Asymptotic behavior of the distribution of the stock price in models with stochastic volatility: the Hull–White model

Probability Theory

Asymptotic behavior of the distribution of the stock price in models with stochastic volatility: the Hull–White model

Archil Gulisashvili

, Elias M. Stein

aDepartment of Mathematics, Ohio University, Athens, OH 45701, USA bDepartment of Mathematics, Princeton University, Princeton, NJ 08540, USA

Received 18 June 2006; accepted 26 September 2006

Presented by Jean-Pierre Kahane

Abstract

In the present Note, we study the asymptotic behavior of the distribution density of the stock price process in the Hull–White model. The leading terms in the asymptotic expansions at zero and infinity are found for such a density and the corresponding error estimates are given. Similar problems are solved for time averages of the volatility process, which are also of interest in the study of Asian options.To cite this article: A. Gulisashvili, E.M. Stein, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

©2006 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Résumé

Comportement asymptotique de la distribution du prix de l’action dans les modèles à volatilité stochastique : le modèle de Hull–White.La présente Note étudie le comportement asymptotique de la densité de distribution du processus du prix de l’action dans le modèle de Hull–White. On determine la partie principale dans le développement asymptotique en zéro et en l’infini pour une telle densité et on estime l’erreur correspondante. Des problèmes similaires se résolvent pour les moyennes temporelles du processus de volatilité qui sont aussi intéressants dans l’étude des options asiatiques.Pour citer cet article : A. Gulisashvili, E.M. Stein, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

©2006 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Version française abrégée

Le modèle de Hull–White est l’un des modèles du prix de l’action à volatilité stochastique. La présente note étudie le modéle suivant qui est équivalent au modèle de Hull–White :

dXt=μX_tdt+Y_tX_tdWt, dYt=νY_tdt+ξ Y_tdZt,

oùμ∈R,ν∈R,ξ >0, W_t etZ_t sont des browniens standards indépendants,Y_t le processus de volatilité etX_t le processus du prix de l’action. Les conditions initiales des processus Xt et Yt sont respectivement notées x0 ety0. On donne l’expression explicite de la partie principale du développement asymptotique de la densité de distribution

E-mail addresses:[email protected] (A. Gulisashvili), [email protected] (E.M. Stein).

1631-073X/$ – see front matter ©2006 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.crma.2006.09.029

Dt(x;μ, ν, ξ, x0, y0) de la variable aléatoire Xt. On étudie aussi le comportement asymptotique de la densité de distributionm_t(y;ν, ξ, y₀)de la variable aléatoire

α_t=α_t(ν, ξ, y₀)=

1 t t

0

Y_s²ds 1/2

.

Par exemple siμ=0,ν=¹₂,ξ=1,x₀=1,y₀=1, on a les expressions asymptotiques suivantes : m_t(y)= 1

√π t yΛ_t(y) 1+O

(logy)^−1/2

, y→ ∞,

oùΛ_t(y)=exp{−^u_2t^2y +^u_2t^y}etu_yvérifient ^sinhu_2u ^y

y =y²;

m_t(y)=

√2

√π texp π²

8t

y⁻¹exp

− 1 2ty²

1+O y²

, y→0;

D_t(x)= 1 2√

π tx⁻²(logx)⁻¹Λ_t 2

t logx 1/2

1+O

(logx)^−δ

, x→ ∞

pour toutδ, 0< δ <¹₂; etD_t(x)=x⁻³D_t(¹_x). On obtient des formules similaires dans le cas général.

1. Introduction

The Hull–White model is one of the standard stock price models with stochastic volatility (see [4]). LetXtandYt

be stochastic processes satisfying the following system of stochastic differential equations:

dXt=μXtdt+YtXtdWt,

dYt=νYtdt+ξ YtdZt, (1)

whereμ∈R,ν∈R,ξ >0, andW_t andZ_t are independent standard Brownian motions. In (1), the processY_t is the volatility process andX_t is the stock price process. The initial conditions for the processesX_t andY_t are denoted byx₀andy₀, respectively. The model described in (1) is equivalent to the Hull–White model. We will denote by D_t(x)=D_t(x;μ, ν, ξ, x₀, y₀)the distribution density of the random variableX_t, and bym_t(y)=m_t(y;ν, ξ, y₀)the distribution density of the random variable

α_t(ν, ξ, y₀)= 1

t t

0

Y_s²ds 1/2

.

The densitym_t is called the mixing distribution density. The densityD_t depends onμ,ν,ξ,x₀, andy₀, while the densitym_t depends onν,ξ, andy₀. It is not hard to see that the following equality holds form_tandD_t:

D(x;μ, ν, ξ, x₀, y₀)= 1 x0e^μt

∞

0

L t, y, x x0e^μt

m_t(y;ν, ξ, y₀)dy, (2)

whereLis the lognormal density given by L(t, y, v)= 1

√2π t yvexp

−(lnv+ty²/2)² 2ty²

.

In the present Note we study the asymptotic behavior of the stock price distribution density D_t and the mixing distribution densitym_t. The mixing densitym_t has numerous applications in the theory of Asian options (see [9]).

We refer the reader to [3] for more information on stock price models with stochastic volatility and to [1,2,5,6,8,9]

regarding the densitym_t. The asymptotics for the mixing density and the resulting asymptotics for the stock price distribution were first studied in [7] for the case of the model whose volatility processY_t is mean-reverting Ornstein–

Uhlenbeck process.

2. The asymptotics of the mixing distribution density

It will be assumed throughout the note that the coefficientμis 0, since one can easily reduce matters to that case.

We shall also set α=2ν−ξ²

2ξ² . (3)

Fory >0, we denote byu_ythe unique positive solution of the equation sinh(2uy)

2uy =y². It is not hard to see that

u_y=logy+1

2log logy+log 2+o(1) asy→ ∞. Put

Λt(y)=exp

−u²_y 2t +u_y

2t

,

wheret, y >0.

The asymptotic behavior ofm_t(y)as y→ ∞ is very roughly like exp{−^(logy)_{2t ξ}2²}. More precisely, the following theorem holds:

Theorem 2.1.If−∞< ν <∞,ξ >0,y₀>0, andt >0, then m_t(y)=c_ty^α−1(logy)^α/2Λ_{t ξ}2

y y₀

1+O

(logy)^−1/2

, y→ ∞, (4)

whereαis given by(3)and c_t= 1

y₀^αξ√ π texp

−α²ξ²t 2

.

We consider first the special case whereν=¹₂,ξ=1, andy0=1, and later the situation for more generalν; then withξ=1, we haveν=α+¹₂. We denote bym^(α)_t the mixing distribution density in this case and keep the notation mt whenα=0 (i.e.ν=¹₂). Formula (4) then becomes

m_t(y)= 1

√π t yΛ_t(y) 1+O

(logy)^−1/2

, y→ ∞. (5)

The proof of (5) and the general case proceeds as follows. First, one has in the special case α=0 an explicit integral formula,

m_t(y)= 1 π texp

π² 8t

y⁻²

∞

−∞

exp

−cosh²u 2ty²

coshuexp

−u² 2t

exp iπ u

2t

du (6)

(see [1] where an equivalent formula is given).

Alternatively, one can derive (6) by using the formula

u(x, t )= ∞

−∞

e^ixsinhyU (y, t )dy,

which transforms a solutionUof the heat equation to a solution of the equation

∂u

∂t =x²∂²u

∂x²+x∂u

∂x −x²u. (7)

Equations similar to that in (7) arise from the Feynman–Kac formula when computing the Laplace transform ofmt(y).

With formula (6) in hand, one relies on the following lemma. Forε >0, we define

I (ε)= ∞

−∞

e^{−ε(coshu)2}e^{−u2/(2t )}coshue^{iπ u/(2t )}du.

Lemma 2.2.The following equality holds:

I (ε)=I₀(ε) 1+ log1 ε

₋1/2

asε→0, where I₀(ε)= π

2 1/2

exp

−π² 8t

exp

−N_ε² 2t +Nε

2t

ε^−1/2,

andN_εis the solution ofεsinh(2Nε)=^N_t^ε.

The proof of the lemma requires that we deform the contour of integration forI_ε(the real one) into the complex u-plane, where the principal contributions are then given on the segments[N_ε, N_ε+iπ]and[−N_ε,−N_ε+iπ]. With this lemma (and the corresponding asymptotics for(_dε^d)^kI (ε)) one obtains the desired result forα=0 andα=2k, respectively, wherekis a positive integer. We next use Dufresne’s recurrence formula (see [2]), which in our notation can be rewritten as follows:

m^(2r_t ^−β)√ y

=C_r,β,ty^{2r−β−1/2}exp

− 1 2ty

^∞

y

(τ−y)^β−r−1τ^−β+1/2exp 1

2t τ

m^(β)_t √ τ

dτ,

where

C_r,β,t=(2t )^r−βexp{(−2r²+2rβ)t} (β−r)

andr < β. This allows us to deduce the asymptotics at∞ofm^(α)_t (y)from those ofm^(2k)_t (y), wheneverα <2k, and thus for allα. Finally, we drop the restrictionξ=1,y₀=1, by observing that

mt(y;ν, ξ, y0)= 1 y₀m_{t ξ}2

y y₀; ν

ξ²,1,1

.

Next, we formulate a theorem describing the asymptotics of the mixing distribution density asy→0. The analysis here is simpler than that fory→ ∞. The result obtained is as follows.

Theorem 2.3.For each realαand positivet, m^(α)_t (y)=b_α,ty^2α−1exp

− 1 2ty²

1+O y²

, y→0, whereb_α,t is an appropriate positive constant.

3. The asymptotics of the stock price distribution density

We can combine Theorem 2.1 with (2) to obtain the asymptotics ofD_t(x)=D_t(x;μ, ν, ξ, x₀, y₀). In stating the result it suffices to consider the case whereμ=0 andx₀=1, since (2) implies that

D_t(x;μ, ν, ξ, x₀, y₀)= 1

x0e^μtD_t x

x0e^μt;0, ν, ξ,1, y0

.

Theorem 3.1.The following formula holds:

D_t(x;0, ν, ξ,1, y0)= 1 2√

π t y₀^αξ t^α/2exp

−α²ξ²t 2

x⁻²(logx)^α/2−1(log logx)^α/2

×Λ_{t ξ}2

1 y₀

2 logx t

1/2 1+O

(logx)^−δ

, x→ ∞,

whereδis any number with0< δ <¹₂ andαis defined by(3). Moreover, Dt(x;0, ν, ξ,1, y0)=x⁻³Dt

1

x;0, ν, ξ,1, y0

.

The proof requires a particular precise version of Laplace’s method. Suppose A(y) is a positive C¹ function on [0,∞)which satisfies |A(y)|cy^−γA(y), y > y₀, for some γ with 0< γ 1. Note that as a consequence _∞

0 A(y)e^−by2dy <∞for allb >0.

Lemma 3.2.Ifc >0, then ∞

0

A(y)exp

− x² y²+c²y²

dy=

√π 2c A

x^1/2c^−1/2 e^−2cx

1+O x^−δ

, x→ ∞,

for anyδwith0< δ <^γ₂.

To apply the lemma we takeA(y)to be the leading term of the asymptotic expansion of the function^mt_y^(y). Theorem 3.1 gives an asymptotic forD_t(x)very roughly of orderx⁻² asx→ ∞, and roughly of orderx⁻¹as x→0. This shows clearly that here the tails are “fatter” than in the case of the model whereY_t is mean-reverting Ornstein–Uhlenbeck process considered in [7]. There the corresponding sizes are of the orderx^−γ forx→ ∞, and x^γ−3whenx→0, for an appropriateγ >2.

References

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