Model Risk in Finance: Some Modeling and Numerical Analysis Issues

Model Risk in Finance: Some Modeling and Numerical Analysis Issues Denis Talay, INRIA 2004 Route des Lucioles, B.P. 93, 06902 Sophia-Antipolis, France.

1. Introduction The impact of erroneous models and measurements is an important issue in all scientific and technological fields: equations and measurement devices provide approximate descriptions of our real world so that one needs to estimate and possibly control the effects of misspecifications during the modeling and calibration process. In fields such as physics, conservation laws constrain the models and the values of the model parameters, even when a part of stochasticity is involved to take uncertainties into account. As well, to solve numerically a partial differential equation (PDE) describing macroscopic quantities whose state space is unbounded, one needs to introduce artificial boundary conditions that allow one to compute the solution within a bounded domain; the design of these boundary conditions is a difficult issue, but one may be helped by intuitive considerations on the physical phenomenon under study; to give an example, if one desires to compute turbulent flows around airplane wings, one may assume that, away from the airplane, the velocity of the flow is equal to the wind velocity, and one thus may derive reasonable approximate Dirichlet conditions from a reasonable physical model. In finance, modeling issues are much more complex than in physics for, at least, the following reasons. First, no physical law helps the modeler to choose a particular dynamics to describe the time evolution of market prices or indices. The real market is incomplete and arbitrages occur. Moreover, no stationarity argument can help justify that parameters estimated from historical data will keep the same values in the next future. Therefore, the modeller has a high degree of freedom to mathematically describe the market in order to compute

Mathematical Modeling and Numerical Methods in Finance Copyright © 2008 Elsevier B.V. Special Volume (Alain Bensoussan and Qiang Zhang, Guest Editors) of All rights reserved HANDBOOK OF NUMERICAL ANALYSIS, VOL. XV ISSN 1570-8659 P.G. Ciarlet (Editor) DOI 10.1016/S1570-8659(08)00001-x 3

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D. Talay

optimal portfolio allocations or risk measures. For example, authors propose to model the volatility of a stock as a deterministic function of the stock (and possibly of exogeneous factors) or as a stochastic process; the stochastic differential equations involved in the models may be driven by Brownian motions or by discontinuous Lévy process; the bond market is modeled by short-term dynamics or by Heath–Jarrow–Morton (HJM) equations. In addition, to compute price options and deltas, practitioners and quants find it convenient to suppose that the no arbitrage and completeness hypotheses prevail: in diffusion models, this assumption constrains the dimension and the algebraic structure of the volatility matrix so that the model used to hedge may not exactly fit the market data. Second, statistical procedures issued from the theory of statistics of random processes and based upon historical data may be extremely inaccurate because of the lack of data. For example, an accurate parametric estimation of a volatility matrix requires that the asset price is observed at very high frequencies. As well, the parametric estimators of a drift parameter may need long-time observations to provide reliable results (see our illustration in Section 2.1). In such a case, one needs to assume that, during the whole period, the model remains relevant and its parameters remain constant. Of course, it would be unclever to use historical data only to calibrate financial models: in order to calibrate a stock price model, the practitioners not only actually consider the past prices of the stock only but also use other available information such as past prices of derivatives on this stock (see, e.g., papers and references in Avellaneda [2001]). However, the stationarity of the market during the observation period remains questionable, and error estimates for complex calibration methods are not available in the literature. Third, in finance one neither can use data issued from experiments repeated independently nor assume a kind of ergodicity in order to increase the set of available observations. The modeler needs to design and calibrate models using one single history of the market. Finally, model uncertainties also occur in the numerical resolution of PDEs related to option pricing or optimal portfolio allocation. Commonly used stochastic models in finance actually lead to consider processes whose time marginal laws have unbounded supports. Consequently, the PDEs are posed in unbounded domains, and artificial boundary conditions are necessary. The situation is quite different from the above example in fluid mechanics: usually one has a little knowledge on the behavior of the solution when the norm of the state variable increases: usually one finds estimates by working with simplified models. For an example of a rigorous procedure to design artificial boundary conditions for European options, see Costantini, Gobet and Karoui [2006]; for an analysis of the error induced by misspecified boundary conditions on American option prices, see Berthelot, Bossy and Talay [2004]. Consequently, model misspecifications cannot be avoided, which leads to model risk. The specificity and definitions of model risk are not universally admitted (see the extended introduction in Cont [2006] for an interesting discussion on this point and an extended list of references). In the present notes, we limit ourselves to a particular restricted family of questions: how to evaluate — and possibly control — the impact of certain model uncertainties on profit and losses (P&Ls) of hedging portfolios or on portfolio management strategies? We do not examine axiomatic questions on risk measures at all, for which we refer to Cheridito, Delbean and Kupper [2005], Barrieu and El Karoui [2005],

Model Risk in Finance: Some Modeling and Numerical Analysis Issues

5

and Föllmer and Schied [2002]. We rather adopt a pragmatic point of view and seek computational means to evaluate the impact of model uncertainties. We start with illustrating the difficulty in constructing a reliable market model by presenting recent results on one of the very first steps of the modeling process, namely, the design of the driving noise of the dynamics of the assets under consideration. We then present some results concerning the numerical approximation of measures of model risk such as Values at Risk (VaR) in diffusion environments. We also present a stochastic game problem related to model risk control. Finally, we propose a tentative methodology to compare the performances of financial strategies derived from (misspecified) mathematical models and strategies, which, derived from technical analysis, avoid modeling and calibration issues.

2. Limitations of statistical procedures based on historical data In the literature, one can find a huge number of papers that propose and analyze parametric and nonparametric estimators for the coefficients of stochastic differential equations. A more specific literature also exists on the statistics of stochastic models in finance (for a survey, see, Aït-Sahalia and Kimmel [2007]). Our purpose here is not to provide a summary of these works, even partially: we limit ourselves to refer to Prakasa Rao [1999a] and Prakasa Rao [1999b] and the references therein for the reader interested by an overview on the subject, and to Jacod [2000] for an advanced result on the identification of the volatility function with kernel estimators. In the latter reference, it is shown that, if a diffusion process is observed at times i/n and if the diffusion coefficient has regularity r, then the accuracy of the estimator is of order 1/nr/(1+2r) , pointwise and uniformly on compact subsets of R. Such a convergence rate is low and illustrates that the design of stochastic models for asset prices or indices from historical data necessarily leads to model risk. We give a few other illustrations below: we will start by an elementary observation that shows that the time scales that are necessary to calibrate stochastic models with good accuracies are often incoherent with the time scales at which the market evolves. We will then examine two questions involved, which, to our knowledge, were recently only tackled in the literature in spite of the fact that they should arise before calibration. They concern the driving noise, more precisely, its continuous or discontinuous nature, and (in the Brownian case) its dimension. 2.1. Cramer–Rao lower bounds Our elementary example concerns maximum likelihood estimators for drift parameters of diffusion processes and therefore the calibration of historical probability measures (e.g., in order to solve optimal porfolio management problems or to simulate benchmark histories of the market). We are given an open set  ⊂ R and a family of real-valued functions {b(θ, ·), θ ∈ }. Suppose that, for each θ ∈ , the function b(θ, ·) is Lipschitz and consider the model  t b(θ, Xsθ )ds + Bt , (2.1) Xtθ = X0 + 0

6

D. Talay

where (Bt ) is a standard Brownian motion. Up to a transformation by  xone-dimensional 1 dz, our situation covers the models with a strictly positive means of the function 0 σ(z) continuous volatility function σ(x). θ θ Let PX be the law of (Xtθ , 0 ≤ t ≤ T), and let EX denote the corresponding expectation. Suppose that the function b(θ, x) is continuously differentiable w.r.t. θ for all x and that 2  T  ∂b  Xθ   IT (θ) := E  ∂θ (θ, πs ) ds < ∞ for all θ ∈ . 0 Under weak additional conditions, for all unbiased estimator θˆ T of θ based upon an observation between times 0 and T such that the function θ QT (θ) := EX (θˆ T − θ)2

(2.2)

is bounded on compact sets, the quadratic estimation error is bounded from below: θ EX (θˆ T − θ)2 ≥

1 IT (θ)

for all θ ∈ .

The right-hand side is the Cramer–Rao lower bound. For a proof of this classical result, see Kutoyants [1984]. For example, consider the model dS θt = μStθ dt + σStθ dBt . Set Xtθ :=

1 log(Stθ ), that is, σ

dXθt = θdt + dBt with

θ :=

μ− σ

σ2 2 .

The Cramer–Rao lower bound implies that all estimator of θ based upon the observation of one trajectory of (Stθ ) — equivalently, of (Xtθ ) – in the time interval [0, T ], has a quadratic estimation error larger than T1 . If the unit of time is 1 year and if one observes the stock prices during 1 year, then the standard deviation of the error cannot be lower than σ. 2.2. Testing whether the noise has jumps In an impressive recent paper, Aït-Sahalia and Jacod [2008] constructed and analyzed a rule to decide whether a price process observed at discrete times is continuous or jumps at least once during the observation time interval. Their paper substantially improves previous works mentioned in its list of references.

Model Risk in Finance: Some Modeling and Numerical Analysis Issues

7

The observed process (Xt ) is supposed to belong to a fairly general class of models, namely, it is supposed to satisfy  Xt = X0 + +

 t 0

t

0

R



t

bs ds +

σs dBs +

0

 t 0

R

κ ◦ δ(s, x)(μ − ν)(ds, dx)

(δ(s, x) − κ ◦ δ(s, x))μ(ds, dx).

(2.3)

Here, B is a Brownian motion, μ is a Poisson random measure with an intensity measure of the form ν(ds, dx) = ds ⊗ dx; the function κ is continuous and locally equal to x around the origin; the processes (bs ) and (σs ) are optional, and the random function δ(s, ·) is predictable and uniformly bounded in ω and time by a deterministic function γ  such that R min(γ(x)2 , 1)dx < ∞. The authors require a few technical conditions that are not limitative for applications in finance (e.g., the process (σt ) is supposed to be of the same type as (Xt ) itself). Now, denote by n a sequence of observation time steps decreasing to 0. Aït-Sahlia and Jacod’s test statistics is t/ n |X2i n − X2(i−1) n |p ˆ . C(p, n )t := i=1 t/ n p k=1 |Xi n − X(i−1) n | Theorem 2.1. Under the above assumptions, for all t > 0 and p > 2, the variables ˆ C(p, n )t converge in probability when n goes to infinity to p

I{ω;s→Xs (ω) is continuous on [o,t]} + 2 2 −1 I{ω;s→Xs (ω) is discontinuous on [o,t]} . Therefore, the decision rule consists in accepting the hypothesis “the process (Xt ) is p/2−1 p/2−1 ˆ ˆ n )t ≥ 1+22 . discontinuous” if C(p, n )t < 1+22 , and rejecting it if C(p, The authors prove several limit theorems that allow them to construct levels of tests based on their tests statistics. In particular, they show the following theorem. −1/2 ˆ n )t − 1), when restricted to the set of disconTheorem 2.2. For p > 3, n (C(p, tinuous paths, converges stably in law. −1/2 ˆ n )t − 2p/2−1 ) converges stably in law. If X is continuous, for p ≥ 2, n (C(p,

In both cases, the limits are constructed on an extension of the original probability space, but their conditional distribution w.r.t. the original filtration is Gaussian; the two conditional variances are explicited in terms of respectively,  s≤t

2 ) |Xs − Xs− |2p−2 (σs2 + σs−  2 p |X − X | s s− s≤t

8

D. Talay

and t

|σs |2p ds  2 . t p ds |σ | 0 s 0

These two asymptotic variances can be estimated by means of the discrete time observations of X. It is consequently possible to construct real tests for the null hypothesis that X is discontinuous as well as for the null hypothesis that X is continuous. For precise critical regions, asymptotic levels and power functions, we refer to Aït-Sahalia and Jacod [2008]. Simulation studies reported in the paper illustrate that observations at high frequencies actually allow one to discriminate continuous and discontinuous models. Similarly, when applied to real historical data (Dow Jones Industrial Average stock prices in 2005), observations each 5 seconds lead to the conclusion that most of the prices should be modeled by models with jumps. However, as predicted by the theoretical results, observations each 30 seconds do not allow one to get a significant information from the test. In conclusion, although Brownian models are commonly used to compute prices and deltas, it seems that driving noises with jumps should also be considered, especially for prices or physical variables observed at low frequencies since, in such a case, it is impossible to test the (dis)continuity hypothesis. 2.3. The explicative Brownian dimension of a stochastic model Suppose now that one observes prices of a basket of d assets and that these prices are Itô processes driven by a q-dimensional Brownian motion. If no arbitrage and completeness are assumed, then d = q. However, it sometimes is useless to calibrate a volatility matrix of dimension d: for example, some components of the noise may play a very small role in the dynamics of the price and, consequently, considering that they are null may not change much the prices of options on the basket under consideration. More generally, one may have to calibrate models for families of processes that do not model prices but indices, meteorological or economical variables, etc, for which the number of random sources is not constrained by no arbitrage or completeness conditions. In all cases, by eliminating “small” noises in the dynamics, one simplifies the calibration of the volatility matrix and decreases the number of operations in the simulations of the model. Jacod, Lejay and Talay [2008] have tackled the question of estimating the “explicative Brownian dimension” of an Itô process from a discrete time observation. By “explicative Brownian dimension rB ,” we (informally) mean that a model driven by rB dimensional Brownian motion satisfyingly fits the information conveyed by the observed path, whereas increasing the Brownian dimension does not bring a better fit. More precisely, suppose that we observe a path of the process 

t

Xt = X0 + 0



t

bs ds +

σs dBs , 0

Model Risk in Finance: Some Modeling and Numerical Analysis Issues

9

where B is a standard q-dimensional Brownian Motion, (bs ) is a predictable Rd -valued locally bounded process, σ is a d × q matrix-valued adapted and càdlàg processes. Set cs := σs σs . Our aim is to estimate the maximal explicative rank of cs on the basis of the observation of XiT/n for i = 0, 1, . . . , n. Of course, a natural candidate should resemble the integer such that, if λ(1)s , . . . , λ(d )s are the eigenvalues of cs in decreasing order, then λ(rB )s is significantly larger than λ(rB + 1)s . However, this sole definition does not lead to a tractable test since one observes a trajectory of (Xt ) and not of (ct ); in particular, this implies that we cannot hope to approximate the eigenvalues of cs with a good accuracy. Therefore, we need to define estimators of the maximal explicative rank or tests based upon observations of (Xt ). Notice also that, as in the preceding section, these observations are at discrete times only. We start with a linear algebra observation. Let Ar be the family of all subsets of {1, . . . , d} with r elements. For all K ∈ Ar and d × d symmetric nonnegative matrix , let determinantK () be the determinant of the r × r submatrix (kl : k, l ∈ K) and set  determinantK (). determinant(r; ) := K∈Ar

It is easy to prove that the eigenvalues λ(1) ≥ . . . λ(d ) ≥ 0 of  satisfy for all r = 1, . . . , d: 1 determinant(r; ) ≤ λ(1)λ(2) . . . λ(r) ≤ determinant(r; ). d(d − 1) . . . (d − r + 1) In addition,



1≤r≤d

=⇒

r ≤ rank() =⇒ determinant(r; ) > 0 r > rank() =⇒ determinant(r; ) = 0,

and 2 ≤ r ≤ d =⇒

r! determinant(r; ) ≤ λ(r) d! determinant(r − 1; )

≤ Now, set



L(r)t :=

d! determinant(r; ) . (r − 1)! determinant(r − 1; )

t

determinant(r; cs )ds. 0

In view of the preceding inequalities, for choosing an explicative Brownian dimension, this quantity plays a role similar to  t ¯ t := L(r) λ(1)s . . . λ(r)s ds. 0

10

D. Talay

We approximate L(r)t by means of our observations of X: denoting by [x] the integer part of x, we set L(r)nt :=

nr−1 T r−1 r

[nt/T ]−r+1 

determinant(r; ζ(r)ni ),

i=1

where ζ(r)ni =

r  ( ni+j−1 X) ( ni+j−1 X)∗ , with n X = XT/n − X(−1)T/n . j=1

Theorem 2.3. The variables L(r)nt converge in probability to L(r)t uniformly in t ∈ [0, T ]. The processes √ V(r)nt := n (L(r)nt − L(r)t ) converge stably in law to a limiting process (V(r)t )1≤r≤d , which is defined on, an extension of the original space and is a nonhomogeneous Wiener process with an “explicit” quadratic variation process. Set R(ω)t := sup rank(cs (ω)). s∈[0,t]

We define a scale invariant estimator of Rt by   Rn,t := inf r ∈ {0 . . . , d − 1} : L(r + 1)nt < ρn t −1/r (L(r)nt )(r+1)/r . The preceding theorem allows one to propose a test based on a scale invariant relative threshold for which we have the following consistency result under reasonably weak assumptions on the coefficients (bs ) and (σs ) (more or less similar to those made in the preceding section): Theorem 2.4. For all r, r in {1, . . . , d}, provided P(Rt = r ) > 0, we have 1 if r  = r ,

P(Rn,t  = r | Rt = r ) −→ 0 if r = r . Empirical studies for this test and a couple of other tests applied to simulations of models with stochastic volatilities are reported in Jacod, Lejay and Talay [2008]. They illustrate that, under circumstances such as observations at low frequencies or systems with strongly oscillating components, the tests may lead to very erroneous conclusions. In any case, the transformation of the real Brownian dimension into an explicative one induces a specific model risk.

Model Risk in Finance: Some Modeling and Numerical Analysis Issues

11

3. On calibration methods in finance Practitioners do not only use estimators based on historical observations of primary assets but also use all the information available on the market, for example, prices of derivatives on the asset under consideration, prices of correlated assets, and forward contracts. Their data set is thus a sample χ of a random vector ξ, which represents market prices of all such products. Various approaches have been developed by various authors: inverse problem techniques applied to the PDEs for option prices, numerical resolution of Dupire’s PDE for the volatility function, optimization techniques to fit the data, entropy minimization techniques, etc. We first briefly describe Avellaneda–Friedman–Holmes–Samperi’s approach for the calibration of volatilities (for more details on this approach and other approaches, see Avellaneda, Friedman, Holmes and Samperi [1997] and the volume edited by Avellaneda [2001], and references therein). Consider an asset whose volatility process (σt ) is progressively measurable and satisfies 0 < σ ≤ σt ≤ σ for some deterministic constants σ and σ. The set of all such processes is denoted by H. Suppose that the market is complete and that various European options are priced on the market, all the maturities belong to the time interval [0, T ]. Avellaneda’s approach consists in choosing a smooth and strictly convex function H defined on R+ with minimal value 0 at a given value σ0 (resulting from statistics based on historical data) and then searching the process (σt ), which solves  T sup −Eσ exp(−rθ)H((σθ )2 )dθ. (σt )∈H

0

Denote the observed option prices by Pk , their maturities by Tk , and their payoff functions by k . Then, set  T f(σ· ) := −Eσ exp(−rθ)H((σθ )2 )dθ, 0

gk (σ· ) := Eσ (exp(−rTk )k (STk )). The calibration procedure consists in solving  sup inf (f(σ· ) + μk (gk (σ· ) − Pk )). (σt )∈H μk

k

For a discussion on the corresponding numerical procedures and a survey on other numerical techniques for calibration, see Achdou and Pironneau [2005]. Another direction has been followed by El Karoui and Hounkpatin (see Hounkpatin [2002]) to calibrate risk premia rather than volatilities. The El Karoui– Hounkpatin’s method is based on a variant of the selection of models by minimizing

12

D. Talay

entropies as introduced in Avellaneda, Friedman, Holmes and Samperi [1997]. Let X be the state space of a random vector ξ, which represents market prices of products related to the asset under consideration (e.g., forward contracts, derivatives, . . .). We observe one sample χ of this random vector. Define the set A of calibration measures as

 Pχ := Q probability on X equivalent to P, EQ [ξ] = χ . How to choose an “optimal” element of Pχ ? Consider the entropy

 H(Q, P) :=

log

dQ dQ dP

if Q << P, + ∞ otherwise.

Observe that H(Q, P) is positive, and that H(Q, P) = 0 iff P = Q. Suppose that the asset price solves dXt = b(t, Xt )dt + σ(t, Xt )dBt . For a vector ξ := (ξ i ) of the form xi = φi (XT ), set  P

h(t, x, λ) := E

N

 i i=1 λi ξ )  Xt  i EP exp( N i=1 λi ξ ) exp(

 =x .

Using results of Csiszar [1975, theorem 3.1], El Karoui and Hounkpatin have shown that there exists a unique Q∗ in the set of calibration measures such that H(Q∗ , P) = inf H(Q, P), Q∈A

and the dynamics of (Xt ) under Q∗ is dXt = (b(t, Xt ) + σ(t, Xt )2 ∂x log h(t, Xt , λ∗ ))dt + σ(t, Xt )dB∗t , t ≤ T, where (Bt∗ ) is a Brownian motion under Q∗ , and λ∗ solves max

λ∈RN

N  i=1

P

λi χi − log E exp(

N 

 i

λi ξ ) .

i=1

∗ The numerical approximation of λ∗ , h(t, x, λ∗ ), and ∂h ∂x (t, x, λ ) is theoretically possible owing to Monte Carlo methods. It is a challenging and interesting question to design an efficient algorithm. It actually appears that all calibration measures lead to difficult numerical problems that getting good accuracies is questionable. We also emphasize that the family of calibration probability measures is arbitrarily chosen. For these reasons, calibration measures, as statistical procedures, cannot cancel model risk.

Model Risk in Finance: Some Modeling and Numerical Analysis Issues

13

4. On Monte Carlo approximations of the VaR of model risk P&Ls 4.1. Model risk P&Ls for misspecified hedging strategies in Markovian markets In this subsection, we follow Bossy, Gibson, Lhabitant, Pistre and Talay [2006] where numerical results for the P&L function below are discussed. Consider a primary asset with price processes S and a saving account (or, more generally, a numéraire) with price process F defining a no arbitrage and complete market. Denote by StF the price of the primary assets expressed in this numéraire. We suppose that, up to a change in probability from P to a new probability PF , the price StF is a F-martingale and defines a no abitrage and complete market. Consider a trader who needs to hedge a European option on the primary assets with maturity T O and payoff function φ. At all time, 0 ≤ t ≤ T O the perfectly hedging portfolio consists in Ht0 units of the saving account and the Ht units of the primary assets: Vt = Ht0 Ft + Ht St . Its value expressed in the numéraire F is VtF = Ht0 + Ht StF . The self-financing condition writes  VtF = V0F +

0

t

Hθ dS Fθ .

As the martingale S F is supposed to define a complete market and thus to satisfy the martingale representation property, the preceding equality provides a characterization of the process H. However, this characterization generally is only implicit, even when Clark– Ocone formula (see Nualart [2006]) applies, and thus when Ht can be expressed by means of conditional expectations of Malliavin derivatives, and its numerical approximation is quite difficult. Thus, would the market asset prices be a general semi-martingale, even if the trader would perfectly know and measure the model, he/she would nevertheless use a simpler model that would allow him/her to easily, and in short computational time, get numerical values for the delta. To this end, most often traders consider Markov models: they are given a filtration F F and a probability P , a F–Markov–Feller process (ρt ), and functions g and h, and they admit that StF = g(t, ρt ), Ft = h(t, ρt ), and dS Ft = (t, ρt )dBFt ,

(4.1)

dρt = β(ρt )dt + γ(t, ρt )dBFt ,

(4.2)

for some functions , β, and γ, and for some Brownian motion BF . The process ρ may be S F itself or the instantaneous rate if S F is a forward bond price in a one-factor

14

D. Talay

short-term rate model or a vector of factors. The model is constrained in such a way that there exists a smooth function π(t, ¯ x) solution to ∂π¯ ρ ¯ x) = 0, T < T O , (t, x) + Lt π(t, ∂t

(4.3) F

ρ

where Lt is the infinitesimal operator of ρ under P . The boundary condition1 is π(T ¯ O , x) =

φ(h(T O , x)g(T O , x)) . h(T O , x)

With the above notation, would the true world be actually governed by (ρt ), then risk ¯t = could be eliminated through the delta-hedge actually used by the trader, that is, H ∂π¯ F ∂x (t, St ). Therefore, the self-financed “pseudoreplicating” portfolio has value F

dV t = H t dS Ft . The model risk P&L function is defined as F

P&LtF = V t − VtF .

(4.4)

Suppose that, in the true world, the process (ρt ) is a (not necessarily Markov) Itô process satisfying under PF : dρt = βt dt + γt dBFt for some adapted processes β and γ. Set ρ

Lt π(t, ρt ) := βt

∂π ∂2 π 1 (t, ρt ) + (γt )2 2 (t, ρt ). ∂x 2 ∂x

(4.5)

A simple calculation shows that, at maturity T O , F

F

P<F O = V T O − VTFO = V 0 − π(0, ρ0 ) +



TO 0

ρ

ρ

(Lθ − Lθ )π(θ, ρθ )dθ.

(4.6)

Notice that, if (ρt ) is a Markov process, that is, if βt = β(t, ρt ) and γt = γ(t, ρt ) ρ for some functions β and γ, then Lt is the classical infinitesimal generator of (ρt ) and ρ ρ F Vt = π(t, ρt )/Ft , where π(t, x) solves a PDE similar to (4.3) with Lt replacing Lt . 4.2. Approximation of quantiles of diffusion processes The discussion in the preceding subsection can obviously be extended to multidimensional market models, where the prices dS F,i are deterministic functions of a vector of Markov factors (ρti ), and basket options based on the prices S i and with maturity T O . 1 Observe that

VTF = HT0 + HT STF =

1 φ(F T STF ). FT

Model Risk in Finance: Some Modeling and Numerical Analysis Issues

15

One desires to get statistical information on P<F O . A commonly used statistics is its VaR. This leads to the question of approximating quantiles of the law of one component j of multidimensional Markov diffusion processes such as (StF,i , ρt , P&Lt ). Consider the fairly general stochastic differential equation  Xt (x) = x +

t

A0 (s, Xs (x))ds +

0

r   i=1

t 0

Ai (s, Xs (x))dBis ,

and suppose that the law of the last component, XTd (x), has a density with respect to Lebesgue’s measure. Here, (Bs ) is an r-dimensional Brownian motion, and the functions A0 , A1 , . . . , Ar are smooth with bounded derivatives. The Euler scheme with step T/n is defined as n n n X(p+1)T/n (x) = XpT/n (x) + A0 (pT/n, XpT/n (x))

+

r 

T n

n i i Ai (pT/n, XpT/n (x))(B(p+1)T/n − BpT/n ).

i=1

We add a small perturbation to XTn (x) in order to get a random variable whose law has a density ˜ Tn (x) = XTn (x) + BT +T/n − BT . X We aim to get error estimates on the approximation, by means of Monte Carlo ˜ n (x), of the quantile of level δ, ρ(x, δ), of the law of Xd (x). simulations of X T T When approximating quantities of the type Ef(XT ), where T is fixed, we have the following result: for functions f with polynomial growth at infinity, Ef(XT ) − Ef(XTn ) = Cf (T, x)

T 1 + Qn (f, T, x) 2 , n n

(4.7)

where |Cf (T, x)| + supn |Qn (f, T, x)| ≤ C(1 + xQ )

1 + K(T ) Tq

for some positive real numbers C, q, and Q and some increasing function K (see Talay and Tubaro [1990] for smooth functions f , Bally and Talay [1995] under a uniform hypoellipticity condition on the fields Ai , Kohatsu-Higa [2001] and Gobet and Munos [2005] for only measurable functions under nondegeneracy conditions on the Malliavin covariance matrix of XT (x)). Thus, Romberg extrapolation techniques can be used to get higher convergence rates (see Talay and Tubaro [1990]). For extensions to barrier options, see Gobet and Menozzi [2004].

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D. Talay

For the quantile approximation problem, the estimates are slightly different. We summarize results in Talay and Zheng [2002]. Suppose first that the stochastic differential equation (SDE) for (Xt ) has time homogeneous coefficients:  t r  t  A0 (Xs (x))ds + Ai (Xs (x))dBis . (4.8) Xt (x) = x + 0

i=1

0

For multiindices α = (α1 , . . . , αk ) ∈ {0, 1, . . . r}k , set A∅i = Ai and for 0 ≤ j ≤ r, (α,j) := [Aj , Aαi ]. Also set Ai VL (x, η) :=

r  

< Aαi (x), η >2

i=1 |α|≤L−1

and VL (x) := 1 ∧ inf VL (x, η). η=1

Suppose (UH) CL := inf x∈Rd VL (x) > 0 for some integer L, j j (C) The coefficients Ai , i = 0, . . . , r, j = 1, . . . , d are of class Cb∞ (Rd ) (the Ai ’s may be unbounded). Under (UH) and (C), the law of XT (x) has a smooth density pT (x, x ), so that the d-th marginal distribution of XT (x) also has a smooth density pdT (x, y), which is strictly positive at all point y in the interior of its support (cf. Nualart [2006]). For 0 < δ < 1, set ρ(x, δ) := inf {ρ ∈ R; P[XTd (x) ≤ ρ] = δ} and ˜ n,d (x) ≤ ρ] = δ}. ρ˜ n (x, δ) := inf {ρ ∈ R; P[X T The discretization error on the quantile ρ(x, δ) is described by the following theorem. Theorem 4.1. Under conditions (UH) and (C), we have |ρ(x, δ) − ρ˜ n (x, δ)| ≤

K(T ) 1 + xQ 1 · d · , Tq pT (ρ(x, δ)) n

(4.9)

where pdT (ρ(x, δ)) =

inf

y∈(ρ(x,δ)−1,ρ(x,δ)+1)

pdT (x, y).

˜ N (for variance reducIn practice, ρ˜ n (x, δ) is estimated by sampling N copies of X T tion techniques, see Kohatsu-Higa and Petterson [2002]). Taking the corresponding

Model Risk in Finance: Some Modeling and Numerical Analysis Issues

17

Monte Carlo error into account, roughly speaking, the global error on the quantile is of order     1 1 , + O n,d O d √ pT (ρ(x, δ))n p˜ T (x, ρ(x, δ)) N ˜ n,d where p˜ n,d T (x, ξ) denotes the density of XT (x). One has (see Bally and Talay [1995], Kohatsu-Higa [2001]) that p˜ n,d T (x, ξ) − d pT (x, ξ) is of order 1/n. For practical applications, one thus needs accurate estimates from below pdT (x, ρ(x, δ)). Such estimates are available when the generator of (Xt ) is strictly uniform elliptic (see Azencott [1984]), but this hypothesis is too stringent j in our context: notice that the law of the above vector (StF,i , ρt , P&Lt ) may not have a density since all its components are driven by the Brownian processes driving the ρj s. Therefore, we now do not suppose that the Malliavin covariance matrix of (Xt (x)) is invertible and return to general inhomogeneous stochastic differential equation. Let (Xst (x ), 0 ≤ s ≤ T − t) be a smooth version of the flow solution to Xst (x )



s

=x +

A0 (t 0

+ θ, Xθt (x ))dθ

+

r   i=1

0

s

Ai (t + θ, Xθt (x ))dBit+θ .

We denote by M(t, s, x ) the Malliavin covariance matrix of Xst (x ). We now suppose (C’) The functions Ai , i = 0, . . . , r, j = 1, . . . , d are of class Cb∞ ([0, T ] × Rd ) (the j Ai ’s may be unbounded). j

(M) For all p ≥ 1, there exists a nondecreasing function K, a positive real number r, and a positive Borel measurable function  such that     K(T ) 1   (t, x )  ≤  d  Md (t, s, x )  sr p

for all t in [0, T ) and s in (0, T − t]. In addition,  satisfies: for all λ ≥ 1, there exists a function λ such that sup E[(t, Xt (x))λ ] < λ (x)

t∈[0,T ]

and sup sup E[(t, Xtn (x))λ ] < λ (x). n>0 t∈[0,T ]

Under condition (M), the d-th marginal distribution of XT (x) has a smooth density. pdT (x, y) is strictly positive at all point y in the interior of its support, and we have the following error estimate.

18

D. Talay

Theorem 4.2. Under conditions (M) and (C’), we have |ρ(x, δ) − ρ˜ n (x, δ)| ≤

K(T ) 1 + xQ 1 · d · λ (x) · , Tq n pT (ρ(x, δ))

where pdT (ρ(x, δ)) =

inf

y∈(ρ(x,δ)−1,ρ(x,δ)+1)

pdT (x, y).

In practice, one needs to check that condition (M) is satisfied. We here give two examples.  Theorem 4.3. Suppose that ri=1 |Adi (t, x)|2 ≥ a > 0 for some t in [0, T ] and x in Rd . Then, the d-th marginal law of Xt (x) has a smooth density, and condition (M) is satisfied. Our second example concerns a model risk problem. The trader wants to hedge a European option (B(T O , T )) on a bond price B(T O , T ), where T O is the option maturity and T > T O is the bond maturity. To hedge, the trader uses bonds with maturities T O and T . Suppose that the bond market is an HJM model. When the HJM model is governed by a deterministic function σ, the delta of the option can be expressed in terms of the solution πσ to the PDE ⎧ 1 ∂ 2 πσ ⎨ ∂πσ (t, x) + x2 (σ ∗ (t, T O ) − σ ∗ (t, T ))2 2 (t, x) = 0, 2 ∂x ⎩ ∂t πσ (T, x) = (x). Suppose that the trader chooses an erroneous deterministic model structure σ(s, T ). Then, for suitable functions u1 (s), u2 (s), and ϕ(s), the forward value of the trader’s P&Ls satisfies an SDE of the type dP&Lt = ϕ(t, Yt )Yt u1 (t)dt + ϕ(t, Yt )Yt u2 (t)dBt , where (Yt ) satisfies dY t = Yt u1 (t)dt + Yt u2 (t)dBt . If |ϕ(t, y)u2 (t)| ≥ a > 0 ∀t, ∀y > 0, then condition (M) is satisfied, and one can get an explicit lower bound estimate for the marginal density.

Model Risk in Finance: Some Modeling and Numerical Analysis Issues

19

5. A stochastic game to face model risk Consider the market model ⎧ d ij j i i i ⎪ ⎪ ⎨dS t = St [bt dt + j=1 σt dBt ] for 0 ≤ i ≤ n,     ⎪    ⎪ ⎩dP t = Pt ni=1 πti bti dt + dj=1 σtij dBjt + rPt 1 − ni=1 πti dt. Here {πi } = set of prescribed strategies. Consider u(·) := (b(·), σ(·)) as the market’s control process. Cvitani´c and Karatzas [1999] have studied the dynamic measure of risks inf

sup Eν (F(Xx,π (T ))),

π(·)∈A(x) ν∈D

where A(x) denotes the class of admissible portfolio strategies issued from the initial wealth x, and Eν denotes the expectation under the probability Pν for all ν in a suitable set. All the measures Pν have the same risk-neutral equivalent martingale measure, which implies that the trader (or the regulator) is concerned by model risk on stock appreciation rates. For numerical methods related to this approach, see Gao, Lim and Ng [2004]. An axiomatic approach to model risk is developed by Cont [2006], who proposes to measure model uncertainty risk by means of a coherent risk measure compatible with market prices of derivatives or of a convex risk measure. The author studies several examples, among them the case where the “real” noise is a linear combination of Poisson and Brownian processes, whereas the trader uses a Brownian model only. We now present a somewhat different approach, based on a PDE and aimed to compute the minimal amount of money and dynamic strategies that allow the financial institution to (approximately) contain the worst possible damage due to model misspecifications for volatilities, stock appreciation rates, and yield curves. Within this approach, we consider that the trader acts as a minimizer of the risk, whereas the market systematically acts as a maximizer of the risk. Thus, the model risk control problem can be set up as a two-player zero-sum stochastic differential game problem. Given a suitable function F , the cost function is J(t, x, p, , u(·)) := Et,x,p F(ST , PT ), and the value function is V(t, x, p) :=

inf

sup

∈Ad (t) u(·)∈Adu (t)

J(t, x, p, , u(·)).

The next theorem shows that this model risk value function solves an Hamilton– Jacobi–Bellman–Issacs equation.

20

D. Talay

Theorem 5.1. Under an appropriate locally Lipschitz condition on F , the value function V(t, x, p) is the unique viscosity solution in the space S := {ϕ(t, x, p) is continuous on [0, T ] × Rn × R; ∃A > 0, lim

|p|2 +x2 →∞

ϕ(t, x, p) exp(−A| log(|p|2 + x2 )|2 ) = 0 for all t ∈ [0, T ]}

to the Hamilton–Jacobi–Bellman–Isaacs equation ⎧ ∂v ⎪ − 2 n+1 ⎪ ⎨ ∂t (t, x, p) + H (D v(t, x, p), Dv(t, x, p), x, p) = 0 in [0, T ) × R , ⎪ ⎪ ⎩ v(T, x, p) = F(x, p), where



−

H (A, z, x, p) := max min

u∈Ku π∈Kπ

 1 (a(x, Tr p, σ, π)A) + z · q(x, p, b, π) . 2

For a proof, see Talay and Zheng [2002]. The numerical resolution of the PDE allows one to compute approximate reserve amounts of money to control model risk. Numerical investigations, undone so far, are necessary to evaluate how large are these provisions. 6. Model risk and technical analysis The practitioners use various rules to rebalance their portfolios. These rules usually come from fundamental economic principles, mathematical approaches derived from mathematical models, or technical analysis approaches. Technical analysis, which provides decision rules based on past prices behavior, avoids model specification and thus model risk (for a survey, see Achelis [2001]). Pastukhov [2004] has studied mathematical properties of volatility indicators used in technical analysis. Blanchet, Diop, Gibson, Talay and Anre [2007] proposed a framework allowing one to compare the performances obtained by strategies derived from erroneously calibrated mathematical models and the performances obtained by technical analysis techniques. Consider an asset whose instantaneous expected rate of return changes at an unknown random time, and a trader who aims to maximize his/her utility of wealth by selling and buying the asset. The benchmark performance results from a strategy that is optimal when the model is perfectly specified and calibrated. To this benchmark we can compare the performances resulting from optimal rules but erroneous parameters, and the performances resulting from technical analysis indicators. The real market is described by 0 dS t = St0 rdt,   dS t = St μ2 + (μ1 − μ2 )I(t≤τ) dt + σSt dBt .

Model Risk in Finance: Some Modeling and Numerical Analysis Issues

21

Here, the Brownian motion (Bt ) and the change time τ are independent, and τ follows an exponential law with parameter λ. One has    t σ2 St = S exp σBt + (μ1 − )t + (μ2 − μ1 ) I(τ≤s) ds =: S 0 exp(Rt ), 2 0 0

where 

σ2 Rt = σBt + μ1 − 2





t

t + (μ2 − μ1 )

I(τ≤s) ds.

0

Suppose μ1 −

σ2 σ2 < r < μ2 − . 2 2

We start with describing one of the technical analysis rules that are applied in the context of instantaneous rates of return changes. Denote by πt ∈ {0, 1} the proportion of the agent’s wealth invested in the risky asset at time t, and by Mtδ the moving average indicator of the prices. Therefore, Mtδ

1 = δ



t

Su du. t−δ

Given a finite set of decision times tn , at each tn the agent invests all his/her wealth into the risky asset if Stn > Mtδn . Otherwise, he/she invests all the wealth into the riskless asset. Consequently, πtn = ISt

δ n ≥Mtn

,

and the wealth at time tn+1 is   St0n+1 Stn+1 Wtn+1 = Wtn πt + 0 (1 − πtn ) , Stn n Stn from which, for T = tM , WT = W0

M−1 

  ! πtn exp(Rtn+1 − Rtn ) − exp(rΔt) + exp(rΔt) .

n=0

t The logarithmic utility of WT can be explicited in terms of the density of ( 0 exp(2Bs ) ds, Bt ): its explicit expression, according to Yor [2001], is interesting by itself: let σ > 0 and ν be real numbers, and let V be the geometric Brownian motion Vt = eσ

2 νt+σB t

.

22

D. Talay

Then,  P 0

t

2)    2 2 z zν−1 − ν 2σ t − (1+z 2σ 2 y i 2 Vs ds ∈ dy; Vt ∈ dz = e dydz, σ t 2y σ2y 2

(6.1)

where 2

zeπ /4y iy (z) := √ π πy



∞

e−z cosh(u)−u

2 /4y

sinh(u) sin(πu/2y)du.

0

The performance of the technical analysis strategy is compared to the benchmark performance: the optimal wealth of a trader who perfectly knows the parameters μ1 , μ2 , λ, and σ. We impose constraints: as a technical analyst is only allowed to invest all his/her wealth in the stock or the bond, the proportions of the benchmark trader’s wealth invested in the stock are constrained to lie within the interval [0, 1]. In addition, the trader’s strategy is constrained to be adapted with respect to the filtration FtS := σ (Su , 0 ≤ u ≤ t) generated by (St ), which because of τ, is different from the filtration generated by (Bt ). Let πt be the proportion of the trader’s wealth invested in the stock at time t; W·x,π denotes the corresponding wealth process. Let A(x) denote the set of admissible strategies, that is, A(x) := {π· − FtS − progressively measurable process such that W0x,π = x, Wtx,π > 0 for all t > 0, π· ∈ [0, 1]}. The value function is V(x) := sup E U(WTπ ). π· ∈A(x)

As in Karatzas and Shreve [1998], we introduce an auxiliary unconstrained market defined as follows. Let D the subset of the {FtS }-progressively measurable processes ν : [0, T ] ×  → R such that  E

T

ν− (t)dt < ∞ , where ν− (t) := − inf (0, ν(t)).

0

The bond price process S 0 (ν) and the stock price S(ν) satisfy  St0 (ν) = 1 +

t

0

 St (ν) = S0 + 0

Su0 (ν)(r + ν− (u))du, t

  Su (ν) (μ1 + (μ2 − μ1 )Fu + ν(u)− + ν(u))du + σdBu ,

Model Risk in Finance: Some Modeling and Numerical Analysis Issues

23

where B· is the innovation process, that is, the FtS Brownian motion defined as    t 1 σ2 Bt = Fs ds , t ≥ 0; Rt − (μ1 − )t − (μ2 − μ1 ) σ 2 0 here, F is the conditional a posteriori probability (given the observation of S) that τ has occurred within [0, t]:   Ft := P τ ≤ t/FtS . For each auxiliary unconstrained market driven by a process ν, the value function is V(ν, x) :=

sup

π· ∈A(ν,x)

Ex U(WTπ (ν)),

where   dW πt (ν) = Wtπ (ν) (r + ν− (t))dt + πt ν(t)dt + (μ2 − μ1 )Ft dt  +(μ1 − r)dt + σdBt . Let the exponential likelihood ratio process (Lt )t≥0 be defined by   μ2 − μ1 1 σ2 2 Rt − 2 (μ2 − μ1 ) + 2(μ2 − μ1 )(μ1 − ) t . Lt = exp 2 σ2 2σ Karatzas and Shreve [1998] have proven the following result. Theorem 6.1. If there exists " ν such that V(" ν, x) = inf V(ν, x), ν∈D

then there exists an optimal portfolio π∗ for which the optimal wealth (for the constrained admissible strategies) is ∗

π Wt∗ = Wt" (" ν ).

An optimal portfolio allocation strategy is   φt ν(t) ∗ −1 μ1 − r + (μ2 − μ1 )Ft +" πt := σ + , t ν− (s)ds ν ∗ −rt− 0 " σ H" t Wt e ν where H" t is the exponential process    t μ1 − r +" ν(s) (μ2 − μ1 )Fs ν + dBs H" = exp − t σ σ 0     1 t μ1 − r +" ν(s) (μ2 − μ1 )Fs 2 − ds , + 2 0 σ σ

24

D. Talay

and φ is a FtS -adapted process, which satisfies E



T

ν −rT − H" Te

0

" ν− (t)dt

−1

(U )

T

ν −rT − (υH" Te

0

" ν− (t)dt

)

S / Ft





t

=x+

φs dBs . 0

Here, v is the Lagrange multiplier, which makes the expectation of the left-hand side equal to x for all x. In addition, Ft satisfies t λeλt Lt 0 e−λs L−1 s ds Ft = . t λt −λs 1 + λe Lt 0 e L−1 s ds The optimal strategies for the constrained problem are the projections on [0, 1] of the optimal strategies for the unconstrained problem. In addition, using again Yor’s Eq. (6.1), one can explicit Wt∗,x and πt∗ in the case of the logarithmic utility. For general utilities, the optimal strategy cannot be explicited. It, thus, is worth considering the case of a trader who chooses to reinvest the portfolio only once, namely at the time when the change time τ is optimally detected owing to the price history. We suppose that the reinvestment rule is the same as the technical analyst’s one: at the detected change time from μ1 to μ2 , all the portfolio is reinvested in the risky asset. The stopping rule K , which minimizes the expected miss E| − τ| over all the stopping rules  with E() < ∞, is as follows:

 t  p∗  K = inf t ≥ 0 λeλt Lt , e−λs L−1 ds ≥ s 1 − p∗ 0 where p∗ is the unique solution in ( 12 , 1) of the equation  0

1/2

(1 − 2s)e−β/s 2−β s ds = (1 − s)2+β



p∗

1/2

(2s − 1)e−β/s 2−β s ds (1 − s)2+β

with β = 2λσ 2 /(μ2 − μ1 )2 (see Shiryaev [2004] and references therein). Up to a numerical approximation of p∗ , this rule can easily be applied. In practice, even if we would be able to estimate μ1 and σ with a good accuracy, the value of μ2 cannot be determined a priori, and the number of observations of τ may be too small to well estimate λ. Therefore, traders believe that the stock price is   dS t = St μ2 + (μ1 − μ2 )It≤τ dt + σSt dBt , where the law of τ is exponential with parameter λ. The above decision rules are then governed by    σ2 1 1 2 Lt = exp 2 (μ2 − μ1 )Rt − 2 (μ2 − μ1 ) + 2(μ2 − μ1 )(μ1 − ) t , 2 σ 2σ t −1 λeλt Lt 0 e−λs Ls ds . Ft = t −1 1 + λeλt Lt 0 e−λs Ls ds

Model Risk in Finance: Some Modeling and Numerical Analysis Issues

25

Actually, the value of a misspecified optimal allocation strategy is π∗t = proj[0,1]

(μ1 − r + (μ2 − μ1 )F t ) σ2

,

and the corresponding wealth is  t  ∗ rt ∗ −ru W t = e exp πu d(e Su ) . 0

Similarly, the erroneous stopping rule is  t K −1  = inf t ≥ 0, λeλt Lt e−λs Ls ds ≥ 0

p∗ , 1 − p∗

where p∗ is the unique solution in ( 12 , 1) of  0

1/2

(1 − 2s)e−β/s (1 − s)2+β

 s

2−β

ds =

p∗

1/2

(2s − 1)e−β/s (1 − s)2+β

s2−β ds,

with β = 2λσ 2 /(μ2 − μ1 )2 . The value of the corresponding portfolio is W T = W0 S 0 K 

ST I K + W0 ST0 I(K >T ) . SK ( ≤T)

In view of the technical analysis technique and misspecified strategies, it is natural to compare them to the benchmark optimal strategyy and to study the following question: Is it better to invest according to a mathematical strategy based a misspecified model or according to a strategy based on technical analysis rules? It appears that, even in the logarithmic utility case, the explicit formulae for the different wealths are too complex to allow analytical comparisons. However, Monte Carlo simulations on study cases show that the technical analyst may overperform misspecified optimal allocation strategies even when for relatively small misspecifications, for example, when the parameter λ is underestimated. Simulations also show that a single misspecified parameter is not sufficient to allow the technical analyst to overperform the traders who use erroneous stopping rules. One can also observe that, when the ratio μ2 /μ1 decreases, the performances of well-specified and misspecified strategies based upon stopping rules decrease. 7. Conclusion We have shown that statistical and calibration procedures can hardly reduce model uncertainties in finance. We have also emphasized that model uncertainties appear in the numerical resolution of PDE related to option pricing or optimal allocation problems. We have reviewed a few approaches to evaluate model risk indicators and to control model risk. We have discussed the accuracy of Monte Carlo methods to approximate

26

D. Talay

VaR statistics in diffusion models. Finally, we have proposed a mathematical framework to compare technical analysis techniques and strategies derived from misspecified mathematical models. Most of the results that are stated above are recent and open new challenging perspectives in financial mathematics and in numerical analysis. Decreasing and controlling model risk is actually an important issue to make financial strategies more reliable.

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Gobet, E., Munos, R. (2005). Sensitivity analysis using Itô-Malliavin calculus and martingales, and application to stochastic optimal control. SIAM J. Control Optim. 43 (5), 1676–1713. Hounkpatin, O. (2002). Volatilité du Taux de Swap et Calibrage d’un Processus de Diffusion, thèse de l’université Paris 6, 2002. Jacod, J. (2000). Non-parametric kernel estimation of the coefficient of a diffusion. Scand. J. Stat. 27 (1), 83–96. Jacod, J., Lejay, A., Talay, D. (2008). Estimation of the Brownian dimension of a continuous Itô process. Bernoulli, 14 (2), 469–498. Karatzas, I., Shreve, S.E. (1998). Methods of Mathematical Finance, Applications of Mathematics, vol. 39 (Springer-Verlag, New York, NY). Kohatsu-Higa, A. (2001). Weak approximations: a Malliavin calculus approach. Math. Compt. 70 (233), 135–172. Kohatsu-Higa, A., Pettersson, R. (2002). Variance reduction methods for simulation of densities on Wiener space. SIAM J. Numer. Anal. 40 (2), 431–450. Kutoyants, Y. (1984). Parameter Estimation for Stochastic Processes (translated and edited by B.L.S. Prakasa Rao), Research and Exposition in Mathematics, vol. 6 (Heldermann Verlag, Berlin, Germany). Nualart, D. (2006). The Malliavin Calculus and Related Topics, Probability and its Applications (New York), second ed. (Springer-Verlag, Berlin, Germany). Pastukhov, S.V. (2004). On some probabilistic-statistical methods in technical analysis. Teor. Veroyatn. Primen. 49 (2), 297–316; translation in Theor. Probab. Appl. 49 (2), 2005, 245–260. Prakasa Rao, B.L.S. (1999a). Semimartingales and Their Statistical Inference (Chapman and Hall, Boca Raton, FL). Prakasa Rao, B.L.S. (1999b). Statistical Inference for Diffusion Type Processes (Arnold, London, UK). Shiryaev, A.N. (2004). A remark on the quickest detection problems. Stat. Decis. 22, 79–82. Talay, D., Tubaro, L. (1990). Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8 (4), 94–120. Talay, D., Zheng, Z. (2002). Worst case model risk management. Financ. Stoch. 6 (4), 517–537. Talay, D., Zheng, Z. (2004). Approximation of quantiles of components of diffusion processes. Stoch. Proc. Appl. 109, 23–46. Yor, M. (2001). Exponential Functionals of Brownian Motion and Related Processes, Springer Finance (Springer, Berlin, Germany).