Numerical Methods for Pricing Exotic Options

1
Numerical Methods for Pricing Exotic Options

• Dimitra Bampou
• Supervisor Dr. Daniel Kuhn
• Second Marker Professor Berç Rustem
• 26/6/2008

2
Options

• Given a stock of Vodafone with price St
• Strike price K 50
• Maturity date T 1 year from now
• Premium 5
• European Call Option value
• Exotic options Asian Barrier
• Payoff of Asian Call option is

3
Profit Diagram Pricing Challenges
4
Standard Pricing Models

• Black Scholes model for European Options
• Assume risk neutral distribution
• Assume Geometric Brownian Motion
• No analytic solution for exotics
• Monte Carlo Simulations for Exotic Options
• Easily adapted
• Heavy computation
• Probabilistic errors

5
aim of the project

• Investigate two numerical methods for pricing
  options based on semidefinite programming
• Gotoh and Konno approach for pricing European
  options (GK)
• Method for pricing European, Asian and Barrier
  options proposed by Lasserre, Prieto-Rumeau and
  Zervos (LPZ)
• The method is based on the Problem of Moments for
  finding max/min of non convex functions
• Numerical results comparison with standard
  pricing models

6
Gotoh And Konno Approach
7
Formulating The Problem

• Want to find upper and lower bounds of a European
  Call option
• Since price of underlying is a process, use risk
  neutral probability distribution to find its
  expected value
• Given the first few moments of that distribution

8
Dual Problems

• Primal problems infinitely many decision
  variables
• Introduce dual variable y
• Dual problems infinitely many constraints

9
Propositions By Bertsimas Popescu

• Proposition 1 Polynomial is
  positive iff there is a positive semidefinite
  matrix
• such that
• Proposition 2 Polynomial is
  positive iff there is a positive
  semidefinite matrix such that

10
Improving The Dual Problems

• Use propositions to derive semidefinite
  optimization problems
• They are convex, with finite number of decision
  variables finite number of constraints
• Use interior point methods

11
The Problem of Moments
12
The Problem Of Moments

• Want to solve
• Or equivalently
• Express constraint using Moment and Localizing
  matrices

13
Moment Matrix

• Introduce sequence
• Moment matrix is defined as
• Example, for n2,k2

14
Localizing Matrix

• Polynomial q with coefficients qa and set R
  given by
• subscript of y in the entry
• Localizing matrix is given by
• Example, for kn2 and

15
MOMENTS OF MEASURE µ

• If the elements of y are moments of measure µ and
  p(x) is of order maximum k then
  because
• If the elements of y are moments of measure µ and
  p(x) is of order maximum k then
  because

16
The Problem of Moments (2)

• Now define the semidefinite optimization problem
• Expresses a non convex optimization problem into
  a convex

17
Lasserre, Prieto-Rumeau and Zervos Approach
18
Asian Call Option

• Payoff function
• Underlying asset is an Ito process of the form
• Define processes
• Exit location measure corresponding to stopping
  time t of Zt

19
Asian Call Option (2)

• Payoff function as a linear combination of the
  moments of exit location measure

20
Asian Call Option (3)

• Formulate the problem
• where

21
European Barrier Options

• Derive similar problems for approximating upper
  and lower bounds of European options
• Can apply same problems for pricing put options
  by changing the objective function
• To calculate the moments of process Z, we use the
  Itos lemma to obtain a system of ordinary
  differential equations
• To price barrier options, we use the
  infinitesimal generator of process Z

22
Numerical Results
23
Lasserre, Prieto-Rumeau and Zervos
24
Lasserre, Prieto-Rumeau and Zervos(2)

• Variance of Monte Carlo simulation is 0.0047
• Distance between bounds for r3 is 0.0013

25
Lasserre, Prieto-Rumeau and Zervos(3)

• Non monotonic sequence of upper bounds
• Variance of Monte Carlo is 0.0056 with confidence
  interval 0.1596,0.1781

26
Lasserre, Prieto-Rumeau and Zervos(4)

• Relative error of 0.24 at 3rd relaxation

27
Pricing Asian Options
28
Gotoh-Konno Approach

• Obtain tight upper and lower bounds in real time
• Must find a way to obtain moments of the
  distribution of the price of underlying asset

29
Lasserre, Prieto-Rumeau and Zervos Method

• Flexible and can be easily extended to price
  other exotic options
• Can be applied to any type of processes for the
  price of the underlying asset if its
  infinitesimal generator maps polynomials into
  polynomials
• Tight bounds and less time required than Monte
  Carlo simulations
• Must ensure that the distribution of the
  underlying asset is moment determinate to obtain
  monotone sequences

30
Future Work

• Adapt the problem of moments for pricing other
  types of exotic options
• Explore the cutting plane algorithm proposed by
  Gotoh and Konno for solving semidefinite
  optimization problems

31
QA