New expansion formulas in local volatility models 

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Application context and overview of different computational ap-proaches

In the two last decades, numerous works have been devoted to designing efficient methods in order to give exact or approximative pricing formulas for many financial products in various models. This quest of efficiency originates in the need for more and more accurate methods, when one takes into account an increasing number of sources of risk, while maintaining a competitive computational time. The current interest in real-time tools (for pricing, hedging, calibration) is also very high.
Let us give a brief overview of different computational approaches. While explicit formulas are available in simple models (Black-Scholes model associated to log-normal distribution [Black 1973], or Bachelier model related to normal distributions [Bachelier 1900] for instance), in general no closed forms are known and numerical methods have to be used. As a numerical method, it is usual to perform PDE solvers for one or two-dimensional sources of risk (see [Achdou 2005] for instance) or Monte Carlo methods for higher dimensional problems [Glasserman 2004]: both approaches are popular, efficiently developed and many improvements have been proposed for years. However, these methods are not intrinsically real-time methods, due to the increasing number of points required in the PDE discretization grid or due to the increasing number of paths needed in the Monte Carlo procedure. Not being real-time method means, for example, that when used for calibration routine based on data consisting of (say) 30 vanilla options, it usually takes more than one minute (in the most favorable situations) to achieve the calibration parameters. The approaches presented below are aimed at reducing this computational time to less than one second.
The class of affine models (such as Heston model, exponential Levy model . . . ) offers an alternative approach related to Fourier computations: on the one hand, in such models the characteristic function of the marginal distribution of the log-asset is explicitly known; on the other hand, there are general relations between Call/Put prices and the characteristic function of the log-asset. These relations write as follows.
• Following Carr and Madan [Carr 1998b], consider the difference zT (k) between the Call price in a given model and that price in an arbitrary Black-Scholes model (with volatility σ), both with maturity T and log-strike k. For zero interest rate (to simplify), it is equal to zT (k) = E(eXT −ek)+ − CallBS(k), where X is the logarithm of the asset. A direct computation gives explicitly the Fourier transform zT (v) in the log-strike variable:
zT (v) = eivkzT (k)dk = ΦTX (1 +iv) −ΦTBS(1 +iv) , where ΦTX,BS(u) :=E(euXT ) is either computed in the X model or in the Black-Scholes model. Since ΦTX (•) is required to be known, we get the X-model Call price zT (k) +CallBS(k) simultaneously for any log-strike using a Fast Fourier Transform.
• Alternatively, following the Lewis approach [Lewis 2000, Chapter 2], let α > 0 be a damping con-stant, set h(y) = (ey −K)+e−(1+α)y which belongs to L2(R,Leb.) and assume that E(e(1+α)XT ) < +∞: from the Parceval-Plancherel identity, assuming that the density pXT of XT w.r.t. the Lebesgue measure exists, we obtain E(eXT −K)+ = R  2π h(y)e(1+α)y pXT (y)dy h(−ξ)[e(1+α)•pXT (•)](ξ)dξ = 1 e−(α+iξ) log(K) ΦX (1 +α+iξ)dξ. 2π R (iξ+α)(iξ+α+1) T
The final identity still holds without assuming the existence of density: this can be proved by adding a small Brownian perturbation (considering XT +εWT instead of XT ), and taking the limit as the perturbation ε goes to 0. From the above formula, using an extra numerical integration method (to compute the ξ-integral), we recover Call prices. For higher numerical performance, Lewis recommends a variant of the formula above, obtained through the decomposition (eXT − K)+ = eXT −min(eXT ,K): it finally writes E(eXT −K)+ = S 0 − 1 e( 21 −iξ) log(K) ΦTX( 1 +iξ)dξ. (1.3)
Regarding computational time, both Fourier-based approaches perform well, since they are essentially reduced to a one-dimensional integration problem. But they can be applied only to specific models for which the characteristic function is given in an explicit and tractable form: in particular, it rules out the local volatility models, the local and stochastic volatility models.
The last approach consists of explicit analytical approximations and this is the main focus of this thesis: it is based on the general principle of expanding the quantity of interest (price, hedge, implied volatility. . . ) with respect to some small/large parameters (possibly multidimensional). The parameters under consideration may be of very different nature: for instance in the case of Call/Put options of strike K and maturity T , it ranges from the asymptotic behavior as K is small or large, to the case of short or long maturity T , passing through coupled asymptotics, or small/fast volatility variations, and so on. . . A detailed description with references is presented in Section 1.3 of this introduction Chapter. Due to the plentiful and recent litterature on the subject, it is likely that we will not be exhaustive in the references. But we will do our best to give the main trends and to expose whenever possible what are the links between different viewpoints; we will compare the mathematical tools to achieve these approximations (rather PDE techniques or stochastic analysis ones), in order to provide to the reader a clarified presentation of this prolific topic.

Notations used throughout the introduction Chapter

⊲ Models. In all this Chapter, financial products are written w.r.t. a single asset, which price at time t is denoted by S t. The dynamics of S is modeled through a filtered probability space (Ω,F ,(Ft)t≥0,P) where (Ft)t≥0 is the natural filtration of a standard linear Brownian motion W, augmented by the P-null sets. The risk-free rate is set1 to 0; most of the time and unless stated otherwise, S follows a local volatility model, i.e. it is solution of the stochastic diffusion equation dS t = S tσ(t,S t)dWt, (1.4) where the dynamics is directly under the pricing measure. Assumptions on the local volatility σ are given later. We assume that the complete market framework holds and that an option with payoff h(S T ) paid at maturity T has a fair value at time 0 equal to E(h(S T )).
For positive S , we define the log-asset X = log(S ) which satisfies
dXt = a(t,Xt)dWt − 1
2 a2(t,Xt)dt, (1.5)
where a(t,x) = σ(t,ex).
⊲ Call options. Let us denote by Call(S 0,T,K) the price at time 0 of a Call option with maturity T and strike K, written on the asset S . « Price » usually means the price given by a model on S , that is Call(S 0,T,K) = E(S T −K)+. (1.6)
This Model Price should equalize the Market Price taken from Market datas (calibration step). As usual, ATM (At The Money) Call refers to S 0 ≈ K, ITM (In The Money) to S 0 ≫ K, OTM (Out The Money) to S 0 ≪ K.

 An overview of approximation results

The increasing need in evaluating financial risks at a very global level and in a context of high-frequency market exchanges is a significant incentive for the computational methods to be efficient in evaluating the exposure of large portfolio to market fluctuations (VaR computations, sensitivity analysis), in quickly calibrating the models to the market data. Hence, in the two last decades, many numerical methods have been developed to meet these objectives: in particular, regarding the option pricing, several approxima-tion results have been derived, following one or another asymptotic point of view. We give a summary of these different approaches, stressing the limits of applicability of the methods.

Large and small strikes, at fixed maturity

The Call price Call(S 0,T,K) as a function of strike is convex and its left/right derivatives are related to the distribution function of S T [Musiela 2005, Chapter 7]: ∂−K Call(S 0,T,K) = −P(S T ≥ K) and ∂+K Call(S 0,T,K) = −P(S T > K). Beyond the important fact that the family of Call prices {Call(S 0,T,K) : K ≥ 0} completely characterizes the marginal distribution of S T , this relation also shows that the tails of the law of S T are intrinsically related to the decay of Call(S 0,T,K) as K → +∞. In terms of implicit volatility, the heuristics is the following: the larger the implied volatility of OTM options, the larger the right tail of S T . This is similar for small strikes K, using Put options. Lee [Lee 2005] has been the first one to quantify these features relating the behavior of implied volatility to the tails of S T , with an encoding of the tails through the existence of positive/negative moments. These are model-free re-lations, that can be applied to any model with E(S T ) < +∞ and not only to local volatility ones like in (1.4). The well-known Lee moment formulas write as follows, using the log-variables x0 = log(S 0) and m = log(S 0/K) = x0 −k.

Long maturities, with large/small strikes

In view of the preceeding results, the asymptotics of the smile for large maturity becomes very simple regarding the strike variable, unless one allows the strike to be large/small together with the maturity. Indeed to recover interesting information at the limit, we should consider strikes of the form K = S 0exT with x , 0, or equivalently k = x0 +xT . From the linearization of the payoff, one obtains
Call(S 0,T,S 0exT ) = E S T 1S T ≥S 0exT −S 0exT P(S T ≥ S 0exT ) =S0PS 1 log(S T /S 0) ≥ x −S 0exT P 1 log(S T /S 0) ≥ x
where the new measure PS is the one associated to the numéraire S . Under this form, it appears clearly that for x large enough (say larger than the asymptotic P-mean or PS -mean of T1 log(S T /S 0) whenever it exists), both probabilities above correspond to the evaluation of large deviation events. The role of Large Deviation Principle satisfied by the sequence ( T1 log(S T /S 0))T ≥0 as T → +∞ has been outlined in [Forde 2011] in the case of Heston model, and in [Jacquier 2011] for more general affine models. Saddle point arguments combined with Lewis formula (1.3) have been performed in [Gatheral 2011] for the He-ston model, to recover the Stochastic Volatility Inspired parameterization of Gatheral [Gatheral 2004]: the squared implied volatility σ2I(x0,T,x0 +xT ) has the simple asymptotic shape σ∞2(x) = ω1 1 +ω2ρx + (ω2 x +ρ)2 +1 −ρ2 . (1.15)
For more general affine models like Heston model, without or with jumps, or Bates model, or Barndorff-Nielsen-Shephard model (see [Duffie 2003] and [Jacquier 2011]), it is possible to derive similar limits. Let Λt(u) = log(E(S tu)) be the exponent of the moment generating function, which is convex in u: in the aforementioned model we can define and compute its asymptotic average Λ(u) = limt→∞ 1t Λt(u), which still satisfies to the convexity feature. We associate its Fenchel-Legendre transform Λ∗(x) = supu∈R(ux −Λ(u)) and it turns out that ( T1 log(S T /S 0))T ≥0 satisfies a LDP under P (resp. PS ) with rate function x Λ∗(x) (resp. x Λ∗(x) − x).
Theorem 1.3.3.1. Under some assumptions (see [Jacquier 2011]), for any x ∈ R, the asymptotic implied volatility σ∞(x) is given by lim σI(x0,T,x0 +xT ) = √2 sgn(Λ′(1) − x) Λ∗(x) − x +sgn(x −Λ′(0)) Λ∗(x) .
In the Black-Scholes model with constant volatility σ, one has Λ(u) = σ22 (u2 −u), Λ∗(x) = 2σ12 x + σ22 2, and we get obviously σ∞(x) =σ. For Heston model, Λ is explicit as well and we finally recover the SVI parsimonious parameterization (1.15). Here again, different models may have the same asymptotic smiles, see [Jacquier 2011].

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Non large maturities and non extreme strikes

To obtain approximation formulas in that situation, the asymptotics should originate from different large/small parameters that are rather related to the model and not to the contract characteristics (matu-rity and strike). These different asymptotics are generally well intuitively interpreted. For the sake of clarity, we spend time to detail a bit the arguments, in order to make clearer the differences between the further expansion results and the tools to obtain them. To the best of our knowledge, such comparative presentation does not exist in the literature and the reader may find it interesting.

Small noise expansion

This is inspired by the Freidlin-Wentzell approach [Freidlin 1998] in which the noise in the Stochastic Differential Equation of interest is small. Denote by Y the scalar SDE under study (which can be X or S in our framework), solution of dYt = µ(t,Yt)dt +ν(t,Yt)dWt, Y0 given. (1.16)
Assume that ν is small, or equivalently that ν becomes εν with ε → 0: after making this small noise parameterization, the model writes dYtε = µ(t,Ytε)dt +εν(t,Ytε)dWt, Y0ǫ = Y0.
The converse result (ν(y0,T ) , 0 implies (1.20)) holds true in the case of time-independent coefficient and in a multidimen-sional setting, see [Watanabe 1987, Theorem 3.4]. Yoshida [Yoshida 1992b, Theorem 2.2] has weakened the assumption (1.20) into a localized version allowing degeneracy on a set of polynomially small prob-ability measure. This approach has been successfully applied to different pricing problems in finance, mainly by Yoshida, Takahashi and their co-authors: see [Yoshida 1992a, Uchida 2004, Kunitomo 2001] or the unpublished work [Osajima 2007]. Their methodology consists in expanding the density of the random variable ZTε = YTε−εy0,T using the Gaussian density as the zero-order term, and then going back to E(h(YTε)) by suitable integration computations. The advantage of this approach is that the expansion result (1.21) holds in a large generality, provided that we assume infinitely differentiable coefficients and uniform non degeneracy. However, observe two difficulties or restrictions:
• within usual financial models like Heston model, the required regularity assumption is not satisfied and we even know that the Malliavin differentiability of high order may fail, see [Alòs 2008].
• the existence of the Malliavin weights (πk)k does not provide an explicit and numerically com-putable expansion: very involved additional computations are required to obtain explicit formu-las. One might compare these tricky computations to those necessary to solve the aforementioned system of PDEs.
Last, this approach usually leads to normal approximations of financial models (Bachelier prices) whereas log-normal approximations (Black-Scholes prices) might be more accurate (numerical evi-dences are given in Chapter 2 Section 2.5).

Table of contents :

1 Introduction 
1.1 Application context and overview of different computational approaches
1.2 Notations used throughout the introduction Chapter
1.3 An overview of approximation results
1.3.1 Large and small strikes, at fixed maturity
1.3.2 Long maturities, at fixed strike
1.3.3 Long maturities, with large/small strikes
1.3.4 Non large maturities and non extreme strikes
1.3.5 Asymptotic expansion versus non-asymptotic expansion
1.4 Structure of thesis and main results
I New expansion formulas in local volatility models 
2 Revisiting the Proxy principle in local volatility models 
2.1 Approximation based on proxy
2.1.1 Notations and definitions
2.1.2 Proxy approximation: a primer using the local volatility at spot
2.1.3 Towards Call option approximations with the local volatility at strike and at midpoint
2.1.4 Second order expansion of the implied volatility
2.2 Proofs: a comparative discussion between stochastic analysis and PDE techniques
2.2.1 A pure stochastic analysis approach
2.2.2 Mixing stochastic analysis and PDE
2.2.3 A pure PDE approach
2.3 Higher-order proxy approximation
2.3.1 Third order approximation with the local volatility at spot and at strike
2.3.2 Third order approximation with the local volatility at mid-point
2.3.3 Third order expansion of the implied volatility
2.4 Approximation of the Delta
2.5 Numerical experiments
2.5.1 The set of tests
2.5.2 Analysis of results
2.5.3 CEV Delta approximations
2.6 Appendix
2.6.1 Computations of derivatives of CallBS w.r.t the log spot, the log strike and the total variance
2.6.2 Derivatives of CallBA w.r.t the spot, the strike and the total variance
2.6.3 Derivatives of δBS w.r.t the log spot, the log strike and the total variance
2.6.4 Proof of Lemma 2.1.2.1
2.6.5 Applications of the expansions for time-independent CEV model
3 Forward implied volatility expansions in local volatility models 
3.1 Introduction
3.2 Notations
3.3 Second and third order forward implied volatility expansions
3.3.1 Second order forward implied volatility expansion of type A
3.3.2 Third order forward implied volatility expansion of type A
3.3.3 Forward implied volatility expansions of type B
3.4 Numerical Experiments
3.5 Appendix
3.5.1 Results on Vega, the Vomma and the Ultima
3.5.2 Proofs of Lemmas 3.3.2.1-3.3.2.2-3.3.2.3
3.5.3 Proof of Theorem 3.3.3.1
3.5.4 Forward implied volatilities in time-independent local volatility models
4 Discussion on the parameterization and on the proxy model 
4.1 Revisiting the parameterization
4.2 Normal proxy on log(S ) or Log-normal proxy on S
4.3 Towards a displaced log-normal proxy
4.4 Numerical experiments
4.4.1 Comparison of the implied volatility behaviors in the displaced log-normal and CEV models
4.4.2 Comparison of the Gaussian, log-normal and displaced log-normal proxys for the CEV model
II Models combining local and stochastic volatility 
5 Price expansion formulas for model combining local and stochastic volatility 
5.1 Introduction
5.2 Main Result
5.2.1 Notations and definitions
5.2.2 Third order approximation price formula
5.2.3 Corollaries and outline of the proof
5.3 Error analysis
5.3.1 Approximation of X, V, and error estimates
5.3.2 Regularization of the function h by adding a small noise perturbation
5.3.3 Malliavin integration by parts formula and proof of estimate (5.40)
5.3.4 Proof of Lemma 5.3.3.2
5.4 Expansion formulas for the implied volatility
5.4.1 Implied volatility expansion at spot
5.4.2 Implied volatility expansion at mid-point
5.5 Numerical experiments
5.6 Appendix
5.6.1 Change of model
5.6.2 Explicit computation of the corrective terms of Theorem 5.2.2.1
5.6.3 Computations of derivatives of CallBS w.r.t the log spot and the volatility
5.6.4 Applications of the implied volatility expansion at mid-point for timeindependent local and stochastic volatility models with CIR-type variance
6 Smile and Skew behaviors for the CEV-Heston model 
III Price approximation formulas for barrier options 
7 Price expansions for regular down barrier options 
7.1 Introduction
7.2 Derivation of the expansion
7.2.1 Notations and definitions
7.2.2 Second order expansion
7.2.3 Third order expansion
7.3 Calculus of the corrective terms and error analysis
7.3.1 Preliminary results
7.3.2 Calculus of vP,φt o,t and estimate of its spatial derivatives
7.3.3 Proof of the error estimate in Theorem 7.2.2.1
7.3.4 Calculus of vP,ψt o,t , v P,ρs,t o,s and estimates of their derivatives
7.3.5 Proof of the error estimate in Theorem 7.2.2.1
7.4 Applications to the pricing of down and in barrier options
7.5 Applications to regular down barrier Call options
7.5.1 Notations
7.5.2 Regular down barrier Call option approximations with the local volatility at midpoint.
7.5.3 Reductions in the time-homogeneous framework
7.5.4 Numerical experiments
7.6 Appendix
7.6.1 Properties of the Gaussian density, the Gaussian cumulative function and the Gaussian hitting times density
7.6.2 Proof of Lemmas 7.3.4.4-7.3.4.5
7.6.3 Proof of Propositions 7.5.2.1-7.5.3.1
IV Efficient weak approximations in multidimensional diffusions 
8 Stochastic Approximation Finite Element method: analytical formulas for multidimensional diffusion process 
8.1 Introduction
8.2 Main results
8.2.1 Second order weak approximation and Monte Carlo simulations on the Proxy
8.2.2 An efficient algorithm using multilinear finite elements
8.2.3 Final approximation and complexity of the SAFE algorithm based on multilinear finite elements
8.2.4 SAFE with multiquadratic finite elements
8.3 Proof of the error estimate in Theorem 8.2.1.1
8.3.1 Lp-norm estimates of Xη − x0, ˙Xη and ¨Xη
8.3.2 Regularization of h with a small noise perturbation
8.3.3 Malliavin integration by parts formula and proof of Proposition (8.3.2.1)
8.3.4 Proof of Proposition 8.3.3.1
8.4 Proof of Theorem 8.2.2.2
8.4.1 Truncation Error ErrorFEL,T
8.4.2 Interpolation Error ErrorFEL,I
8.5 Numerical experiments
8.5.1 Model and set of parameters
8.5.2 Results
8.6 Appendix
8.6.1 Computation of the correction terms in Theorems 8.2.1.1 and representation as sensitivities
9 Price approximations in multidimensional CEV models using the SAFE methods 239
List of Figures
List of Tables
Bibliography

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