What Is That – Monte Carlo?

Supposing we have an integral

)()()( gEgdyyfygA

where g(y) is an arbitrary function, f(y) is a probability density function, and A is the range of integration.

To obtain an estimate of E(g), we pick a number N of sample values (yt) at random from the probability density function f(y).

The estimate of E(g) is then given by

This simple equation is the quintessence of Monte Carlo method:

)(1

)(1

N

iiygN

gE

An integral is interpreted as the average value of a function over an interval that contains a fixed number of points chosen in random.

History of The Method

The Monte Carlo method was first utilized by scientists at Los Alamos to study some neutron characteristics during the Manhattan Project in 1940s.

With the progress in computing technology and development of random number generating algorithms, simplicity of the method has led to its wide spread application to a variety of problems where a closed-form solution is either unattainable or extremely hard to find, especially in evaluating multi-dimensional integrals.

Although Monte Carlo is applicable to a wide range of deterministic problems, it comes as a natural choice in modeling stochastic processes which explains why the method is so popular in financial industry. Indeed, the method’s very name implies its origin in games of chance and random numbers.

In this project, I have attempted to demonstrate how the method can simplify pricing of certain derivative securities.

Example: European Call Option

To price a European call option on a stock, we first the stock price equation under Q-measure in the discretized manner:

Si+1 = Si exp(r – σ2/2 + σZ)

Where Si is the current stock price; r is the risk-free rate;

σ2 is the constant variance of the return; Z is a normally distributed random number

Let S0 = 75, the strike price K = 50, σ2 = 0.025 (per time interval), r = 0.015 (per interval), and number of intervals equal to 20.

For this data, a simulation with a number of paths N = 75000 produces call option price X = 39.62 as the next slide illustrates. The option price is computed by averaging (SN – K)+ over the number of paths and discounting result to the present time.

(For comparison, a 20-period binomial tree yields price for the same option X = 40.99)

Simulation of A Stock Option Price

38.8

39

39.2

39.4

39.639.8

40

40.2

40.4

40.6

0 10000 20000 30000 40000 50000 60000 70000 80000

Pricing A Caplet

The option pricing theory tells us that a caplet is a put option expiring at time Ti-1 on a bond that expires at time Ti and can therefore be priced by the following equation:

)|)),((()1( t11

1FTTBKBEBkV iiTQtt i

Where k is the fixed rate;

δ is the a time interval (LIBOR tenor);

Bt is a risk-free bond (for t = 0, Bt = 1);

B-1Ti-1 is a discounting factor;

K = 1/(1+kδ) is the strike price;

B(Ti-1, Ti) is the underlying asset (bond)

Computing Bond Prices From CIR Model

Cox Ingersoll Ross model belongs to the family of affine models, and this circumstance allows for bond prices to be expressed in a relatively simple closed form, such as the following:

)(),(),(),( trTtZeTtYTtB

Where r(t) is a short term rate obtained from the model; Y(t,T) and Z(t,T) are deterministic functions defined as follows:

1)(2

)1(2),(

)}({

)(

tTh

htT

ehh

eTtZ

2/2

)}({

}2/))({(

1)(2

2),(

tTh

tTh

ehh

heTtY

222 h

The Discounting Factor BTi-1-1

The factor which discounts the option price to time 0 is determined from the following equation:

10

1

)(1T

i

dssrT eB

Which can be approximated using the trapezoid method as

sT

ssss

i

srr

T eB 0

1

)(2

1

1

Generating Short Term Rates Simulation from CIR Model

11 )())(()()( iiiii Wtrtrtrtr

In its discretized form, the Cox Ingersoll Ross model looks the following way:

Where σ, θ, and κ are deterministic parameters, and W is a normally distributed random number.

Given the parameters σ, θ, κ, and r(t= 0) one can easily construct an iteration procedure to obtain short term rates in a given interval starting from zero.

Having generated a series of short term rates for a period Ti

by using Monte Carlo simulation, one can compute the underlying bond price from the rate r(Ti-1) as

)(),(11

11),(),( iii TrTTZ

iiii eTTYTTB

and the discounting factor as

1

01

1

)(2

1

1

T

iii

i

rr

T eB

Then it becomes possible to compute the discounted put option price by averaging

)),(( 11

1 iiT TTBKBi

over ALL the sample paths (5000,10000,…75000?) That gives us our EQ!

The following two slides display a few sample paths chosen in random (by me) with their corresponding short rate term structures and bond prices

Effect Of CIR Parameters On Short Term Rates

From the discretized version of CIR model, it follows that a change in parameter κ changes the short rate for one unit of discretization in time by

))(()( 1 ii trtr

The following slide shows sensitivity of short rates to changes in Kappa (all other parameters fixed)

If we change θ by Δθ, the effect on short rates will be

)( 1itr

and the following slide displays the same sample path with three different Thetas

Following the same approach with the volatility, we obtain the following relationship between a change in σ and the corresponding change in short rates:

11 )()( iii Wtrtr

The following slide displays the same path with three different Sigmas

References

• Boyle, Phelim P. (1976) Options: A Monte Carlo Approach, The University of British Columbia, Vancouver, BC, Canada

• Baxter, M. and Rennie A. (1996) Financial Calculus, Cambridge University Press

• Zeytun, S. and Gupta, A. (2007) A Comparative Study of the Vasicek and the CIR Model of the Short Rate, Berichte des Fraunhofer ITWM, Nr. 124

Thank you very much !