Agenda Introduction to plain vanilla and exotic options Crude Monte Carlo method

1. Numerical Methods in Finance An introduction to Financial mathematics (Abdus Salam - ICTP,Trieste, 17-19 December 2007) Marco Airoldi marco.airoldi@mediobanca.it • Agenda • Introduction to plain vanilla and exotic options • Crude Monte Carlo method • Lab on Monte Carlo • Monte Carlo improvements • Lab on Monte Carlo with antithetic variables • Binomial Tree and lab • Finite Difference method (optional)

2. Lecture 1 – Options • Plain vanilla options • Exotic options • The Black & Scholes equations • The option pricing problem: possible strategies

3. Plain vanilla options • Plain vanilla option: it provides the right to buy or sell a specified quantity of a security (e.g. a stock) at a set strike price at some time on (european) or before (american) expiration: • Options can be used to hedge a position or for speculative reasons. • An option is based on the random behavior of the underlying, therefore stochastic calculus is required for its valuation.

4. Exotic options • They are a variations on the pay-off profiles of the plain vanilla options • They are traded over the counter (no regular market) • Due to their complexity, a part some exception, no closed formula are available to price them. • Digital options • Cliquet options • Lookback options • Basket options • Asian options • Barrier options • Reverse cliquet options

5. Asian option • An option where the pay-off depends on the average value of the underlying over a specified set of dates (the so called fixing dates). average price call: • The asian options are chipper than plain vanilla options because the averaging mechanism reduces the effective underlying volatility

6. Barrier option • An option with a pay-off that depends on whether or not the underlying price through a barrier level. call down-out: • The down-out barrier options are cheaper than corresponding plain vanilla options (with same characteristics) because in some situation (the touch of barrier) the option is knocked out, without paying any pay-off.

7. Reverse cliquet option • An option with a pay-off that depends on the average of underlying negative performances over some periods

8. The log-normal model for stock prices Evolution of stock prices: Wiener stochastic process / geometric brownian motion The discrete version of this model is the famous random walk model Einstein 1905 -- Bachelier 1900! Return distributionGaussian distribution

9. Black & Scholes PDE equation • Basing on equity price log normal model, the option pricing problem can be reduced to the solution of a Partial Differential Equation (PDE): the Black & Scholes equation • A closed form solution can be derived for plain vanilla options (Black & Scholes formula) and some simple exotic options. • No closed form solutions are available for more structured exotic options.

10. The option pricing problem • Closed form solution Usually exotic option pricing requires numerical algorithms: • Binomial or trinomial trees • Monte Carlo methods • Finite difference methods (based on PDE) • Quadrature algorithms B&S plain vanilla, Barrier option (with continuous monitoring) …… !

11. Pricing methods:a general view