Variational formulation of American option prices in the Heston Model

# Variational formulation of American option prices in the Heston Model

## Abstract

We give an analytical characterization of the price function of an American option in Heston-type models. Our approach is based on variational inequalities and extends recent results of Daskalopoulos and Feehan (2011). We study the existence and uniqueness of a weak solution of the associated degenerate parabolic obstacle problem. Then, we use suitable estimates on the joint distribution of the log-price process and the volatility process in order to characterize the analytical weak solution as the solution to the optimal stopping problem. We also rely on semi-group techniques and on the affine property of the model.

Keywords: American options; degenerate parabolic obstacle problem; optimal stopping problem.

## 1 Introduction

The model introduced by S. Heston in 1993 () is one of the most widely used stochastic volatility models in the financial world and it was the starting point for several more complex models which extend it. The great success of the Heston model is due to the fact that the dynamics of the underlying asset can take into account the non-lognormal distribution of the asset returns and the observed mean-reverting property of the volatility. Moreover, it remains analytically tractable and provides a closed-form valuation formula for European options using Fourier transform.

These features have called for an extensive literature on numerical methods to price derivatives in Heston-type models. In this framework, besides purely probabilistic methods such as standard Monte Carlo and tree approximations, there is a large class of algorithms which exploit numerical analysis techniques in order to solve the standard PDE (resp. the obstacle problem) formally associated with the European (resp. American) option price function. However, these algorithms have, in general, little mathematical support and in particular, as far as we know, a rigorous and complete study of the analytic characterization of the American price function is not present in the literature.

The main difficulties in this sense come from the degenerate nature of the model. In fact, the infinitesimal generator associated with the two dimensional diffusion given by the log-price process and the volatility process is not uniformly elliptic: it degenerates on the boundary of the domain, that is when the volatility variable vanishes. Moreover, it has unbounded coefficients with linear growth. Therefore, the existence and the uniqueness of the solution to the pricing PDE and obstacle problem do not follow from the classical theory, at least in the case in which the boundary of the state space is reached with positive probability, as happens in many cases of practical importance (see ). Moreover, the probabilistic representation of the solution, that is the identification with the price function, is far from trivial in the case of non regular payoffs.

It should be emphasized that a clear analytic characterization of the price function allows not only to formally justify the theoretical convergence of some classical pricing algorithms but also to investigate the regularity properties of the price function (see  for the case of the Black and Scholes models).

Concerning the existing literature, E. Ekstrom and J. Tysk in  give a rigorous and complete analysis of these issues in the case of European options, proving that, under some regularity assumptions on the payoff functions, the price function is the unique classical solution of the associated PDE with a certain boundary behaviour for vanishing values of the volatility. However, the payoff functions they consider do not include the case of standard put and call options.

Recently, P. Daskalopoulos and P. Feehan studied the existence, the uniqueness, and some regularity properties of the solution of this kind of degenerate PDE and obstacle problems in the elliptic case, introducing suitable weighted Sobolev spaces which clarify the behaviour of the solution near the degenerate boundary. Again, as regards the probabilistic representation, they only treat the case with heavy regularity assumptions on the payoff function (see ).

The aim of this paper is to give a precise analytical characterization of the American option price function for a large class of payoffs which includes the standard put and call options. In particular, we give a variational formulation of the American pricing problem using the weighted Sobolev spaces and the bilinear form introduced in . The paper is organized as follows. In Section 2 we introduce our notations and we state our main results. Then, in section 3 we study the existence and uniqueness of the solution of the associated variational inequality, extending the results obtained in  in the elliptic case. The proof essentially relies on the classical penalization technique introduced by Bensoussan and Lions  with some technical devices due to the degenerate nature of the problem. We also establish a Comparison Theorem. Finally, in section 4, we prove that the solution of the variational inequality with obstacle function is actually the American option price function with payoff , with conditions on which are satisfied, for example, by the standard call and put options. In order to do this, we use the affine property of the underlying diffusion given by the log price process and the volatility process . Thanks to this property, we first identify the analytic semigroup associated with the bilinear form with a correction term and the transition semigroup of the pair with a killing term. Then, we prove regularity results on the solution of the variational inequality and suitable estimates on the joint law of the process and we deduce from them the analytical characterization of the solution of the optimal stopping problem, that is the American option price.

## 2 Notations and main results

### 2.1 The Heston model

We recall that in the Heston model the dynamics under the pricing measure of the asset price and the volatility process are governed by the stochastic differential equation system

 {dStSt=(r−δ)dt+√YtdBt,S0=s>0,dYt=κ(θ−Yt)dt+σ√YtdWt,Y0=y≥0,

where and denote two correlated Brownian motions with

 d⟨B,W⟩t=ρdt,ρ∈(−1,1).

Here and are respectively the risk free rate of interest and the continuous dividend rate. The dynamics of follows a CIR process with mean reversion rate and long run state . The parameter is called the volatility of the volatility. Note that we do not require the Feller condition : the volatility process can hit (see, for example, [2, Section 1.2.4]).

We are interested in studying the price of an American option with payoff function . For technical reasons which will be clarified later on, hereafter we consider the process

 Xt=logSt−¯ct,with ¯c=r−δ−ρκθσ, (2.1)

which satisfies

 (2.2)

Note that, in this framework, we have to consider payoff functions which depend on both the time and the space variables. For example, in the case of a standard put option (resp. a call option) with strike price we have (resp. ). So, the natural price at time of an American option with a nice enough payoff is given by , with

 P(t,x,y)=esssupθ∈Tt,TE[e−r(θ−t)ψ(θ,Xt,x,yθ,Yt,yθ)],

where is the set of all stopping times with values in and denotes the solution to (2.2) with the starting condition .

Our aim is to give an analytical characterization of the price function . We recall that the infinitesimal generator of the two dimensional diffusion is given by

 L=y2(∂2∂x2+2ρσ∂2∂y∂x+σ2∂2∂y2)+κ(θ−y)∂∂y+(ρκθσ−y2)∂∂x,

which is defined on the open set . Note that has unbounded coefficients and is not uniformly elliptic: it degenerates on the boundary .

### 2.2 American options and variational inequalities

#### Heuristics

From the optimal stopping theory, we know that the discounted price process is a supermartingale and that its finite variation part only decreases on the set . We want to have an analytical interpretation of these features on the function . So, assume that . Then, by applying Ito’s formula, the finite variation part of is

 (∂~P∂t+L~P)(t,Xt,Yt).

Since is a supermartingale, we can deduce the inequality

 ∂~P∂t+L~P≤0

and, since its finite variation part decreases only on the set , we can write

 (∂~P∂t+L~P )(ψ−P)=0.

This relation has to be satisfied along the trajectories of . Moreover, we have the two trivial conditions and .

The previous discussion is only heuristic, since the price function is not regular enough to apply the Ito’s formula. However, it suggests the following strategy:

1. Study the obstacle problem

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩∂u∂t+Lu≤0,u≥ψ,in [0,T]×O,(∂u∂t+Lu)(ψ−u)=0,in [0,T]×O,u(T,x,y)=ψ(T,x,y). (2.3)
2. Show that the discounted price function is equal to the solution of (2.3) where is replaced by .

We will follow this program providing a variational formulation of system (2.3).

#### Weighted Sobolev spaces and bilinear form associated with the Heston operator

We consider the measure first introduced in :

 mγ,μ(dx,dy)=yβ−1e−γ|x|−μydxdy,

with . It will be clear later on that this measure in some sense describes the qualitative behaviour of the process near the degenerate boundary. For we denote by the standard euclidean norm of in . The relevant Sobolev spaces are defined as follows (see ).

###### Definition 2.1.

For every let be the space of all measurable functions for which

 ∥u∥pLp(O,mγ,μ):=∫O|u|pdmγ,μ<∞,

and denote

1. If and , are defined in the sense of distributions, we set

 H1(O,mγ,μ):={u∈L2(O,mγ,μ):√1+yu and √y|∇u|∈L2(O,mγ,μ)},

and

 ∥u∥2H1(O,mγ,μ):=∫O(y|∇u|2+(1+y)u2)dmγ,μ.
2. If and all derivatives of are defined in the sense of distributions, we set

 H2(O,mγ,μ):={u∈L2(O,mγ,μ):√1+yu, (1+y)|∇u|, y|D2u|∈L2(O,mγ,μ)}

and

 ∥u∥2H2(O,mγ,μ):=∫O(y2|D2u|2+(1+y)2|∇u|2+(1+y)u2)dmγ,μ.

For brevity and when the context is clear, we shall often denote

 H:=H0(O,mγ,μ),V:=H1(O,mγ,μ)

and

 ∥u∥H:=∥u∥L2(O,mγ,μ),∥u∥V:=∥u∥H1(O,mγ,μ).

Note that the spaces , for are Hilbert spaces with the inner products

 (u,v)H=(u,v)L2(O,mγ,μ)=∫Ouvdmγ,μ,
 (u,v)V=(u,v)H1(O,mγ,μ)=∫O(y(∇u,∇v)+(1+y)uv)dmγ,μ

and

 (u,v)H2(O,mγ,μ):=∫O(y2(D2u,D2v)+(1+y)2(∇u,∇v)+(1+y)uv)dmγ,μ,

where denotes the standard scalar product in . Moreover, note that

 H2(O,mγ,μ)⊂H1(O,mγ,μ).

We can now introduce the bilinear form associated with the differential operator .

###### Definition 2.2.

For any we define the bilinear form

 aγ,μ(u,v)= 12∫Oy(uxvx(x,y)+ρσuxvy(x,y)+ρσuyvx(x,y)+σ2uyvy(x,y))dmγ,μ +∫Oy(jγ,μ(x)ux(x,y)+kγ,μ(x)uy(x,y))v(x,y)dmγ,μ,

where

 jγ,μ=12(1−γsgn(x)−μρσ),kγ,μ=κ−γρσ2sgn(x)−μσ22.

We will prove that for every and for every , we have

 (Lu,v)H=−aγ,μ(u,v).

In order to simplify the notation, from now on we fix and and we write and instead of and .

### 2.3 Variational formulation of the American price

Fix . We consider an assumption on the payoff function which will be crucial in the discussion of the penalized problem.

Assumption . We say that a function satisfies Assumption if , , and there exists such that .

We will also need a domination condition on by a function which satisfies the following assumption.

Assumption . We say that a function satisfies Assumption if , and for some .

The domination condition is needed to deal with the lack of corercivity of the bilinear form associated with our problem. Similar conditions are also used in .

The first step in the variational formulation of the problem is to introduce the associated variational inequality and to prove the following existence and uniqueness result.

###### Theorem 2.3.

Assume that satisfies Assumption together with , where satisfies Assumption . Then, there exists a unique function such that and

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩−(∂u∂t,v−u)H+a(u,v−u)≥0,a.e. in [0,T]v∈V, v≥ψ,u≥ψ a.e. in [0,T]×R×(0,∞),u(T)=ψ(T),0≤u≤Φ. (2.4)

The proof is presented in Section 3 and essentially relies on the penalization technique introduced by Bensoussan and Lions (see also ) with some technical devices due to the degenerate nature of the problem. We extend in the parabolic framework the results obtained in  for the elliptic case.

The second step is to identify the unique solution of the variational inequality (2.4) as the solution of the optimal stopping problem, that is the (discounted) American option price.

Recall that an adapted right continuous process is said to be of class if the family , where is the set of all stopping times with values in , is uniformly integrable. We introduce the following further assumption:

Assumption . We say that a function satisfies Assumption if is continuous and, for all , the process is of class .

Assumption is crucial in order to get the following identification result.

###### Theorem 2.4.

Fix . Assume that, in addition to the assumptions of Theorem 2.3, there exists a sequence of continuous functions on which converges uniformly to and satisfies the following properties for each :

1. satisfies Assumption and for some satisfying Assumption , Assumption and ;

2. and .

Then, the solution of the variational inequality (2.4) associated with is continuous and coincides with the function defined by

 u∗(t,x,y)=supτ∈Tt,TE[ψ(τ,Xt,x,yτ,Yt,x,yτ)].

We conclude this overview with a natural remark. The assumptions on in Theorem 2.3 and Theorem 2.4 seem to be very stringent but we will see that, by choosing large enough, they are satisfied by the class of payoff functions , where as defined in (2.1), is continuous, positive and such that

 |¯ψ|+∣∣¯ψ′∣∣≤C(ex+1),

with . Note that the standard call and put payoff functions fall into this category (see Remark 4.17).

## 3 Existence and uniqueness of solutions to the variational inequality

### 3.1 Integration by parts and energy estimates

The following result justifies the definition of the bilinear form .

###### Proposition 3.1.

If and , we have

 (Lu,v)H=−a(u,v). (3.1)

Before proving Proposition 3.1, we show some preliminary results. The first one is about the standard regularization of a function by convolution.

###### Lemma 3.2.

Let be a function with compact support in and such that . For we set . Then, for every function u locally square-integrable on and for every compact set , we have

 limj→∞∫∫K(φj∗u−u)2(x,y)dxdy=0.
###### Proof.

We first observe that

 ∫∫K(φj∗u)2(x,y)dxdy ≤∫∫Kdxdy∫∫φj(ξ,ζ)u2(x−ξ,y−ζ)dξdζ =∫∫φj(ξ,ζ)dξdζ∫∫\mathbbm1K(x+ξ,y+ζ)u2(x,y)dxdy.

We deduce, for large enough,

 ∫∫K(φj∗u)2(x,y)dxdy≤∫∫¯Ku2(x,y)dxdy,

where . Let be a positive constant and be a continuous function such that . We have

 ∫∫K (φj∗u−u)2(x,y)dxdy ≤3∫∫K(φj∗u−φj∗v)2(x,y)dxdy+3∫∫K(φj∗v−v)2(x,y)dxdy+3∫∫K(v−u)2(x,y)dxdy ≤3∫∫¯K(v−u)2(x,y)dxdy+3∫∫K(φj∗v−v)2(x,y)dxdy+3∫∫¯K(v−u)2(x,y)dxdy ≤6ϵ+3∫∫K(φj∗v−v)2(x,y)dxdy.

Since is continuous, we have and on . Therefore, by Lebesgue Theorem, we can pass to the limit in the above inequality and we get

 limsupj→∞∫∫K(φj∗u−u)2(x,y)dxdy≤6ϵ,

which completes the proof. ∎

Then, the following two propositions justify the integration by parts formulas with respect to the measure .

###### Proposition 3.3.

Let us consider locally square-integrable on , with derivatives and locally square-integrable on as well. Moreover, assume that

 ∫O(|ux(x,y)v(x,y)|+|u(x,y)vx(x,y)|+|u(x,y)v(x,y)|)dm<∞.

Then, we have

 (3.2)
###### Proof.

First we assume that has compact support in . For any we consider the functions and , with as in Lemma 3.2. Note that and so, for large enough, . For any , integrating by parts, we have

 ∫∞−∞(uj)x(x,y)vj(x,y)e−γ√x2+ϵdx=−∫∞−∞uj((vj)x(x,y)−γx√x2+ϵvj(x,y))e−γ√x2+ϵdx,

and, letting ,

 ∫∞−∞(uj)x(x,y)vj(x,y)e−γ|x|dx=−∫∞−∞uj((vj)x(x,y)−γsgn(x)vj(x,y))e−γ|x|dx.

Multiplying by and integrating in we obtain

 ∫O(uj)x(x,y)vj(x,y)dm=−∫Ouj((vj)x(x,y)−γsgn(x)vj(x,y))dm.

Recall that, for large enough, has compact support in and is bounded on this compact. By using Lemma 3.2, letting we get

 ∫Oux(x,y)v(x,y)dm=−∫Ou(vx(x,y)−γsgn(x)v(x,y)dm.

Now let us consider the general case of a function without compact support. We introduce a function with values in , for all , for all and a function with values in , for all , for all We set

 Aj(x,y)=α(xj,yj)χ(jy),j∈N.

For every , has compact support in and we have

 ∫Oux(x,y)Aj(x,y)v(x,y)dm
 =−∫Ou(x,y)(ux(x,y)−γsgn(x)v(x,y))Aj(x,y)dm−∫Ou(x,y)v(x,y)(Aj)x(x,y)dm.

The function is bounded by and for every . Moreover , so that

 ∣∣∣∫Ou(x,y)v(x,y)(Aj)x(x,y)dm∣∣∣≤Cj∫O\mathbbm1{|x|≥j}|u(x,y)v(x,y)|dm,

where . Therefore, we obtain (3.2) letting . ∎

###### Proposition 3.4.

Let us consider locally square-integrable on , with derivatives and locally square-integrable on as well. Moreover, assume that

 ∫Oy(|uy(x,y)v(x,y)|+|u(x,y)vxy(x,y)|)+|u(x,y)v(x,y)|dm<∞.

Then, we have

 ∫Oyuy(x,y)v(x,y)dm=−∫Oyu(x,y)vydm−∫O(β−μy)u(x,y)v(x,y)dm. (3.3)
###### Proof.

If has compact support in , we obtain (3.3) as in the proof of Proposition 3.3. On the other hand, if has not compact support,

 ∫Oyuy(x,y) v(x,y)Aj(x,y)dm=−∫Oyu(x,y)vy(x,y)Aj(x,y)dm −∫O(β−μy)u(x,y)v(x,y)Aj(x,y)dm−∫Oyu(x,y)v(x,y)(Aj)y(x,y)dm,

where , as in the proof of Proposition 3.3 but choosing such that, moreover, . We have . Note that

 ∣∣∣∫Oyu(x,y)v(x,y)jα(xj,yj)χ′(jy)dm∣∣∣≤∫O\mathbbm1y≤1j|u(x,y)v(x,y)|∥α∥∞supζ>0|ζχ′(ζ)|dm.

The last expression goes to 0 as since . The assertion follows by passing to the limit . ∎

We can now prove Proposition 3.1.

###### Proof of Proposition 3.1.

By using Proposition 3.3 and Proposition 3.4 we have

 ∫Oy∂2u∂x2vdm=−∫Oy∂u∂x(∂v∂x−γsgn(x)v)dm,
 ∫Oy∂2u∂y2vdm=−∫Oy∂u∂y∂v∂ydmγ,μ+∫O(μy−β)∂u∂yvdm,
 ∫Oy∂2u∂x∂yvdmγ,μ=−∫Oy∂u∂y(∂v∂x−γsgn(x)v)dm

and

 ∫Oy∂2u∂x∂yvdmγ,μ=−∫Oy∂u∂x∂v∂ydm+∫O(μy−β)∂u∂xvdm.

Recalling that

and using the equality , we have

 (Lu,v)H = −∫Oy2(∂u∂x∂v∂x+σ2∂u∂y∂v∂y+ρσ∂u∂x∂v∂y+ρσ∂u∂y∂v∂x)dm +∫O12∂u∂x(yγsgn(x)+ρσ(μy−β))vdm +∫O12∂u∂y(μσ2y−βσ2+ρσyγsgn(x))vdm +∫O[(ρκθσ−y2)∂u∂x+κ(θ−y)∂u∂y]vdm =−a(u,v).

###### Remark 3.5.

It is now clear why we have considered the process instead of the standard log-price process . Actually, the choice of allows to avoid terms of the type in the associated bilinear form . This trick will be crucial in order to obtain suitable energy estimates.

Recall the well known inequality

 bc=(√ζb)(1√ζc)≤ζ2b2+12ζc2, b,c∈R, ζ>0. (3.4)

Hereafter we will often apply (3.4) in the proofs even if it is not explicitly recalled each time.

###### Proposition 3.6.

For every , the bilinear form satisfies

 |a(u,v)|≤C1∥u∥V∥v∥V, (3.5)
 a(u,u)≥C2∥u∥2V−C3∥(1+y)12u∥2H, (3.6)

where

 C1=δ0+K1,C2=δ12,C3=δ12+K212δ1,

with

 δ0=sups21+t21>0,s22+t22>0|s1s2+ρσs1t2+ρσs2t1+σ2t1t2|2√(s21+t21)(s22+t22), (3.7)
 δ1=infs2+t2>0s2+2ρσst+σ2t22(s