[Titolo]

1
Numerical Methods in Finance
An introduction to Financial mathematics (Abdus
Salam - ICTP, Trieste, 17-19 December 2007)
Marco Airoldi marco.airoldi_at_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

!

• BS plain vanilla, Barrier option (with
  continuous monitoring)

11
Pricing methodsa general view