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Stochastic Calculus for Finance

Contents :

# Basic examples of financial derivatives: Examples of financial instruments, a first example of `arbitrage pricing’.
# Discrete time models I: Single period models, pricing a European option, characterising no arbitrage, risk neutral probabilities.
# Discrete time models II: Multiperiod binary models, discrete parameter martingales, risk-neutral pricing, Cox-Ross-Rubinstein.
# Brownian motion: Definition of Brownian motion (motivated via a rescaling of simple random walk), Lévy’s construction.
# The reflection principle and hitting times: reflection principle, hitting times, scaling properties.
# Martingales in continuous time: filtrations, adapted processes, Optional Sampling Theorem.
# Stochastic integration and Itô’s formula: variation and quadratic variation, quadratic variation of Brownian motion, outline of construction of the Itô stochastic integral and the Itô isometry, the chain rule and integration by parts for stochastic calculus. The Martingale Representation Theorem, Lévy’s characterisation of Brownian motion, Girsanov’s Theorem.
# The Black-Scholes model: self-financing strategies, equivalent martingale measures, the risk-neutral pricing formula.
# Pricing and hedging European options: Examples of European options. Explicit pricing formula. Evaluation of price and hedging strategies for European calls and puts.
# Valuation of some exotic options. Digital options, barrier options etc

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