Stochastic processes and financial derivatives

E. Gobet - N. El Karoui - C. Medalie

This course runs from September to December, 3 hours per week + exercise sessions.

This course provides an overview of derivatives from a modeling and risk management perspective. The markets considered range from traditional markets (stocks/equity indices, interest rates, foreign exchange) to the new crypto-currency markets. Three speakers are planned for this course:

The derivatives market is an important element in the transfer of market risk between investors (banks, hedge funds, financial institutions,…). The objective of the course is to describe the financial products offered, and the theoretical and practical methods used in the market to value and hedge these financial products. The topics covered are the following.

Part A: the basics

I - Introduction to Financial markets, Black-Scholes formula, market conventions

  • Introduction to risk management: actors, products, data, risks
  • Valuation without model, no arbitrage and static replication, Carr’s formula
  • Fair pricing in discrete time-space market
  • Continuous-time market with log-normal dynamics (risk management, dynamic portfolio, vanilla option)
  • Black-Scholes formula, valuation PDE, computation of greeks

II - Some alternatives to lognormal modeling

  • Implicit distribution from data
  • Implied volatility
  • Displaced log-normal model and Bachelier model
  • Heston model and other Fourier-based pricing model

III - Back to hedging risks

  • Refinement of Black-Scholes formula
  • Barrier options
  • Delta-hedging and Delta-gamma hedging

Part B: towards more advanced topics

IV - Hedging risk with several assets in the same currency market

  • Modelling the volatility
  • Self-financing portfolio and no arbitrage
  • Complete market
  • Markets with friction and non-linear valuation
  • Change of numéraire and applications
  • Future markets
  • Implicit diffusion à la Dupire, Black-Scholes robustness

V - Foreign Exchanges

  • Market convention
  • Arbitrage
  • Principles of option valuation
  • Garman-Kohlhagen formula

VI - Introduction to Cryptocurrency Market Structure (Clara Medalie)

  • Understanding the Basics

    • 2021: Traditional Finance Meets Crypto
    • A Brief History of Crypto
    • Blockchain Technology and Digital Assets
    • Bitcoin, Ethereum, Altcoins, and Stablecoins
    • Key Players in the Crypto Industry
    • The Institutionalization of Crypto
  • Exchanges and Market Data

    • Cryptocurrency Exchange Architecture
    • The Crypto Exchange Landscape
    • Derivatives vs. Spot Exchanges
    • Centralized vs. Decentralized Exchanges
    • Exchange Regulation
    • Cryptocurrency Market Data
    • Defining Market Data
  • Liquidity and Market Structure

    • Defining Liquidity
    • Liquidity Analysis: Exchanges
    • Price Discovery
    • Market Crashes
    • Market Manipulation on Exchanges
    • Key Crypto Market Trends

VII - Interest rate (Nicole El Karoui)

  • Introduction to the interest rate market and interest rate derivatives

  • Definition and construction of the yield curve:

    • The classical models: Vasicek, C.I.R, Longstaff and Schwarz, affine models.
    • Multifactor models. HJM models: Interest rate structure equations derived from arbitrage.
  • The BGM or market model. Approximations

  • Interest rate options and hybrid instruments: valuation and hedging

  • Swaps, variable rate bonds

  • Caps, floors, swaptions, boosts

  • Volatility matrices and calibration problems