Pricing perpetual American options under multiscale stochastic elasticity of variance

Chaos, Solitons & Fractals 70 (2015) 14–26

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Pricing perpetual American options under multiscale stochastic elasticity of variance Ji-Hun Yoon ⇑ Department of Mathematics, Seoul National University, Seoul 120-749, Republic of Korea

a r t i c l e

i n f o

Article history: Received 9 August 2014 Accepted 17 October 2014

a b s t r a c t This paper studies pricing the perpetual American options under a constant elasticity of variance type of underlying asset price model where the constant elasticity is replaced by a fast mean-reverting Ornstein–Ulenbeck process and a slowly varying diffusion process. By using a multiscale asymptotic analysis, we find the impact of the stochastic elasticity of variance on the option prices and the optimal exercise prices with respect to model parameters. Our results enhance the existing option price structures in view of flexibility and applicability through the market prices of elasticity risk. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction The constant elasticity of variance (CEV in brief) model introduced by Cox [3] and Cox and Ross [4] has been one of successful alternative models replacing the classical Black–Scholes model [1] in view of capturing the empirical evidence of the geometric (decreasing and convex) structure of the implied volatility curves. However, the CEV model has revealed its own limitations. One of them is well documented by Hagan et al. [8] showing that the dynamic nature of the implied curves has an opposite direction to market’s behavior causing the instability of delta and vega hedging. The other one lies in a direct observation that the variance of the elasticity does not seem to be a fixed constant and it rather demonstrates fast fluctuation around a mean level with a small amplitude. Based upon this observation, a ‘stochastic elasticity variance’ (SEV) model has been proposed and applied to European vanilla option pricing and portfolio optimization problems in Kim et al. [9] and Yang et al. [14], respectively. In this model the constant elasticity is replaced by a small function of a fast mean-reverting process in a perturbative form to ⇑ Tel.: +82 10 2658 1179. E-mail (J.-H. Yoon).

addresses:

[email protected],

http://dx.doi.org/10.1016/j.chaos.2014.10.012 0960-0779/Ó 2014 Elsevier Ltd. All rights reserved.

[email protected]

the Black–Scholes case which corresponds to the zero elasticity. Although the perpetual American option is not a tradable option in the market, its mathematical properties well deserve to be investigated as it can be applied to several real option problems as well as it may become a building block for pricing American option with finite expiration. This paper’s contribution is to obtain a comprehensive option price formula which can reduce to the renowned option price of the Black–Scholes model or the CEV model and so make it easy to give a direct comparison analysis of the price structures. Particularly, our SEV option price formula demonstrates well its flexibility of the price structure in terms of the market price of elasticity risk and so enhances its applicability to option market. The market price of elasticity risk here is an analogue of the market price of volatility risk in stochastic volatility models. As well-known, a distinctive feature of the stochastic volatility models is that asset prices are driven by volatility risk in addition to market risk. This additional risk factor implies that a market is incomplete. That is, there is no unique equivalent martingale measure that can be used to price derivatives from the principle of no-arbitrage so that the choice of the relevant equivalent martingale measure is fully determined by the market price of volatility risk (the risk preference of agents in the market). So, if

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J.-H. Yoon / Chaos, Solitons & Fractals 70 (2015) 14–26

the market price of volatility risk is simply assumed to be zero, then substantial discrepancy may exhibit between option prices derived from an estimated stochastic volatility model using historical stock return data and observed market prices. Such pricing errors may reflect a violation of the assumption, which is actually taken in the literatures quite often, that the market prices of volatility risk are zero. So, incorporating the relevant market price of risk in pricing instrument is essential when one models quantities such as elasticity that is not directly tradable and further option prices dependence on the non-zero market price of elasticity risk should be well analyzed. This paper addresses an extensional work on the pricing of a perpetual American put option under stochastic elasticity of variance (SEV) model driven by a fast mean-reverting process given by Yoon et al. [15]. By adding a slow time-scale diffusion process to the elasticity part of the established SEV model, we study a multiscale stochastic elasticity of variance model to analyze the change of option price and the optimal exercise boundary against model parameters. In fact, the introduction of a slowly varying diffusion process under multiscale stochastic volatility model has very significant results economically since it gives a much better fit for implied volatility of options with longer maturities as seen in Fouque et al. [6]. Based upon the method of Fouque et al. [7] for pricing European derivative, we obtain a sequence of more complicated option pricing systems with three small parameters for adjusting the scales. By constructing perpetual American option price under a multiscale SEV model, we are trying to observe the pricing impact of the correction term of the slow factor in comparison with that of the fast factor. In fact, Chen and Zhu [2] studied the effect of perpetual American put price under multiscale stochastic volatility. They demonstrated that the influence of the slow volatility factor on the option prices is more substantial than that of the fast volatility factor because of the ‘‘time accumulative effect’’. Also, Kim et al. [10] investigated an investment timing problem under a real option model where the factors of volatility are driven by a fast-mean reverting factor and a slow-mean reverting factor. They found that the effect of the slow mean reverting factor on the option prices is a much larger than the effect of the fast mean reverting factor in terms of model parameters. In other words, it implies that the existence of slow factor Z t in dealing with stochastic volatility model on option price is very significant. In view of the observation above, we observe the importance of slow elasticity factor on option price under multiscale stochastic elasticity of variance model. In fact, this paper also shows that the impact of the slow elasticity factor on option price is superior to that of the fast elasticity factor with respect to model parameters. This means that the slow factor has a greater effect than the fast factor on option price based on stochastic elasticity of variance model, in common with the results of the stochastic volatility model [2] and the hybrid stochastic and local volatility model [10]. Therefore, the option pricing under our extended model is vital.

To analyze the solution of the perpetual American option price and the free boundary value, Chen and Zhu [2] have used an asymptotic expansion in terms of two small parameters corresponding to the fast-mean reversion rate and the slow-mean reversion rate, but our work exploits three small parameters corresponding to a small scale in order to make the Black–Scholes dynamics as a dominant one, the fast-mean reversion rate and the slow-mean reversion rate. The rest of this paper is organized as follows. In Section 2, we formulate an extended SEV model driven by multiscale stochastic elasticity of variance and establish a singularly-and-regularly perturbed partial differential equation for the perpetual American option price based on the formulation. In Section 3, we employ a multiscale asymptotic analysis to obtain an option price formula by solving the free boundary value problems term by term. Section 4 is devoted to finding the implications of the pricing formula by numerical computation. Finally, we give some concluding remarks in Section 5. 2. Model formulation In this section, we construct an extended SEV model under asset price formulation where the elasticity of variance takes a perturbative form in terms of a fast meanreverting OU process Y t and a slowly varying diffusion process Z t . This choice of stochastic process gives us enormous analytic advantage as effectively used by Fouque et al. [7] for pricing derivatives on a stochastic volatility model. By adding the model proposed by Kim et al. [9] to the slow-varying diffusion process, the dynamics of the underlying asset price is given by the following stochastic differential equations x

dX t ¼ lX t dt þ rX t1þnf ðY t ; Zt Þ dBt ; y

dY t ¼ aðm  Y t Þdt þ bdBt ; dZ t ¼ dcðZ t Þdt þ

pffiffiffi z dgðZ t ÞdBt ;

ð1Þ ð2Þ ð3Þ

where f is a smooth positive function that is bounded away from zero, the coefficients c and g are smooth and at most linearly increase as z goes near infinity, l; n; m; a; b and d are some constants, and Bxt ; Byt and Bzt are correlated standard Brownian motions with correlation coefficients qxy 2 ½1; 1; qxz 2 ½1; 1 and qyz 2 ½1; 1, respectively. We note that the OU process Y t in (2) is a Gaussian process which has an invariant distribution given by Nðm; m2 Þ, 2

where m2 :¼ 2ba. Later, we will have notation hi for the expectation with respect to the invariant distribution and ðymÞ2 R  1 ffi 1 it is denoted by hgi :¼ pffiffiffiffiffiffiffi gðyÞe 2m2 dy for arbitrary 2pm2 1 function g. Next, the factor Z t denoted by (3) is a slowly varying diffusion process. The SDE (3) is obtained by using the simple time change t ! dt of a given diffusion process as mentioned in Fouque et al. [6]. Here, d > 0 is a small parameter. Meanwhile, from the above diffusion model (1)–(3), we

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J.-H. Yoon / Chaos, Solitons & Fractals 70 (2015) 14–26

have a general correlation structure between Bxt ; Byt and Bzt as follows

0 1 1 Bxt B B y C B qxy @ Bt A ¼ B @ Bzt q 0

0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1  qxy

0

C 0 C CB ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A t y 2 2 1  qxz  qyz

y yz

q

xz

1

ð4Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

q2xz þ qyyz < 1; qyz ¼ qxy qxz þ qyyz 1  q2xy :

Now, we should build up the model of new dynamics under risk-neutral measure. We exploit arbitrage pricing theory so that option prices are expectations of discount payoffs under an equivalent martingale measure. Using notation f and j (the market prices of elasticity risk), we can write new Brownian motions under a risk-neutral world as follows

Bt

¼ Bt þ

where

0 ðlrÞ

Z

t

0



ðlrÞ

r

r

B @

ðY s ;Z s Þ X nf s

fðY s ; Z s Þ

1

ð5Þ

jðY s ; Z s Þ

 ðY s ;Z s Þ X nf ; fðY t ; Z t Þ; jðY t ; Z t Þ satisfies the Novikov s

j are smooth

bounded functions of y and z only. We denote the combined market prices of elasticity risk K and C

Cðy; zÞ ¼

qxy ðl  rÞ f ðy; zÞ

qxz ðl  rÞ f ðy; zÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ fðy; zÞ 1  q2xy ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ fðy; zÞqyyz þ jðy; zÞ 1  q2xz  q2 yz :

From the process Y t and Z t given by (2,3), if a goes to infinity and if d goes to zero, then the process X t approaches the CEV diffusion and then by taking n as a small parameter, our model becomes the well-known geometric Brownian motion. This means that if fast mean reversion is extremely fast, slow mean-reversion rate is very slow and simultaneously n are very small, then the Black–Scholes model would become a good approximation. Therefore, we introduce three small parameters designating the inverse of the fast mean reversion rate , which is denoted by  ¼ a1, the slow mean reversion rate d and the approximation parameter . given by . ¼ n2 , respectively. Hence, our new model under a risk-neutral probability measure Q is given by the following SDEs 1þ

dX t ¼ rX t dt þ rX t

pffiffi

.f ðY t ;Z t Þ

ðxÞ

dBt ;

pffiffiffi !

  pffiffiffi pffiffiffi ðzÞ dZ t ¼ dcðZ t Þ  dgðZ t ÞC dt þ dgðZ t ÞdBt ; 0 where

BðxÞ t B ðyÞ @ Bt BtðzÞ

1

0

1

0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1  qxy

qxz

qyyz

qxy C B A¼B @

A pricing of perpetual American options is simpler than a pricing of American options since the perpetual American options has a closed solution so that we can analyze the price of option more easily. If X t is an underlying asset price under the classical Black–Scholes model, then the expectation representation of American Perpetual put is defined as follows:

P ðxÞ ¼ sups2! EH ferðstÞ ðK  X s Þþ 1fs<1g jF t g;

ð9Þ

where ! is the set of all stopping times for the filtration F generated by the underlying asset process. Here, we define optimal exercise boundary

where X 1 is the critical asset price that separates the stopping and continuation regions of the perpetual American option. Then, we obtain the price of perpetual American put P  ðxÞ and the free-boundary, X 1 , expressed by

 c X1 P ðxÞ ¼ ðK  X 1 Þ ; Xt

and X 1 :¼

c K; cþ1

ð10Þ

where r; K and r are the interest rate, strike price and volatility of the classical Black–Scholes model, respectively, and c ¼ r2r2 . If r ¼ 0, then the perpetual American option price P  ðxÞ ¼ K (K is a strike price) and the free boundary value X 1 ¼ 0 for all x. However, since this situation is realistically impossible, we assume r > 0 here. 2.2. Problem formulation Now, let Pðx; y; zÞ be the value of a perpetual American put option for the given underlying asset price X t ¼ x, the fast mean-reverting stochastic elasticity correction level Y t ¼ y and the slow mean-reverting stochastic elasticity correction level Z t ¼ z. Then, under the risk-neutral measure Q, the price of perpetual American put option is given by

Pðx; y; zÞ ¼ sups2! EH ferðstÞ ðK  X s Þþ 1fs<1g jX t ¼ x; Y t ¼ y; Z t ¼ zg:

ð6Þ

Then, following the procedure of Tao [13], one can obtain the solution of the free boundary value problem as follows:

ð7Þ

1 2 2ð1þf ðy;zÞpffiffi.Þ @ 2 Pðx; y; zÞ @Pðx; y; zÞ rx þ rx 2 @x2 @x

pffiffiffi

m 2 m 2 ðyÞ dY t ¼ ðm  Y t Þ  pffiffiffi K dt þ pffiffiffi dBt ;    1

prices of volatility risk) are assumed to be independent of x; y and z as seen in Yoon et al. [15].

sH ¼ inf fq > t : X q ¼ X 1 g;

C Ads;

condition. Also, we assume that f and

Kðy; zÞ ¼

qxz dt and dhBðyÞ ; BðzÞ it ¼ qyz dt. Also, K and C (the market

2.1. A review of the perpetual American options

where Bt is a three-dimensional Brownian motion which is mutually independent and the constants qxy ; qxz ; qyz and qyyz satisfy the following relations

jqxy j < 1;

BtðzÞ are given by dhBðxÞ ; BðyÞ it ¼ qxy dt , dhBðxÞ ; BðzÞ it ¼

ð8Þ 1

0 C  0 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ABt 2 y 2 1  qxz  qyz

and the correlation of Brownian motions BðxÞ ; BðyÞ and t t

pffiffi @ 2 Pðx; y; zÞ 1 pffiffiffi @Pðx; y; zÞ þ pffiffiffi m 2 qxy rxð1þf ðy;zÞ .Þ K @x@y @y 

!

! pffiffiffi pffiffi @ 2 Pðx; y; zÞ @Pðx; y; zÞ  gðzÞC d qxz gðzÞrxð1þf ðy;zÞ .Þ @x@z @z rffiffiffi pffiffiffi @ 2 Pðx; y; zÞ d qyz gðzÞm 2 þ @y@z e

þ

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J.-H. Yoon / Chaos, Solitons & Fractals 70 (2015) 14–26

! @ 2 Pðx; y; zÞ @Pðx; y; zÞ þ ðm  yÞ @y2 @y  ! 1 2 @ 2 Pðx; y; zÞ @Pðx; y; zÞ ¼ 0; þd g ðzÞ þ cðzÞ 2 @z2 @z

þ

1

;d;. @P;d;. ;d;. u ðuf ; y; zÞ ¼ e f ; @u

m2

xf ðy; zÞ 6 x < 1;

1 < y < 1;

L;d;. :¼

lim Pðx; y; zÞ ¼ 0;

ð11Þ

From now on, we use the independent variable u transformed by u ¼ ln x instead of x for analytic convenience. We use notation P;d;. for the option price and the corre;d;. sponding free boundary uf to designate the dependence of the small parameters. Also, we use differential operators defined by

L0 :¼ m2

@2 @ þ ðm  yÞ ; @y @y2

L10



3. Option price approximation We cannot solve the singularly-and-regularly perturbed partial differential equation problem (13) directly. So, in order to exploit the small parameters ; d and . in (13) we take the asymptotic expansion

L11

pffiffiffi @ :¼ quy m 2rf ðy; zÞu ; @u@y

L12

pffiffiffi 2 u2 @ 2 :¼ quy m 2rf ðy; zÞ ; 2 @u@y

;d;.

uf

ð12Þ

P;d;. ¼ P0;d þ

0

@ @ :¼ quz gðzÞr  gðzÞC ; @z @u@z

1



pffiffiffi ;d pffiffiffi .P1 þ .P2;d þ . .P3;d þ   

ð16Þ

pffiffiffi 1 L0 þ pffiffiffi L10 þ L20 þ dM10 þ dM2 þ



! rffiffiffi d M3 P0;d



ð17Þ

@2 ; @u@z

u2 @ 2 ; 2 @u@z

M2 :¼

1 2 @2 @ g ðzÞ 2 þ cðzÞ ; 2 @z @z

1 2

. :

1



pffiffiffi 1 L0 þ pffiffiffi L10 þ L20 þ dM10 þ dM2 þ



  pffiffiffi 1 þ pffiffiffi L11 þ L21 þ dM11 P0;d ¼ 0;

! rffiffiffi d M3 P1;d



ð18Þ



Here, the operator 1 L0 is the infinitesimal generator of the OU process Y t and acts only on the y variable. The operator L20 corresponds to the Black–Scholes operator for the perpetual American option so that it is denoted by LpBS . Considering the order structure of the free boundary value problem (11), it can be expressed as

P

pffiffiffi

  .  ;

¼ 0;

pffiffiffi @ 2 M3 :¼ qyz gðzÞm 2 : @y@z

 1 < z < 1;

ð15Þ

Plugging (16) into (13), it leads to the following hierarchy:

. :

;d;.

uf

;d;.

.i=2 dj=2 k=2 ui;j;k ðy; zÞ;

First, we expand P ;d;. with respect to the small parameter . as follows.

2

L;d;. P;d;. ðu; y; zÞ ¼ 0;

1 X

ðy; zÞ ¼

3.1. The leading price

! @2 @  ; @u2 @u ! @2 @ 2 ; :¼ r2 u2 f ðy; zÞ  @u2 @u

M12 :¼ quz gðzÞr

ð14Þ

where we conditions pffiffiffiassume the ordering pffiffiffi d  .  d and   d  .

L21 :¼ r2 uf ðy; zÞ

M11 :¼ quz gðzÞru

.i=2 dj=2 k=2 Pi;j;k ðu; y; zÞ;

i¼0;j¼0;k¼0

!   1 2 @2 @ @ :¼ LpBS ; :¼ r    þ r 2 @u @u2 @u

M10

1 X i¼0;j¼0;k¼0

2

L22



P;d;. ðu; y; zÞ ¼

pffiffiffi @ 2 pffiffiffi @ :¼ quy m 2r  m 2K ; @y @u@y

L20

pffiffiffi pffiffiffi 1 L0 þ pffiffiffi ðL10 þ .L11 þ .L12 þ Oð. .ÞÞ

pffiffiffi pffiffiffi  þ LpBS þ .L21 þ .L22 þ Oð. .Þ pffiffiffi pffiffiffi pffiffiffi þ dðM10 þ .M11 þ .M12 þ Oð. .ÞÞ þ dM2 rffiffiffi d M3 : þ ð13Þ

1 < z < 1;

x!1





@P ðxf ðy; zÞ; y; zÞ ¼ 1; @x

Pðxf ðy; zÞ; y; zÞ ¼ K  xf ;

1

lim P;d;. ðu; y; zÞ ¼ 0;

u!1

6 u < 1;

;d;.

ðuf

1 < y < 1; u

; y; zÞ ¼ K  e

;d;. f

;

.:

1



pffiffiffi 1 L0 þ pffiffiffi L10 þ L20 þ dM10 þ dM2 þ



  pffiffiffi 1 þ pffiffiffi L11 þ L21 þ dM11 P1;d    pffiffiffi 1 þ pffiffiffi L12 þ L22 þ dM12 P0;d ¼ 0;



rffiffiffi d



! M3 P2;d

ð19Þ

where P 0;d ; P1;d and P 2;d are functions of ðu; y; zÞ satisfying the free boundary conditions

18

J.-H. Yoon / Chaos, Solitons & Fractals 70 (2015) 14–26 ;d

P0;d ðu0;d ; y; zÞ ¼ K  eu0 ; ;d

lim P u!1 0

P0;1 ðu0;0 ; y; zÞ ¼ 0;

;d @P0;d ;d ðu0 ; y; zÞ ¼ eu0 ; @u

ðu; y; zÞ ¼ 0;

ð20Þ



P1;d ðu0;d ; y; zÞ ¼ 0;

lim P ðu; y; zÞ u!1 0;1

!

;d @P1;d ;d @ 2 P0;d ;d ðu0 ; y; zÞ ¼ u1;d ðu0 ; y; zÞ þ eu0 ; @u @u2

;d

lim P u!1 1



P0;2

ðu; y; zÞ ¼ 0;

ð21Þ !

;d 1 ;d @ 2 P0;d ;d u ðu0 ; y; zÞ þ eu0 ; 2 1 @u2

P2;d ðu0;d ; y; zÞ ¼ lim P;d ðu; y; zÞ u!1 2

 u1;d

!

@ 2 P1;d ;d ðu0 ; y; zÞ; @u2

ð22Þ

pffiffiffi  pffiffiffi dP0;1 þ dP0;2 þ d dP0;3 þ   

ð23Þ

 1 L0 þ pffiffiffi L10 þ L20 P0;0     pffiffiffi 1 1 þ d L0 þ pffiffiffi L10 þ L20 P 0;1        1 1 1 L0 þ pffiffiffi L10 þ L20 P0;2 þ M10 þ pffiffiffi M3 P0;0 þ d       1   þ M10 þ pffiffiffi M3 P0;1 þ M2 P 0;0 þ    ¼ 0:



Then, it implies

1

d2 :

 1



 1 L0 þ pffiffiffi L10 þ L20 P 0;0 ¼ 0;

ð24Þ



   1 1 L0 þ pffiffiffi L10 þ L20 P0;1 þ M10 þ pffiffiffi M3 P0;0 ¼ 0;





ð25Þ d:

 1



   1 1 L0 þ pffiffiffi L10 þ L20 P0;2 þ M10 þ pffiffiffi M3 P0;1





þ M2 P0;0 ¼ 0;



P0;0 ðu0;0 ; y; zÞ ¼ K  eu0;0 ; lim P ðu; y; zÞ u!1 0;0

¼ 0;

!

@P0;0   ðu0;0 ; y; zÞ ¼ eu0;0 ; @u ð27Þ

@ 2 P0;1  ðu0;0 ; y; zÞ; @u2

!

ð29Þ

respectively. Similarly, we consider the expansion P0;0 with respect to  as follows

P0;0 ¼ P0;0;0 þ

pffiffiffi

pffiffiffi

P0;0;1 þ P0;0;2 þ  P0;0;3 þ   

ð30Þ

By substituting (30) into (24), it leads to a sequence of PDEs in the following form:

L0 P0;0;k þ L10 P0;0;k1 þ L20 P0;0;k2 ¼ 0; P0;0;2 :¼ 0;

ð31Þ

P0;0;1 :¼ 0; where k ¼ 0; 1; 2;   . The free boundary conditions of P0;0;k in Eq. (31) are obtained by substituting the expansion (30) pffiffiffi pffiffiffi and u0;0 ¼ u0;0;0 þ u0;0;1 þ u0;0;2 þ  u0;0;3 þ    into (27). If k ¼ 0, we obtain the ODE L0 P0;0;0 ¼ 0 and impose that the ODE admits only solutions that do not grow so much as

@P0;0;0 @y

y2

 e2 ;

y ! 1. Therefore, P 0;0;0 must be a

function of u and z only so that P 0;0;0 is of the form P0;0;0 ¼ P0;0;0 ðu; zÞ. Also, for k ¼ 1, the ODE L0 P 0;0;1 ¼ 0 is satisfied owing to the fact that L10 contains the y-derivative. Similarly, from the same growth condition, P0;0;1 is independent of y and has the form P0;0;1 ¼ P0;0;1 ðu; zÞ. For k ¼ 2, the above (31) becomes the PDE

L0 P 0;0;2 þ L20 P0;0;0 ¼ 0

ð32Þ

since L10 P0;0;1 ¼ 0 holds. Then, the PDE is a Poisson equation for P0;0;2 with respect to L0 (Fredholm alternative [12]), which is the infinitesimal generator of the OU process Y t . Hence, it must satisfy the following equation

hL20 P0;0;0 i ¼ 0 ð26Þ

with the free boundary conditions



¼ 0;

 u0;1

 1





lim P ðu; y; zÞ u!1 0;2

3  1 2 @ P 0;0 u  u0;1 ðu0;0 ; y; zÞ þ e 0;0 2 @u3

If we substitute (23) into (17), then we obtain the following equation:

1

! 2  1  @ P0;0  u0;0 ; u ðu0;0 ; y; zÞ þ e 0;0 ; y; zÞ ¼ 2 0;1 @u2

@P0;2  @ 2 P0;0   ðu0;0 ; y; zÞ ¼ u0;2 ðu0;0 ; y; zÞ þ eu0;0 @u @u2

respectively. Now, we expand P0;d in (17) with respect to the parameter d as follows:



ð28Þ

ðu

!

3 ;d ;d 1 2 @ P0  u1;d ðu0;d ; y; zÞ þ eu0 2 @u3

d0 :

¼ 0;

¼ 0;

;d @P2;d ;d @ 2 P0;d ;d ðu0 ; y; zÞ ¼ u2;d ðu0 ; y; zÞ þ eu0 @u @u2

P0;d ¼ P0;0 þ

! @P 0;1  @ 2 P0;0  u0;0 ðu0;0 ; y; zÞ ¼ u0;1 ðu ; y; zÞ þ e ; 0;0 @u @u2

ð33Þ

in order for (32) to have a solution. However, since L20 does not contain y or y-derivative terms, it is required to have hL20 P 0;0;0 i ¼ L20 P 0;0;0 ¼ 0. Similarly, by using the expansion of u0;0 ðy; zÞ in (27), u0;0;0 and u0;0;1 are all independent of y and we obtain the free-boundary problem of P0;0;0 and u0;0;0

J.-H. Yoon / Chaos, Solitons & Fractals 70 (2015) 14–26

LpBS P0;0;0 ðu; zÞ ¼ 0;

u0;0;0 6 u < 1;

1 < z < 1

19

u!1

However, since M10 contains the derivative terms in terms of z and P 0;0;0 has the independence of z; M10 P0;0;0 ¼ 0 must be satisfied. Therefore, from the boundary conditions of (39), the solutions of the PDE (39) have P 0;1;0 ðu; zÞ ¼ 0 and pffiffiffi u0;1;0 ðzÞ ¼ 0. Similarly, from Oð Þ term in (25), we obtain the following equation

which has the solutions

L0 P0;1;3 þ L10 P0;1;2 þ L20 P0;1;1 ¼ 0

P0;0;0 ðu0;0;0 ; zÞ ¼ K  eu0;0;0 ; @P0;0;0 ðu0;0;0 ; zÞ ¼ eu0;0;0 ; @u lim P0;0;0 ðu; zÞ ¼ 0;

PpBS ðuÞ ¼

upBS ¼ ln

K

cþ1

ecðuupBS Þ ;

ð34Þ

c :¼

2r

ð35Þ

r2



 Kc ; cþ1

ð36Þ

where LpBS ¼ L20 ; P pBS ¼ P0;0;0 and upBS ¼ u0;0;0 . However, as you can see in (35,36), P pBS and upBS are all independent of z since the operator LpBS is independent of z. Also, in (32), L20 P 0;0;0 ¼ 0 leads to L0 P0;0;2 ¼ 0, and from the argument of the above growth condition, P 0;0;2 is also independent of y.

and by the centering condition mentioned above, it leads to P0;1;1 ðu; zÞ ¼ 0 and u0;1;1 ðzÞ ¼ 0. Also, by plugging the expansion

P0;2 ¼ P0;2;0 þ

LpBS P0;0;1 ðu; zÞ ¼ 0;

u0;0;0 6 u < 1;

1 < z < 1

pffiffiffi pffiffiffi P0;2;1 þ P0;2;2 þ  P0;2;3 þ   

into (26), we obtain the free boundary value problem for P0;2;0 , which is independent of y. We have the solutions P0;2;0 ðu; zÞ ¼ 0 and u0;2;0 ðzÞ ¼ 0. Similarly, we plug the asymptotic expansions

pffiffiffi  pffiffiffi dP1;1 þ dP1;2 þ d dP1;3 þ    ; pffiffiffi pffiffiffi P2;d ¼ P2;0 þ dP2;1 þ dP2;2 þ d dP2;3 þ   

P1;d ¼ P1;0 þ

3.2. The correction terms Next, for k ¼ 3, the Eq. (31) has the PDE L0 P 0;0;3 þ L20 P0;0;1 ¼ 0 since L10 P 0;0;2 ¼ 0 holds by the same reason as above. Then again the centering condition yields the free-boundary value problem

and the expansion (23) into (18) and (19). From the same arguments mentioned above, P 1;0;0 ; P 1;0;1 ; P 1;1;0 and P 2;0;0 are all independent of y, and so are the free boundary values u1;0;0 ; u1;0;1 ; u1;1;0 and u2;0;0 . pffiffiffi Consequently, from Oð .Þ term, we have the free boundary value problem for P1;0;0 and u1;0;0

P0;0;1 ðu0;0;0 ; zÞ ¼ 0;

LpBS P1;0;0 ðu; zÞ ¼ hL21 iP 0;0;0 ðuÞ;

! 2 @P0;0;1 d P0;0;0 u0;0;0 ; ðu0;0;0 ; zÞ ¼ u0;0;1 ðu0;0;0 Þ þ e 2 @u du

u0;0;0 6 u < 1; ð37Þ

u!1

However, the solutions of the PDE (37) are P 0;0;1 ðu; zÞ ¼ 0 and u0;0;1 ðzÞ ¼ 0 owing to the boundary conditions. For k ¼ 4, one can use the same method as above and obtain P 0;0;2 ðu; zÞ ¼ 0 and u0;0;2 ðzÞ ¼ 0. Now, we observe (25) with respect to the parameter . By plugging the expansion

P0;1 ¼ P0;1;0 þ

pffiffiffi pffiffiffi P0;1;1 þ P0;1;2 þ  P0;1;3 þ   

P1;0;0 ðu0;0;0 Þ ¼ 0;

L0 P0;1;2 þ L20 P 0;1;0 þ M10 P0;0;0 ¼ 0

lim P1;0;0 ðu; zÞ ¼ 0;

pffiffiffi pffiffiffi . . and if we set P 1;0;0 ¼ .P1;0;0 and u1;0;0 ¼ .u1;0;0 , the closed-form solution is given by the following equation .

ð38Þ

u0;0;0 6 u < 1;

1
lim P0;1;0 ðu; zÞ ¼ 0:

u!1

( ! . V 1;0;0 ðzÞc u20;0;0 1 þ u0;0;0 2 cþ1 cþ1  2  u 1 þ u Kecðuu0;0;0 Þ ;  2 cþ1

P1;0;0 ðu; zÞ ¼

and from the centering condition, we have the free-boundary value problem for P0;1;0 as follows:

! 2 @P0;1;0 d P0;0;0 u0;0;0 ; ðu0;0;0 ; zÞ ¼ u0;1;0 ðu Þ þ e 0;0;0 2 @u du

ð41Þ

u!1

into the PDE (25), we obtain the y-independence of P 0;1;0 and P 0;1;1 with the same arguments as before. In addition, the PDE (25) yields the following equation

LpBS P0;1;0 ðu; zÞ ¼ M10 P0;0;0 ;

1 < z < 1

! 2 @P1;0;0 d P0;0;0 u0;0;0 ; ðu0;0;0 ; zÞ ¼ u1;0;0 ðu0;0;0 Þ þ e 2 @u du

lim P0;0;1 ðu; zÞ ¼ 0:

ð40Þ

ð39Þ

.

u1;0;0 ðzÞ ¼ .

  . V 1;0;0 ðzÞ 1 þ u0;0;0 ; ðc þ 1Þ c þ 1

V 1;0;0 ðzÞ :¼ 2

ð42Þ

ð43Þ

pffiffiffi

.hf i:

pffiffiffiffiffiffi pffiffiffiffiffiffi Also, from Oð .Þ; Oð .dÞ and Oð.Þ terms, the free boundary value problems for (P 1;0;1 ; u1;0;1 ), (P1;1;0 ; u1;1;0 ) and (P 2;0;0 ; u2;0;0 ) are expressed as follows:

LpBS P1;0;1 ðu; zÞ ¼ hL10 L1 0 ðL21  hL21 iÞiP 0;0;0 ðuÞ; u0;0;0 6 x < 1;

1 < z < 1

20

J.-H. Yoon / Chaos, Solitons & Fractals 70 (2015) 14–26

P1;0;1 ðu0;0;0 ; zÞ ¼ 0;

and /ðy; zÞ is defined by the solution of the Poisson equation L0 / ¼ f  hf i,

!

2

@P1;0;1 d P 0;0;0 ðu0;0;0 ; zÞ ¼ u1;0;1 ðu0;0;0 Þ þ eu0;0;0 ; 2 @u du

ð44Þ

.;d



P1;1;0 ðu; zÞ ¼ ða u3 þ b u2 þ c u þ g  ÞK cecðuu0;0;0 Þ ;    .;d u1;1;0 ðzÞ ¼  3a u20;0;0 þ 2b u0;0;0 þ c ;

ð48Þ

lim P1;0;1 ðu; zÞ ¼ 0;

u!1



where a ; b ; c and g  are given by

LpBS P1;1;0 ðu; zÞ ¼ M10 P1;0;0 ðu; zÞ  hM11 iP0;0;0 ðuÞ; u0;0;0 6 x < 1;

(

P1;1;0 ðu0;0;0 ; zÞ ¼ 0;



b :¼

!

2

@P1;1;0 d P 0;0;0 ðu0;0;0 ; zÞ ¼ u1;1;0 ðu0;0;0 Þ þ eu0;0;0 ; 2 @u du

ð45Þ

" c :¼

limu!1 P1;1;0 ðu; zÞ ¼ 0; and

u0;0;0 6 x < 1;

! 2 1 2 d P0;0;0 u0;0;0 ; P2;0;0 ðu0;0;0 ; zÞ ¼ u1;0;0 ðu0;0;0 Þ þ e 2 2 du 2

3

 u1;0;0

2ðc þ 1Þ

.;d

V 1;1;0 ðzÞ

.;d

.;d

V ðzÞ þ 3 1;1;0

V 1;1;0 ðzÞ

;

ðc þ 1Þ3

(

u2 1 c 0;0;0 þ u 2 c þ 1 0;0;0

.;d V 1;1;0 ðzÞ 2

)

! þ

ðc þ 1Þ2 !#

u20;0;0 1 2 þ u  2 c þ 1 0;0;0 ðc þ 1Þ2

ðc þ 1Þ

)

2

;

pffiffiffiffiffiffi .drquz gðzÞhf ð; zÞi0z ; p ffiffiffiffiffiffi .;d V 1;1;0 ðzÞ :¼ 2 .dgðzÞChf ð; zÞi0z ; .;d

V 1;1;0 ðzÞ :¼ 2

!

1 d P 0;0;0  u21;0;0 ðu0;0;0 Þ þ eu0;0;0 3 2 du

ðc  1Þ

   g  :¼  a u30;0;0 þ b u20;0;0 þ c u0;0;0 ;

1 < z < 1

@P2;0;0 d P 0;0;0 ðu0;0;0 ; zÞ ¼ u2;0;0 ðu0;0;0 Þ þ eu0;0;0 2 @u du

2

ðc þ 1Þ2



LpBS P2;0;0 ðu; zÞ ¼ hL21 iP1;0;0 ðu; zÞ  hL22 iP0;0;0 ðuÞ;

6ðc þ 1Þ





  .;d .;d V 1;1;0 ðzÞc  V 1;1;0 ðzÞ ;

1

a :¼ 

1 < z < 1

  @ where ðHÞ0z ¼ H @z

!

and .

@ 2 P1;0;0 ðu0;0;0 ; zÞ; @u2

ð46Þ



P2;0;0 ðu; zÞ ¼ ðA u4 þ B u3 þ C  u2 þ D u þ d ÞK cecðuu0;0;0 Þ ; 2

lim P2;0;0 ðu; zÞ ¼ 0:

.

u!1

  pffiffiffiffiffiffi pffiffiffiffiffiffi .; .; Then, by setting P 1;0;1 ¼ .P1;0;1 ; u1;0;1 ¼ .u1;0;1 ,    pffiffiffiffiffiffi pffiffiffiffiffiffi .;d .;d . P 1;1;0 ¼ .dP 1;1;0 ; u1;1;0 ¼ .du1;1;0 and P2;0;0 ¼

u2;0;0 ðzÞ ¼

.P2;0;0 ; u.2;0;0 ¼ .u2;0;0 Þ, we can have the exact solutions

ðud1;0;0 ðzÞÞ ð2c  1Þ 2    . V 2;0;0 ðzÞK c 1 c1 2 u þ c u þ  0;0;0 0;0;0 cþ1 cþ1 ðc þ 1Þ2    3   2   4A u0;0;0 þ 3B u0;0;0 þ 2C u0;0;0 þ D ; ð49Þ

.;

P1;0;1 ðu; zÞ ¼ ðau2 þ bu þ dÞK cecðuu0;0;0 Þ ;   .; .; V 1;0;1 ðzÞ  V 1;0;1 ðzÞ 1 .; ; u1;0;1 ðzÞ ¼  u0;0;0 þ cþ1 ðc þ 1Þ



.;

.;

.;

b :¼

.;

V 1;0;1 ðzÞ  V 1;0;1 ðzÞ ðc þ 1Þ

2



ðV 1;0;1 ðzÞc þ

2

 pffiffiffiffiffiffiffiffiffi @/ .; V 1;0;1 ðzÞ :¼ 2 2.mrquy ; @y

 pffiffiffiffiffiffiffiffiffi @/ .; ; V 1;0;1 ðzÞ :¼ 2 2.mK @y

.;

þ

V 1;0;1 ðzÞ 

.; V 1;0;1 ðzÞ 2

ðc þ 1Þ

V 2;0;0 ðzÞc ; 8ðc þ 1Þ



.;

.; V 1;0;1 ðzÞÞu20;0;0



! . V 2;0;0 ðzÞðc  2Þ 1 . B :¼ þ V 2;0;0 ðzÞ ; 3ðc þ 1Þ 2ðc þ 1Þ

;

u1;0;1 ðzÞ 1 þ d :¼ cþ1 cþ1 .;



.

A :¼ V 1;0;1 ðzÞc þ V 1;0;1 ðzÞ ; 2ðc þ 1Þ



where A ; B ; C ; D and d are given by

ð47Þ

where

a :¼ 



! ;

( . . V 2;0;0 ðzÞðc  2Þ V 2;0;0 ðzÞ 1 þ cþ1 cþ1 2ðc þ 1Þ2 ! ) 2 u0;0;0 u0;0;0 1 ecu0;0;0 ; þ . þ 4 2ðc þ 1Þ 2ðc þ 1Þ2 Kc

C  :¼

D :¼

2 C; ðc þ 1Þ

21

J.-H. Yoon / Chaos, Solitons & Fractals 70 (2015) 14–26 2

.



d :¼

ðu1;0;0 ðzÞÞ  ðA u30;0;0 þ B u20;0;0 þ C  u0;0;0 þ D Þu0;0;0 ; 2

.

V 2;0;0 ðzÞ :¼ 4.hf ð; zÞi2 ;

.

2

V 2;0;0 ðzÞ :¼ 2.hf ð; zÞi:

;d;.

R1

.;d;

In this section, we use approximations in terms of P .;d; and uf to analyze the change of the option prices and the free boundaries. Considering (P 0;1;0 ¼ 0; u0;1;0 ¼ 0) and (P 0;0;1 ¼ 0; u0;0;1 ¼ 0) mentioned before, we first define the e ;d;. ) and free first-order approximation for option price ( P ð1Þ  ;d;. ~ ð1Þ ) as follows: boundary (u ;d;.

uf

;d;.

.

~ð1Þ :¼ u0;0;0 þ u1;0;0 : u

¼

pffiffiffi

.ðL22 P1;0;0 þ L21 P2;0;0 Þ þ .ðL22 P2;0;0 þ L21 P3;0;0 Þ;

;d;.

pffiffiffiffiffiffi ¼ M10 P 1;0;0 þ O .d ;

;d;.

pffiffiffiffiffiffi ¼ L10 P1;0;2 þ LpBS P1;0;1 þ O . ;

R2

4. Accuracy and numerical implication

e ;d;. :¼ P0;0;0 þ P. ; P;d;. P 1;0;0 ð1Þ

by the centering condition of the Poisson equation stated before. Here, L;d;. is the operator given by (13) and ;d;. ;d;. ;d;. ;d;. R1 ; R2 ; R3 and R4 are given by

R3

;d;.

R4

pffiffiffiffiffiffiffiffi ¼ M10 P 1;0;1 þ LpBS P1;1;1 þ M3 P1;0;2 þ O .d ;

respectively and they are uniformly bounded in u and z and at most linearly growing in jyj (see Chapter 4 in Fouque et al. [7] for details). Note next that, at the optimal stopping time s, we can write

ð50Þ The proof of this accuracy is similar to the procedure of Fouque et al. [6]. However, compared to the work of Fouque et al. [6] which deals with two small parameters  and d, our work is the approximation of the extended version for three small parameters .;  and d. We establish the following Lemma.

;d;.

R;d;. ðuf

Lemma 1. When the price of perpetual American put option P ;d;. given by (11) has a smooth payoff function, for fixed ðx; y; zÞ and for any . 6 1; d 6 1;  6 1, there exists a constant C > 0 such that ;d;.

e jP;d;.  P ð1Þ j 6 Cð. þ

pffiffiffiffiffiffi

pffiffiffiffiffiffi

pffiffiffiffiffiffiffiffi

. þ .d þ .dÞ:

d ;d;. ;d;. ; y; zÞ ¼ P;d;. ðuf ; y; zÞ  P ;d;. ðuf ; y; zÞ pffiffiffi pffiffiffi ¼ .ðP2;0;0 þ .P3;0;0 þ dP2;1;0 pffiffiffi ;d;. þ P2;0;1 Þðuf ; y; zÞ pffiffiffiffiffiffi ;d;. þ .dP1;1;0 ðuf ; y; zÞ pffiffiffi pffiffiffiffiffiffi  þ . P1;0;1 ðxf ; y; zÞ þ P1;0;2 ðxf ; y; zÞ pffiffiffiffiffiffiffiffi pffiffiffi þ .dðP 1;1;1 þ .P2;1;1 Þðxf ; y; zÞ pffiffiffiffiffiffi ;d;. ;d;. ¼ .G1 ðuf ; y; zÞ þ .dG2 ðuf ; y; zÞ pffiffiffiffiffiffi ;d;. þ .G3 ðuf ; y; zÞ pffiffiffiffiffiffiffiffi ;d;. þ .dG4 ðuf ; y; zÞ: ð54Þ

Proof. The proof of the accuracy of this approximation is similar to that given by Fouque et al. [6]. In order to demonstrate this Lemma we consider the following higher order approximation for P;d;.

pffiffiffi d e ;d;. þ .ðP2;0;0 þ pffiffiffi P ;d;. ¼ P .P3;0;0 þ dP2;1;0 ð1Þ pffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi þ P 2;0;1 Þ þ .dðP1;1;1 þ .P2;1;1 Þ pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi þ .dP1;1;0 þ .P1;0;1 þ  .P 1;0;2 p ffiffiffiffiffi ffi pffiffiffiffiffiffi pffiffiffi ¼ P0;0;0 þ .P1;0;0 þ .dP1;1;0 þ .P 1;0;1 pffiffiffi pffiffiffi 3 þ  .P1;0;2 þ .P2;0;0 þ .2 P3;0;0 þ . dP2;1;0 pffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffi þ . P2;0;1 þ .dP1;1;1 þ . dP2;1;1 ;

R;d;. ¼ .sups2! EH erðstÞ G1 ðuf ;y; zÞ1fs<1g Z s .;d;  erðstÞ R1 ðus ; ys ; zs Þ1fs<1g dsjut ¼ u;Y t ¼ y;Z t ¼ z t

pffiffiffiffiffiffi þ .dsups2! EH erðstÞ G2 ðuf ;y; zÞ1fs<1g Z s .;d;  erðstÞ R2 ðus ; ys ; zs Þ1fs<1g dsjut ¼ u;Y t ¼ y;Z t ¼ z t

ð51Þ

where P 0;0;0 ¼ P pBS is the Black Scholes price of perpetual American put defined by (35) and P1;0;0 ; P1;1;0 ; P1;0;1 and P 2;0;0 are defined by (41), (44)–(46), respectively. Also, P 1;0;2 and P 1;1;1 are the terms obtained by the expansion of P1;d given by (41) and P 2;1;0 and P 2;1;1 are the terms obtained by the expansion of P 2;d given by (41). P3;0;0 is given by the expansion of P3;d . If we introduce the residual

d R;d;. ¼ P;d;.  P ;d;. ;

Similarly, G1 ; G2 ; G3 and G4 are uniformly bounded in u and z and at most linearly growing in jyj. It follows from (53) and (54) that

ð52Þ

pffiffiffiffiffiffi þ .sups2! EH erðstÞ G3 ðuf ;y;zÞ1fs<1g Z s .;d;  erðstÞ R3 ðus ; ys ; zs Þ1fs<1g dsjut ¼ u;Y t ¼ y;Z t ¼ z t

pffiffiffiffiffiffiffiffi þ .dsups2! EH erðstÞ G4 ðuf ; y;zÞ1fs<1g Z s .;d;  erðstÞ R4 ðus ; ys ; zs Þ1fs<1g dsjut ¼ u;Y t ¼ y;Z t ¼ z : t

ð55Þ Consequently, from (51) and (52), we obtain

 pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffiffi R;d;. ¼ O .; .d; .; .d :



ð56Þ

then it satisfies ;d;.

L;d;. R;d;. ¼ .R1

þ

pffiffiffiffiffiffi ;d;. pffiffiffiffiffiffi ;d;. pffiffiffiffiffiffiffiffi ;d;. .dR2 þ .R3 þ .dR4 ð53Þ

However, since the first-order perturbation mentioned in Lemma 1 does not contain the d-dependent term and the -dependent term, we cannot capture the effect of

22

J.-H. Yoon / Chaos, Solitons & Fractals 70 (2015) 14–26

fast-mean reverting process and slowly varying diffusion process against the option price and the free boundary. The d-dependence and the -dependence appear in the second-order approximation. Therefore, by using (P0;1;1 ¼ 0; u0;1;1 ¼ 0), (P 0;2;0 ¼ 0; u0;2;0 ¼ 0) and (P 0;0;2 ¼ 0; u0;0;2 ¼ 0) stated above, we can define the following   e ;d;. and second-order approximation for option price P ð2Þ   ;d;. ~ ð2Þ free boundary u

e ;d;. :¼ P 0;0;0 þ P. þ P .;d þ P.; þ P. ; P ;d;. P 1;0;0 1;0;1 2;0;0 1;1;0 ð2Þ ;d;.

uf

~ ;d;.

.

.;

.;d

ð57Þ

.

uð2Þ :¼ u0;0;0 þ u1;0;0 þ u1;1;0 þ u1;0;1 þ u2;0;0 :

zero. Hence, as stated in [2], we obtain the following equality

1 t!1 t

lim f ðY t ; Z t Þ ¼ lim

!0;d!0

Z

t

f ðY t ; zÞds:

ð59Þ

0

Meanwhile, the volatility hf ð; zÞi is almost surely the same as the long-run time average of the function f by using the ergodic theorem [5]. Then, the equality follows.

Z

1 t!1 t

hf ð; zÞi ¼ lim

t

f ðY t ; zÞds:

ð60Þ

0

Therefore, combining (59) and (60) yields the following equality (61).

lim f ðY t ; Z t Þ ¼ hf ð; zÞi: ðalmost surelyÞ:

ð61Þ

Now, we consider the following notation to analyze the above approximated results and observe the effect of the correction terms

!0;d!0

P pBS ¼ P0;0;0 ;

ple 1. The initial level value of Z t has been chosen within their reasonable ranges satisfying model parameters estimated from Kim et al. [11] by using S&P 500 market data. The initial level value used for the numerical comparison is z ¼ 0:01.

upBS ¼ u0;0;0 ; .

.;d

.;

.

.

.;d

.;

.

P pSEV ¼ P0;0;0 þ P1;0;0 þ P 1;1;0 þ P1;0;1 þ P 2;0;0 ; upSEV ¼ u0;0;0 þ u1;0;0 þ u1;1;0 þ u1;0;1 þ u2;0;0 ; where pBS and pSEV signify Black–Scholes and stochastic elasticity of variance for perpetual American option, respectively. For the numerical comparison analysis of the approximated option prices and free boundaries, we choose the 1 yþz function f ðy; zÞ ¼  10 e (negative elasticity), which has appropriate cutoffs so as not to affect the calculations within the accuracy of our comparisons. This is well laid out in Yoon et al. [15]. The perpetual American option is a nontradable option in the market. Therefore, there is no available perpetual American option data to assess the chosen parameters in the paper. The author decided to pick up model parameters by using parameter values estimated from the December 10, 2010 S&P 500 implied volatility surface in Kim et al. [11] dealing with multiscale stochastic volatility model based on European option price. For the model parameters, r; r and K are taken from Kim et al. [11] directly, and the rest of the value are assumed within their reasonable ranges satisfying the values of group parameters for fast-mean reverting factors and slow-mean reverting factors estimated from Kim et al. [11]. The model parameters that we select are as follows:

. ¼ 0:007; d ¼ 0:001;  ¼ 0:0001; m ¼ 0; r ¼ 0:01;

r ¼ 0:15; K ¼ 100;

1 m ¼ pffiffiffi ; 2

quy ¼ 0:2;

quz ¼ 0:2; gðzÞ ¼ 2:

ð58Þ

Then, we can choose the initial level value z to compute 2

hf ð; zÞi; hf ð; zÞi and h@/ ð; zÞi by using the following Exam@y

Example 1. Let /ðy; zÞ be the solution of the Poisson 1 yþz e , then we first equation L0 / ¼ f  hf i. If f ðy; zÞ ¼  10 R1 z compute hf ð; zÞiou . Since hf ð; zÞiou ¼  peffiffiffiffiffiffiffiffi2 1 ey 10 2pm



e

ðymÞ2 10m2

mþm2 =2þz

e

10

R 1 ðymm pffiffiffiffiffiffiffiffi e 2m2 10 2pm2 1 R 1 y2 using 1 e dy ¼ mþm2 =2þz

dy ¼  e by

2mþ2m2 þ2z

2

hf ð; zÞiou ¼ e h@/ @y ð; zÞiou .

50

2Þ

dy, we obtain hf ð; zÞiou ¼ pffiffiffiffi p. By the similar method,

is derived. Also, we should find

From the definition of L0 , the Poisson equation

is written by

  @/ 2 2 @ hðyÞ @y @ / @/ m L0 / ¼ m2 2 þ ðm  yÞ ¼ ¼ f  hf i; @y @y hðyÞ @y 2

1 ffi where hðyÞ ¼ pffiffiffiffiffiffiffi expð ðymÞ Þ. Therefore, we deduce 2m2 2pm2

@/ 1 ¼ @y m2 hðyÞ

Z

y

ðf ðk; zÞ  hf ð; zÞiÞhðkÞdk

1

and the definition of hð; zÞiou leads to the following relation

 Z 1 Z y @/ 1 ¼ 2 ðf ðk; zÞ  hf ð; zÞiÞhðkÞdkdy: @y m 1 1 Here, by using f ðk; zÞ ¼ 

1

ekþz , the above equation yields

10

 Z 1 Z y  k  @/ ez pffiffiffiffiffiffiffiffiffiffiffi ¼ e  hek i hðkÞdkdy @y 10m2 2pm2 1 1

ez hek i pffiffiffiffiffiffiffiffiffiffiffi 10m2 2pm2 Z 1 Z y Z 2 ðkmm2 Þ  e 2m2 dk  

¼ 2

Also, it implies hf ð; zÞi ¼ 0:1296; hf ð; zÞi ¼ 0:0272 and 2

ð; zÞi ¼ 0:1296, where each function hf ð; zÞi; hf ð; zÞi h@/ @y and h@/ ð; zÞi contain the slow factor z. From a stochastic @y point of view, we can observe that the distribution of the fast elasticity factor Y t depends only upon the parameter of the mean-reversion rate and the time. It implied that if the mean-reversion rate is very big, it must equal to the distribution of the large time [5]. Whereas for the slow elasticity factor Z t , it should be ‘‘frozen’’ at its initial level z when the deviation of the volatility part goes near

1

1

y

e

ðkmÞ2 2m2



 dk dy:

1

ð62Þ Now, we use the change of variable and Fubini’s theorem of the double integral in (62) and obtain the results

 Z 1 Z kþm2 ðkmÞ2 @/ ez hek i  pffiffiffiffiffiffiffiffiffiffiffi ¼ e 2m2 dydk ¼ hf ð; zÞiou : @y 10m2 2pm2 1 k ð63Þ

23

J.-H. Yoon / Chaos, Solitons & Fractals 70 (2015) 14–26 80 BS

70

P

fastSEV

: Λ = 10

PSEV : Λ = 10, Γ = −10

60

PSEV : Λ = 10, Γ = 0

50 Option Price

well brought out in the Table 1. Table 1 exhibits the impact of the free boundary against the parameters K; C; quy and quz . However, as seen in the table, note that the effect of slow factor of the stochastic elasticity term in terms of parameters (C; quz ) is bigger than that of fast factor of the stochastic elasticity term in terms of parameters (K; quy ) on the free boundary. The same is true for the option price. Fig. 2 shows the option price correction term structure .; .; depending on the group parameters ðV 1;0;1 ; V 1;0;1 Þ and

P

P

: Λ = 10, Γ = 10

P

: Λ = 10, Γ = 20

SEV

40

SEV

30 20

.;d

.;d

ðV 1;1;0 ; V 1;1;0 Þ corresponding to the term of the fast factor .;

.;d

10

(P1;0;1 ) and the term of the slow factor (P 1;1;0 ), respectively.

0

These four group parameters ðV 1;0;1 ; V 1;0;1 ; V 1;1;0 ; V 1;1;0 Þ are

.;



−10

.;

.;d

.;d



ðV 3 ; V 2 ; Bd1 ; Bd2 Þ

3

4

5 6 Stock Price u=ln(X)

7

8

Fig. 1. The comparison of the Black–Scholes-Merton price P BS , the pffiffiffi second-order approximation with the only fast scale P BS þ .P 1;0;0 þ pffiffiffiffiffiffi .P1;0;1 þ .P2;0;0pffiffiffiffiffi (the SEV price) and the second-order approximation ffi pffiffiffiffiffiffi pffiffiffi P BS þ .P 1;0;0 þ .dP 1;1;0 þ .P 1;0;1 þ .P 2;0;0 (the multiscale SEV price) 1 yþz for f ðy; zÞ ¼  10 e . 2

These computations of hf ð; zÞi; hf ð; zÞi and

D

E

@/ ð; zÞ @y

under the given function f ðy; zÞ and the given parameters of (58) help us to analyze and understand the change of option price and free boundary more clearly. Yoon et al. [15] studied the behaviors of the perpetual American option price and the optimal exercise boundary with respect to the parameters of the fast-mean reverting factor, emphasizing the importance of the presence of a stochastic elasticity term on the option price and the free boundary. In this article, by adding slow-mean reverting factor to the stochastic elasticity term of [15], we will compare the influence of slow mean reverting factor of the stochastic elasticity term with that of fast mean reverting factor of the term in terms of parameters on option price and analyze the movements of the option price and the free boundary. Fig. 1 plots a behavior of the Black–Scholes-Merton price, the SEV option price (with only the fast-mean reverting factor) and the multiscale SEV option prices against the market price of elasticity risk C. Here, we vary the market price of elasticity risk C for Z t , fixing the market price of elasticity risk K for Y t . As shown in Yoon et al. [15], it could be seen that the SEV price tends to enhance the BS price when K becomes larger. Fig. 1 shows that the multiscale SEV price tends to outgrow the reduced SEV price from K as C increases. From this phenomenon, we can observe that the high market prices of elasticity risk of the fast factor and the slow factor increase the price of American perpetual put option. Also, the change of free boundary with respect to the market prices of elasticity risk K and C is

estimated from Kim et al. [11]. taken from It implies that the slow scale correction has a more significant influence than the fast scale correction on the option price as the stock price changes. Fig. 3 exhibits the sensitivity of the correction terms .; .;d P1;0;1 and P 1;1;0 to volatility (r) according to the change of .; .; .;d .;d the group parameters ðV 1;0;1 ; V 1;0;1 Þ and ðV 1;1;0 ; V 1;1;0 Þ, respectively. One can observe that the slow factor of the stochastic elasticity creates a more crucial impact on option price than the fast factor of the stochastic elasticity. Particularly, it is quite remarkable that the fast factor has little influence on the option price as r decreases, but the slow factor has a larger influence on the option price as r gets smaller. @u @upSEV Now, we consider the two Greeks @pSEV against C and @ q uz

the value of an underlying asset, where quz is the correlation between the underlying asset ut ¼ ln X t and the slow-factor Z t . As seen in Fig. 1 and Table 1, one can observe that the SEV option price is inverse proportional to C and the free boundary is in proportion to C. In order to account for this phenomenon clearly, we use a direct computation for the free boundary upSEV as follows:

pffiffiffiffiffiffi   .dgðzÞhf ð;zÞi0z 2 u0;0;0 þ 3 cþ1 ðc þ 1Þ pffiffiffiffiffiffi  0  p ffiffiffiffiffi ffi 2 .dgðzÞrhf ð; zÞiz @upSEV @u1;1;0 2 : ¼ .d ¼ u þ 0;0;0 @ quz @ quz cþ1 ðc þ 1Þ3 @upSEV pffiffiffiffiffiffi @u1;1;0 2 ¼ .d ¼ @C @C

ð64Þ It can be easily observed that in (64), the free boundary is in proportion to C and inversely proportional to quz on condition that hf ð; zÞi0z > 0 and vice versa if hf ð; zÞi0z < 0. As the Table 1 shows, if hf ð; zÞi0z < 0; @upSEV @ quz

@upSEV @C

has a negative sign and

has a positive sign. Therefore, as C rises or quz

decreases, upSEV decreases. This implies that, in the SEV model scenario, the bigger market price of elasticity risk is, the more delayed its optimal exercise time is, when the underlying price is falling.

Table 1 1 yþz Free boundary movement for f ðy; zÞ ¼  10 e with respect to the parameters K; C; quy and quz . upSEV upSEV upSEV

3.4825 ðK ¼ 20Þ 3.7020 ðC ¼ 20Þ 3.45226 ðquy ¼ 0:9Þ

3.4725 ðK ¼ 10Þ 3.6188 ðC ¼ 10Þ 3.45233 ðquy ¼ 0:4Þ

3.4624 ðK ¼ 0Þ 3.5356 ðC ¼ 0Þ 3.45239 ðquy ¼ 0Þ

3.4524 ðK ¼ 10Þ 3.4524 ðC ¼ 10Þ 3.45245 ðquy ¼ 0:4Þ

3.4423 ðK ¼ 20Þ 3.3692 ðC ¼ 20Þ 3.45253 ðquy ¼ 0:9Þ

upSEV

3.45145 ðquz ¼ 0:9Þ

3.45187 ðquz ¼ 0:4Þ

3.45220 ðquz ¼ 0Þ

3.45253 ðquz ¼ 0:4Þ

3.45294 ðquz ¼ 0:9Þ

J.-H. Yoon / Chaos, Solitons & Fractals 70 (2015) 14–26 0.5

0

0

−0.5

−0.5 1,1,0

0.5

−1

Correction Price Pρ, δ

ε Correction Price Pρ, 1,0,1

24

−1.5 −2 −2.5 (V −3

*ρ, ε ) 1,0,1

ρ, ε , 1,0,1

V

= (0.000091, −0.000480)

= (0.00484, −0.00751) = (0.00167, −0.00996)

−1 −1.5 −2 −2.5

1,0,1

1,0,1

4

= (0.00802, −0.00507)

−3.5

(Vρ, ε , V*ρ, ε ) = (0.000176, 0.00161)

−4 3.5

*ρ, δ ) 1,1,0 *ρ, δ V ) 1,1,0 V*ρ, δ ) 1,1,0

V

−3

ρ, ε

ε (V1,0,1, V*ρ, ) = (0.000133, 0.000563) 1,0,1

−3.5

ρ, δ , 1,1,0 ρ, δ (V , 1,1,0 ρ, δ (V1,1,0,

(V

4.5

5

5.5 6 6.5 Stock Price u=ln(X)

7

7.5

−4 3.5

8

(a) The price change of the correction term P1,0,1

4

4.5

5

5.5 6 6.5 Stock Price u=ln(X)

7

7.5

8

(b) The price change of the correction term P1,1,0

Fig. 2. The change of the correction term prices against the stock price for the group parameters ðV 1;0;1 ; V 1;0;1 Þ and ðV 1;1;0 ; V 1;1;0 Þ, respectively.

−4

0

−3

x 10

1.2

x 10

ρ, δ

(V1,1,0, V*ρ, δ ) = (0.00802, −0.00507)

−0.2 1

−0.4

(V

−0.6

ρ, δ

Correction Price P1,1,0

ε Correction Price Pρ, 1,0,1

ρ, δ

−0.8 −1 −1.2 −1.4

ρ, ε , 1,0,1 ρ, ε (V , 1,0,1 ρ, ε (V1,0,1,

(V

−1.6 −1.8 −2 0.05

0.1

*ρ, ε ) 1,0,1 *ρ, ε V ) 1,0,1 *ρ, ε V1,0,1)

V

0.15

1,1,0 *ρ, δ

(V1,1,0, V1,1,0) = (0.00484, −0.00751) ρ, δ , 1,1,0

*ρ, δ ) 1,1,0

V

= (0.00167, −0.00996)

0.8

0.6

0.4

= (0.000091, −0.000480) 0.2

= (0.000133, 0.000563) = (0.000176, 0.00161) 0.2

σ

0.25

0.3

0.35

0.4

0 0.05

(a) The price change of the correction term P1,0,1

0.1

0.15

0.2

σ

0.25

0.3

0.35

0.4

(b) The price change of the correction term P1,1,0

Fig. 3. The sensibility of the correction term prices with respect to the parameter r with the group parameters ðV 1;0;1 ; V 1;0;1 Þ and ðV 1;1;0 ; V 1;1;0 Þ, respectively.

Also, to observe the sensitivity of the value of the perpetual American put option to the parameters C and quz , we have the Greeks for the approximation price PpSEV

pffiffiffiffiffiffi .dgðzÞhf ð; zÞi0z K c @P pSEV pffiffiffiffiffiffi @P 1;1;0 ¼ .d ¼ @C @C 3ðc þ 1Þ2

  6 2  ðu  u0;0;0 Þðu þ 2u0;0;0 Þ þ uþ cþ1 cþ1

and

ðu  u0;0;0 Þe

@P pSEV @C

with respect to the stock price ut ¼ lnðX t Þ.

ð; zÞi and hf ð; zÞi0z , Without reference to the signs of h@/ @y

ðu  u0;0;0 Þecðuu0;0;0 Þ ; pffiffiffiffiffiffi .dgðzÞrhf ð; zÞi0z K c @P pSEV pffiffiffiffiffiffi @P 1;1;0 ¼ .d ¼ @ quz @ quz 3ðc þ 1Þ2 ( ) 3ðc  1Þ 12  cðu2 þ uu0;0;0  2u20;0;0 Þ þ u  3u0;0;0  ðc þ 1Þ ðc þ 1Þ2 cðuu0;0;0 Þ

In above Eq. (65), if hf ð; zÞi0z < 0, the SEV price grows for u > upBS ¼ u0;0;0 when C becomes larger. The change of the option price is clearly recognized with the naked eye in Fig. 1. @P Fig. 4 shows the difference between the two greeks @pSEV K

: ð65Þ

the sensitivity of the option price to the market price of elasticity risk K is nonmonotonic. When the underlying @P

asset grows, @pSEV is increasing, but dropping back at a K certain point of the stock price. However, the sensitivity of the option price to the market price of elasticity risk C is monotonic as the underlying asset increases. @P

@P

Figs. 5 and 6 plot the two greeks @qpSEV and @ qpSEV with uy uz respect to the stock price ut ¼ lnðX t Þ. Here, the sensitivities of the SEV price to the two parameters quy and quz are both

25

J.-H. Yoon / Chaos, Solitons & Fractals 70 (2015) 14–26 0.3

1

〈 ∂ φ / ∂ y 〉 >0 〈 ∂ φ / ∂ y 〉 <0

0.8

〈 f(⋅, z) 〉z <0

0.4

0.1

pSEV

/∂Γ

0.2 0

0

∂P

∂ PpSEV / ∂ Λ

〈 f(⋅, z) 〉z > 0

0.2

0.6

−0.2

−0.1

−0.4 −0.6

−0.2

−0.8 −1

2

4

6

8 10 Stock Price u=ln(X)

12

−0.3

14

2

4

6

8 10 Stock Price u=ln(X)

Fig. 4. The sensitivity of the second approximated price P pSEV with respect to K and C. The parameters we used in this figure are  ¼ 0:0001; m ¼ 0; r ¼ 0:01; r ¼ 0:3; K ¼ 100; m ¼ p1ffiffi2 ; quy ¼ 0:2; quz ¼ 0:2; gðzÞ ¼ 2.

1.5

〈 f(⋅, z) 〉 >0 〈 f(⋅, z) 〉 < 0

0.01 /∂ρ

uz

uy

∂P

pSEV

0

0

∂P

/∂ρ

. ¼ 0:007; d ¼ 0:001;

0.02

0.5

pSEV

14

0.03 〈 ∂ φ / ∂ y 〉 >0 〈 ∂ φ / ∂ y 〉 <0

1

−0.5

−0.01

−1

−0.02

−1.5

2

4

6

8 10 Stock Price u=ln(X)

12

14

−0.03

2

4

6

8 10 Stock Price u=ln(X)

Fig. 5. The sensitivity of the second approximated price P pSEV with respect to quy and quz . The parameters we used in this figure are  ¼ 0:0001; m ¼ 0; r ¼ 0:01; r ¼ 0:15; K ¼ 100; m ¼ p1ffiffi2 ; quy ¼ 0:2; quz ¼ 0:2; gðzÞ ¼ 2.

non-monotonic. Moreover, we have critical points of the underlying asset at which

@P pSEV @ quy

and

Interestingly, the critical value of

@P pSEV @ quz

@PpSEV @ quz

switch its sign.

is always further

away from the free-boundary upBS ¼ u0;0;0 than that of @PpSEV . @ quy

The direct computation of the critical values of

and

@P pSEV @ quz



2 cðcþ1Þ

@PpSEV @ quy

given by uquy ¼  upBS and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  r 2  

3ðc1Þ

 cupBS þ cþ1

@P pSEV @ quy

demonstrates this phenomenon. From Yoon

et al. [15] and (65), the critical values of are 

12

þ

1Þ cupBS þ3ðccþ1

2c

12 ðcþ1Þ2

þ4c 2cu2pBS þ3upBS þ

and

@P pSEV @ quz 

uquz ¼

for u P upBS ,

12

14

. ¼ 0:007; d ¼ 0:001;

respectively. By a straightforward calculation, uquz P uquy is always satisfied for all upBS and c. If we use parameters given by (58), then the optimal values have uquz 5:2294 and uquy 2:6602, where upBS 3:8154. In this case, there is not any critical value of

@PpSEV @ quy

as shown

instead of

r ¼ 0:3 r ¼ 0:15 in (58), as seen in Fig. 6, then both

@PpSEV @ quy

@PpSEV @ quz



in Fig. 5 since uquy < upBS . However, if we take and

uquz > upBS ,

have critical values as uquy > upBS and

where

uquy 4:4632.

upBS ¼ 2:9004; uquz 12:8565

and

26

J.-H. Yoon / Chaos, Solitons & Fractals 70 (2015) 14–26 0.3

0.15 〈 ∂ φ / ∂ y 〉 >0 〈 ∂ φ / ∂ y 〉 <0

0.2

〈 f(⋅, z) 〉z < 0

0.05 ∂ PpSEV / ∂ ρ

uy

uz

0.1 ∂ PpSEV / ∂ ρ

〈 f(⋅, z) 〉z > 0

0.1

0

0

−0.1

−0.05

−0.2

−0.1

2

4

6

8 10 Stock Price u=ln(X)

12

14

2

4

6

8 10 Stock Price u=ln(X)

12

14

Fig. 6. The sensitivity of the second approximated price P pSEV with respect to quy and quz . The parameters we used in this figure are . ¼ 0:007; d ¼ 0:001;  ¼ 0:0001; m ¼ 0; r ¼ 0:01; r ¼ 0:3; K ¼ 100; m ¼ p1ffiffi2 ; quy ¼ 0:2; quz ¼ 0:2; gðzÞ ¼ 2.

5. Conclusion

References

Based upon the underlying asset price formulation driven by a perturbative form of the stochastic elasticity of variance with a fast-mean reverting process and a slowly varying process, an explicit comprehensive closed-form formula is obtained for the perpetual American put option prices. By the numerical computation and comparison of the option price formula and its reduced forms, we observe both the quantitative and qualitative correction effects to the established SEV model [9]. Our results implicate quite interesting and delicate impacts of the addition of a slow mean-reverting stochastic factor on both the perpetual option prices and the free boundary values. In the extended multiscale SEV model, we find that a slow scale mean-reverting factor on stochastic elasticity term makes a more significant impact on the option pricing than a fast scale mean-revering factor. In other words, the role of the slow factor of the stochastic elasticity is very important on the option pricing in common with the cases of the stochastic volatility model [2] and the hybrid stochastic and local volatility model [10]. In conclusion, our paper emphasizes on the necessity of studying the multiscale SEV model on the option pricing.

[1] Black F, Scholes M. The pricing of options and corporate liabilities. J Political Econ 1973;81:637–54. [2] Chen W-T, Zhu S-P. Pricing perpetual American puts under multiscale stochastic volatility. Asympt Anal 2012;80:133–48. [3] Cox J. Notes on option pricing I: Constant elasticity of variance diffusions. Working paper, Stanford University; 1975 (Reprinted in J. Portf. manage 1996; 22: 15–17.). [4] Cox J, Ross S. The valuation of options for alternative stochastic processes. J Financial Econ 1976;3:145–66. [5] Fouque J-P, Papanicolaou G, Sircar R, Solna K. Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press; 2000. [6] Fouque J-P, Papanicolaou G, Sircar R, Solna K. Multiscale stochastic volatility asymptotics. SIAM J Mutiscale Model Simul 2003;2(1): 22–42. [7] Fouque J-P, Papanicolaou G, Sircar R, Solna K. Multiscale Stochastic Volatility for Equity, Interest Rate and Credit Derivatives. Cambridge University Press; 2011. [8] Hagan PS, Kumar D, Lesniewski AS, Woodward DE. Managing Smile Risk. Wilmott magazine, September; 2002: 84–108, [9] Kim J-H, Lee J, Zhu S-P, Yu S-H.A multiscale correction to the Black– Scholes formula. Appl Stoch Model Bus, Article first published online: 19 JAN 2014, http://dx.doi.org/10.1002/asmb.2006. [10] Kim J-H, Lee M-G, Sohn S. Investment timing under hybrid stochastic and local volatility. Chaos, Solitons Fractals 2014;67:58–72. [11] Kim J-H, Yoon J-H, Yu S-H. Multiscale stochastic volatility with the Hull–White rate of interest. J Futures Markets 2014;34(9):819–37. [12] Ramm AG. A simple proof of the Fredholm alternative and a characterization of the Fredholm operators. Math Assoc Am 2001;108:855–60. [13] Tao LN. The analyticity and general solution of the Cauchy–Stefan problem. Quart J Mech Appl Math 1983;36(4):487–504. [14] Yang S-J, Lee M-K, Kim J-H. Portfolio optimization under the stochastic elasticity of variance. Stoch Dyn 2014;14(3):1350024. http://dx.doi.org/10.1142/S021949371350024X. [15] Yoon J-H, Kim J-H. A closed-form analytic correction to the Black– Scholes–Merton price for perpetual American options. Appl Math Lett 2013;26:1146–50.

Acknowledgments The author thanks anonymous reviewers for valuable comments on the improvement of the paper. The research of J.-H. Yoon was supported by BK21 PLUS SNU Mathematical Sciences Division.