Probabilistic approach will be discussed in MA5248 (Stochastic Analysis in Mathematical Finance).

We shall simply present explicit solutions, if available, instead of the details of solution process.
Long Position / Short Position



Linear Payoff
Forward contracts

At the initial time, the delivery price is chosen such that it costs nothing for both sides to take a long or short position.

Contents of this course

Preliminary: Black-Scholes model and Binomial model
Pricing exotic options under Black-Scholes Framework American options and early exercise Multi-asset options Path-dependent options Beyond Black-Scholes world Interest rate derivatives

Matlab coding is preferable but not required.
Basic concepts



A derivative is a security whose value depends on the values of other more underlying variables underlying: stocks, indices, commodities, exchange rate, interest rate derivatives: futures, options, forward contracts, bonds, swaps, swaptions, convertible bonds
A question: how to determine the delivery price?

Options

A call option is a contract which gives the holder the right to buy an asset (known as the underlying asset) by a certain date (expiration date or expiry) for a predetermined price (strike price).
Stochastic control

Option Pricing Theory
Black-Scholes (1973), R. Merton (1973)
Mathematical Instruments

Probabilistic methods stochastic calculus, martingale theory, stochastic control, backward stochastic differential equations, Monte-Carlo simulation Partial differential equations, numerical methods and binomial tree model

As for how to find the explicit solutions, please refer to MA4221 (Partial Differential Equations).

We do not fear numerical solutions. We shall mainly focus on the binomial tree method which is easy to implement and can be regarded as a certain explicit finite difference scheme.

Monte-Carlo simulation will be briefly introduced. Other numerical methods for partial difference equations will be discussed in MA4255 (Numerical Partial Differential Equations).
Forward contracts

An agreement between two parities to buy or sell an asset (known as the underlying asset) at a future date (expiry) for a certain price (delivery price) Contrasted to the spot contract.



(The pricing of credit derivatives is not included.)
Our philosophy

We present a unified approach to modeling those derivative products as partial differential equations (PDEs), or equivalently, binomial models.
Modern Finance

Modern Portfolio Theory
Hale Waihona Puke single-period models: H. Markowitz (1952) Optimization problem

continuous-time finance: R. Merton, P. Samuelson