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An analytic pricing formula for lookback options under stochastic volatility
Applied Mathematics Letters 26 (2013) 145–149
Contents lists available at SciVerse ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
An analytic pricing formula for lookback options under stochastic volatility Kwai Sun Leung ∗ Division of Science and Technology, Beijing Normal University–Hong Kong Baptist University, United International College, Zhuhai, 519085, China
article
info
Article history: Received 31 May 2012 Received in revised form 2 July 2012 Accepted 2 July 2012 Keywords: Lookback options Stochastic volatility Pricing options Homotopy analysis method
abstract In this work, an analytic pricing formula for floating strike lookback options under Heston’s stochastic volatility model is derived by means of the homotopy analysis method. The fixed strike lookback options can then be priced on the basis of the results of floating strike and the put–call parity relation for lookback options. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The payoffs of path dependent options that depend on the extreme (maximum or minimum) value of the underlying asset prices over a certain period of time (called the lookback period) are known as lookback options. Standard lookback options can be classified into two types: fixed strike strike.Fixed strike and floating lookback call and put options with fixed strike price X have the terminal payoffs of max M0T − X , 0 and max X − mT0 , 0 , respectively, where M0T and mT0 denote the realized maximum and minimum asset prices over the lookback period [0, T ], respectively. In general, M0T and mT0 can be monitored discretely or continuously over the lookback period. In the case of a floating strike lookback option, its terminal payoff is given by ST − mT0 and M0T − ST for the case of call and put options, respectively, where ST is the terminal asset price. Goldman et al. [1] introduce several desirable features of lookback options. First, the floating strike lookback option could allow the option trader to buy or sell the underlying asset at the lowest or highest price achieved during the life of the option. Also, lookback options offer opportunities for investors who only have a view on the underlying asset’s price fluctuations during the life of the option instead of its price at the option’s maturity. Under the Black and Scholes [2] model, closed-form pricing formulas for continuously monitored lookback options were derived by Goldman et al. [1] and Conze and Viswanathan [3]. Heynen and Kat [4] derived the analytical formulas for discretely monitored lookback options under the Black–Scholes setting and the corresponding formulas were further generalized to the general Lévy setting by Agliardi [5]. In the Black–Scholes model, the underlying asset price process is assumed to follow the geometric Brownian process, in which the volatility of the asset price is a constant. This assumption is inconsistent with the phenomena of implied volatility smiles observed in the market data. Thus, stochastic volatility models are proposed (see [6–10] for instance) to fix this problem.1 The pricing of lookback options under the stochastic volatility model poses interesting mathematical challenges. Wong and Chan [12] derive semi-analytical pricing formulas for lookback
∗
Fax: +86 756 3620888. E-mail address: [email protected].
1 The general Lévy setting in Agliardi [5] can also capture the implied volatility smiles in the market data. In this work, we just focus on the stochastic volatility models since the stochastic volatility models are found empirically to have first-order importance in option pricing by Bakshi et al. [11]. 0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.07.008
146
K.S. Leung / Applied Mathematics Letters 26 (2013) 145–149
options under the two-factor stochastic volatility model and both stochastic volatility factors are driven by mean-reverting processes. Their results provide a good approximation of the price for both fixed strike and floating strike lookback options when the mean-reverting rate of one stochastic volatility factor is large and the mean-reverting rate of the second factor is small. In this work, we use the homotopy analysis method to derive the analytic pricing formulas for lookback options under Heston’s stochastic volatility model. In contrast to those of Wong and Chan [12], our formulas are free from the assumption of the relative magnitudes of all the model parameters. The homotopy analysis method, as initially suggested by Ortega and Rheinboldt [13], has been used by Liao [14] to solve many nonlinear problems in heat transfer and fluid mechanics. Recently, Zhu [15] was the first to apply this method to derive the closed-form solution for the valuation of American options and his work has been further extended to different asset price processes and different types of options (see [16–19]). The work is organized as follows. In Section 2, we introduce Heston’s stochastic volatility model and derive the governing partial differential equation (PDE) for the lookback options with the payoffs that satisfy the linear homogeneous property. In Section 3, the homotopy analysis method is used to derive the analytic pricing formula for the floating strike lookback option. Using the put–call parity relation of lookback options of Wong and Kwok [20], the pricing formula for the fixed strike lookback option is derived in Section 4. The conclusions are presented in Section 5. 2. Problem formulation Let (Ω , F , {Ft }t ≥0 , Q) be a filtered probability space where Q is the risk-neutral measure. Under the risk-neutral measure Q, the dynamics of the underlying asset price process St and its instantaneous variance vt are assumed to be governed by dSt St
= (r − q)dt +
√ vt dZtS
dvt = κ(θ − vt )dt + σ
√ vt dZtv ,
(1)
where the constants κ , θ and σ are respectively the rate of mean reversion, the long-term mean and the volatility of the instantaneous variance, the constants r and q are respectively the risk-free interest rate and dividend yield of the asset price and {ZtS }t ≥0 and {Ztv }t ≥0 are two correlated Wiener processes with the correlation ρ and are also assumed to be adapted to the filtration {Ft }t ≥0 . Under continuous monitoring, the minimum and maximum asset prices during the given time interval [t1 , t2 ] are denoted by t
mt21 = min Sξ , t1 ≤ξ ≤t2
t Mt12
= max Sξ .
(2)
t1 ≤ξ ≤t2
Under the risk-neutral valuation approach, the fair price at time t of a lookback option with the payoff function f (S , S ∗ ) can be represented as
V (t , S , S ∗ , v) = E Q e−r (T −t ) f (ST , ST∗ )|St = S , St∗ = S ∗ , vt = v ,
(3)
where E Q [·] denotes the expectation taken under the risk-neutral measure Q and St∗ represents either mt0 or M0t . The Feynman–Kac formula in [21] shows that the governing equation of V (t , S , S ∗ , v) is given by
∂V ∂V 1 ∂ 2V ∂ 2V 1 ∂ 2V ∂V + (r − q)S + κ(θ − v) + v S 2 2 + ρσ v S + σ 2 v 2 − rV = 0, ∂t ∂S ∂v 2 ∂S ∂ S ∂v 2 ∂v V (T , S , S ∗ , v) = f (S , S ∗ ), ∂ V (t , S , S ∗ , v) ∗ = 0, ∂ S∗
(4)
S =S
where 0 ≤ t ≤ T , S > 0, β S ≤ S ∗ and β takes the value of 1 or −1 if St∗ = M0t or mt0 respectively. 3. Floating strike lookback options In this section, we first consider a lookback option with the payoff function f (S , S ∗ ) that satisfies the linear homogeneous property, which is represented as f (S , S ∗ ) = Sg
ln
S∗ S
for some function g (·).
(5)
K.S. Leung / Applied Mathematics Letters 26 (2013) 145–149
The linear homogeneous property together with the transformations x = ln
∗ S S
and U =
147 V S
transforms the governing
equation (4) into
(L + M)U (t , x, v) = 0, U (T , x, v) = g (x), ∂U (t , x, v) = 0, ∂x x =0
0 ≤ t < T , β x > 0,
(6)
where
1 ∂2 v ∂ ∂ + v 2 − r −q+ − q, ∂t 2 ∂x 2 ∂x ∂2 ∂ 1 ∂2 ∂ − ρσ v + κ(θ − v) + σ 2v 2 . M = ρσ v ∂v ∂ x∂v ∂v 2 ∂v L=
As compared with (4), (6) is much easier to solve because its dimension is reduced by 1. The payoffs of floating strike lookback options satisfy the above mentioned linear homogeneous property. Without loss of generality, a floating strike lookback put option is used to illustrate the details of our approach. The payoff of the floating strike lookback put option is given by f (ST , M0T ) = M0T − ST
= ST (eXT − 1) , ST g (XT ),
(7)
M0t
where Xt = ln S . t Following the same vein as Park and Kim [18], the homotopy analysis method is applied to solve U (t , x, v) from (6). Let p ∈ [0, 1] be an embedding parameter. The zeroth-order deformation of (6) is given by
(1 − p)(LU¯ (t , x, v, p) − LU0 (t , x, v)) = −p(L + M)U¯ (t , x, v, p), U¯ (t , x, v, p) = g (x), ∂ U0 (t , 0, v) ∂ U¯ (t , 0, v, p) = (1 − p) . ∂x ∂x With p = 1, we have
(8)
(L + M)U¯ (t , x, v, 1) = 0, U¯ (t , x, v, p) = g (x), ∂ U¯ (t , x, v, 1) = 0. ∂x x =0
(9)
Comparing with (6), it is obvious that U¯ (t , x, v, 1) is equal to our searched solution U (t , x, v). If p = 0, (8) becomes
LU¯ (t , x, v, p) = LU0 (t , x, v), U¯ (t , x, v, 0) = g (x),
∂ U¯ (t , 0, v, 0) ∂ U0 (t , 0, v) = . ∂x ∂x
(10)
U¯ (t , x, v, 0) will be equal to U0 (t , x, v) when U0 (T , x, v) = g (x). U0 (t , x, v) is known as the initial guess of U (t , x, v). Following an idea similar to that in [18], U0 (t , x, v) is chosen as the solution of the following PDE:
LU0 (t , x, v) = 0, U0 (0, x, v) = g (x),
∂ U0 (t , x, v) = 0. ∂x x =0
(11)
SU0 (t , x, v) is the price of the floating strike lookback put option under the Black–Scholes model with the constant volatility being equal to v . Its analytic formula is given in Goldman et al. [1]: −q(T −t ) U0 (t , x, v) = ex e−r (T −t ) N (−d− N (−d+ M) − e M) +
v −r (T −t ) 2x(rv−q) e−q(T −t ) N (d+ e N (drM ) , M) − e 2(r − q)
(12)
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K.S. Leung / Applied Mathematics Letters 26 (2013) 145–149
where
−x + r − q ± v2 (T − t ) 2(r − q) √ dm = , drm = d+ T − t. √ √ m − v v(T − t ) Consider the Taylor expansion of U¯ (t , x, v, p) with respect to p, ±
U¯ (t , x, v, p) =
∞
U¯ n (t , x, v)pn ,
(13)
n =0
where U¯ n (t , x, v) =
¯ (t , x, v, p)
1 ∂n U n! ∂ pn
. p=0
To find U¯ n (t , x, v) in (13), we substitute (13) into (8) and obtain the following recursive relation:
LU¯ n + MU¯ n−1 = 0,
n = 1, 2, . . . ,
U¯ n (T , x, v) = 0,
∂ U¯ n (t , x, v) = 0. ∂x x =0
(14)
We define the following transformations:
τ = T − t, U¯ n (τ , x, v) = η(τ , x, v)Uˆ n (τ , x, v), −q− 18 v(α(v)+1)2 τ + 21 (α(v)+1)x
(15) 2(r −q)
where η(τ , x, v) = e and α(v) = v in the form of a standard nonhomogeneous diffusion equation:
. With the transformations in (15), (14) can be written
1 ∂ 2 Uˆ n ∂ Uˆ n − v 2 = η(τ , x, v)MU¯ n−1 (T − τ , x, v), ∂τ 2 ∂x ˆ Un (0, x, v) = 0,
∂ Uˆ n 1 (τ , 0, v) + (α(v) + 1)Uˆ n (τ , 0, v) = 0. ∂x 2
(16)
The PDE (16) has a well-known closed-form solution (see [22] for details): Uˆ n (τ , x, v) =
0
τ
∞
η(s, ξ , v)MU¯ n−1 (T − s, ξ , v)G(τ − s, x, ξ , v)dξ ds,
(17)
0
where 1 G(t , x, ξ , v) = √ 2π v t 1 k = (α(v) + 1). 2
exp −
∞ (x − ξ )2 ( x + ξ )2 (x + ξ + η)2 + exp − + 2k exp − + kη dη 2v t 2v t 2v t 0
and
4. Fixed strike lookback options In the case of fixed strike lookback options, their payoffs do not satisfy the linear homogeneous property (5). Hence, we do not have the benefit of the nice feature of the PDE dimension reduction as in (6). Although we could apply the homotopy analysis method directly to (4), it would be more difficult to solve. To circumvent this difficulty, we use the model independent put–call parity relation for lookback options, developed by Wong and Kwok [20], to derive the price of fixed strike lookback options. Eberlein and Papapantoleon [23] also derived a symmetry relationship between floating strike and fixed strike lookback options for Lévy-driven asset prices. However, their symmetry relationship is model dependent since it is derived on the basis of the property that the Lévy process remains a Lévy process, but with different parameters, after changing the numéraire to the underlying asset price. Hence, we find that the put–call parity relation by Wong and Kwok is more useful to our stochastic volatility model in (1). Consider a floating strike lookback put option and a fixed strike lookback call option; their prices are connected by the put–call parity relation cfix (t , S , M0t , K ) = pfl (t , S , max(M0t , K )) + Se−q(T −t ) − Ke−r (T −t ) ,
(18)
where pfl (t , S , max( , K )) denotes the price of the floating strike lookback put option at time t with the realized maximum equal to max(M0t , K ). Now, the price of the fixed strike lookback call option can be obtained by substituting the pricing formula for pfl from Section 3 into (18). M0t
K.S. Leung / Applied Mathematics Letters 26 (2013) 145–149
149
5. Conclusion By means of the homotopy analysis method, we derive the closed-form analytical pricing formula for the floating strike lookback option under Heston’s stochastic model. To avoid complications in solving the corresponding governing equation for the fixed strike lookback option, the put–call parity for lookback options is then used to derive the pricing formula of the fixed strike lookback. Besides appearing in financial derivatives, the features of lookback options have also commonly appeared in many insurance products such as dynamic fund protection. Hence, our pricing formulas can offer a useful analytical tool for the study of the effect of stochastic volatility in those insurance products with lookback features. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
M.B. Goldman, H.B. Sosin, M.A. Gatto, Path dependent options: buy at the low, sell at the high, Journal of Finance 34 (1979) 1111–1127. F. Black, M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy 81 (1973) 637–659. A. Conze, R. Viswanathan, Path dependent options: the case of lookback options, Journal of Finance 46 (1991) 1893–1907. R.C. Heynen, H.M. Kat, Lookback options with discrete and partial monitoring of the underlying price, Applied Mathematical Finance 2 (1995) 273–284. R. Agliardi, A comprehensive mathematical approach to exotic option pricing, Mathematical Methods in the Applied Sciences 35 (2012) 1256–1268. C.A. Ball, A. Roma, Stochastic volatility option pricing, Journal of Financial and Quantitative Analysis 29 (1994) 589–607. S.L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies 6 (1993) 327–343. J.C. Hull, A. White, The pricing of options on assets with stochastic volatilities, Journal of Finance 42 (1987) 281–300. L.O. Scott, Option pricing when the variance changes randomly: theory, estimation and an application, Journal of Financial and Quantitative Analysis 22 (1987) 419–438. E.M. Stein, J.C. Stein, Stock price distributions with stochastic volatility: an analytic approach, The Review of Financial Studies 4 (1991) 727–752. G. Bakshi, C. Cao, Z. Chen, Empirical performance of alternative option pricing models, Journal of Finance 52 (1997) 2003–2049. H.Y. Wong, C.M. Chan, Lookback options and dynamic fund protection under multiscale stochastic volatility, Insurance: Mathematics and Economics 40 (2007) 357–385. J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992. S.-P. Zhu, An exact and explicit solution for the valuation of American put option, Quantitative Finance 6 (2006) 229–242. S. Gounden, J.G. O’Hara, An analytic formula for the price of an American-style Asian option of floating strike type, Applied Mathematics and Computation 217 (2010) 2923–2936. S. Gounden, J.G. O’Hara, An analytic formula for the price of American exchange options, working paper. S.-H. Park, J.-H. Kim, Homotopy analysis method for option pricing under stochastic volatility, Applied Mathematics Letters 24 (2011) 1740–1744. J. Zhao, H.Y. Wong, A closed-form solution to American options under general diffusion processes, Quantitative Finance 12 (2012) 725–737. H.Y. Wong, Y.K. Kwok, Sub-replication and replenishing premium: efficient pricing of multi-state lookbacks, Review of Derivatives Research 6 (2003) 83–106. I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, 1991. A.D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, 2002. E. Eberlein, A. Papapantoleon, Equivalence of floating and fixed strike Asian and lookback options, Stochastic Processes and their Applications 115 (2005) 31–40.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
An analytic pricing formula for lookback options under stochastic volatility Kwai Sun Leung ∗ Division of Science and Technology, Beijing Normal University–Hong Kong Baptist University, United International College, Zhuhai, 519085, China
article
info
Article history: Received 31 May 2012 Received in revised form 2 July 2012 Accepted 2 July 2012 Keywords: Lookback options Stochastic volatility Pricing options Homotopy analysis method
abstract In this work, an analytic pricing formula for floating strike lookback options under Heston’s stochastic volatility model is derived by means of the homotopy analysis method. The fixed strike lookback options can then be priced on the basis of the results of floating strike and the put–call parity relation for lookback options. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The payoffs of path dependent options that depend on the extreme (maximum or minimum) value of the underlying asset prices over a certain period of time (called the lookback period) are known as lookback options. Standard lookback options can be classified into two types: fixed strike strike.Fixed strike and floating lookback call and put options with fixed strike price X have the terminal payoffs of max M0T − X , 0 and max X − mT0 , 0 , respectively, where M0T and mT0 denote the realized maximum and minimum asset prices over the lookback period [0, T ], respectively. In general, M0T and mT0 can be monitored discretely or continuously over the lookback period. In the case of a floating strike lookback option, its terminal payoff is given by ST − mT0 and M0T − ST for the case of call and put options, respectively, where ST is the terminal asset price. Goldman et al. [1] introduce several desirable features of lookback options. First, the floating strike lookback option could allow the option trader to buy or sell the underlying asset at the lowest or highest price achieved during the life of the option. Also, lookback options offer opportunities for investors who only have a view on the underlying asset’s price fluctuations during the life of the option instead of its price at the option’s maturity. Under the Black and Scholes [2] model, closed-form pricing formulas for continuously monitored lookback options were derived by Goldman et al. [1] and Conze and Viswanathan [3]. Heynen and Kat [4] derived the analytical formulas for discretely monitored lookback options under the Black–Scholes setting and the corresponding formulas were further generalized to the general Lévy setting by Agliardi [5]. In the Black–Scholes model, the underlying asset price process is assumed to follow the geometric Brownian process, in which the volatility of the asset price is a constant. This assumption is inconsistent with the phenomena of implied volatility smiles observed in the market data. Thus, stochastic volatility models are proposed (see [6–10] for instance) to fix this problem.1 The pricing of lookback options under the stochastic volatility model poses interesting mathematical challenges. Wong and Chan [12] derive semi-analytical pricing formulas for lookback
∗
Fax: +86 756 3620888. E-mail address: [email protected].
1 The general Lévy setting in Agliardi [5] can also capture the implied volatility smiles in the market data. In this work, we just focus on the stochastic volatility models since the stochastic volatility models are found empirically to have first-order importance in option pricing by Bakshi et al. [11]. 0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.07.008
146
K.S. Leung / Applied Mathematics Letters 26 (2013) 145–149
options under the two-factor stochastic volatility model and both stochastic volatility factors are driven by mean-reverting processes. Their results provide a good approximation of the price for both fixed strike and floating strike lookback options when the mean-reverting rate of one stochastic volatility factor is large and the mean-reverting rate of the second factor is small. In this work, we use the homotopy analysis method to derive the analytic pricing formulas for lookback options under Heston’s stochastic volatility model. In contrast to those of Wong and Chan [12], our formulas are free from the assumption of the relative magnitudes of all the model parameters. The homotopy analysis method, as initially suggested by Ortega and Rheinboldt [13], has been used by Liao [14] to solve many nonlinear problems in heat transfer and fluid mechanics. Recently, Zhu [15] was the first to apply this method to derive the closed-form solution for the valuation of American options and his work has been further extended to different asset price processes and different types of options (see [16–19]). The work is organized as follows. In Section 2, we introduce Heston’s stochastic volatility model and derive the governing partial differential equation (PDE) for the lookback options with the payoffs that satisfy the linear homogeneous property. In Section 3, the homotopy analysis method is used to derive the analytic pricing formula for the floating strike lookback option. Using the put–call parity relation of lookback options of Wong and Kwok [20], the pricing formula for the fixed strike lookback option is derived in Section 4. The conclusions are presented in Section 5. 2. Problem formulation Let (Ω , F , {Ft }t ≥0 , Q) be a filtered probability space where Q is the risk-neutral measure. Under the risk-neutral measure Q, the dynamics of the underlying asset price process St and its instantaneous variance vt are assumed to be governed by dSt St
= (r − q)dt +
√ vt dZtS
dvt = κ(θ − vt )dt + σ
√ vt dZtv ,
(1)
where the constants κ , θ and σ are respectively the rate of mean reversion, the long-term mean and the volatility of the instantaneous variance, the constants r and q are respectively the risk-free interest rate and dividend yield of the asset price and {ZtS }t ≥0 and {Ztv }t ≥0 are two correlated Wiener processes with the correlation ρ and are also assumed to be adapted to the filtration {Ft }t ≥0 . Under continuous monitoring, the minimum and maximum asset prices during the given time interval [t1 , t2 ] are denoted by t
mt21 = min Sξ , t1 ≤ξ ≤t2
t Mt12
= max Sξ .
(2)
t1 ≤ξ ≤t2
Under the risk-neutral valuation approach, the fair price at time t of a lookback option with the payoff function f (S , S ∗ ) can be represented as
V (t , S , S ∗ , v) = E Q e−r (T −t ) f (ST , ST∗ )|St = S , St∗ = S ∗ , vt = v ,
(3)
where E Q [·] denotes the expectation taken under the risk-neutral measure Q and St∗ represents either mt0 or M0t . The Feynman–Kac formula in [21] shows that the governing equation of V (t , S , S ∗ , v) is given by
∂V ∂V 1 ∂ 2V ∂ 2V 1 ∂ 2V ∂V + (r − q)S + κ(θ − v) + v S 2 2 + ρσ v S + σ 2 v 2 − rV = 0, ∂t ∂S ∂v 2 ∂S ∂ S ∂v 2 ∂v V (T , S , S ∗ , v) = f (S , S ∗ ), ∂ V (t , S , S ∗ , v) ∗ = 0, ∂ S∗
(4)
S =S
where 0 ≤ t ≤ T , S > 0, β S ≤ S ∗ and β takes the value of 1 or −1 if St∗ = M0t or mt0 respectively. 3. Floating strike lookback options In this section, we first consider a lookback option with the payoff function f (S , S ∗ ) that satisfies the linear homogeneous property, which is represented as f (S , S ∗ ) = Sg
ln
S∗ S
for some function g (·).
(5)
K.S. Leung / Applied Mathematics Letters 26 (2013) 145–149
The linear homogeneous property together with the transformations x = ln
∗ S S
and U =
147 V S
transforms the governing
equation (4) into
(L + M)U (t , x, v) = 0, U (T , x, v) = g (x), ∂U (t , x, v) = 0, ∂x x =0
0 ≤ t < T , β x > 0,
(6)
where
1 ∂2 v ∂ ∂ + v 2 − r −q+ − q, ∂t 2 ∂x 2 ∂x ∂2 ∂ 1 ∂2 ∂ − ρσ v + κ(θ − v) + σ 2v 2 . M = ρσ v ∂v ∂ x∂v ∂v 2 ∂v L=
As compared with (4), (6) is much easier to solve because its dimension is reduced by 1. The payoffs of floating strike lookback options satisfy the above mentioned linear homogeneous property. Without loss of generality, a floating strike lookback put option is used to illustrate the details of our approach. The payoff of the floating strike lookback put option is given by f (ST , M0T ) = M0T − ST
= ST (eXT − 1) , ST g (XT ),
(7)
M0t
where Xt = ln S . t Following the same vein as Park and Kim [18], the homotopy analysis method is applied to solve U (t , x, v) from (6). Let p ∈ [0, 1] be an embedding parameter. The zeroth-order deformation of (6) is given by
(1 − p)(LU¯ (t , x, v, p) − LU0 (t , x, v)) = −p(L + M)U¯ (t , x, v, p), U¯ (t , x, v, p) = g (x), ∂ U0 (t , 0, v) ∂ U¯ (t , 0, v, p) = (1 − p) . ∂x ∂x With p = 1, we have
(8)
(L + M)U¯ (t , x, v, 1) = 0, U¯ (t , x, v, p) = g (x), ∂ U¯ (t , x, v, 1) = 0. ∂x x =0
(9)
Comparing with (6), it is obvious that U¯ (t , x, v, 1) is equal to our searched solution U (t , x, v). If p = 0, (8) becomes
LU¯ (t , x, v, p) = LU0 (t , x, v), U¯ (t , x, v, 0) = g (x),
∂ U¯ (t , 0, v, 0) ∂ U0 (t , 0, v) = . ∂x ∂x
(10)
U¯ (t , x, v, 0) will be equal to U0 (t , x, v) when U0 (T , x, v) = g (x). U0 (t , x, v) is known as the initial guess of U (t , x, v). Following an idea similar to that in [18], U0 (t , x, v) is chosen as the solution of the following PDE:
LU0 (t , x, v) = 0, U0 (0, x, v) = g (x),
∂ U0 (t , x, v) = 0. ∂x x =0
(11)
SU0 (t , x, v) is the price of the floating strike lookback put option under the Black–Scholes model with the constant volatility being equal to v . Its analytic formula is given in Goldman et al. [1]: −q(T −t ) U0 (t , x, v) = ex e−r (T −t ) N (−d− N (−d+ M) − e M) +
v −r (T −t ) 2x(rv−q) e−q(T −t ) N (d+ e N (drM ) , M) − e 2(r − q)
(12)
148
K.S. Leung / Applied Mathematics Letters 26 (2013) 145–149
where
−x + r − q ± v2 (T − t ) 2(r − q) √ dm = , drm = d+ T − t. √ √ m − v v(T − t ) Consider the Taylor expansion of U¯ (t , x, v, p) with respect to p, ±
U¯ (t , x, v, p) =
∞
U¯ n (t , x, v)pn ,
(13)
n =0
where U¯ n (t , x, v) =
¯ (t , x, v, p)
1 ∂n U n! ∂ pn
. p=0
To find U¯ n (t , x, v) in (13), we substitute (13) into (8) and obtain the following recursive relation:
LU¯ n + MU¯ n−1 = 0,
n = 1, 2, . . . ,
U¯ n (T , x, v) = 0,
∂ U¯ n (t , x, v) = 0. ∂x x =0
(14)
We define the following transformations:
τ = T − t, U¯ n (τ , x, v) = η(τ , x, v)Uˆ n (τ , x, v), −q− 18 v(α(v)+1)2 τ + 21 (α(v)+1)x
(15) 2(r −q)
where η(τ , x, v) = e and α(v) = v in the form of a standard nonhomogeneous diffusion equation:
. With the transformations in (15), (14) can be written
1 ∂ 2 Uˆ n ∂ Uˆ n − v 2 = η(τ , x, v)MU¯ n−1 (T − τ , x, v), ∂τ 2 ∂x ˆ Un (0, x, v) = 0,
∂ Uˆ n 1 (τ , 0, v) + (α(v) + 1)Uˆ n (τ , 0, v) = 0. ∂x 2
(16)
The PDE (16) has a well-known closed-form solution (see [22] for details): Uˆ n (τ , x, v) =
0
τ
∞
η(s, ξ , v)MU¯ n−1 (T − s, ξ , v)G(τ − s, x, ξ , v)dξ ds,
(17)
0
where 1 G(t , x, ξ , v) = √ 2π v t 1 k = (α(v) + 1). 2
exp −
∞ (x − ξ )2 ( x + ξ )2 (x + ξ + η)2 + exp − + 2k exp − + kη dη 2v t 2v t 2v t 0
and
4. Fixed strike lookback options In the case of fixed strike lookback options, their payoffs do not satisfy the linear homogeneous property (5). Hence, we do not have the benefit of the nice feature of the PDE dimension reduction as in (6). Although we could apply the homotopy analysis method directly to (4), it would be more difficult to solve. To circumvent this difficulty, we use the model independent put–call parity relation for lookback options, developed by Wong and Kwok [20], to derive the price of fixed strike lookback options. Eberlein and Papapantoleon [23] also derived a symmetry relationship between floating strike and fixed strike lookback options for Lévy-driven asset prices. However, their symmetry relationship is model dependent since it is derived on the basis of the property that the Lévy process remains a Lévy process, but with different parameters, after changing the numéraire to the underlying asset price. Hence, we find that the put–call parity relation by Wong and Kwok is more useful to our stochastic volatility model in (1). Consider a floating strike lookback put option and a fixed strike lookback call option; their prices are connected by the put–call parity relation cfix (t , S , M0t , K ) = pfl (t , S , max(M0t , K )) + Se−q(T −t ) − Ke−r (T −t ) ,
(18)
where pfl (t , S , max( , K )) denotes the price of the floating strike lookback put option at time t with the realized maximum equal to max(M0t , K ). Now, the price of the fixed strike lookback call option can be obtained by substituting the pricing formula for pfl from Section 3 into (18). M0t
K.S. Leung / Applied Mathematics Letters 26 (2013) 145–149
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5. Conclusion By means of the homotopy analysis method, we derive the closed-form analytical pricing formula for the floating strike lookback option under Heston’s stochastic model. To avoid complications in solving the corresponding governing equation for the fixed strike lookback option, the put–call parity for lookback options is then used to derive the pricing formula of the fixed strike lookback. Besides appearing in financial derivatives, the features of lookback options have also commonly appeared in many insurance products such as dynamic fund protection. Hence, our pricing formulas can offer a useful analytical tool for the study of the effect of stochastic volatility in those insurance products with lookback features. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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