Monte Carlo Methods in Finance - Stony Brookyiyang/research/presentation/MonteCarloFinan… ·...

IntroductionImplement of Monte Carlo Method

Techniques for Elaborate SimulationExample of Pricing European Options

Monte Carlo Methods in Finance

Author: Yiyang YangAdvisor: Pr. Xiaolin Li, Pr. Zari Rachev

Department of Applied Mathematics and StatisticsState University of New York at Stony Brook

October 2, 2012

Yiyang Yang (Advisor: Pr. Xiaolin Li and Pr. Zari Rachev) Monte Carlo Methods in Finance



Outline

1 Introduction

2 Implement of Monte Carlo MethodGenerating Random NumbersGenerating Random VariablesGenerating Sample Path

3 Techniques for Elaborate SimulationVariance Reduction TechniquesQuasi Monte Carlo Method

4 Example of Pricing European OptionsBlack-Scholes EquationsMonte Carlo Simulations for Option Pricing




Introduction

Background

History:

John von Neumann, Stainslaw Ulam and Nicholas MetroplisManhattan Project in Los Alamous National LaboratoryMonte Carlo Casino, Monaco

Monte Carlo methods:

experimental mathematicslarge number of random variable simulationstrong law of large numbers

the sample average converges almost surely to the expectedvalue

Xn =X1 + · · ·+ Xn

na.s.→ µ n→∞ i .e.

P(

limn→∞

Xn = µ)

= 1.




Introduction (Cont.)

Two broad classes of Monte Carlo methods:

Direct simulation of a naturally random system

Operations research (inventory control, hospital management)Statistics: properties of complicated distributionFinance: models for stock prices, credit riskPhysical, biology and social science: models with complexnondeterministic time evolution

Adding artificial randomness to a system, then simulating thenew system

Solving some partial differential equationsMarkov chain Monte Carlo methods: for problems in statisticalphysics and in Bayesian statisticsOptimization: “travelling salesman”, “genetic algorithms”




Example

Objective: integral

α =

ˆ 1

0f (x) dx = E [f (U)]

U uniformly distributed between 0 and 1

Get points U1,U2, · · · independently and uniformly from [0, 1]

The Monte carlo estimate

αn =1

n

n∑i=1

f (Ui )

If f is integrable over [0, 1], by strong law of large numbers

αn → α with probability 1 as n→∞

The error αn − α is approximately normally distributed with mean 0 andstandard deviation σf/√n, where σ2

f =´ 10 (f (x)− α)2 dx and can be

approximated by smaple standard deviation sf =√

1n−1

∑ni=1 (f (Ui )− αn)

2




Principles of Derivative Pricing

Principles of theory for Monte Carlo

If a derivative security can be perfectly replicated throughtrading in other assets, then the price of the derivativesecurity is the cost of the replicating trading strategy.

Discounted asset prices are martingales under a probabilitymeasure associated with the choice of discount factor. Pricesare expectation of discounted payoffs under such martingalemeasure.

In a complete market, any payoff can be realized through atrading strategy and the martingale measure associated withthe discount rate is unique.




Generating Random NumbersGenerating Random VariablesGenerating Sample Path

Random Number Generator

A generator of genuinely random numbers has the mechanismfor producing random variables U1,U2, · · · such that

each Ui is uniformly distirbuted between 0 and 1the Ui are mutually indepedent

A random number generator produces a finite sequence ofnumbers u1, u2, · · · , uK in the unit interval

pseudorandom number generatornot real random number, only mimics randomness





Linear Congruential Gnerators

Definition

The general linear congruential generator proposed by Lehmertakes the form

xi+1 = (axi + c) mod m,

ui+1 =xi+1

m.

a, c,m are integer constants that determine the valuegenerated.

Initial value x0 is called seed.





Inverse Transform Method

Definition

In order to sample from a cumulative distribution function F , i.e.generate a random variable X with the property that P (X ≤ x) = F (x)for all x . The inverse transform method sets

X = F−1 (U) , U ∼ Unif [0, 1]

where F−1 is the inverse of F and Unif [0, 1] denotes the uniformdistribution on [0, 1].

Proof sketch: P (X ≤ x) = P(F−1 (U) ≤ x

)= P (U ≤ F (x)) = F (x)

Example: The exponential distribution with mean θ has distribution

F (x) = 1− e−x/θ, x ≥ 0.

Inverting the exponential distirbution yields X = −θ log (1− U) and canbe implemented as

X = −θ log (U) .





Acceptance-Rejection Method

Definition

In order to generate random variable X with density function f (x),we first generate X from distribution g (x). Then generate U fromUnif [0, 1]. If U ≤ f (X )/cg(X ), this is the expected X ; else, repeatabove steps.

Proof sketch:

Let Y be a sample returned by the algorithm and observe that Y has thedistribution of X conditional on U ≤ f (X )/cg(X ). For any A ⊆ R

P (Y ∈ A) = P (X ∈ A |U ≤ f (X )/cg(X ) )

=P (X ∈ A,U ≤ f (X )/cg(X ))

P (U ≤ f (X )/cg(X ))

P (X ∈ A,U ≤ f (X )/cg(X )) =Á

f (x)cg(x)

g (x) dx = 1c

Á

f (x) dx

P (U ≤ f (X )/cg(X )) =´R

f (x)cg(x)

g (x) dx = 1c

P (Y ∈ A) =Á

f (x) dx conclusion proved.�





Brownian Motion

Definition

A standard one-dimensional Brownian motion on [0,T ] is a stochastic process{W (t) , 0 ≤ t ≤ T} with following properties:

W (0) = 0;

The mapping t →W (t) is a continuous function;

The incrementsW (t1)−W (t0) ,W (t2)−W (t1) , · · · ,W (tk)−W (tk−1) areindependent for any k and any 0 < t0 < t1 < · · · < tk ≤ T ;

W (t)−W (s) ∼ N (0, t − s) for any 0 < s < t < T .

Simulation of Brownian Motion

Based on the stationary and independent increment properties, generate nindependent and identically distributed random variable B1, · · · ,Bn suchthat Bi ∼ N

(0, T

n

), i = 1, · · · , n.

Define W(kTn

)=∑k

i=1 Bi , then as n→∞, W is an appropriate of W .





Brownian Motion (Cont.)

0 20 40 60 80 100 120−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4





Geometric Brownian Motion

Definition

A stochastic process S (t) is a geometric Brownian motion if log S (t) is aBrownian motion with initial value log S (0).

Geometric Brownian motion is the most fundamental model of the valueof a financial asset.Suppose W is a standard Brownian motion and a geometric Brownianmotion process is often specified by an SDE

dS (t)

S (t)= µdt + σdW (t)

By Ito formula, we have

d (log S (t)) =

(µ−

1

2σ2

)dt + σdW (t)

and if S has initial value S (0) then

S (t) = S (0) exp

[(µ−

1

2σ2

)t + σW (t)

].





Geometric Brownian Motion (Cont.)

0 20 40 60 80 100 1200.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6




Variance Reduction TechniquesQuasi Monte Carlo Method

The Stratified Sampling

Definition

Stratified sampling refers broadly to any sampling mechanism thatconstrains the fraction of observations drawn from specific subsetsof the sample space.

Let A1, · · · ,Ak be disjoint subsets of the real line for whichP (Y ∈ ∪iAi ) = 1, then

E [Y ] =K∑i=1

P (Y ∈ Ai ) E [Y |Y ∈ Ai ] =K∑i=1

piE [Y |Y ∈ Ai ]

Decide in advance what fraction of the sample should be drawn from eachstratum Ai and the theoretical probability pi = P (Y ∈ Ai )

An unbiased estimator of E [Y ] is provided by

Y =K∑i=1

pi1

ni

ni∑j=1

Yij =1

n

K∑i=1

ni∑j=1

Yij





Stratified Sampling (Cont.)

comparison of stratified sample (left) and random sample (right)





Antithetic Variates

Definition

The method of antithetic variates attempts to reduce variance byintroducing negative dependence between pairs of replications.

U and 1− U are both uniformly distributed over [0, 1]F−1 (U) and F−1 (1− U) both have distribution F and are mutuallyantithetic.Implement of the antithetic variates method:

the pairs(

Y1, Y1

),(

Y2, Y2

), · · · ,

(Yn, Yn

)are i .i .d .

each Yi and Yi have same distribution and Cov(

Yi , Yi

)< 0

Monte Carlo estimate is Y = 12n

∑ni=1

(Yi + Yi

)Var

(Y)

=1

nVar

n∑i=1

(Yi + Yi

2

)=

1

2

(Var (Y1) + Cov

(Y1, Y1

))Yiyang Yang (Advisor: Pr. Xiaolin Li and Pr. Zari Rachev) Monte Carlo Methods in Finance




Importance Sampling

Definition

Compute an expectation under a given probability measure Q of a randomvariable X , there is another measure Q equivalent to Q such that

EQ [X ] = EQ

[X

dQ

dQ

].

define Radon-Nikodym derivative L = dQdQ

and measure Q is called animportance measure which give more weight to important outcomes.

Theorem

Let Q∗ be define bydQ∗

dQ=|X |

E |X |the importance sampling estimator ZL∗ under Q∗ has a smaller variance thanthe estimator ZL under any other Q.





Quasi Monte Carlo Method

Definition

Quasi Monte Carlo method is a method numerical integration and solving someproblems using low-discrepancy sequences (or quasi-random sequence orsub-random sequences)

Properties of Quasi Monte Carlo Method

Quasi Monte Carlo method make no attempt to mimic randomness, butto generating points evenly distributed.

Accelerate convergence of ordinary Monte Carlo method from O(

1√n

)to

Quasi Monte Carlo method O(1n

).

Example: Suppose the objective is to calculate

E [f (U1, · · · ,Ud)] =

ˆ[0,1)d

f (x) dx ≈ 1

n

n∑i=1

f (xi )

for carefully and deterministically chosen points x1, · · · , xn in [0, 1)d .




Black-Scholes EquationsMonte Carlo Simulations for Option Pricing

European Option

Definition

An option is a derivative financial instrument that specifies a contract betweentwo parties for future transaction on an asset at a reference price (the strike).The buyer of the option gains the right, but not the obligation, to engage inthe transaction, while the seller incurs the corresponding obligation to fulfill thetransaction.

An option conveys the right to buy something is called a call option; anoption conveys the right to sell something is called a put option.

The reference price at which the underlying asset may be traded is calledstrike price or exercise price.

Most options have an expiration date and if it is not exercised by theexpiration date, it becomes worthless.

A European option may be exercised only at the expiration date of theoption.





Black-Scholes Equations

Black-Scholes model of the market follows these assumptions:

There is no arbitrage opportunity.The stock price follows a geometric Brownian motion with constantdrift and volatility.It is possible to borrow and lend cash at a known constant risk freeinterest rate.It is possible to but and sell any amount, even fractional of stock.The transactions do not incur any fees or costs and underlyingsecurity does not pay a dividend.

Definition

The Black-Scholes equation is a partial differential equation which describesthe price of the option over time:

∂V

∂t+

1

2σ2S2 ∂

2V

∂S2+ rS

∂V

∂S− rV = 0.





Black-Scholes Solution

The value of a call option for a non-dividend paying underlying stock interms of Black-Scholes parameter is

C (S, t) = N (d1)S − N (d2)Ke−r(T−t)

d1 =ln(

SK

)+(r + σ2

2

)(T − t)

σ√T − t

d2 =ln(

SK

)+(r − σ2

2

)(T − t)

σ√T − t

= d1 − σ√T − t.

The price of a corresponding put option based on put-call parity is

P (S, t) = Ke−r(T−t) − S + C (S, t) = N (−d2)Ke−r(T−t) − N (−d1) S.

N (·) is the cumulative distribution function of the standard normaldistribution.

T − t is the time to maturity.

S is spot price of the underlying asset and K is strike price.

r is the risk free rate and σ is the volatility of the returns.





Numerical Scheme

With boundary conditions

C (0, t) = 0 for all t, C (S , t)→ S as S →∞, C (S ,T ) = max {S − K , 0}corresponding numerical schemes can be developed.





No-Arbitrage Pricing Formula

Non-Arbitrage Pricing Formula

Price of a European option can be obtained by the expectation of thepresent value of the payoff for the options under the equivalent martingalemeasure Q. That is, at time t < T , the non-arbitrage price of a Europeanoption Vt with the payoff Π(T ) and the maturity T is obtained by

Vt = e−r(T−t)EQ [Π(T )|Ft] .

Consider European call option, then Π(T ) = (ST − K)+ .

Then we have

C (St , t) = e−r(T−t)EQ[(ST − K)+ |Ft

]and the stock price follows geometric Brownian motion

dSt

St= µdt + σdWt .





Monte Carlo Simulation

Simulate N (about ten thousands scale) paths of Si , i = 1, · · · ,NEvery path Si is genererated from time t to T step by step

Sk = Sk−1 exp

[(µ− 1

2σ2

)4t + σW4t

], W4t ∼ N (0,4t)

Start from S0i = St , i = 1, · · · ,N we have N paths ended with S

T−t4t

i ,then the simulated Monte Carlo result of call option can be approximatedas

C (St , t) ≈ 1

N

N∑i=1

(S

T−t4t

i − K

)+





Monte Carlo Simulation (Cont.)

0

0.2

0.4

0.6

0.8

1

11.2

1.41.6

1.82

0

0.2

0.4

0.6

0.8





References

Paul Glasserman, Monte Carlo Methods in FinancialEngineering, Springer, ISBN-10: 0387004513, ISBN-13:978-0387004518, August 2003

Phelim P. Boyle, Options: A Monte Carlo Approach, Journalof Financial Economics 4(1977) 323-338

Phelim Boyle, Mark Broadie, Paul Glasserman, Monte CarloMethods for Security Pricing, Journal of Economic Dynamicsand Control 21(1997) 1267-1321


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