and 3rd derivatives are given by (t =, η ( η ( Be ɛ μ S (t = Beɛ S ɛ [Be ɛ βμ + S αμ S β], and η (3 (t = 3η( (t, respectively. Also, in this case, the biggest difference between η(x and η(ɛ, x occurs at x = and max{ η(ɛ, x η(x} = η(ɛ, η( x = β B e μb/s (e ɛ = O(ɛ. S In table 6, as for the CEV cases, we compare our approximation method with the benchmark Monte Carlo simulation results. While MC means the benchmark results, (3.9 indicates the prices calculated by our approximation method. Also, in order to check the convergence, the columns Approx(/m are appended to show the approximated prices when approximated volatility functions (5. are used with ɛ = /m, m =, 3,...,9. It is interesting to point out that the sequence of Approx(/m is monotonically increasing in m, in contrast to the CEV cases shown in tables 5; cf. remark 4.. As a comparison purpose, we also list the results obtained by the Brownian-bridge Monte Carlo simulation as in the CEV cases. The number in the parenthesis represents the relative error (% from the benchmark in order to check the accuracy of these approximations. As the base case parameters, we set S =, B = 95, α =.5, β =.75 and μ =.. Also, we consider short maturity case (A T =.5 year and long maturity case (B T =. year. We observe that, in the entire range of strikes, the relative errors of our approximation method calculated by (3.9 are small enough for practical uses. The relative error becomes larger for the OTM strikes and long maturity; however, even in these cases, the errors of our approximation method are smaller than those of daily step Brownian-bridge Monte Carlo simulation. 5. Discussions This section is devoted to discussions about some important topics that relate to the pricing of barrier options. 5.. Akahori Imamura symmetrization Carr and Lee (9 define the arithmetic put-call symmetry (APCS by E τ [ f (S T B] =E τ [ f (B S T ], τ < T, (5. for any bounded, measurable function f (S. Imamura et al. ( have applied the APCS to down-and-out options as follows. First note that E[G(S T χ {τ>t} ]=E[G(S T χ {τ>t, ST >B}] = E[G(S T χ {ST >B}] E[G(S T χ {τ T, ST >B}]. But, because of the APCS (5., one has E[G(S T χ {τ T, ST >B}] =E [ ] E τ [G(S T χ {ST >B}]χ {τ T } = E [ ] E τ [G(B S T χ {ST <B}]χ {τ T } = E [ ] G(B S T χ {ST <B, τ T } = E [ G(B S T χ {ST <B}], where we take f (x = G(x + Bχ {x>} in (5.. It follows that E[G(S T χ {τ>t } ]=E[G(S T χ {ST >B}] E[G(B S T χ {ST <B}]. (5. This idea is totally different from that of the geometric PCA studied in Carr and Lee (9. In fact, the term APCS is misleading. Equation (5. may be better interpreted as the reflection principle. Also, any security price that sounds reasonable economically never satisfies the APCS (5., as we shall see later. The corresponding down-and-in options can be evaluated by (..

14 Analytical pricing of single barrier options under local volatility models 879 Downloaded by [Wagner College] at :8 6 September 7 Table. Down-and-in call option prices for the CEV model with β =.75. MC means the benchmark prices calculated by the Monte Carlo simulation, and (3.9 indicates the prices calculated by our approximation method. While Approx (/m show the approximated prices when (5. is used with ɛ = /m, MC (daily, MC (weekly and MC (monthly represent the option prices calculated by the daily, weekly and monthly time steps Monte Carlo simulation with Brownian-bridge technique, respectively. The number in the parenthesis represents the relative error (% from the exact solution. Other parameters are set as S = and B = 95. K MC (3.9 Approx (/ Approx (/3 Approx (/4 Approx (/5 Approx (/6 Approx (/7 Approx (/8 Approx (/9 MC (daily MC (weekly MC (monthly (A σ =.5.5, T = (.4. (.4. (.4. (.4. (.4. (.4. (.4. (.4. (.4.99 ( ( ( (.6.86 (..85 (..84 ( ( (..83 (.5.8 (.6.8 (.5.7 ( (.7.6 ( (..64 ( (.6.59 ( ( ( (.6.56 (..56 (.3.46 ( ( ( (.9.44 (8..39 ( (.7.35 (.6.34 ( (.7.33 (.8.3 (.55.4 ( (.48.8 ( (.36.7 (.4. (6.8.8 (4..6 (.4.4 (.3.3 (.36.3 (.33. (.9.5 ( (.63. ( (.66. (7.3.4 (9.57. ( ( ( ( (.6.94 (.4.88 ( ( ( (.8.98 (3.6.9 (.8.86 ( (4.7.8 (.98.8 (.6.79 (.9.78 (..74 ( (.98.7 ( ( ( ( ( ( (.5.67 (.3.66 ( ( ( (.9.59 ( ( ( (.9.6 ( ( (.4.55 ( ( ( ( ( ( ( ( ( ( ( (.4.45 ( ( ( ( (.54.4 ( ( ( ( (7.3.4 ( (..37 ( ( ( ( ( (.7 (B σ =.5.5, T = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (. 7.4 ( ( ( ( ( ( ( (. 7.4 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (.5 6. ( ( ( ( ( ( ( ( ( ( (. 5.7 ( ( ( ( ( ( ( ( ( ( ( (. 5.4 ( ( ( ( ( ( ( ( ( ( ( (.4 5. (. 5.8 ( ( (. 5. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (.7 (C σ =.3.5, T = (.3 6. (.3 6. (.3 6. (.3 6. (.3 6. (.3 6. (.3 6. (.3 6. ( ( ( ( ( ( ( ( ( ( ( ( ( ( (.4 5. ( ( ( ( (. 5.7 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (. 4.9 ( ( ( ( ( ( (. 4.6 ( ( ( ( ( ( ( ( ( ( ( ( ( ( (. 4.5 ( ( (. 4.5 ( ( ( ( ( (.4 4. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (.3.94 ( ( ( ( (..96 (.6.93 ( ( ( ( ( (..73 ( 9.7