1. Numerical Methods in Finance
An introduction to Financial mathematics
(Abdus Salam - ICTP, Trieste, December 2007)
Marco Airoldi
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)
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2. Lecture 1 – Options Plain vanilla options Exotic options
The Black & Scholes equations
The option pricing problem: possible strategies
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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.
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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
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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
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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.
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7. Reverse cliquet option
An option with a pay-off that depends on the average of underlying negative performances over some periods
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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 distribution Gaussian distribution
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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.
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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) ……
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11. Pricing methods: a general view
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