Robustness of the Hobson-Rogers model with respect to the offset function

Vera Blaka Hallulli and Tiziano Vargiolu

Abstract. In this paper we analyse the robustness of the Hobson–Rogers model with respect to the offset function, which depends on the whole past of the risky asset and is thus not fully observable. We prove that, if the offset function is the realisation of a stationary process, then the error in pricing a derivative asset decreases exponentially with respect to the observation window. We present sufficient conditions on the volatility in order to characterise the invariant density and three examples.

Mathematics Subject Classification (2000). Primary 91B28; Secondary 60G10.

Keywords. Hobson–Rogers model, robustness, invariant measures.

1. Introduction

The year 1973 is a milestone in the modeling of financial markets: in fact, in that year the papers of Black and Scholes [2] and Merton [15], where an explicit formula for the price of call and put options was present, saw the light. The formula now known universally as “the Black and Scholes formula” links the price of a call option to quantities which are observed in the market (current price, strike price, time to maturity) and a parameter, the volatility, which gives an idea of how rapidly the asset prices can change. The two papers cited above influenced financial markets so deeply that every investment bank today has to deal with “the Black and Scholes approach”: this is also witnessed by the Nobel prize in 1997.

The so-called “Black and Scholes model” is however valid only as a first approximation: in fact, it was soon realised that the assumption of a constant volatility was in contrast with the empirical observations of derivative prices in real markets, which suggest that the volatility is not constant, but rather depends both on time to maturity and on the strike price.

Both the authors gratefully acknowledge financial support from the CNR Strategic Project Modellizzazione matematica di fenomeni economici.

In the last years a growing interest has been raised for models where the asset prices’ dynamics do not depend only on their current values, but also on past values: these models can be usually seen as generalisations of the so-called leveldependent volatility models, where the volatility is usually a function of time and current price level, and the market is complete. By making the volatility depend also on the past prices of the risky assets, it is also possible to enrich the model by reproducing correlations and dependencies which are observed in practice. Among these models, the one proposed by Hobson–Rogers [12] is the only case (to the authors’ knowledge) where the model is equivalent to a 2-dimensional Markov model, thus the problem of pricing and hedging a derivative asset is led to the solution of a linear PDE. In particular, one component of this Markov process represents the price and the other one represents the so-called offset function of order 1, which is an integral depending on all the past history of the asset price, and is thus not fully observable.

There are two ways of using the Hobson–Rogers model in practice. One is to consider a finite horison approximation, where the offset function is defined only on a finite observation interval of the past price. Unfortunately, the authors proved in a previous paper [10] that it is impossible to obtain a Markov system in this way. The other way is to use the pricing PDE with a misspecified initial offset function, thus making a mistake both on the path of the process as on the calculation of the price of the derivative assets. This approach is studied in detail in this paper. One can then search for the initial offset value which minimises this error. We find out that, for all the contingent claims which are Lipschitz continuous functions of the log-price of the asset, this error is proportional to the variance of the offset function at time 0. By assuming that we can observe the past prices of the risky asset on an interval of length R, this variance decreases exponentially with respect to R, and is proportional to the variance of the offset function at time −R. If we also assume that the offset function is a stationary process, we can calculate this variance, which does not depend on R: in this way, if one wants an error less than a given ε in pricing a derivative asset, one only has to observe the past price for a sufficient time R.

The paper is organised as follows. In Section 2 we present the Hobson–Rogers model. In Section 3 we make a survey, based on [10], on the reasons why a version of the Hobson–Rogers model with finite observation horizon loses Markovianity. In Section 4 we study the robustness of the Hobson–Rogers model with respect to the misspecification of the offset function, and in Section 5 we provide an estimate of the minimum observation horizon required for having an error less than a given threshold. In Section 6 we provide a way to calculate the variance of the offset function at the beginning of the observation window in terms of the invariant measure of the offset function, and provide sufficient conditions on the volatility in order to have a characterisation of the invariant density. Section 7 presents three examples.

2. The Hobson–Rogers model

We define the discounted log-price process Z(t) at time t as

Z(t) = log(S(t)e^{−rt})

where r is the (constant) risk-free interest rate, and the offset function of order m, denoted by P^{(m)}(t), by

P^{(m)}(t) = ∫_{0}^{∞} λe^{−λu}(Z(t) − Z(t − u))^{m}du for m = 0, …, n  (2.1)

the constant λ being a parameter of the model which describes the rate at which past information is discounted. Then, for some value n, we assume the following.

Assumption 2.1. Z(t) satisfies the SDE

dZ(t) = −\frac{1}{2}σ^{2}(P^{(1)}(t), …, P^{(n)}(t))dt + σ(P^{(1)}(t), …, P^{(n)}(t)) dW (t) (2.2)

where σ(·) and σ^{2}(·) are globally Liptschitz, σ(·) is strictly positive and (W_{t})_{t∈ℝ} is a so-called two-sided Brownian motion [3] under a probability measure ℙ, which is chosen such that (S(t)e^{−rt})_{t} is a ℙ-martingale.

This probability measure ℙ is in fact known as risk-neutral probability or martingale measure), and the existence of such a ℙ is equivalent to the non-existence of arbitrage opportunities in the market (see [10, 12] and the references therein for details).

This model can be seen as a ”good” model because no new Brownian motions (or other source of uncertainty) have been introduced in the specification of the price process. This means that the market is complete and any contingent claim is hedgeable (see [10] for details). On the other hand, it is possible to allow σ(·) to be a function of the price level S(t) also. So, this model can be extended to include the class of level-dependent volatility processes as a special case. The reason for the definition of the processes P^{(m)}(t), m = 0, …, n, is seen in the following lemma.

Lemma 2.2. (Z, P^{(1)}, …, P^{(n)}) is a (n + 1)-dimensional Markov process, and the offset processes P^{(m)}(t) satisfy the coupled SDEs

dP^{(m)}(t) = mP^{(m−1)}(t) dZ(t) + \frac{m(m − 1)}{2}P^{(m−2)}(t) d⟨Z⟩(t) − λP^{(m)}(t) dt,
m > 1  (2.3)
dP^{(1)}(t) = dZ(t) − λP^{(1)}(t) dt  (2.4)

Proof. See [12] □

Being (Z, P^{(1)}, …, P^{(n)}) a (n+1)-dimensional Markov process, we can easily employ the Kolmogorov equation when pricing a contingent claim with final payoff h(S(T)). In fact, (for sake of simplicity consider from now on the case n = 1 and denote P (t) ≡ P^{(1)}(t)) its price V (t) = 𝔼[h(S(T))|F_{t}] is of the form V (t) = F (t, S(t), P (t)), where F is the solution of the Kolmogorov equation

F_{t} + rsF_{s} − λpF_{p} + (\frac{1}{2}s^{2}F_{ss} + sF_{ps} + \frac{1}{2}F_{pp} − \frac{1}{2}F_{p}) σ^{2}(p) = rF  (2.5)

subject to the boundary condition

F (p, s, T) = h(s)  (2.6)

Besides, the solution of the hedging problem is a closed formula: it is enough to use the Itô formula on F and to make some calculations to obtain that the hedging strategy at time t is given by

Δ(t) = F_{s}(t, S(t), P (t)) + \frac{F_{p}(t, S(t), P (t))}{S(t)}

In conclusion this model allows to construct a process for the price, but we can see that some difficulties arise. In fact, for the computation of P (0) (or in general P (t)), we need to know the path of S on all its past (−∞, 0) (or (−∞, t)). This requirement is unusual in the modelisation of financial markets, where one usually meets models that start from a certain moment in time (usually 0). In fact, the requirement of an infinite horizon in the past raises mathematical and “practical” (or better economical) complications. From the mathematical side, we would have to define a stochastic calculus with time ranging on all the real line. Once that this is done, we would have to establish that P is well defined: in fact, remember that P is the integral of a process on (−∞, 0), so one must also prove that this integral is well defined. From the economical side, assets that ”existed forever” do not exist in the real market. Thus, one has to establish what can be used instead of the price path of S when the asset still did not exist.

While these problems seem less worrying than stated, mainly due to the exponential weight in (2.1), still theoretical (and practical) solutions to these issues are not present in literature, at least to the authors’ knowledge. For this reason, we will explore two different approaches to avoid these problems.

The first one consists in specifying a model with finite horizon and to make the volatility depend on integrals of the price path. Unfortunately up to now all the models of this kind present in literature [1, 8] do not give a Markovian structure as the Hobson-Rogers model does, unless one uses from the beginning a leveldependent volatility model: in the next Section 3 we present a survey, based on [10], of these results.

The second one is the following. The problem of pricing a contingent claim with the Hobson-Rogers model is equivalent to solve the PDE (2.5), once the initial conditions S(0) = s, P (0) = p are specified. While the price S(0) is observed in the market, in order to calculate the true value P (0) one would have to observe the asset in all its past. Since this is impossible, one has to use the model with a misspecification P̃(0). Our aim will be then to search for the initial condition P̃(0) which minimizes the error of pricing the contingent claim h(S(T)). This will be done from Section 4 on.

3. A finite delay model

Now we analyse a modification of the Hobson-Rogers model where we consider a finite time horizon and we make the risky asset’s dynamics depend on integrals of the price path. Inspired by a model in [8], the model that we study is

dS(t) = S(t)σ(Y (t), Z(t)) dW (t)

where the processes Y and Z are defined as

Y (t) = ∫_{0}^{τ} e^{−λv}f(S(t − v)) dv = ∫_{t−τ}^{t} e^{λ(u−t)}f(S(u)) du, Z(t) = S(t − τ)

where f is a strictly monotone function and τ is a given finite delay. Notice that for f(x) = log x and τ = +∞ one has that λY (t) = log S(t) − P^{(1)}(t), P^{(1)} being the first offset function of the Hobson-Rogers model. Our scope is now to find a selffinancing portfolio V which replicates the option with payoff h(S(T)) (or more generally h(S(T), Y (T))). Unlike in the Hobson-Rogers model, here the process (S, Y) is not Markov, and this is more due to the finite horizon nature of Y rather then to the specification of the volatility, more general than the Hobson-Rogers’ one.

One can immediately think to use the state variables (S(t), Y (t), Z(t)), but this entails the use of anticipative stochastic calculus. In fact, by making use of the Itô formula on a deterministic function of (S(t), Y (t), S(t − τ)), we end up with stochastic differentials of the kind G(t, S(t), Y (t), S(t − τ))dS(t − τ), where G(t, S(t), Y (t), S(t − τ)) is not adapted to the filtration of the differential dS(t − τ) but ”anticipates” (see [14] and the references therein). Conversely, we would have to prove that the portfolio dynamics could be written in the form dV (t) = Δ(t) dS(t), with Δ adapted to the filtration of S. In doing this, we will surely lose the Markovianity of the original Hobson-Rogers model.

One can be tempted to explore the following shortcut: though (S, Y) is not in general a Markov process, we make the strong assumption that for every final payoff of the form h(S(T), Y (T)) there exists a deterministic function F such that

V (t) = 𝔼[h(S(T), Y (T))|F_{t}] = F (t, S(t), Y (t))  (3.1)

If this assumption is true, then the self-financing portfolio depends in a deterministic way only on the current values of S and Y. Unfortunately, the next result states that the assumption (3.1) is equivalent to σ not depending on y, z, that is to S to be Markov; moreover, in this case, (3.1) is only true for h not depending on Y and the function F depends on t, s only.

Theorem 3.1. If assumption (3.1) is true, then σ_{z} = σ_{y} = 0.

The interested reader can find the proof in [10].

Remark 3.2. In this failed try, we were inspired by the positive results in [8]. We however have to say that in that paper the authors analyse a controlled system (which gives more degrees of freedom in reaching Markovianity), and also in that situation the authors succeed in reducing the system to the current values of S and Y only when the dynamics of S is linear and with some restriction on the coefficients.

4. Robustness of the Hobson-Rogers model

As already announced, now we focus ourselves in establishing what happens if our Markov process (P, Z) starts from a misspecified initial condition (P̃(0), Z(0)) instead of the true initial condition (P (0), Z(0)).

From now on, denote with Σ := (P, Z) the process with the correct (but not known) initial conditions and by Σ̃ = (P̃, Z̃) the process starting from the misspecified initial conditions (P̃(0), Z(0)). Then the evolution of (both Σ and) Σ̃ is given by

\{ \begin{matrix}dP̃(t) & = & −(\frac{1}{2}σ^{2}(P̃(t), Z̃(t)) + λP̃(t)) dt + σ(P̃(t), Z̃(t)) dW (t), \\ P̃(0) & ≠ & P (0) \\ dZ̃(t) & = & −\frac{1}{2}σ^{2}(P̃(t), Z̃(t)) dt + σ(P̃(t), Z̃(t)) dW (t), \\ Z̃(0) & = & Z(0) \end{matrix} (4.1)

the dynamics of Σ being driven by the same differential equation with the “right” initial conditions.

Now we present two estimates on the dependence of the process Σ (or Σ̃) on the initial condition: the first one is an L^{2}-estimate on sup_{0≤t≤T} |Σ(u) − Σ̃(u)|, and the second one is an L^{2}-estimate on |Σ(T)−Σ̃(T)|. Assume that the functions σ(p, z) and σ^{2}(p, z) are globally Lipschitz in (p, z) with respect to the Euclidean norm, in the sense that for f = σ, σ^{2} there exists K ≥ 0 (called Lipschitz constant of f) such that

|f(p, z) − f(p̃, z̃)| ≤ K|(p, z) − (p̃, z̃)| = K\sqrt{(p − p̃)^{2} + (z − z̃)^{2}} ∀(p, z), (p̃, z̃)

Theorem 4.1. If σ, σ^{2} are globally Lipschitz with Lipschitz constants respectively L, M, then for t ∈ [0, T] we have

𝔼 [ sup_{0≤u≤t} |Σ(u) − Σ̃(u)|^{2}] ≤ 3𝔼[|P (0) − P̃(0)|^{2}]e^{c(L,M,T)t}

where c(L, M, T) = 2M^{2}T + 6λ^{2}T + 20L^{2}, and

𝔼[|Σ(t) − Σ̃(t)|^{2}] ≤ 3𝔼[|P (0) − P̃(0)|^{2}]e^{C(L,M,T)t}

where C(L, M, T) = 2M^{2}T + 6λ^{2}T + 5L^{2}.

Results of this kind are classical in the theory of SDEs: we present the proof in order to show that the constants C(L, M, T) and c(L, M, T) are the best possible for our equations.

Proof. We have that

sup_{0≤u≤t} |Σ(u) − Σ̃(u)|^{2} ≤ sup_{0≤u≤t} |Z(u) − Z̃(u)|^{2} + sup_{0≤u≤t} |P (u) − P̃(u)|^{2}

which yields

𝔼 [ sup_{0≤u≤t} |Σ(u) − Σ̃(u)|^{2}] ≤

≤ 𝔼 [ sup_{0≤u≤t} |Z(u) − Z̃(u)|^{2}] + 𝔼 [ sup_{0≤u≤t} |P (u) − P̃(u)|^{2}] = (1) + (2)

For the first term in the right hand side, applying Doob’s inequality and the Lipschitz property of σ and σ^{2}, we have

(1) = 𝔼 [ sup_{0≤u≤t} |Z(0) − Z̃(0)) + ∫_{0}^{u} \frac{1}{2}(σ^{2} − σ̃^{2}) ds + ∫_{0}^{u}(σ − σ̃) dW (s)|^{2}] ≤

≤ 2𝔼 [ sup_{0≤u≤t} |∫_{0}^{t} \frac{1}{2}(σ^{2} − σ̃^{2})ds|^{2}] + 2𝔼 [ sup_{0≤u≤t} |∫_{0}^{u}(σ − σ̃)dW (s)|^{2}] ≤

≤ \frac{1}{2}T ∫_{0}^{t} 𝔼|σ^{2} − σ̃^{2}|^{2}ds + 8 ∫_{0}^{t} 𝔼|σ − σ̃|^{2}ds ≤
≤ (\frac{1}{2}M^{2}T + 8L^{2}) ∫_{0}^{t} 𝔼 [|Z − Z̃|^{2} + |P − P̃|^{2}] ds

where σ, σ̃ is a shorthand notation for σ(P (s), Z(s)), σ̃(s) = σ(P̃(s), Z̃(s)). For the second term we have

(2) ≤ 3𝔼|P (0) − P̃(0)|^{2} + 3T ∫_{0}^{t} 𝔼 [|\frac{1}{2}(σ^{2} − σ̃^{2}) + λ(P − P̃)|^{2}] ds +

+3𝔼 [ sup_{0≤u≤t} |∫_{0}^{u}(σ − σ̃)dW (s)|^{2}] ≤

≤ 3𝔼|P (0) − P̃(0)|^{2} + 3T ∫_{0}^{t} 𝔼 [\frac{1}{2}|σ^{2} − σ̃^{2}|^{2} + 2λ^{2}|P − P̃|^{2}] ds +

+12 ∫_{0}^{t} 𝔼|σ − σ̃|^{2}ds ≤

≤ 3𝔼|P (0) − P̃(0)|^{2} + (\frac{3}{2}M^{2}T + 12L^{2}) ∫_{0}^{t} 𝔼|Z − Z̃|^{2}ds +

+(3M^{2}T + 6λ^{2}T + 12L^{2}) ∫_{0}^{t} 𝔼|P − P̃|^{2}ds ≤

≤ 3𝔼|P (0) − P̃(0)|^{2} + (\frac{3}{2}M^{2}T + 6λ^{2}T + 12L^{2}) ∫_{0}^{t} 𝔼 [|P − P̃|^{2} + |Z − Z̃|^{2}] ds

then

𝔼 [ sup_{0≤u≤t} |Σ(u) − Σ̃(u)|^{2}] ≤

≤ 3𝔼|P (0) − P̃(0)|^{2} + (2M^{2} + 6λ^{2}T + 20L^{2}) ∫_{0}^{t} 𝔼|Σ(s) − Σ̃(s)|^{2}ds

and the theorem follows from the Gronwall lemma applied to v(t) = 𝔼[sup_{0≤u≤t} |Σ(u) − Σ̃(u)|^{2}]. For the estimate on 𝔼[|Σ(t) − Σ̃(t)|^{2}], the proof proceeds in a similar way, without applying Doob’s inequality to the term 𝔼[sup_{0≤u≤t} | ∫_{0}^{u}(σ − σ̃)dW (s)|^{2}]. □

Corollary 4.2. If h : C^{0}[0, T] → ℝ is the payoff of a path-dependent claim such that the function z(·) → h(e^{z(·)}) is globally Lipschitz, then

|𝔼[h(S_{T})] − 𝔼[h(S̃_{T})]|^{2} ≤ 3J^{2}𝔼|P (0) − P̃(0)|^{2}e^{c(L,M,T)T}  (4.2)

where J is the Lipschitz constant of z(·) → h(e^{z(·)}). If h is a simple European claim, then an analogous estimate holds, with C(L, M, T) instead of c(L, M, T) and J the Lipschitz constant of z → h(e^{z}).

Proof. We have that

𝔼|h(S(T)) − h(S̃(T))|^{2} ≤ J^{2}𝔼∥Z(·) − Z̃(·)∥_{C^{0}}^{2} ≤
≤ J^{2}𝔼 [ sup_{0≤t≤T}(|Z(t) − Z̃(t)|^{2} + |P (t) − P̃(t)|^{2})]

and from Theorem (4.1) we obtain Equation (4.2). □

We can see that the difference between the processes Σ and Σ̃ depends on the difference between the initial conditions P (0) and P̃(0). Unfortunately, we cannot obtain any improvement on the coefficients c(L, M, T) or C(L, M, T) in the case σ = σ(P).

Remark 4.3. Notice that in Corollary 4.2 the function z → h(e^{z}) is required to be globally Lipschitz, so a little caution must be used. For example, if the function h : ℝ → ℝ is globally Lipschitz and piecewise C^{1}, then

\frac{∂h(e^{z})}{dz} = e^{z}h′(e^{z})

is bounded (thus z → h(e^{z}) is globally Lipschitz) if and only if h′ decreases faster than e^{z}.

Consider now some examples.

Example (European put). The payoff is h(s) = (K − s)^{+}. We have

\frac{∂h(e^{z})}{∂z} = −e^{z}I_{z<log K}

then the Lipschitz constant in this case is less or equal than

I = sup_{z} |\frac{∂h(s)}{∂s}| = e^{log K} = K

Example (European call). The payoff is now given by h(s) = (s − K)^{+}. We can write this expression as h(s) = s − K − (K − s)^{+}, so the error is the same as when pricing the put.

Example (Asian put). The payoff is now given by h(s(·)) = (K − ∫_{0}^{T} s(t)dt)^{+}. For two generic paths z, z̄ ∈ C^{0}([0, T]), if both ∫_{0}^{T} e^{z(u)} du, ∫_{0}^{T} e^{z̄(u)} du are less than K, then

|h(e^{z(·)}) − h(e^{z̄(·)})| = |(K − ∫_{0}^{T} e^{z(u)} du)^{+} − (K − ∫_{0}^{T} e^{z̄(u)} du)^{+}| ≤

≤ | ∫_{0}^{T} e^{z(u)} du − ∫_{0}^{T} e^{z̄(u)} du| ≤
≤ | ∫_{0}^{T} |z(u) − z̄(u)|e^{sup(z(u),z̄(u))} du| ≤
≤ ∥z − z̄∥_{C^{0}} | ∫_{0}^{T} e^{sup(z(u),z̄(u))} du| ≤ 2K∥z − z̄∥_{C^{0}} (4.3)

where in the last line we applied the inequality

∫_{0}^{T} e^{sup(z(u),z̄(u))} du = ∫_{{u:z(u)>z̄(u)}} e^{z(u)} du + ∫_{{u:z̄(u)>z(u)}} e^{z̄(u)} du ≤ 2K

If (say) ∫_{0}^{T} e^{z̄(u)} du > K and ∫_{0}^{T} e^{z(u)} du ≤ K, then we can choose z̃ ∈ C^{0} such that ∫_{0}^{T} e^{z̃(u)} du = K and ∥z̃ − z∥_{C^{0}} ≤ ∥z̄ − z∥_{C^{0}} (for example, z̃:= tz + (1 − t)z̄ for a suitable t ∈ (0, 1)). Then

|h(e^{z(·)}) − h(e^{z̄(·)})| = K − ∫_{0}^{T} e^{z(u)} du = | ∫_{0}^{T} e^{z̃(u)} du − ∫_{0}^{T} e^{z(u)} du| ≤
≤ 2K∥z̃ − z∥_{C^{0}} ≤ 2K∥z̄ − z∥_{C^{0}}

by Equation (4.3). If both ∫_{0}^{T} e^{z(u)} du, ∫_{0}^{T} e^{z̄(u)} du are greater than K, there is nothing to prove. Then the Lipschitz constant in this case is equal to 2K.

Example (Lookback put). The payoff is now given by h(s(·)) =

(K − max_{0≤t≤T} s(t))^{+}. As above, if both max e^{z(·)}, max e^{z̄(·)} are less than K,

then we calculate

|h(e^{z(·)}) − h(e^{z̄(·)})| ≤ | max_{0≤u≤T} e^{z(u)} − max_{0≤u≤T} e^{z̄(u)}| ≤ ∥e^{z(·)} − e^{z̄(·)}∥_{C^{0}} ≤
≤ ∥z − z̄∥_{C^{0}}∥e^{max(z,z̄)}∥_{C^{0}} ≤ K∥z − z̄∥_{C^{0}}

If at least one of the quantities max e^{z(·)}, max e^{z̄(·)} is greater than K, an argument similar to the one of the previous example applies. Thus, in this case the Lipschitz constant is equal to K.

5. Using past information

We have seen in Section 4 that the error in pricing derivative assets depends on the difference between the true offset function P (0) and the misspecified value P̃(0), which we can choose. Of course, our aim will be to choose it in order to minimise the final error. In doing this, we are entitled to use not only the current value of S(0), but also past values.

More in detail, we assume (as it is reasonable) that we know all the past values of the price S(t) (thus, of Z(t)) for t ∈ [−R, 0], where R > 0 is a given real number which represents the width of an observation window in the past. As before, the process P (t) remains unobserved also in the past. However, it turns out that we can make the uncertainty on P decay exponentially with respect to the width R of the observation window. Again, we represent this uncertainty by defining the process P̃, starting from the misspecified condition P̃(−R) and following the dynamics

dP̃(t) = −λP̃(t) dt + dZ(t), t ∈ (−R, 0]  (5.1)
P̃(−R) ≠ P (−R)  (5.2)

while the process P always follows the dynamics given by Equation (2.4). Notice that this time, as we can observe Z in the interval [−R, 0], we have no uncertainty on this process.

The following lemma shows that, as both the dynamics of P̃ and P depend on the known values of Z, the difference between P (0) and P̃(0) decays exponentially with respect to the width R, as announced.

Lemma 5.1. For every choice of P̃(−R), we have

|P (0) − P̃(0)| = e^{−λR}|P (−R) − P̃(−R)|  (5.3)

Proof. By calculating the Itô differential of the process (e^{λt}P (t))_{t}, we have

d(e^{λt}P (t)) = e^{λt} dP (t) + λe^{λt}P (t) dt =
= e^{λt}(dZ(t) − λP (t) dt) + λe^{λt}P (t) dt = e^{λt} dZ(t)

and, analogously,

de^{λt}P̃(t) = e^{λt} dZ(t)

This means that, calculating the two processes in the two points t = −R, 0, we have

P (0) = e^{−λR}P (−R) + ∫_{−R}^{0} e^{λt} dZ(t),
P̃(0) = e^{−λR}P̃(−R) + ∫_{−R}^{0} e^{λt} dZ(t).

The lemma follows by calculating the difference. □

Remark 5.2. Notice that Equation (5.1) entails

P̃(0) = e^{−λR}P̃(−R) + Z(0) − e^{−λR}Z(−R) − ∫_{−R}^{0} λe^{λt}Z(t) dt =
= ∫_{0}^{R} λe^{−λu}(Z(0) − Z(−u)) du + e^{−λR}(Z(0) − Z(−R) + P̃(−R))

This can be seen by the properties of stochastic integrals of deterministic functions, or directly from Equation (2.1) (which obviously extends to P̃).

Now we are in the position of solving the following problem: for a given ε > 0 we want to find a minimum observation time R such that the error when pricing a contingent claim h is less than ε.

Corollary 5.3. If h is a general path-dependent claim as in Corollary 4.2 and

R > \frac{log(\frac{3J^{2}E|P (−R)−P̃(−R)|^{2}}{ε^{2}} ) + c(L, M, T)T}{2λ}  (5.4)

then

|E[h(S_{T})] − E[h(S̃_{T})]| < ε  (5.5)

Moreover, if h(S(T)) is the payoff of a simple European claim, then to obtain the same estimate it is sufficient that

R > \frac{log(\frac{3I^{2}E|P (−R)−P̃(−R)|^{2}}{ε^{2}} ) + C(L, M, T)T}{2λ}  (5.6)

Proof. From (5.4) we have

2λR > log (\frac{3J^{2}𝔼|P (−R) − P̃(−R)|^{2})}{ε^{2}}) + c(L, M, T)T

that yields

[c(L, M, T)T − 2λR] + log(3J^{2}𝔼|P (−R) − P̃(−R)|^{2}) < log ε^{2}

By taking the exponential of both the members we obtain

3J^{2}𝔼|P (−R) − P̃(−R)|^{2}e^{c(L,M,T)T −2λR} < ε^{2}

From (4.2) and (5.3) we have

|𝔼[h(Z_{T})] − 𝔼[h(Z̃_{T})]|^{2} ≤ 3J^{2}𝔼|P (−R) − P̃(−R)|^{2}e^{C(L,M,T)T −2λR}  (5.7)

this implies that (5.5) is verified. For the case of a European claim, the proof is the same with c(L, M, T) instead of C(L, M, T). □

6. Stationariety

So far, we have seen that the problem of estimating the pricing error when we misspecify the offset function P̃ is led to the knowledge of 𝔼[|P (−R) − P̃(−R)|^{2}], which is in general not allowed as we do not know the initial distribution of P (−R), even if we can decide the value P̃(−R).

The situation can be simplified by much if we make the crucial assumption that the 2-dimensional process (P, Z) is stationary, or that the process P itself is stationary. In this case, if we want the error to be (for example) less than a given ε > 0, it is sufficient to fix P̃(−R) be equal to the mean of the invariant measure of P (this minimises the quantity 𝔼[|P (−R) − P̃(−R)|^{2}], which is thus equal to the variance of P (−R)) and to observe the risky asset in the past for a sufficiently long time R. In fact, if the process P is stationary and admits a unique invariant measure, under suitable assumptions the marginal distribution of P (t) converges, for t → +∞, to the invariant measure, regardless of the initial condition of P. This means that, if we assume that the process P started in the past at a time T << −R from an arbitrary initial condition, the distribution of P (−R) can be approximated very well by the invariant measure. Thus, the situation of finding 𝔼[|P (−R) − P̃(−R)|^{2}] boils down to finding the variance of the invariant measure for P, provided we let P̃(−R) be equal to the mean of the invariant measure.

While the general case when the volatility σ depends on both P and Z seems more difficult to analyse, much can be said in the case when σ depends only on P. In this case the process P is a Markov process with the following evolution

dP (t) = m(P (t)) dt + σ(P (t)) dW (t)  (6.1)

where m(x) = −\frac{1}{2}σ^{2}(x) − λx. We now give sufficient conditions for the existence and uniqueness for the invariant distribution. For this purpose we use the following theorem from [11], that gives condition for the existence of the invariant measure.

Theorem 6.1. Assume that there exists a function V ∈ C^{2}(ℝ) such that

V (x) ≥ 0,  sup_{|x|>R} LV (x) := −A_{R} → −∞ as R → ∞

where LV (x) := m(x)V′(x) + \frac{1}{2}σ^{2}(x)V′′(x) and R is arbitrary. Then there exists a solution of Equation (6.1) which is a stationary Markov process.

Take V (x) = x^{2}, then

LV (x) = (−\frac{1}{2}σ^{2}(x) − λx) x + \frac{1}{2}σ^{2}(x) = \frac{1}{2}(1 − x)σ^{2}(x) − λx^{2}

Now if we assume

σ^{2}(x) ≤ a|x| + b  (6.2)

it follows that

LV (x) ≥ −\frac{1}{2}(x − 1)(a|x| + b) − λx^{2} =
= −\frac{1}{2}ax|x| − \frac{1}{2}bx + \frac{1}{2}(a|x| + b) − λx^{2}  (6.3)

If x > 0 then LV (x) → −∞ when R → ∞. If x < 0 then

LV (x) ≥ (\frac{1}{2}a − λ) x^{2} − \frac{1}{2}bx + \frac{1}{2}(a|x| + b) → −∞

if a < 2λ. We can thus conclude with the following result.

Theorem 6.2. If Assumption (6.2) holds with a < 2λ, there exists an invariant measure for the process (6.1).

In order to obtain also uniqueness results, we will need additional assumptions. If the process P has an invariant probability with density μ(x), from the backward Kolmogorov equation we have

0 = −\frac{d[m(x)μ(x)]}{dx} + \frac{1}{2} \frac{d^{2}[σ^{2}(x)μ(x)]}{dx^{2}}
0 = \frac{d}{dx} [−m(x)μ(x) + \frac{1}{2} \frac{dσ^{2}(x)μ(x)}{dx}]  (6.4)

this implies that

\frac{1}{2} \frac{dσ^{2}(x)μ(x)}{dx} = m(x)μ(x) + c

Assume that c = 0 and y(x) = σ^{2}(x)μ(x): then we have

∫ \frac{dy}{y} = ∫ \frac{2m(x)}{σ^{2}(x)}dx
ln y = ∫_{x_{0}}^{x} \frac{2m(u)}{σ^{2}(u)}du + ln C
y(x) = Ce^{∫{x \atop x_{0}} \frac{2m(u)}{σ^{2}(u)}} du  (6.5)

where C is an arbitrary constant and x_{0} is an arbitrary point. If the following relation

μ(x) = C \frac{e^{G(x)}}{σ^{2}(x)}  (6.6)

where G(x) = ∫_{x_{0}}^{x} \frac{2m(u)}{σ^{2}(u)}du, gives a density, this is the invariant density for our process P.

Now we study the conditions for existence and uniqueness of the invariant measure for the process P when σ satisfies the following assumption:

Assumption 6.3. There exist a ∈ [0, 2λ), b, ε > 0 such that

ε ≤ σ^{2}(x) ≤ a|x| + b

Theorem 6.4. If σ satisfies Assumption (6.3), then there exists an unique invariant measure for P, with density given by (6.6). Moreover, if P^{−T,η} follows the dynamics (6.1) with initial condition P^{−T,η}(−T) = η with −T < −R, then for every initial distribution η and E ∈ ℝ, we have

lim_{T →∞} 𝔼[(P^{−T,η}(−R) − E)^{2}] = ∫_{ℝ}(x − E)^{2}μ(x) dx

Proof. By results contained in [11], it is sufficient to prove that

∫_{−∞}^{∞} \frac{e^{G(x)}}{σ^{2}(x)} dx < ∞

and that

∫_{−∞}^{0} \frac{e^{−G(x)}}{σ^{2}(x)} dx = ∫_{0}^{∞} \frac{e^{−G(x)}}{σ^{2}(x)} dx = +∞

where

G(x) = ∫_{0}^{x} (−1 − \frac{2λu}{σ^{2}(u)}) du = −x − ∫_{0}^{x} \frac{2λu}{σ^{2}(u)} du

If x ≥ 0,

G(x) ≤ −x − \frac{2λ}{a} ∫_{0}^{x} \frac{au + b − b}{au + b}du + C = −x − \frac{2λ}{a}x + \frac{2λ}{a^{2}} ∫_{0}^{x} \frac{−b}{au + b}du + C
= −x − \frac{2λ}{a}x + \frac{2λb}{a^{2}} ln(ax + b) + C_{1} =: n_{1}(x)

If x < 0,

G(x) ≤ −x − 2λ ∫_{0}^{x} \frac{u}{ε} du = −x − \frac{λ}{ε}x^{2} =: n_{2}(x)

where as usual C, C_{1}, are some constants. Then e^{G(x)} ≤ e^{n_{1}(x)} if x ≥ 0 and e^{G(x)} ≤ e^{n_{2}(x)} if x < 0. So, we can write

∫_{−∞}^{∞} \frac{e^{G(x)}}{σ^{2}(x)}dx ≤ ∫_{−∞}^{∞} \frac{e^{G(x)}}{ε} dx ≤

≤ K_{1} ∫_{−∞}^{0} \frac{e^{−x− \frac{λ}{ε} x^{2}}}{ε} dx + K_{2} ∫_{0}^{+∞} \frac{e^{−x(1+ \frac{2λ}{a})}(ax + b)^{\frac{2λb}{a^{2}}}}{ε} dx < +∞

where K_{1} and K_{2} are constants. Besides,

∫_{−∞}^{0} e^{−G(x)} dx ≥ ∫_{−∞}^{0} e^{−n_{2}(x)} dx = K_{1} ∫_{−∞}^{0} e^{x+ \frac{λ}{ε} x^{2}} dx = +∞,
∫_{0}^{∞} e^{−G(x)} dx ≥ ∫_{0}^{+∞} e^{−n_{1}(x)} dx = K_{2} ∫_{0}^{+∞} e^{x(1+ \frac{2λ}{a})}(ax + b)^{− \frac{2λb}{a^{2}}} dx = +∞
□

7. Some examples

Now we analyse some particular specifications for σ. The first two are present in the original Hobson-Rogers paper and in other related papers (see [12]), while the third is suggested by the fact that affine processes are very often used in Mathematical Finance, and they have a well-established theory.

7.1. The case σ(P) = min\{\sqrt{a + bP^{2}}, N\}. This example comes from the original Hobson-Rogers paper [12]:

σ(P) = min\{\sqrt{a + bP^{2}}, N\}  (7.1)

where a > 0, b > 0 and N > 0 are some constants. As σ satisfies Assumption 6.3 for each possible value of a, b, N > 0, we can calculate the function G(x):

G(x) = −(x − x_{0}) − ∫_{x_{0}}^{x} \frac{2λu}{σ^{2}(u)}du

When x < \sqrt{\frac{N^{2}−a}{b}} the function G(x) becomes

G(x) = −(x − x_{0}) − ∫_{x_{0}}^{x} \frac{2λu}{N^{2}} du = (x_{0} − x) − \frac{2λ}{N^{2}} (\frac{x^{2}}{2} − \frac{x_{0}^{2}}{2} ) = −x − \frac{λ}{N^{2}} x^{2} + L_{1}

where L_{1} is a constant. In this case the function μ(x) is equal to

μ(x) = C \frac{e^{G(x)}}{N^{2}} = K \frac{e^{− \frac{λ}{N^{2}} x^{2}−x}}{N^{2}} = K_{1} \frac{e^{− \frac{λ}{N^{2}} (x+ \frac{N^{2}}{2λ})^{2}}}{N^{2}}

where K and K_{1} are constants. When x ∈ [−\sqrt{\frac{N^{2}−a}{b}}, +\sqrt{\frac{N^{2}−a}{b}}], the function

G(x) is

G(x) = −(x − x_{0}) − ∫_{−\sqrt{\frac{N^{2}−a}{b}}}^{x} \frac{2λu}{a + bu^{2}} du = −x − \frac{λ}{b} ln(a + bx^{2}) + L_{2}

where L_{2} is a constant. The function μ(x) is equal to

μ(x) = K_{2} \frac{e^{−x}(a + bx^{2})^{− \frac{λ}{b}}}{a + bx^{2}} = K_{2}e^{−x}(a + bx^{2})^{− \frac{λ}{b} −1}

where K_{2} is a constant. Now we see the case when x > \sqrt{\frac{N^{2}−a}{b}}. In this case the function G(x) is

G(x) = −(x − x_{0}) − ∫_{√ \overline{\frac{N^{2}−a}{b}}}^{x} \frac{2λu}{N^{2}}du = −x − \frac{λ}{N^{2}} x^{2} + L_{3}

for some constant L_{3}. Then

μ(x) = K_{3} \frac{e^{− \frac{λ}{N^{2}} (x+ \frac{N^{2}}{2λ})^{2}}}{N^{2}}

where K_{3} is a constant. The function μ(x) must be continuous at the points

x_{1} = −\sqrt{\frac{N^{2}−a}{b}} and x_{2} = \sqrt{\frac{N^{2}−a}{b}}, so that at this point we have

lim_{x→x_{1}^{−}} μ(x) = lim_{x→x_{1}^{+}} μ(x), and lim_{x→x_{2}^{−}} μ(x) = lim_{x→x_{2}^{+}} μ(x)

that implies

K_{2} = K_{1}e^{− \frac{λ(N^{2}−a)}{bN^{2}} − \frac{N^{2}}{4λ}} N^{\frac{2λ}{b}} , K_{3} = K_{1}

In conclusion, the invariant density is

μ(x) = \{ \begin{matrix}K_{1}e^{− \frac{λ(N^{2}−a)}{bN^{2}} − \frac{N^{2}}{4λ}} N^{\frac{2λ}{b}} e^{−x}(a + bx^{2})^{− \frac{λ}{b} −1} & |x| ≤ \sqrt{\frac{N^{2}−a}{b}} \\ K_{1}^{\frac{e^{− \frac{λ}{N^{2}} (x+ \frac{N^{2}}{2λ} )^{2}}}{N^{2}}} & |x| ≥ \sqrt{\frac{N^{2}−a}{b}} \end{matrix}

For the mean and the covariance of the process P under the invariant measure, there is not an explicit form. For this reason, a numerical calculation is required.

Example. As in [9], we take

a = 0.04, b = 0.2, λ = 1, N = 1

so we have

L = sup_{x∈ℝ} |\frac{∂σ}{∂x} | = \frac{\sqrt{b(N^{2} − a)}}{N} = 0.438178

and

M = sup_{x∈ℝ} |\frac{∂σ^{2}}{∂x} | = 2\sqrt{b(N^{2} − a)} = 0.876356

then we have

𝔼[P] = −0.022293, Var[P] = 0.022437

We want to find R such that (5.5) is verified for ε = 10^{−2}. If J = 1 (as is often the case), by taking different maturities, we find these results both for a general path-dependent claim as for a European one:

path-dependent claim  European claim
T c(L, M, T)  R  c(L, M, T)  R
0.25 5.724000  3.971457 2.844000 3.611457
0.5 7.608000 5.157957 4.728000 4.437957
1.0 11.376000  8.943957 8.496000 7.503957
2.0 18.912000 22.167957 16.032000 19.287957
3.0 26.448000 42.927957 23.568000 38.607957
4.0 33.984000 71.223957 31.104000 65.463957
5.0 41.520000 107.055957 38.640000 99.855957

In this case, if we want to make an error of less than ε = 10^{−2} in pricing (for example) a 6-months contingent claim, we have to observe the underlying asset for at least 5.15 years in the case of a path-dependent contingent claim and at least 4.43 years in the case of a European contingent claim.

Of course the situation can change, depending on the parameters. Take for example (always from [9])

a = 0.49, b = 2.45, λ = 1, N = 2.236068

Now we have

L = sup_{x∈ℝ} |\frac{∂σ}{∂x} | = 1.486573, M = sup_{x∈ℝ} |\frac{∂σ^{2}}{∂x} | = 6.648158

and

𝔼[P] = 1.281530, Var[P] = 2.674600

If again we want to find R such that (5.5) is verified for ε = 10^{−2} and J = 1, this time we find these results both for a general path-dependent claim as for a European one:

path-dependent claim  European claim
T c(L, M, T)  R  c(L, M, T)  R
0.25 67.797000  14.121001 34.648500  9.977439
0.5  91.396000  28.495376 58.247500  20.208251
1.0 138.594000  74.943376 105.445500  58.369126
2.0 232.990000 238.636376 199.841500 205.487876
3.0 327.386000 496.725376 294.237500 447.002626
4.0 421.782000 849.210376 388.633500 782.913376
5.0 516.178000 1296.091376 483.029500 1213.220126

In this case, if we want to make an error of less than ε = 10^{−2} in pricing (for example) a 6-months contingent claim, we have to observe the underlying asset for at least 28.49 years in the case of a path-dependent contingent claim and at least 20.20 years in the case of a European contingent claim.

7.2. The case σ^{2}(P) = \frac{a+bP^{2}}{c+d′P^{2}} Consider σ of the form

σ^{2}(P) = \frac{a + bP^{2}}{c + d′P^{2}}

where a, b, c, d′ are some positive numbers. As σ satisfies Assumption 6.3 for each possible value of a, b, c, d′ > 0, as in the previous section we calculate the function G:

G(x) = −(x − x_{0}) − 2λ ∫_{x_{0}}^{x} \frac{(c + d′u^{2})u}{a + bu^{2}}du = −x − λ ∫_{x_{0}^{2}}^{x^{2}} \frac{c + d′u^{2}}{a + bu^{2}} du^{2} + c_{0} =

= −x − \frac{λc}{b} ∫_{bx_{0}^{2}+a}^{bx^{2}+a} \frac{1}{y} dy − \frac{λd′}{b^{2}} ∫_{bx_{0}^{2}+a}^{bx^{2}+a} \frac{a + bu^{2} − a}{a + bu^{2}}d(bu^{2} + a) + c_{1} =
= −x − \frac{λc}{b} ln(bx^{2}) − \frac{λd′}{b^{2}} (bx^{2} + a) + \frac{λd′a}{b^{2}} ln(bx^{2} + a) + c_{2} =
= −x − \frac{λ(bc − ad′)}{b^{2}} ln(bx^{2} + a) − \frac{λd′}{b^{2}} (bx^{2} + a) + c_{2}  (7.2)

The function μ is

μ(x) = C \frac{e^{G(x)}}{σ^{2}(x)} = C \frac{e^{−x}(bx^{2} + a)^{− \frac{λ}{b^{2}} (bc−ad′)}e^{− \frac{λd′}{b^{2}} (bx^{2}+a)+c_{1}}}{\frac{a+bx^{2}}{c+d′x^{2}}}

= K \frac{e^{− \frac{λd′}{b} (x+ \frac{b}{2λd′})^{2}}(bx^{2} + a)^{− \frac{λ}{b^{2}} (bc−ad′)−1}}{c + d′x^{2}}  (7.3)

and it is the density of the unique invariant measure of the process P. Also in this case, we cannot calculate explicitly the mean and the variance of the process P, so a numerical integration is again required.

Example. We take

a = 0.452, b = 3.012, c = 1.0, d′ = 0.261, λ = 1.02

We calculate the Lipschitz constants L and M for the functions σ and σ^{2}. We have

L = sup_{x∈ℝ} |\frac{∂σ}{∂x} | = 1.22302 and M = sup_{x∈ℝ} |\frac{∂σ^{2}}{∂x} | = 3.67938

In fact, denote

k(x) := \frac{∂σ(x)}{∂x} = \frac{(bc − ad′)x}{(c + d′x^{2})^{\frac{3}{2}} \sqrt{a + bx^{2}}}

which reaches its maximum for x = ±\sqrt{\frac{−d′a+\sqrt{d^{′2}a^{2}+4abcd′}}{4d′b}}. Then

L = |k( ± \sqrt{\frac{−d′a + \sqrt{d^{′2}a^{2} + 4abcd′}}{4d′b}} )| = 1.22302

Similarly, let us denote

g(x) := \frac{∂σ^{2}}{∂x} = \frac{2(bc − ad′)x}{(c + d′x^{2})^{2}}

which reaches its maximum for x = ±\sqrt{\frac{c}{3d′}}. Then

M = |g( ± \sqrt{\frac{c}{3d′}} )| = \frac{2|bc − ad′|\sqrt{\frac{c}{3d′}}}{(c + d′ \frac{c}{3d′})} | = 3.67938

We obtain

𝔼[P] = −0.324053, Var[P] = 0.612203

and we have these results respectively for a path-dependent and for a European contingent claim:

path-dependent claim  European claim
T c(L, M, T)  R  c(L, M, T)  R
0.25 38.245077  9.499770 15.808408  6.750178
0.5  46.574596 16.228215 24.137927 10.729032
1.0  63.233633 35.809752 40.796964 24.811385
2.0  96.551707 99.471410 74.115038 77.474675
3.0 129.869782 195.797846 107.433113 162.802745
4.0 163.187856 324.789061 140.751187 280.795593
5.0 196.505930 486.445055 174.069261 431.453220

7.3. The case σ^{2}(P) = a + bP. Suppose that the process P is a so-called affine process [5], i.e. σ is given by

σ^{2}(P) = a + bP  (7.4)

where a and b are two arbitrary constants. So, Equation (6.1) becomes:

dP (t) = (− (\frac{b}{2} + λ) P (t) − \frac{a}{2}) dt + \sqrt{a + bP (t)}dB(t)  (7.5)

Clearly, there is a solution to (7.5) when the process a + bP (t) is non-negative for all t. So, the domain D implied by the non-negativity is

D = {x ∈ R : a + bx > 0}

We will therefore need to assume, in effect, that the process a + bP (t) has a sufficiently strong positive drift on the boundary point x = −\frac{a}{b}. Under the following assumption, we have a unique (strong) solution for the stochastic equation (7.5).

Assumption 7.1. We assume that 2λa > b^{2}.

In fact, for x such that a + bx = 0, b[−(\frac{1}{2}b + λ)x − \frac{1}{2}a] > \frac{b^{2}}{2} , i.e. equivalently (1 + \frac{2λ}{b})x + (1 + \frac{a}{b}) < 0, this implies 2λa > b^{2}. See [5].

Theorem 7.2. Under Assumption (7.1), there is a unique (strong) solution P to the stochastic differential equation (7.5) in the domain D. Moreover, for this solution P, we have a + bP (t) > 0 for all t almost surely.

Since σ is not Lipschitz, we cannot apply Theorem 4.1, but we have to formulate an analogous result here.

Theorem 7.3. If the coefficient σ satisfies (7.4), then for t ∈ [0, T] we have

𝔼 [ sup_{0≤u≤t} |Σ(u) − Σ̃(u)|^{2}] ≤ [3𝔼|P (0) − P̃(0)|^{2} + \frac{10b^{2}}{θ} t] e^{c(θ,T)t}  (7.6)

20 V. Hallulli and T. Vargiolu where θ is an arbitrary parameter and c(θ, T) = [3(\frac{b}{2} + λ)^{2} + \frac{b^{2}}{2} ] T + 10b^{2}θ, and

𝔼|Σ(t) − Σ̃(t)|^{2} ≤ [3𝔼|P (0) − P̃(0)|^{2} + \frac{5b^{2}}{2θ′} t] e^{C(θ′,T)t}  (7.7)

where θ′ is an arbitrary parameter and C(θ′, T) = \frac{b^{2}}{2} (T + 5θ′) + 3(\frac{b}{2} + λ)^{2}T. Proof. We have that

𝔼 [ sup_{0≤u≤t} |Σ(u) − Σ̃(u)|^{2}] ≤

≤ 𝔼 [ sup_{0≤u≤t} |Z(u) − Z̃(u)|^{2}] + 𝔼 [ sup_{0≤u≤t} |P (u) − P̃(u)|^{2}] = (1) + (2)

For the first term we have

(1) = 𝔼 [ sup_{0≤u≤t} |−\frac{1}{2} ∫_{0}^{u} b(P (s) − P̃(s))ds+

+ ∫_{0}^{u} (\sqrt{a + bP (s)(s)} − \sqrt{a + bP̃(s)}) dW (s)|^{2}] ≤

≤ 2𝔼 [ sup_{0≤u≤t} |\frac{1}{2} ∫_{0}^{u} b(P (s) − P̃(s)) ds|^{2}] +

+2𝔼 [ sup_{0≤u≤t} |∫_{0}^{u} (\sqrt{a + bP (s)} − \sqrt{a + bP̃(s)})dW (s)|^{2}]

≤ \frac{b^{2}}{2} T ∫_{0}^{t} 𝔼|P (s) − P̃(s)|^{2} ds + 8b^{2} ∫_{0}^{t} 𝔼|P (s) − P̃(s)| ds ≤
≤ (\frac{b^{2}}{2} T + 4b^{2}θ) ∫_{0}^{t} 𝔼|P (s) − P̃(s)|^{2} ds + \frac{4b^{2}}{θ} t.

where in the third line we apply the inequality

|\sqrt{a + bP} − \sqrt{a + bP̃}| ≤ b\sqrt{|P − P̃|}

and in the last line the inequality

|P − P̃| ≤ \frac{θ}{2}|P − P̃|^{2} + \frac{1}{2θ}

which holds for any real number θ > 0. Then,

(2) = 𝔼 [ sup_{0≤u≤t} |P (0) − P̃(0) − (\frac{b}{2} + λ) ∫_{0}^{u}(P − P̃)ds+

+ ∫_{0}^{u} (\sqrt{a + bP} − \sqrt{a + bP̃})dW (s)|^{2}] ≤

≤ 3𝔼|P (0) − P̃(0)|^{2} + 3 (\frac{b}{2} + λ)^{2} T ∫_{0}^{t} 𝔼|P − P̃|^{2}ds +

+12 ∫_{0}^{t} 𝔼|\sqrt{a + bP} − \sqrt{a + bP̃}|^{2}ds ≤

≤ 3𝔼|P (0) − P̃(0)|^{2} + 3 (\frac{b}{2} + λ)^{2} T ∫_{0}^{t} 𝔼|P − P̃|^{2} ds +

+6b^{2}θ ∫_{0}^{t} 𝔼|P − P̃|^{2}ds + \frac{6b^{2}}{θ} t =

= 3𝔼|P (0) − P̃(0)|^{2} + [3 (\frac{b}{2} + λ)^{2} T + 6b^{2}θ] ∫_{0}^{t} 𝔼|P − P̃|^{2}ds + \frac{6b^{2}}{θ} t.

then

𝔼 [ sup_{0≤u≤t} |Σ(u) − Σ̃(u)|^{2}] ≤ 3𝔼|P (0)−P̃(0)|^{2}+ \frac{10b^{2}}{θ} t+c(θ, T) ∫_{0}^{t} 𝔼|Σ(s)−Σ̃(s)|^{2}ds

Similarly as in Theorem (4.1), the result follows from Gronwall lemma applied to v(t) = 𝔼 [sup_{0≤u≤t} |Σ(u) − Σ̃(u)|^{2}] . □

The parameters θ and θ′ which minimize the right hand side of Equation (7.6) and Equation (7.7) are

θ = \frac{−5b^{2}t + \sqrt{25b^{4}t^{2} + 3𝔼|P (0) − P̃(0)|^{2}}}{3𝔼|P (0) − P̃(0)|^{2}} , θ′ = \frac{1}{4}θ.

Now we calculate the function G(x). The inequality a + bx ≥ 0 is equivalent to x ≥ −\frac{a}{b} if b > 0 and to x ≤ −\frac{a}{b} if b < 0. Consider the case b > 0.

G(x) = − (x + \frac{a}{b}) − ∫_{− \frac{a}{b}}^{x} \frac{2λu}{a + bu}du =

= − (1 + \frac{2λ}{b}) x + \frac{2λa}{b^{2}} ln(a + bx) − \frac{4λa}{b^{2}} − \frac{a}{b}

So, the function μ(x) is

μ(x) = C \frac{e^{G(x)}}{σ^{2}(x)} = K \frac{e^{−(1+ \frac{2λ}{b})x}(a + bx)^{\frac{2λa}{b^{2}}}}{a + bx}

where K = Ce^{− \frac{4λa}{b^{2}} − \frac{a}{b}} is constant. For μ(x) to be a density, the quantity

∫_{− \frac{a}{b}}^{∞} e^{−(1+ \frac{2λ}{b})x}(a + bx)^{\frac{2λa}{b^{2}} −1}dx

must be finite. This is true if (1 + \frac{2λ}{b}) > 0 which is always true, and \frac{2λa}{b^{2}} − 1 > −1 i.e. a > 0.

Now we analyze the case b < 0. In this case x ≤ −\frac{a}{b} then,

G(x) = −(x − x_{0}) − ∫_{x_{0}}^{x} \frac{2λu}{a + bu} du =
= (\frac{2λ}{b} − 1)x − \frac{2λa}{b^{2}} ln(bx + a) + C_{1}

where in the first line we change the variable of integration to y = a + bu and C_{1} is some constant. Similarly, as in the case b > 0, the function μ(x) is a density if

∫_{−∞}^{− \frac{a}{b}} e^{−(1− \frac{2λ}{b})x}(a + bx)^{− \frac{2λa}{b^{2}} −1}dx

is finite. This is true when (1 − \frac{2λ}{b}) < 0 (is equivalently when b > 2λ > 0), and −\frac{2λa}{b^{2}} − 1 > −1. But this is absurd because we supposed that b < 0. In conclusion,

μ(x) = K′e^{−(1+ \frac{2λ}{b})x}(a + bx)^{\frac{2λa}{b^{2}} −1}

is an invariant density for our process P in (−\frac{a}{b}, +∞) if and only if a > 0 and b > 0. In this case we can calculate the marginal mean and variance for the process P under the invariant measure. For the mean we have that for all t ∈ ℝ,

E[P (t)] = ∫_{− \frac{a}{b}}^{∞} xKe^{−(1+ \frac{2λ}{b})x}(a + bx)^{\frac{2λa}{b^{2}} −1}dx =
= \frac{1}{b} ∫_{− \frac{a}{b}}^{∞} K(bx + a − a)e^{−(1+ \frac{2λ}{b})x}(a + bx)^{\frac{2λa}{b^{2}} −1}dx =
= \frac{1}{b} ∫_{− \frac{a}{b}}^{∞} Ke^{−(1+ \frac{2λ}{b})x}(a + bx)^{\frac{2λa}{b^{2}}} dx − \frac{a}{b} =
= \frac{2λa}{b(b + 2λ)} ∫_{− \frac{a}{b}}^{∞} Ke^{−(1+ \frac{2λ}{b})x}(a + bx)^{\frac{2λa}{b^{2}} −1}dx − \frac{a}{b} =
= \frac{2λa}{b(b + 2λ)} − \frac{a}{b} = −\frac{a}{b + 2λ}

Robustness of the Hobson-Rogers model 23

Now we calculate E[P^{2}(t)]:

E[P^{2}(t)] = ∫_{− \frac{a}{b}}^{∞} Kx^{2}e^{−(1+ \frac{2λ}{b^{2}} )x}(a + bx)^{\frac{2λa}{b} −1}dx =

= \frac{1}{b^{2}} ∫_{− \frac{a}{b}}^{∞} K(bx + a − a)^{2}e^{−(1+ \frac{2λ}{b^{2}} )x}(a + bx)^{\frac{2λa}{b} −1}dx =

= \frac{1}{b^{2}} ∫_{− \frac{a}{b}}^{∞} Ke^{−(1+ \frac{2λ}{b^{2}} )x}(a + bx)^{\frac{2λa}{b} +1}dx −

−\frac{2a}{b^{2}} ∫_{− \frac{a}{b}}^{∞} Ke^{−(1+ \frac{2λ}{b^{2}} )x}(a + bx)^{\frac{2λa}{b}} dx + \frac{a^{2}}{b^{2}} =

= \frac{2λa + b^{2}}{b^{2}(b + 2λ)(1 + \frac{2λ}{b})} \frac{2λa}{b^{2}}b − \frac{4λa^{2}}{b^{2}(b + 2λ)} + \frac{a^{2}}{b^{2}} =

= \frac{a^{2} + 2λa}{(b + 2λ)^{2}} = \frac{a(a + 2λ)}{(b + 2λ)^{2}}

So that, the variance of the invariant measure of the process P is equal to

Var[P (t)] = E[P^{2}(t)] − E[P (t)]^{2} = \frac{2λa}{(b + 2λ)^{2}}

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24 V. Hallulli and T. Vargiolu

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Acknowledgment Both the authors gratefully acknowledge financial support from the CNR Strategic Project Modellizzazione matematica di fenomeni economici. The authors also wish to thank Paolo Dai Pra, Gianni Di Masi, Franco Flandoli, Alessandro Platania, Chris Rogers and Giovanna Villani for useful discussions on this topic, and an anonymous referee for having pointed out some ambiguities on the first draft of the manuscript.

Vera Blaka Hallulli Department of Pure and Applied Mathematics University of Padova via Belzoni 7 I-35131 Padova Italy

e-mail: hallulli@math.unipd.it

Tiziano Vargiolu Department of Pure and Applied Mathematics University of Padova via Belzoni 7 I-35131 Padova Italy

e-mail: vargiolu@math.unipd.it