| Lecturers | S. Crépey |
| Tutorials | M.C. Quenez |
| Périod | Terms 1 and 2 |
| Nombre de crédits: | 3+3 |
| Hourly Volume |
3 hours per week + tutorials |
This course bears on the fundamentals of financial derivatives modeling, with volatility as a core underlying concept. These are applied to market models of FX, interest rates, credit, and volatility derivatives.
- Revision week: European and American options in the Cox-Ross-Rubinstein model
1. The probabilistic characterizations of no arbitrage and completeness (FTAPs):
in a finite setup (with full proofs)
in continuous time (the easy parts)
2. Model-free results in continuous time:
Q-prices of options
Forwards and call-put parity
Call-put bound and shape constraints
The Breeden-Litzenberger formula
3. The Black-Scholes model:
Calls and puts pricing and Greeking formulas
Dynamic hedging and the Black-Scholes equation
Black-Scholes Implied volatility and the Black-Scholes model in practice
Binomial approximation to Black-Scholes
4. The local volatility model:
Dupire equation and formula
Gatheral equations and formulas, Berestycki-Busca-Florent short maturity asymptotics
Local volatility in stochastic volatility models: Gyönfi’s formula
Static arbitrage and Kellerer’s theorem
Pricing with a local volatility: implicit finite differences scheme vs trinomial approximation
Extracting the local volatility from an SVI interpolation of the implied total variance
5. Affine models:
The Heston model
The Merton Model
Fourier pricing schemes in affine models
6. Multivariate continuous Itô processes market model:
Change of numéraire
Exchange and FX options
7. Libor Market Model (LMM) of interest-rate derivatives:
Short Rate (CIR/Vasicek) and HJM Models in a Nutshell
Caps/floors and swaptions in the LMM
Pricing a swaption by Monte Carlo simulation in the Libor market model
8. Local stochastic volatility models:
The SABR model
A SABR/Bergomi-Type Model of Rough Volatility
Volatility derivatives: variance and volatility swaps, VIX options
9. One-factor Gaussian copula model of portfolio credit risk:
CDSs, CDOs
Implied correlation.
10. Nonparametric model calibration techniques:
Tikhonov regularization
Introduction to martingale optimal transport and applications to calibration
Case studies: (the yield curve, risk-neutral density,) local volatility, stochastic local volatility, the VIX/SPX joint calibration problem, calibration to path-dependent options
Main reference
Crépey, S. (2022-23), Derivatives Modeling, M2MO Lecture Notes.
Also
Bouchard, B. and J.-F. Chassagneux (2016), Fundamentals and Advanced Techniques in Derivatives Hedging, Springer Universitext.
Gatheral, J. (2011), The volatility surface: a practitioner's guide, Wiley.
Hull, J. (2021), Options, Futures, and Other Derivative Securities, 11e edition, Pearson [il existe une version française mais préférer la version anglaise].
Karatzas, I. and S. Shreve (1991), Brownian Motion and Stochastic Calculus, Springer Graduate Texts in Mathematics.
Lamberton, D. and Lapeyre B. (2013), Introduction au calcul stochastique appliqué à la finance. 3ème édition, Ellipses.
Shreve, S.: Stochastic Calculus for Finance II: Continuous-Time Models, Springer, 2004 or later.
Tankov, P. (2015 et 2008), Mathématiques financières et Calibration de modèles, polycopiés M2MO.