Analysis of an uncertain volatility model

We examine, from both analytical and numerical viewpoints, the uncertain volatility model by Hobson-Rogers in the framework of degenerate parabolic PDEs of Kolmogorov type.


Introduction
Several extensions of the Black-Scholes model [3] have appeared in the literature (see, for a survey, Epps [6]) aiming to capture the characteristic observed patterns of the implied volatility given by the market.Here we are concerned with the seemingly promising model proposed by Hobson and Rogers [8] who assume that the volatility is a deterministic function of the history of the spot process.The merit of the model is twofold: first, it is potentially able to reproduce smiles, skews of different directions, and volatility term structures.Second, it preserves the completeness of the market since no exogenous source of risk is added, so that the classical arbitrage pricing and hedging theory apply.In particular, there are unique preference-independent prices for claims given in terms of the expectation under an equivalent martingale measure or as solutions to a PDE in three variables.Indeed incorporating the dependence on past prices entails an additional state variable on which the derivative's price depends: then, as in the case of Asian or look-back options, the associated PDE is of degenerate parabolic type.
In this paper we focus on the analytical and numerical treatment of the Hobson-Rogers model in the framework of Kolmogorov PDEs.Degenerate equations of Kolmogorov type naturally arise in the problem of pricing path-dependent contingent claims.The simplest significant example is given by Asian-style derivatives: if we assume that the stock price S t is a standard geometric Brownian motion with volatility σ, then the price U of a geometric average Asian option is a solution to the equation where A denotes the path-dependent variable and r is the risk-free rate (see, e.g., [2,18]).By an elementary change of variables, (1.1) can be reduced to the following PDE in R 3 : Although (1.2) is strongly degenerate due to the lack of diffusion in the y-direction, Kolmogorov [9] constructed an explicit fundamental solution to (1.2) of Gaussian type, which is a C ∞ function outside the diagonal (cf.(2.11)).Consequently (1.2) has a closed form solution and is hypoelliptic, that is, every distributional solution to (1.2) is a C ∞ function.
Starting from these considerations, two of the authors proved in [4] theoretical results about the convergence of non-Euclidean finite difference schemes for the pricing PDE in the Hobson & Rogers model.The aim of this paper is to refine and improve those results: we recall that the lack of a homogeneous structure for the Hobson & Rogers PDE was one of the main concerns in [4] (see the comments after Remark 3.7 below).In the next section we propose an ad hoc change of variables which solves that problem: as a major benefit we obtain efficient finite-differences schemes.Secondly, we study the discretization of the PDE in a bounded domain and we show that no boundary conditions on the y-variable are required to uniquely solve the related Cauchy-Dirichlet problem.This is a considerable advantage since there are no clear financial motivations for imposing conditions on the option price u(x, y,t) for fixed x.In the forthcoming paper [7], these techniques are used to test the performance of the Hobson & Rogers model on a set of S & P 500 options data.
This paper is organized as follows: in the next section, we briefly introduce the theory of Kolmogorov PDEs and describe their link with uncertain volatility models.Then, in Section 3 we consider the numerical solution to the option pricing problem in the Hobson-Rogers model by finite-differences schemes: our main goal is to provide some new non-Euclidean schemes which seem to be particularly efficient when compared with the classical ones.Finally in Section 4 we present some empirical results.

Hobson-Rogers model and Kolmogorov equations
In this section we cast the Hobson & Rogers model in the framework of Kolmogorov PDEs.In order to make the presentation self-contained we briefly review some of the main results regarding this class of evolutionary equations.
For a fixed maturity T and a probability space (Ω,Ᏺ,P) with one-dimensional Brownian motion B, we denote by S t the underlying price and by D t the deviation of prices from the trend, defined by where Z t = log(e −rt S t ).In (2.1), the parameter λ amounts to the rate at which past prices are discounted.In [8] Hobson and Rogers assume that S t is an Itô process satisfying where μ and σ are deterministic functions and σ is positive.Existence and uniqueness of the solution (S t ,D t ) to the system of SDEs (2.1)-(2.2) are guaranteed by the usual Lipschitz assumptions on μ and σ.
The main advantage of the Hobson-Rogers model is that no exogenous source of risk has been included.Then the complete market setting is preserved and usual no-arbitrage arguments provide unique option prices.In particular the derivative's price can be represented as the conditional expectation of the payoff under the (unique) P-equivalent risk-neutral measure.On the other hand, the deviation (2.1) enters as an additional state variable on which the option price depends.Then the drawback is that the augmented PDE associated to the model is of degenerate type.Indeed, let denote the price of a contingent claim at time T − t.By the Feynman-Kac formula the function f satisfies the PDE in R 3 : In the case of a European put option with strike K, we also have the condition Here we make the additional remark that (2.4) belongs to the noteworthy subclass of Hörmander PDEs today called of Kolmogorov or Ornstein-Uhlenbeck type.For this class a very satisfactory theory has been developed and many sharp results are available even under few regularity assumptions (see [10] for an exhaustive survey on this topic).We aim to show that the theory of Kolmogorov PDEs provides the natural framework for the study of the Hobson-Rogers model from both analytical and a numerical viewpoints.
For what follows, it is convenient to rewrite (2.4) in the following form: where u = u(x, y,τ) is determined by the transformation and we have set (2.9) By this change of variables, problem (2.4)-(2.5) is equivalent to the forward Cauchy problem for (2.7) in the strip (2.10) Transformation (2.8) seems to be more convenient than the one proposed by Hobson and Rogers (cf.[8,Section 4.2]).This will be apparent in the next section where we investigate the numerical approximation.From a qualitative viewpoint, we note that the derivative ∂ x u in (2.7) does not affect the main properties of the equation since it is, roughly speaking, "dominated" by the second-order part ∂ xx u.Then we may consider (2.7) as a perturbation of the Kolmogorov-type equation (1.2) with constant diffusion term.We recall that Kolmogorov [9] constructed explicitly a fundamental solution to (1.2).Precisely, we have for t > τ and Γ (z;ζ) = 0 for t ≤ τ is the fundamental solution to with pole at ζ = (ξ,η,τ) and evaluated in z = (x, y,t).We also recall that operator has the remarkable property of being invariant with respect to the non-Euclidean left translations in the law and homogeneous of degree two with respect to the dilations in the sense that Then it is natural to introduce in R 3 the δ s -homogeneous norm and the following notion of -Hölder continuity.Let α ∈]0, 1[ and Q be an open subset of R 3 ; we say that a function u : is finite.We remark explicitly that where C α (Q) denotes the space of Hölder continuous functions in the usual sense (see, e.g., [11]).Moreover we say that where Yu = x∂ y u − ∂ t u is the first-order part of (1.2).
Remark 2.1.Roughly speaking, one should consider ∂ x and Y , respectively, as a first-and second-order derivatives intrinsic to .These are the main directional derivatives of the degenerate (1.2) in the sense that they allow to recover the other lacking directions.For instance, ∂ y can be obtained as the commutator of ∂ x and Y : therefore it should be considered as a third-order derivative in the intrinsic sense.
In some way plays for (2.7) a role analogous to that of constant-coefficients operators in the classical theory of elliptic or parabolic PDEs (actually, a constant coefficient operator is nothing more than a translation-invariant operator).Then many results can be extended from (1.2) to (2.7) by perturbation arguments.In particular the next theorem, proved in [5,15], ensures the existence of a unique classical solution to the Cauchy problem for (2.7) under minimal regularity assumptions.
and a ≥ c for some positive constant c.Then (2.7) has a fundamental solution Γ and for every bounded ) and is the unique bounded, classical solution to the Cauchy problem for (2.7) with initial condition u(x, y,0) = φ(x, y).
We also recall that global estimates of Γ (and its derivatives) in terms of Γ are proved in [12,16]; then sharp estimates of the solution u in (2.20) and of its derivatives are available.In particular we recall the following Gaussian-type estimate of Γ: where C, c are positive constants and Γ c denotes the (explicitly known) fundamental solution to the Kolmogorov operator c∂ xx u + x∂ y u − ∂ t u.
The non-Euclidean differential-geometrical structure naturally associated to the Kolmogorov equation (1.2) also gives some insight for the numerical approximation of the solution.We recall that in the paper by Barucci et al. [2] the price of a geometric average Asian option is represented in terms of the solution to the Kolmogorov equation (1.2) and the non-Euclidean finite-differences scheme proposed in [17] is used for the numerical solution.In the next section we adapt this approach to the Hobson-Rogers model and perform a comparison among different numerical methods.
We close this section recalling that nonlinear Kolmogorov equations have been considered for pricing options with memory feedback by Peszek [14] and for a stochastic differential utility problem by Antonelli and Pascucci [1], Pascucci and Polidoro [13].

Finite-difference schemes for the Hobson-Rogers model
We consider the numerical solution of the option pricing PDE (2.4) by finite-difference methods.For simplicity we assume r = 0.A standard approach consists in approximating the derivatives ∂ S , ∂ D , and ∂ t by finite differences: we refer to this as an Euclidean finitedifference scheme.In view of Remark 2.1, it seems very natural to study the Kolmogorov equation (2.7) by approximating its main directional derivatives ∂ x and Y = x∂ x − ∂ t rather than the usual Euclidean derivatives: we call this a Kolmogorov finite-difference scheme.It turns out that this last scheme allows a better comprehension of the discrete structure of the equation and provides very efficient approximations: we refer to the next section for a comparison of numerical methods.We recall that Kolmogorov finitedifference schemes for (1.2) were first proposed by Polidoro and Mogavero [17] and applied to the Hobson-Rogers model by Di Francesco and Pascucci [4].
Solving options pricing PDEs by finite-difference methods requires the discretization of the equation in a bounded region.Therefore a primary question arises about the choice of the initial-boundary conditions.This specification seems to be unavoidable in the case of an Euclidean finite-difference scheme and it is usually made relying upon financial considerations.On the contrary, as we will see in Section 3.1, it is possible to derive boundary conditions for Kolmogorov schemes in bounded domains by purely analytical considerations.This is the first apparent advantage of Kolmogorov schemes over Euclidean schemes.

3.1.
A Euclidean finite-difference scheme.We consider problem (2.4)-(2.5)for the price of a European put option with strike K.We solve (2.4) in the bounded cylinder for some μ,ν > 0, subject to the following set of initial-boundary conditions: if f denotes the solution, we obviously impose the initial condition Moreover we assume Finally, let p BS (S,σ,T − t,K) denote the Black-Scholes price of a put option with time to expiry T − t, strike K, underlying price S and volatility σ; we assume that f (S,±ν,t) = p BS S,σ(±ν),T − t,K for (S,t) ∈ Ke −μ ,Ke μ ×]0, T[. (3.4)We consider the following transformation which slightly simplifies the problem: Then (2.4) (for r = 0) becomes for u = u(x, y,t) defined in the cylinder (3.8) Moreover, in view of the classical Black-Scholes formula, (3.4) corresponds to for (x,t) ∈] − μ,μ[×]0,λT[, where Φ(•) is the standard normal cumulative distribution function.We explicitly remark that the function u is independent of K, therefore by (3.5) it provides option prices for different strikes.
We consider an explicit finite-differences scheme on the uniform grid and we approximate L in (3.6) by the following discrete operator: where z = (x, y,t) and Operator L G is well defined on the grid G with x = y and approximates L in the sense that for some positive constant C depending on the L ∞ -norms of σ, ∂ θ has to be considered as a derivative of order twelve (in the intrinsic sense) whose existence is guaranteed assuming that σ has derivatives of order ten -Hölder continuous functions.
By standard arguments, we can prove the following convergence result.
Theorem 3.1.Let u be the solution to the Cauchy-Dirichlet problem for (3.6) in the cylinder Q subject to conditions (3.8)- (3.10).Let u G denote the solution to the correspondent discrete problem for L G in Q ∩ G with the same initial-boundary conditions.Assume Δ x ≤ 2 and the following stability condition: (3.15) 2. An implicit method can be developed by using the backward differences instead of the forward differences D t, t u(z) in L G .Even if this approach has the advantage of being unconditionally stable (i.e., the time step t can be chosen independently of x ), however it is very computationally expensive.Indeed the numerical computation of u G requires, at each time step, the solution of a linear system of order (2I + 1) 2 , where I = 2μ/ x .This becomes onerous in terms of memory and computational time when the spatial grid spacing diminishes.

An explicit Kolmogorov finite-difference scheme. We approximate problem (2.7)-(2.10) by a Cauchy-Dirichlet problem for
with a defined in (2.9), in the cylinder where μ,ν > 0 and τ 0 = 1 − e −λT .By transformation (2.8), this corresponds to the initialboundary value problem for (2.4) in the twisted region In the sequel we denote by the parabolic boundary of Q.
We consider an explicit Kolmogorov scheme on the uniform grid and we discretize ᏸ in (3.18) by approximating the main directional derivatives as follows: since in this case the discrete operator defined by is well defined on the grid G. Moreover ᏸ G approximates ᏸ in the sense that for some positive constant C depending on the L ∞ -norms of a, ∂ 3 x u, ∂ 4 x u, and Y 2 u on Q. Therefore estimate (3.26) involves the intrinsic derivatives of u up to the fourth order: this should be compared with the analogous result for the Euclidean scheme in the previous section.
We first show a general result about the convergence of the scheme (3.23) and then address the problem of the specification of the initial-boundary conditions for (3.18).By Remark 3.5, it is possible to assume that the vertices (±μ,±ν,0) belong to the grid G so that the discrete Cauchy-Dirichlet problem is well posed.Following [17], we prove the following.
Lemma 3.3 (maximum principle).Assume Δ x ≤ 2 and the stability condition we have that (3.28) is equivalent to The thesis follows by an elementary inductive argument.
By means of the maximum principle it is standard to prove the following convergence result.
Theorem 3.4.Let u (resp., u G ) denote the solution to the (discrete) Cauchy-Dirichlet problem for (3.18) (resp., (3.25)) in Q (resp., Q ∩ G) with given initial-boundary conditions.Assume the stability condition (3.27).Then It is easy to choose μ, ν, τ 0 , and a grid G such that conditions (3.24) and (3.27) hold and the vertices (±μ,±ν,0) belong to G. Suppose that μ and τ 0 are given: we fix a natural number m and set ν = mμτ 0 .Moreover we put x = μ/I and τ = τ 0 /N where I,N ∈ N and N ≥ 2τ 0 supa/ 2 x so that the stability condition holds.Finally, we set y = x τ .Then we have Finally we show a simple way to avoid giving conditions on the lateral boundary {y = ±ν} of Q.Let us consider the grid in (3.33).We first remark that for a solution v to i,k only depends on v n−1 i,k+i and v n−1 i±1,k+i (see (3.31)).More generally, it is straightforward to determine the domain of dependence of the set of values Indeed, if we put then, by (3.31), we have In particular k (0) = mNI, therefore V m is independent on the value of v at the lateral boundary y = ±ν.Let us note that this is true for every refinement of the grid, that is, for every choice of N and I.In view of these remarks, in order to approximate the solution u(x, y,τ 0 ) for |y| ≤ (m − 1)μτ 0 , conditions on the lateral boundary {y = ±ν} are superfluous.Alternatively, one can solve (3.18) in the prism .40)rather than in the whole cylinder Q.
Remark 3.7.Although attractive from an analytical viewpoint, the explicit Kolmogorov scheme has a high computational cost.Indeed the stability condition (3.27) combined with assumption (3.24) implies y = O( 3 x ) as x → 0. Thus, for instance, in a grid with 10 2 x -nodes we have about 10 4 τ -nodes and 10 6 y -nodes.For this reason, in the next section, we propose a first-order implicit scheme which combines the fine analytical properties of Kolmogorov schemes with the numerical efficiency.
Let us recall that an explicit Kolmogorov difference scheme for the Hobson-Rogers model was first proposed in [4].The starting point was the following PDE given in [8,Section 4.3]: Following the same approach used in [4], one may approximate the first-order term ∂ θ u, where θ = (0,x − y,−1), by the difference On the other hand, the "correction" introduces an additional error of order Δ x .Specifically, they show that Here, since transformation (2.8) leads to a homogeneous Kolmogorov equation, we considerably improve the results in [4].We close this section by proving Proposition 3.6.
Proof of Proposition 3.6.We only study the first limit, the second one being analogous.We note that for every ε > 0 there exists a positive constant

3.
3. An implicit Kolmogorov finite-difference scheme.In view of the previous section, it is quite simple to derive an implicit Kolmogorov finite-difference scheme for (3.18).We set Q as in (3.19) and consider a grid as in Remark 3.5.Then using the same notations of Section 3.2, the discrete operator is well defined on G (remember that y = x τ ) and approximates ᏸ in the sense that ᏸu − ᏸ G u L ∞ (Q∩G) ≤ C x + τ . (3.52) We may consider the same set of initial-boundary conditions used in the previous section: in particular it is straightforward to show that conditions at the lateral boundary {y = ±ν} do not affect the value of the solution in the prism (3.40).Finally, by means of the following maximum principle it is not difficult to prove a convergence result for ᏸ analogous to Theorem 3.4.

Lemma 3.8 (maximum principle). Suppose that
(3.54) Proof.By contradiction, suppose that M = max G∩Q v = v n i,k > 0. Then denoting price due to the influence of the boundary data.On the other hand, the Kolmogorov scheme demonstrates a hint of convergence to the solution, even if not so rapidly.
Regarding the computational cost of the methods, it should be noticed that the number of steps (and thus the computational cost) required by the Euclidean scheme depends on the volatility as well as on the time to expiration.Then, for simple problems like the one reported in Table 4.1, an Euclidean scheme is quite acceptable (even if it should be noted that, by reducing the grid spacing, we do not obtain any significant improvement of precision).On the contrary, the other examples show that the Euclidean scheme is more computationally expensive than the implicit Kolmogorov and not reliable (let us also recall Remark 3.2 and that implicit schemes of Euclidean type are particularly onerous).This is particularly apparent in Table 4.4 which shows a persistent error of nearly 30% in the Euclidean scheme, in addition to a high computational cost.This example enlightens the advantage of the Kolmogorov scheme in not requiring the boundary conditions on the lateral boundary {y = ±ν}.Experiments on a Crank-Nicholson method based on the Kolmgorov scheme have also been performed, but are not reported here because of their numerical unreliability when dealing with discontinuous payoff.
by (3.53), we have Note that, in view of Remark 2.1, ∂4 .42) This approach has the drawback that the two points z and z + Δ t θ cannot belong both to a uniform grid G like in(3.22): this is essentially due to the lack of a homogeneous structure for (3.41) analogous to that of the Kolmogorov equation (1.2).To overcome this problem, Di Francesco and Pascucci propose to approximate ∂ θ u by the "corrected difference" The thesis is a simple consequence of (3.45) and the fact that, by Theorem 2.2, c (x,τ;ξ,0)dη ≤ ε (3.47) if x ≤ −δ − Rand R is suitably large.This proves (3.45).