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Towards Call option approximations with the local volatility at strike and at midpoint
For general payoff functions, the most natural choice seems to choose a proxy with the local volatility frozen at spot. When we are dealing with Call or Put payoffs, the spot and strike variables play a symmetrical role [Dupire 1994], and there is a priori no reason to advantage one or the other one. A first attempt to exploit this duality in proxy expansion is analysed in [Gobet 2012b]. In this subsection, we briefly recall the expansion formulas with a local volatility at strike and then we present new expansion formulas with a local volatility at mid-point xavg = (x0 + k)/2 = log √S 0K or S avg = (S 0 + K)/2. We detail the analysis only for the log-normal proxy. The proofs for the normal proxy are very similar and are left as an exercise to the reader. To directly obtain expansions formulas with local volatility frozen at strike, the idea is to follow the Dupire approach [Dupire 1994], using explicitly the PDE satisfied by the Call price function (T,K) 7→ Call(S 0,T,K) = E[(S T −K)+]. Indeed we have that:
Higher-order proxy approximation
In this section, we give several expansions formulas with a third order accuracy. First, we recall without proof results obtained in [Benhamou 2010a] and [Gobet 2012b] for expansions based on local volatility at spot and at strike. Second we introduce a new expansion with local volatility frozen at mid-point. Finally new expansions of implied volatility are provided.
Towards a displaced log-normal proxy
So far we only have considered Gaussian and log-normal proxys. Such proxys are obtained with a zero order approximation of the diffusion coefficients. In order to reduce the number of corrective terms in the expansions, we could envisage higher order proxys. In the context of time-homogeneous1 local volatility, a natural surrogate is the displaced log-normal proxy obtained with a first order expansion of the diffusion coefficients. We consider the solution of the following SDE: dS t = (S t)dWt, S 0 > 0,
Comparison of the Gaussian, log-normal and displaced log-normal proxys for the CEV model
In this Subsection we compare the performance of the approximations at spot using a normal, a lognormal and a displaced log-normal proxy for the pricing in the CEV model. Note that for any β ∈ [0, 1], any S 0 ∈]0, 1] and any K > 0 we have (S 0) = S β 0, (1)(S 0) = βS β−1 0 > 0 and Kd = (S 0) + (1)(S 0)(K − S 0) = βK + S β 0 − βS 0 > βK + (1 − β)S 0 > 0 as required in the hypotheses of Theorem 4.3.0.3. For the numerical experiments, we set again ν = 0.25, S 0 = 1 and we allow β to vary by choosing the two values β = 0.8 and β = 0.2. We keep the same sets of maturities and strikes and report in Tables 4.9-4.10-4.11-4.12-4.13-4.14-4.15-4.16:
• The true value of the implied volatility in the CEV model denoted by True CEV.
• The implied volatility approximations obtained with the second order price approximations using the normal, the log-normal and the displaced log-normal proxys (see Theorem 4.3.0.3 equation (4.14)) denoted respectively by AppNCEV 2 , AppLNCEV 2 and AppDLNCEV 2 .
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 h
8.4.2 Interpolation Error ErrorFEL,I h
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
List of Figures
List of Tables
Bibliography
