Numerical Methods in Finance

The majority of the financial derivatives pricing problems can not be exactly solved; accordingly, one needs to consult numerical methods. The state of-the-art numerical schemes for option pricing problems can be based on either simulation processes (e.g. Monte Carlo), or lattice methods (e.g. Binomial Trees), or (3) solutions of the partial differential equations (e.g. Finite Difference or Finite Element Methods).

1. Monte Carlo Method

The name “Monte Carlo” comes from the resemblance to casino games. It is a method, which can generate lots of data & scenarios. Each simulation describes a randomly chosen path of the underlying.

Monte Carlo simulation can deal with path dependent options (e.g. Asians, barriers,…), options dependent on several underlying state variables (e.g. Forex & interest rates), options with complex payoffs.

Key Features:

• Relatively easy to implement
• Can be applied practically on all payouts, with all models
• Can be applied on payouts with several underlyings
• Easy to parallelize computations
• Slow
• More difficult to implement on options with American exercise
• Calculation of greeks is not easy

A practical alternative is pricing via simulations:

• We simulate the evolution of the underlying a large number of times (~10000)
• For every simulation we calculate the expected gain for the owner of the option
• Option price = (average of gains) x (disc-fact)

2. Binomial Tree

Binomial trees are frequently used to approximate the movements of an underlying. In each small interval of time the stock price can move up by a proportional amount u, move down by a proportional amount d.

Key Features:

• Relatively easy to implement
• Exists for many payouts (barriers), with only some models
• Very stable for the calculation of the greeks
• Fast
• Difficult to parallelise computations

SEE ALSO: Stochastic Modeling – Definition, History, Applications

3. Finite Difference Methods

Finite difference methods represent the differential equation as a difference equation. The Finite Difference Method has been very fashionable due to its ease of implementation. Utilization
of Finite Difference Method requires the effective use of linear system solvers.

Key Features:

• Can be applied on many payouts, with most models
• Very stable for the calculation of the greeks
• Fast
• Difficult to parallelise computations
• Relatively difficult to implement