Analytical Pricing of Asian Options under a Hyper-Exponential Jump - - PowerPoint PPT Presentation

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

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Outline

Introduction of the hyper-exponential jump diffusion model (HEM) Analytical pricing of Asian options Numerical examples Summary

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The HEM

Under the HEM the asset return process {Xt : t ≥ 0} follows Xt = X0 + µt + σWt +

Nt

• i=1

Yi, for any t ≥ 0, where {Yi} assume a hyper-exponential distribution with pdf fY (x) =

m

• i=1

piηie−ηixI{x≥0} +

n

• j=1

qjθjeθjxI{x<0}. 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), · · · · · ·

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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.

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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.

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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.

The continuous Asian option has a payoff S0At

t

− K +, where

At = t

0 eXudu.

{Xt := log(St/S0)} is the asset return process. Cheaper than European options. Reduce the risk of market manipulation. A suitable hedging instrument in reality.

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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); · · ·

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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) ATµ

d

= 2 σ2 Z(1, −γ1) Z(β1) , (1) where At = t

0 eXsds, Tµ ∼ Exp(µ), and γ1 < 0 < β1 are two

roots of the exponent equation G(x) = µ. (G(x) satisfies EX0[exXt ]= exX0+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), · · ·

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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).

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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.

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Pricing Asian options step by step

Step (I): Derive a closed-form expression for E[Aν

Tµ] by

studying the distribution of ATµ.

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.

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Distribution of ATµ under the BSM

Consider a nonhomogeneous ODE Ly(s) = (s + µ)y(s) − µ, for s ≥ 0, (2) where L is the infinitesimal generator of {St = S0eXt} Lf(s) = σ2 2 s2f ′′(s) + rsf ′(s). (2) may have infinitely many solutions. However, the solution of (2) is unique if we impose an additional condition.

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Distribution of ATµ 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 sups∈[0,∞) |a(s)| ≤ C < ∞ for some constant C > 0. Then we must have a(s) = E

• exp
• −sATµ
• for any s ≥ 0.

(3) Thus the bounded solution is unique. A typical argument by constructing a martingale.

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Proof of Theorem 1

Sketch of Proof: Assume S0 = s. Consider Mt :=a(St) exp

• −

t [µ + Su]du

• +

t µ exp

• −

v [µ + Su]du

• dv.

By Itô’s formula, {Mt} is a true martingale. Thus, a(s) = a(S0) = Es[M0] = Es[Mt]. Letting t → +∞, by the dominated convergence theorem, a(s) =Es[ lim

t→∞ Mt] = Es

∞ µ exp

• −

v {µ + Su}du

• dv
• =E[exp
• −sATµ
• ].
• Ning Cai and Steven Kou, HKUST and Columbia University

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Distribution of ATµ 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, β1) h(ν)H(ν) = νH(ν − 1) for any ν ∈ (0, β1) H(0) = 1 (4) where h(ν) ≡ µ − G(ν) = − σ2

2 (ν − β1)(ν − γ1).

(4) has infinitely many solutions, however · · ·

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Distribution of ATµ 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 = ATµ. Question: Does there exist such a nonnegative random variable X with a simple distribution?

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Distribution of ATµ under the BSM

Consider X d = 2 σ2 Z(1, −γ1) Z(β1) . Easily verify that H(ν) := E[X ν] satisfies the recursion (4). Theorem (Cai and Kou (2011), originally proved by Geman and Yor (1993) in another way) Under the BSM, we have ATµ

d

= 2 σ2 Z(1, −γ1) Z(β1) and therefore E[Aν

Tµ] =

2 σ2 ν Γ(ν + 1)Γ(β1 − ν)Γ(1 − γ1) Γ(β1)Γ(−γ1 + ν + 1) , for any ν ∈ (−1, β1).

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Distribution of ATµ under the HEM

Distribution of ATµ under the HEM Theorem (Cai and Kou 2011) Under the HEM, we have ATµ

d

= 2 σ2 Z(1, −γ1) n

j=1 Z(θj + 1, −γj+1 − θj)

Z(βm+1) m

i=1 Z(βi, ηi − βi)

and therefore for any ν ∈ (−1, β1), E[Aν

Tµ]

= 2 σ2 ν Γ(1 + ν)Γ(1 − γ1) Γ(1 − γ1 + ν) ·

n

• j=1

Γ(θj + 1 + ν)Γ(1 − γj+1) Γ(1 − γj+1 + ν)Γ(θj + 1)

• ·

m

• i=1

Γ(βi − ν)Γ(ηi) Γ(ηi − ν)Γ(βi)

• · Γ(βm+1 − ν)

Γ(βm+1) .

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Pricing Asian options via double Laplace transforms

A double Laplace transform of the Asian option price Theorem (Cai and Kou 2011) Define L(µ, ν) = ∞ ∞

−∞

e−µte−νkE(S0At − e−k)+dkdt, where k := − ln(Kt). Then we have that L(µ, ν) = Sν+1 µν(ν + 1)E[Aν+1

Tµ ],

µ > 0, ν > 0. Two-sided, two dimensional Euler inversion algorithms apply; see Petrella (2004).

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Numerical results under the BSM

Comparison of accuracy with other existing methods.

Case Cai-Kou Linetsky GY-Shaw Vecer 1 0.0559860415 0.0559860415 0.0559860415 0.055986 2 0.2183875466 0.2183875466 0.2183875466 0.218388 3 0.1722687410 0.1722687410 0.1722687410 0.172269 4 0.1931737903 0.1931737903 0.1931737903 0.193174 5 0.2464156905 0.2464156905 0.2464156905 0.246416 6 0.3062203648 0.3062203648 0.3062203648 0.306220 7 0.3500952190 0.3500952190 0.3500952190 0.350095 Table: These seven cases are commonly used in the literature for

testing the pricing algorithms of Asian options under the BSM; e.g., Fu et al. (1999), Craddock et al. (2000), Vecer (2001), Linetsky (2004), and Deywnne and Shaw (2008).

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Numerical results under Kou’s model

Numerical prices of Asian options under Kou’s model: λ = 3, S0 = 100, r = 0.09, t = 1.0, p1 = 0.6, q1 = 0.4, and η1 = θ1 = 25.

σ K DL Prices MC Prices Std Err Abs Err Rel Err 0.05 95 8.98812 8.98730 0.00095 0.00082 0.0091% 0.05 105 2.13611 2.13453 0.00208 0.00158 0.0740% 0.1 95 9.20478 9.20559 0.00135

• 0.00081

0.0088% 0.1 105 2.88896 2.88890 0.00249 0.00006 0.0021% 0.2 95 10.32293 10.32461 0.00276

• 0.00168

0.0163% 0.2 105 4.78516 4.78822 0.00380

• 0.00306

0.0638% 0.3 95 11.92926 11.93168 0.00431

• 0.00242

0.0203% 0.3 105 6.86049 6.86412 0.00533

• 0.00363

0.0529% 0.4 95 13.73384 13.73747 0.00586

• 0.00363

0.0264% 0.4 105 8.99114 8.99645 0.00692

• 0.00531

0.0590% 0.5 95 15.62810 15.63389 0.00738

• 0.00579

0.0370% 0.5 105 11.13944 11.14602 0.00853

• 0.00658

0.0590%

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Proof of Theorem 2

Sketch of Proof: Define y(s) = E[e−sX], for s ≥ 0. Note that for any a ∈ (0, min(α1, 1)), we have +∞ s−ae−sXds = Γ(1 − a)X a−1 +∞ s−a−1 e−sX − 1

• ds = −Γ(1 − a)

a X a. Taking expectations on both sides and applying Fubini’s theorem yields E[X a−1] = 1 Γ(1 − a) ∞ s−ay(s)ds E[X a] = − a Γ(1 − a) ∞ s−a−1 (y(s) − 1) ds.

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Proof of Theorem 2

Thus, by the recursion (4), we have − ah(a) Γ(1 − a) ∞ s−a−1 (y(s) − 1) ds = a Γ(1 − a) ∞ s−ay(s)ds, i.e. 0 = ∞ s−a−1 [sy(s) + h(a)(y(s) − 1)] ds. Setting s = e−x and z(x) = y(s) − 1, we have that for any a ∈ (0, min(α1, 1)) 0 = ∞

−∞

eax e−x(z(x) + 1) + h(a)z(x)

• dx.

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Proof of Theorem 2

Rewrite h(a) as h(a) = h0a2 + h1a + h2, with h0 = − σ2

2 , h1 = −r + σ2 2 , and h2 = µ.

Note that integration by parts yields ∞

−∞

eaxaz(x)dx = − ∞

−∞

eaxz′(x)dx ∞

−∞

eaxa2z(x)dx = ∞

−∞

eaxz′′(x)dx.

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Proof of Theorem 2

Then for any a ∈ (0, min(α1, 1)), = ∞

−∞

eax e−x(z(x) + 1) +

• h0a2 + h1a + h2
• z(x)
• dx

= ∞

−∞

eax e−x(z(x) + 1) + h0z′′(x) − h1z′(x) + h2z(x)

• dx.

By the uniqueness of the moment generating function, we have an ODE h0z′′(x) − h1z′(x) + h2z(x) + e−x(z(x) + 1) = 0.

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Proof of Theorem 2

Now transferring the ODE for z(x) back to one for y(s), with s = e−x we have z(x) = y(s) − 1, z′(x) = −sy′(s), and z′′(x) = sy′(s) + s2y′′(s). Then the ODE becomes h0s2y′′(s) + (h1 + h0)sy′(s) + (h2 + s)y(s) = h2. Substituting h0, h1, and h2 into above, we have a nonhomogeneous ODE as follows σ2 2 s2y′′(s) + rsy′(s) − (s + µ)y(s) = −µ.

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Summary

Analytical pricing of Asian options under the HEM

Step (I): Derive a closed-form expression for E[Aν

Tµ] by

studying the distribution of ATµ.

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 28/ 29

Thank you!

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