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Explicit Formula for Conditional Expectations of Product of Polynomial and Exponential Function of Affine Transform of Extended Cox-IngersollRoss Process To cite this article: Phiraphat Sutthimat et al 2018 J. Phys.: Conf. Ser. 1132 012083

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3rd International Conference on Mathematical Sciences and Statistics IOP Conf. Series: Journal of Physics: Conf. Series 1132 (2019) 012083

IOP Publishing doi:10.1088/1742-6596/1132/1/012083

Explicit Formula for Conditional Expectations of Product of Polynomial and Exponential Function of Affine Transform of Extended Cox-Ingersoll-Ross Process Phiraphat Sutthimat1,a , Khamron Mekchay1,b and Sanae Rujivan2,c 1

Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand 2 Division of Mathematics, School of Science, Walailak University, Nakhon Si Thammarat, 80161, Thailand E-mail: a [email protected], b [email protected] and [email protected]

c

Abstract. In this study, an explicit formula for conditional expectations of the product of polynomial and exponential function of an affine transform is derived under the extended Cox-Ingersoll-Ross (ECIR) process. Moreover, we simplify the result to derive an explicit formula for the CIR process.

1. Introduction For bond pricing, to improve accuracy in calculation of the present value of all cash flows, equilibrium models are offered to be more suitable than models based on arbitrage arguments. Since equilibrium models come from price behavior, one of the interesting things of construction of the models is to predict a long term equilibrium. Vasicek [4] introduced equilibrium model for interest rate based on the idea of the Black-Scholes model by considering arbitrage argument for option pricing, which is drt = κ(θ − rt )dt + σdWt ,

(1)

where κ is the means reversion rate, θ is the equilibrium interest rate, σ is the volatility and Wt refers to the standard Brownian motion. The Vasicek model is a straightforward model to illustrate the equilibrium as the mean and variance of rt that converge to finite values when t approaches infinity. One of the earliest way to control positivity of rt and to fix undesirable feature of (1) was introduced by Cox, Ingersoll and Ross [2], known as CIR model, which is suitable for describing the evolution and structure of interest rate, namely: √ (2) drt = κ(θ − rt )dt + rt σdWt . To address the inconveniences of the CIR model, the extended Cox-Ingersoll-Ross (ECIR) model [3] is considered for time dependent parameters. In addition, the ECIR model is Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1

3rd International Conference on Mathematical Sciences and Statistics IOP Conf. Series: Journal of Physics: Conf. Series 1132 (2019) 012083

IOP Publishing doi:10.1088/1742-6596/1132/1/012083

frequently used to explain behavior of structure term of interest rates, especially prices of zero-coupon bonds, where the price varies over time. The general form of ECIR is: drt = κ(t)(θ(t) − rt )dt +

√ rt σ(t)dWt ,

(3)

where κ, θ and σ are functions depending on t. The expectation rate at time t for maturity date T ≥ t, E P [rT |Ft ], is useful in areas of finance. In particular, in estimating fair price, fitting interest rates structure, forecasting, as well as pricing coupon bonds. In practice, Monte Carlo simulation is used to obtain the expectation which requires a lot of computation time, see more in [1]. For the ECIR model (3) with positive real-valued, bounded, continuous functions, κ(t), θ(t), σ(t), Maghsoodi [6] derived the closed form formula of the expectation of rt based on the assumption to guarantee positive rt : 2κ(t)θ(t) ≥ σ 2 (t). (4) Furthermore, the derived closed form was applied to price the T -maturity discount bond: P (t, T ) = A(t, T )e−B(t,T )rt .

(5)

Rujivan [5] applied (4) with Feynman-Kac theorem to obtain a closed-form formula for the conditional moments for the ECIR model: E P [rTγ |Ft ] = E P [rTγ |rt = r],

(6)

for 0 ≤ t ≤ T , and any real γ and positive r. In this study, we extend the result of [5] with a suitable construct of transformation to obtain explicit formulas for conditional expectations of the product of polynomial and exponential function of an affine transform, i h E P rTγ eαrT +β | rt = r . (7) The rest of this paper is structured as follows. The explicit formulas for conditional expectations of the product of polynomial and exponential function of an affine transform is introduced in Section 2. Moreover, its consequences are explored, especially in some cases, the explicit formula can be reduced to a closed-form formula. The aim of the paper are recapitulated in Section 3. 2. Main results This section derives explicit formula for conditional expectations of product of polynomial and exponential function of affine transform of ECIR process for parameters γ, α, β ∈ R. Furthermore, the result is simplified to standard CIR model with constants κ, θ and σ. In order to guarantee rt ≥ 0 for all t ∈ [0, ∞], the condition of Maghsoodi [6] is needed. Assumption. The parameter functions θ(t), κ(t) and σ(t) are positive and continuous on [0, T ] such that the dimension parameters δ(t) := 4θ(t)κ(t) of the ECIR process (3) is σ 2 (t) bounded and δ(t) ≥ 2 for all t ∈ [0, T ]. Theorem 1. Suppose that Vt follows the ECIR process (3) and let γ, α, β ∈ R. By assuming the above assumption holds, we obtain i h (γ) (8) UE (v, τ ) := E P VTγ eαVT +β | Vt = v

2

3rd International Conference on Mathematical Sciences and Statistics IOP Conf. Series: Journal of Physics: Conf. Series 1132 (2019) 012083

IOP Publishing doi:10.1088/1742-6596/1132/1/012083

(γ)

for v > 0 and τ = T − t ≥ 0. Then, UE can be written as (γ)

UE (v, τ ) =

∞ X

Aγ−k (τ )v γ−k eB(τ )v+β ,

(9)

k=0

for (v, τ ) ∈ D(y) ⊂ (0, ∞) × [0, ∞), where Z τ  2 Aγ (τ ) = exp γσ (T − u)B(u) + κ(T − u)θ(T − u)B(u) − γκ(T − u)du , 0 Z τ Z τ  Z s  Aγ−k (τ ) = exp Qγ−k (T − u)du exp − Qγ−k (T − u)du 0

0

(10)

0

Pγ−k+1 (T − s)Aγ−k+1 (s)ds,

(11)

for k ∈ N, and  Pγ−k+1 (τ ) = (γ − k + 1)

 1 (γ − k)σ 2 (τ ) + κ(τ )θ(τ ) , 2

(12)

Qγ−k (τ ) = (γ − k)σ 2 (τ )B(T − τ ) + κ(τ )θ(τ )B(T − τ ) − (γ − k)κ(τ ),

(13)

In addition: τ

 Z α exp −

κ(T − u)du  Z s  . 1 2 σ (T − s) exp − κ(T − u)du ds 2 0 0

B(τ ) =

τ

Z 1−α 0

Proof

 (14)

Given the expectation: i h (γ) UE (v, τ ) := E P VTγ eαVT +β | Vt = v ,

(15) (γ)

for v > 0 and τ = T − t ≥ 0. Applying the Feynman-Kac Theorem, U := UE the PDE: 0

= =

satisfies

∂U 1 ∂2U ∂U + σ 2 (t, v) 2 + µ(t, v) ∂t 2 ∂v ∂v  ∞  X d d B(τ )v+β γ−k γ−k+1 Aγ−k (τ )v B(τ )Aγ−k (τ )v + −e dτ dτ k=0 " ∞ X 1 2 + σ (T − τ )veB(τ )v+β Aγ−k (τ )(γ − k)(γ − k − 1)v γ−k−2 2 k=0 # +2B(τ )Aγ−k (τ )(γ − k)v γ−k−1 + B 2 (τ )Aγ−k (τ )v γ−k +κ(T − τ ) [θ(T − τ ) − v] eB(τ )v+β

∞ h X Aγ−k (τ )(γ − k)v γ−k−1 k=0

i +B(τ )Aγ−k (τ )v γ−k .

(16)

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3rd International Conference on Mathematical Sciences and Statistics IOP Conf. Series: Journal of Physics: Conf. Series 1132 (2019) 012083 (γ)

IOP Publishing doi:10.1088/1742-6596/1132/1/012083

(γ)

Since UE (v, 0) = v γ eαv+β , we consider the form of UE (v, τ ) as: (γ)

UE (v, τ ) =

∞ X

Aγ−k (τ )v γ−k eB(τ )v+β .

(17)

k=0

with condition: B(0) = α, Aγ (0) = 1, Aγ−k (0) = 0,

(18)

for all k ∈ N. Then, given eB(τ )v+β > 0 and conditional on Aγ 6= 0 for all γ ∈ R, the coefficient of γ+1 v can be written as a deterministic PDE: d 1 B(τ ) = σ 2 (T − τ )B 2 (τ ) − κ(T − τ )B(τ ), dτ 2

(19)

where the solution for initial condition (18) is:   Z τ κ(T − u)du α exp − 0  Z s  . Z τ B(τ ) = 1 2 σ (T − s) exp − 1−α κ(T − u)du ds 0 2 0

(20)

Given (16), we obtain functional relationship between Aγ (τ ), Aγ−1 (τ ) and B(τ ), which is the term structure coefficient of v γ as follows: d Aγ (τ ) dτ

d B(τ )Aγ−1 (τ ) dτ 1 1 + σ 2 (T − τ )B(τ )Aγ (τ )(γ) + σ 2 (T − τ )B(τ )Aγ (τ )(γ) 2 2 1 2 2 + σ (T − τ )B (τ )Aγ−1 (τ ) + κ(T − τ )θ(T − τ )B(τ )Aγ (τ ) 2 −κ(T − τ )Aγ (τ )(γ) − κ(T − τ )B(τ )Aγ−1 (τ ). −

=

(21)

Using (20) and initial condition (18) yields: Z τ  2 γσ (T − u)B(u) + κ(T − u)θ(T − u)B(u) − γκ(T − u)du . Aγ (τ ) = exp

(22)

0

Moreover by (16) and initial conditional (18), the coefficients of v γ−k+1 for all k = 2, 3, 4, . . . are found as follows: d Aγ−k+1 (τ ) = Qγ−k+1 (T − τ )Aγ−k+1 (τ ) + Pγ−k+2 (T − τ )Aγ−k+2 (τ ), dτ

(23)

where: 

Pγ−k+2 (τ )

=

Qγ−k+1 (τ )

=

 1 2 (γ − k + 2) (γ − k + 1)σ (τ ) + κ(τ )θ(τ ) , 2

(24)

(γ − k + 1)σ 2 (τ )B(T − τ ) + κ(τ )θ(τ )B(T − τ ), −(γ − k + 1)κ(τ ).

(25) (26)

This results to: Z

τ

τ

Z Qγ−k+1 (T − u)du

Aγ−k+1 (τ ) = exp 0

0

Pγ−k+2 (T − s)Aγ−k+2 (s)ds,

4

 Z exp −

s

 Qγ−k+1 (T − u)du

0

(27)

3rd International Conference on Mathematical Sciences and Statistics IOP Conf. Series: Journal of Physics: Conf. Series 1132 (2019) 012083

IOP Publishing doi:10.1088/1742-6596/1132/1/012083

which completes the proof. The result of Theorem 1 can be simplified into a finite sum in the case when γ is a non-negative integer, as stated in the following result. Theorem 2. Suppose that Vt follows the ECIR process (3) with γ, α, β ∈ R. Let n be a non-negative integer. Then: n i h X (n) UE (v, τ ) := E P VTn eαVT +β | Vt = v = eB(τ )v+β Aj (τ )v j ,

(28)

j=0

for (v, τ ) ∈ D(y) ⊂ (0, ∞) × [0, ∞), where Z τ  2 An (τ ) = exp nσ (T − u)B(u) + κ(T − u)θ(T − u)B(u) − nκ(T − u)du , (29) 0 Z τ Z τ  Z s  Aj (τ ) = exp Qj (T − u)du exp − Qj (T − u)du Pj+1 (T − s) 0

0

0

Aj+1 (s)ds,   1 2 Pj+1 (τ ) = (j + 1) jσ (τ ) + κ(τ )θ(τ ) , 2

(30) (31)

Qj (τ ) = jσ 2 (τ )B(T − τ ) + κ(τ )θ(τ )B(T − τ ) − jκ(τ ),

(32) (n)

for j = n − 1, ..., 0, and B(τ ) satisfies equality (14). In addition, UE (v, τ ) is strictly increasing with respect to v for any positive τ and j = n − 1, n − 2, ..., 0.. Proof Let γ = n be a non-negative integer and k = n + 1. Given P0 (τ ) = 0, from (12), we have A−1 (τ ) = 0, and An−k (τ ) = 0 for all k = n + 1, n + 2, .... Then, (9) can be written as: n X (n) B(τ )v+β UE (v, τ ) = e An−k (τ )v n−k . (33) k=0

Setting k = n − j, the exact term can be rewritten as: (n)

UE (v, τ ) = eB(τ )v+β

n X

Aj (τ )v j .

(34)

j=0

Furthermore, since from (31) Pj+1 (τ ) > 0 for all τ > 0, and (29) and (30) guarantee that (n,α,β) Aj (τ ) > 0 for j = 0, 1, . . . , n, therefore, UE (v, τ ) is strictly increasing with respect to v for τ > 0 (for v > 0). Derivation of (7) when κ(t), θ(t) and σ(t) are constant for all 0 ≤ t ≤ T , reduces the ECIR model (3) to the CIR model (2) as stated in Theorems 3 and 4. The proofs of Theorems 3 and 4 are similar to Theorems 1 and 2 thus omitted. Theorem 3. Suppose that Vt follows the CIR process such that κ(t) = κ, θ(t) = θ and σ(t) = σ. Let γ, α, β ∈ R. Then:

5

3rd International Conference on Mathematical Sciences and Statistics IOP Conf. Series: Journal of Physics: Conf. Series 1132 (2019) 012083 (γ)

UC (v, τ )

:= =

IOP Publishing doi:10.1088/1742-6596/1132/1/012083

h i E P VTγ eαVT +β | Vt = v ,   2ακ 2θκ2 τ exp v + β + γκτ + ασ 2 + eκτ (2κ − ασ 2 ) σ2   22 (γσ2 +κθ) σ 2κ vγ 2 κτ 2 ασ + e (2κ − ασ ) (   ∞ X 2ακ 2θκ2 τ + exp v + β + (γ − k)κτ + ασ 2 + eκτ (2κ − ασ 2 ) σ2 k=1 !   22 ((γ−k)σ2 +κθ) k Y σ 2κ 2k ¯ Pγ−m+1 2 κτ 2 k! ασ + e (2κ − ασ ) m=1 )   k eκτ − 1 v γ−k (35) ασ 2 + eκτ (2κ − ασ 2 )

for all (v, τ ) ∈ D(γ) where D(γ) is a subset of (0, ∞) × [0, ∞) and   1 2 ¯ Pγ−m+1 = (γ − m + 1) (γ − m)σ + κθ 2

(36)

for m = 1, 2, ..., k. Theorem 4. Suppose that Vt follows the CIR process (2) such that κ(t) = κ, θ(t) = θ and σ(t) = σ. Let n be a non-negative integer. Then: h i (n) UC (v, τ ) := E P VTn eαVT +β | Vt = v ,   2ακ 2θκ2 τ = exp v + β + nκτ + ασ 2 + eκτ (2κ − ασ 2 ) σ2   22 (nσ2 +κθ) σ 2κ vn 2 κτ 2 ασ + e (2κ − ασ ) (   n−1 X 2ακ 2θκ2 τ + exp v + β + jκτ + ασ 2 + eκτ (2κ − ασ 2 ) σ2 j=0 !   22 (jσ2 +κθ) n−j Y σ 2κ 2n−j ¯ Pn−m+1 2 κτ 2 (n − j)! ασ + e (2κ − ασ ) m=1 )   n−j eκτ − 1 vj (37) ασ 2 + eκτ (2κ − ασ 2 ) for all v > 0 and τ = T − t ≥ 0 where P¯n−m+1 = (n − m + 1){ 12 (n − m)σ 2 + κθ} for m = 1, 2, ..., n − j. Remark. Following Theorem 1, the affine transform eαrT +β is removed by substituting α = β = 0. Theorems (1-4) corresponds to modification of Rujivan [5]. 3. Conclusion For real parameters γ, α and β, this work extends the result of Rujivan [5] to obtain explicit formulas for conditional expectations of the product of polynomial and exponential function of an affine transform (7) for ECIR and CIR processes. The results are simplified into exact formulas for the case that γ is a non-negative integer, and by taking α = β = 0, the results reduced to the results in [5].

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3rd International Conference on Mathematical Sciences and Statistics IOP Conf. Series: Journal of Physics: Conf. Series 1132 (2019) 012083

IOP Publishing doi:10.1088/1742-6596/1132/1/012083

Acknowledgments Financial support provided by the Development and Promotion of Science and Technology Talents Project (DPST), the Institute for the Promotion of Teaching Science and Technology (IPST), Thailand. References [1] E Higham and X Mao 2005 Convergence of Monte Carlo simulations involving the mean reverting square root process Journal of Computational Finance 8 35-62 [2] J Cox E Ingersoll and S Ross 1985 A theory of the term structure of interest rates Econometrica 53 385-487 [3] J Hull and A White 1990 Pricing interest-rate derivative securities The Review of Financial Studies 3 573-392 [4] O A Vasicek 1977 An equilibrium characterization of the term structure Journal of Financial Economics 5 177-188 [5] S Rujivan 2016 A closed-form formula for the conditional moments of the extended CIR process Journal of Computational and Applied Mathematics 297 75-84 [6] Y Maghsoodi 1996 Solution of the extended CIR term structure and bond option valuation Mathematical Finance 6 89-109

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