Option PricingMontecarlo Method, Path Integrals

A.Chiarugi, G. Cipriani, M.Rosa-Clot, S.Taddei Firenze

E. Bennati Dip Scienze Econ. (Pisa)

G.Lotti Dip. Matematica (Parma)

M.Cerchiai, G. Einaudi, P. Rosa-Clot (Pisa)

S.R. Amendolia, B.Golosio (Sassari)

Tossing a Coin:100, 1000, 10000, 50000 Trials

Montecarlo Method: a simple barrier test

0 2 4 6 8 10 12-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1 monte1.m

Probability distribution

• How we can check a distribution law ? • Looking at many trials !• How many ? A lot ! ! !

Tossing a coin we assume

x= 1 x =w

This is a Wiener process

Wiener Process

In general we write

x =x,t)t + (x,t) w

As a particular case we have

r =a (b - r ) t + w Vasicek

or

r =a (b - r ) t + r w CIR

General case: stochastic equation

This equation can be solved with several techniques

• analytical methods

• differential equations (Fokker Plank)

• tree discretized steps

• Montecarlo method !!!!

• path integral approach !!!!

dWtxdttxdx ),(),(

Analytical approach: the CIR model

A CIR model realisation

0 2 4 6 8 10 120

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Cox, Ingersoll, Ross a= 0.5 b= 0.1 sigma= 0.075

monte3.m

Why realistic models?

• Vasicek model has serious drawbacks (it allows negative interest rate values)

• CIR (Cox Ingersoll Ross) model looks more realistic and it allows analytical solutions.

• However the first target is to do without “analytical models” and to use real rates.

• The second target is to work with a general “functional” (???)

What “Functional” means?

0 2 4 6 8 10 120

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Cox, Ingersoll, Ross a= 0.5 b= 0.1 sigma= 0.075

In the left plot , all the paths which overcome

the black line are weighted with the

corresponding interest rate

Functionals can be very complicated especially for exotic options We have to deal with barriers, look back options, and with option price depending on past averaged quantities

A functional evaluation requires :

• To average on all the possible paths

• However a path can depend on the functional

• So we have to perform a huge number of trials

• Then => MONTECARLO

• To convert the continuos process in a set of finite steps

• To know the probability distribution at any time

• To integrate numerically on the prob. distributions

• Then => PATH INTEGRAL

Convolution and composition law

• The density (y,t,x,0) gives the probability to get the y value at a time t’, once the distribution is known a time t=0.

• Such a density satisfies the convolution law

dzxzztyxty )0,,,(),,,()0,,,(

Composition law for a short time t

• For a short time we get

ttxxLetxt

txtty

),,(

),(21),,,(

with

txyx

ttxxtx

txxL

)(

),(),(2

1),,(2

2

The numerical problem: N t = T

• We have to perform N numerical convolutions i.e. N matrix products.

• Matrices are exponentials with a lagrangian L(x,v,t) as exponent

• where v= (y-x)/t is the “velocity” of the system.

Tred.m

Some paths through finite steps

0 2 4 6 8 10-4

-3

-2

-1

0

1

2

3

4 We show 5 realisations of a

stochastic process

The transfer function

is given for each step t

path1.m

Feynman Approach: the P.I.• Wiener creates the theory of stochastic processes in 1921

• Feynman introduces the path integral concept in physics with his master thesis in 1942.

• The computational problems are too big. In fact only in 1981 Kreutz e Freedman are able to perform a first numerical calculation of the “Harmonic Oscillator”

• 90th Huge explosion of Montecarlo approaches to P.I.

• Recently: deterministic approaches (Rosa-Clot and Taddei). Very quick but low dimension (<4).

• Which is enough for financial markets.

Numerical and theoretical improvements

• Well based theory• All the analytical cases

are under control• All the results in the

literature are “easily” reproduced

• Approximation techniques are well known

• Numerically stable• It is quick as and it

looks like tree approach• It allows the evaluation

of functional of arbitrary complexity in one, two or more dimensions

The functional

• In the more general case it is necessary to evaluate quantities which depends on the process itself.

• Two typical examples are the barrier options and the put American options

• This is “impossible” with Montecarlo but “easy” with Path Integrals

The put American option

0 2 4 6 8 10-4

-3

-2

-1

0

1

2

3

4

This option is exercised when its value is below a minimum which depends on the path and on the market model

What is available

• A set of Montecarlo codes for different models 1-2-3 D with and without stochastic volatility and for vanilla, barrier, swap options

• A corresponding set of Path Integral codes

• A set of Path integral codes for path dependent options (American, look back and exotic)

What of new

• CPU time for PI codes is remarkably shorter than for MC. The reduction factor ranges between 10 and 1000.

• The PI can be extended to a larger set of functional.

• The CPU time does not depend on the model complexity

Open Problem: DATA ANALYSIS

Open Problem: DATA ANALYSIS

METHODS•Statistical analysis•Autoregressive models

•Spectral test and wavelets

•Neural networks

TARGETS• Models for the Option

Pricing • Optimisation of Hedging

strategy• Short Forecasting (minutes)• Long Forecasting (months)

A short look to FIB30

price distribution with delay of

1 4 16 64 256 1024 tic

Levy.m

Real % Gaussian distribution

This work can be improved but

NEVER ENDS

See Amendolia Slides