# M2MO: Modélisation Aléatoire, Finance et Data Science

Master en statistique, probabilités et finance - Université Paris 7 - Paris Diderot

## Derivatives modeling

 Lecturers S. Crépey Tutorials M.C. Quenez and Z. Grbac 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. Advanced interest rates modeling or the analysis of counterparty credit risk, as well as numerical (including machine learning) methods, are developed in optional courses.

I  No arbitrage and hedging principles: introduction in a discrete setup

1. Derivative cash flows, call/put parity, arbitrage bounds and shape constraints

2.  The fundamental theorems of asset pricing in a discrete setup: no arbitrage and completeness

3.   Illustration in the Cox-Ross-Rubinstein model (European and American options)

1. The Black-Scholes model

2. Calls and puts pricing and Greeking formulas

3. Dynamic hedging and  the Black-Scholes equation

4. Implied volatility and the Black-Scholes model in practice

5.   Binomial approximation  for European and American options

III. Local volatility

1.  The Local volatility model

2.  Dupire equation and formula

3.  Gatheral equations and formulas,  Berestycki-Busca-Florent short maturity asymptotics

4.  Local volatility in stochastic volatility models: Gyönfi’s formula

6. Pricing with a local volatility: implicit finite differences scheme vs trinomial approximation

7. Extracting the local volatility from an SVI interpolation of the implied total variance

IV. Stochastic volatility and Jumps

16. The Heston model

17. The Merton model

18. Fourier pricing schemes

V.  Exotic options

1.  Change of numéraire and application to exchange options, quanto options

2.  Dynamic vs. static replication of barrier options,

3. Static replication of variance swaps, VIX index

VI.  Interest rates derivatives (swaps, swaptions, caps and floors): an introduction

1.  Short rate models (Vacicek, Cox-Ingersoll-Ross)

2.  Interest rate market models

3.  Pricing a swaption by simulation in the Libor market model.

Main references

Crépey, S. (2021), Derivatives Modeling, M2MO Lecture Notes

Tankov, P. (2015), Mathématiques financières, polycopié M2MO.

Lamberton, D. and Lapeyre B. (2013), Introduction au calcul stochastique appliqué à la finance.  3ème édition, Ellipses.

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.

Also

Shreve, S.: Stochastic Calculus for Finance II: Continuous-Time Models, Springer, 2004 or later.

Karatzas, I. and S. Shreve (1991),  Brownian Motion and Stochastic Calculus, Springer Graduate Texts in Mathematics.

Hull, J. (2014), Options, Futures, and Other Derivative Securities, 10e edition, Prentice Hall. [Il existe une version française mais préférer la version anglaise].