Analytical Pricing of Asian Options under a Hyper-Exponential Jump Diffusion Model Ning Cai Joint work with Steven Kou, Columbia University Department of Industrial Engineering and Logistics Management The Hong Kong University of Science and Technology March 8, 2012 Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 1 / 29
Outline Introduction of the hyper-exponential jump diffusion model (HEM) Analytical pricing of Asian options Numerical examples Summary Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 2 / 29
The HEM Under the HEM the asset return process { X t : t ≥ 0 } follows N t � for any t ≥ 0 , X t = X 0 + µ t + σ W t + Y i , i = 1 where { Y i } assume a hyper-exponential distribution with pdf m n � � p i η i e − η i x I { x ≥ 0 } + q j θ j e θ j x I { x < 0 } . f Y ( x ) = i = 1 j = 1 An extension of the Black-Scholes model and the double-exponential jump diffusion model. Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 3 / 29
The HEM Literature on the HEM (or various generalizations) and related option pricing (to name just a few): Barrier-type options : Boyarchenko and Levendorski˘ ı (2002), Boyarchenko and Levendorski˘ ı (2009), Jeannin and Pistorius (2010), Crosby, Saux, and Mijatovic (2010), Cai and Kou (2011), · · · · · · American-type options : Boyarchenko and Levendorski˘ ı (2002), Mordecki (2002), Asmussen, Avram, and Pistorius (2004), Avram, Kyprianou, and Pistorius (2004), Alili and Kyprianou (2005), Boyarchenko (2006), Chen, Lee and Sheu (2007), · · · · · · Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 4 / 29
Motivations I: Approximation to other models The hyper-exponential distribution can approximate any completely monotone distributions (Bernstein’s Theorem). The HEM can be used to approximate exponential Lévy models with completely monotone Lévy densities such as CGMY, NIG, and VG model. Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 5 / 29
Motivation II: Analytical Tractability Additionally, HEM can lead to analytical solutions to pricing problems for many path-dependent options. This is primarily because we can obtain distributions of related random variables. This paper is focused on analytical pricing of Asian options. Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 6 / 29
Asian options Asian options are among the most popular options actively traded in financial markets. E.g., foreign exchange markets, equity derivative markets, commodity markets, etc. � S 0 A t � + , The continuous Asian option has a payoff − K t where � t 0 e X u du . A t = { X t := log ( S t / S 0 ) } is the asset return process. Cheaper than European options. Reduce the risk of market manipulation. A suitable hedging instrument in reality. Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 7 / 29
Asian options - literature review Monte Carlo simulation: e.g. Kemna and Vost (1990), Broadie and Glasserman (1996), Glasserman (2000), Lapeyre and Temam (2001), · · · Lower and upper bounds under the BSM: e.g. Rogers and Shi (1995), Thompson (1998), · · · PDE approach: e.g. Ingersoll (1987), Rogers and Shi (1995), Vecer (2001), Zhang (2001, 2003), Dubois and Lelièvre (2004), · · · Distribution approximation: e.g. Turnbull and Wakeman (1991), Curran (1992), Milevsky and Posner (1998), Ju (2002), · · · · · · Spectral expansions: Linetsky (2004); · · · Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 8 / 29
Geman and Yor’s method (1993) A breakthrough for analytical pricing of Asian options under the BSM: A closed-form result for one-dimensional Laplace transform: Geman and Yor (1993); One crucial observation is (see, e.g., Yor 1992, 2001) = 2 Z ( 1 , − γ 1 ) d A T µ (1) , σ 2 Z ( β 1 ) � t 0 e X s ds , T µ ∼ Exp ( µ ) , and γ 1 < 0 < β 1 are two where A t = roots of the exponent equation G ( x ) = µ . ( G ( x ) satisfies E X 0 [ e xX t ]= e xX 0 + G ( x ) t . ) The literature along this line includes Carmona et al. (1994), Geman and Eydeland (1995), Fu et al. (1999), Carr and Schroder (2004), Fusai (2004), Dewynne and Shaw (2008), · · · Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 9 / 29
Geman and Yor’s method (1993) Moreover, various proofs for (1) were proposed; see, e.g., Dufresne (2001), Yor (2001), Matsumoto and Yor (2005), · · · Two analytical approaches for (1) in the literature: Advanced math tools: Lamperti’s representation and Bessel process. See, e.g., Yor (1992, 2001). Complicated computations: solve PDE or ODE using special functions such as Bessel and hypergeometric functions. See, e.g., Dufrense (2001). Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 10/ 29
Our contribution It is very hard, if not impossible, to extend these two approaches to models beyond the Black-Scholes. Counterpart of Bessel process has not been studied extensively. The corresponding OIDE is too complicated to solve. In comparison, our approach for (1) in Cai and Kou (2011) has two advantages: First, our approach is much simpler. Only use Itô’s formula; No need to solve ODE or PDE explicitly. Second, our approach is more robust in that it can be extended to the HEM easily. Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 11/ 29
Pricing Asian options step by step Step (I): Derive a closed-form expression for E [ A ν T µ ] by studying the distribution of A T µ . Our argument is much simpler and more robust. Step (II): Derive a closed-form double Laplace transform for Asian option price. Step (III): Numerically inverting the double Laplace transform. Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 12/ 29
Distribution of A T µ under the BSM Consider a nonhomogeneous ODE for s ≥ 0 , L y ( s ) = ( s + µ ) y ( s ) − µ, (2) where L is the infinitesimal generator of { S t = S 0 e X t } L f ( s ) = σ 2 2 s 2 f ′′ ( s ) + rsf ′ ( s ) . (2) may have infinitely many solutions. However, the solution of (2) is unique if we impose an additional condition. Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 13/ 29
Distribution of A T µ under the BSM Uniqueness of the ODE (2) via a stochastic representation. Theorem 1 (Cai and Kou 2011) There is at most one bounded solution to the ODE (2). More precisely, suppose a ( s ) solves the ODE (2) and sup s ∈ [ 0 , ∞ ) | a ( s ) | ≤ C < ∞ for some constant C > 0 . Then we must have � � �� for any s ≥ 0 . a ( s ) = E exp − sA T µ (3) Thus the bounded solution is unique. A typical argument by constructing a martingale. Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 14/ 29
Proof of Theorem 1 Sketch of Proof: Assume S 0 = s . Consider � t � � M t := a ( S t ) exp − [ µ + S u ] du 0 � t � v � � + µ exp [ µ + S u ] du dv . − 0 0 By Itô’s formula, { M t } is a true martingale. Thus, a ( s ) = a ( S 0 ) = E s [ M 0 ] = E s [ M t ] . Letting t → + ∞ , by the dominated convergence theorem, � v �� ∞ � � � a ( s ) = E s [ lim t →∞ M t ] = E s µ exp { µ + S u } du dv − 0 0 � � = E [ exp − sA T µ ] . � Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 15/ 29
Distribution of A T µ under the BSM We start to look for a bounded solution. Consider a difference equation (or a recursion) for a function H ( ν ) defined on ( − 1 , β 1 ) � h ( ν ) H ( ν ) = ν H ( ν − 1 ) for any ν ∈ ( 0 , β 1 ) (4) H ( 0 ) = 1 where h ( ν ) ≡ µ − G ( ν ) = − σ 2 2 ( ν − β 1 )( ν − γ 1 ) . (4) has infinitely many solutions, however · · · Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 16/ 29
Distribution of A T µ under the BSM A particular bounded solution to the ODE (2) Theorem 2 (Cai and Kou 2011) If there exists a nonnegative random variable X such that H ( ν ) = E [ X ν ] satisfies the difference equation (4), then the Laplace transform of X, i.e. E [ e − sX ] , solves the nonhomogeneous ODE (2). Theorem 1 implies X d = A T µ . Question: Does there exist such a nonnegative random variable X with a simple distribution? Ning Cai and Steven Kou, HKUST and Columbia University Workshop on Stochastic Processes & Applications 17/ 29
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