Monte Carlo Methods in Finance

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Monte Carlo Methods in Finance

Author: Yiyang Yang

Advisor: Pr. Xiaolin Li, Pr. Zari Rachev

Department of Applied Mathematics and Statistics State 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

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Introduction Implement of Monte Carlo Method Techniques for Elaborate Simulation Example of Pricing European Options

Outline

1

Introduction

2

Implement of Monte Carlo Method Generating Random Numbers Generating Random Variables Generating Sample Path

3

Techniques for Elaborate Simulation Variance Reduction Techniques Quasi Monte Carlo Method

4

Example of Pricing European Options Black-Scholes Equations

Monte Carlo Simulations for Option Pricing

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Introduction

Background History:

John von Neumann, Stainslaw Ulam and Nicholas Metroplis Manhattan Project in Los Alamous National Laboratory Monte Carlo Casino, Monaco

Monte Carlo methods:

experimental mathematics

large number of random variable simulation strong law of large numbers

the sample average converges almost surely to the expected value

X¯n= X1+ · · · + Xn

n

a.s.→ µ n → ∞ i .e.

P

n→∞lim X¯n= µ

= 1.

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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 distribution

Finance: models for stock prices, credit risk

Physical, biology and social science: models with complex nondeterministic time evolution

Adding artificial randomness to a system, then simulating the new system

Solving some partial differential equations

Markov chain Monte Carlo methods: for problems in statistical physics and in Bayesian statistics

Optimization: “travelling salesman”, “genetic algorithms”

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Example

Objective: integral

α = ˆ1

0

f (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

ˆ αn=1

n

X

i =1

f (Ui)

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

αn→ α with probability 1 as n → ∞

The error αn− α is approximately normally distributed with mean 0 and standard deviation^σ^f/^√n, whereσ_f²=´1

0(f (x ) − α)²dx and can be approximated by smaple standard deviationsf =

q 1 n−1

Pn

i =1(f (Ui) − ˆαn)²

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Principles of Derivative Pricing

Principles of theory for Monte Carlo

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

Discounted asset prices are martingales under a probability measure associated with the choice of discount factor. Prices are expectation of discounted payoffs under such martingale measure.

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

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Random Number Generator

A generator of genuinely random numbers has the mechanism for producing random variables U

₁

, U

₂

, · · · such that

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

A random number generator produces a finite sequence of numbers u

1

, u

2

, · · · , u

K

in the unit interval

pseudorandom number generator

not real random number, only mimics randomness

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Generating Random Numbers Generating Random Variables Generating Sample Path

Linear Congruential Gnerators

Definition

The general linear congruential generator proposed by Lehmer takes the form

x

_{i +1}

= (ax

_i

+ c) mod m, u

i +1

= x

i +1

m .

a, c, m are integer constants that determine the value generated.

Initial value x

0

is called seed.

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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 uniform distribution 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 can be implemented as

X = −θ log (U) .

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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 from Unif [0, 1]. If U ≤

^{f (X )}

/

cg (X )

, this is the expected X ; else, repeat above steps.

Proof sketch:

Let Y be a sample returned by the algorithm and observe that Y has the distribution 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 )) =´

A f (x )

cg (x )g (x ) dx =_c¹´

Af (x ) dx P (U ≤^{f (X )}/cg (X )) =´

R f (x )

cg (x )g (x ) dx = ¹_c P (Y ∈ A) =´

Af (x ) dx conclusion proved.

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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 increments

W (t1) − W (t0) , W (t2) − W (t1) , · · · , W (tk) − W (tk−1) are independent 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 n independent and identically distributed random variable B1, · · · , Bnsuch that Bi ∼ N 0,^T_n , i = 1, · · · , n.

Define ˆW ^kT_n = P^k_{i =1}Bi, then as n → ∞, ˆW is an appropriate of W .

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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

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Geometric Brownian Motion

Definition

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

Geometric Brownian motion is the most fundamental model of the value of a financial asset.

Suppose W is a standard Brownian motion and a geometric Brownian motion process is often specified by an SDE

dS (t)

S (t) = µdt + σdW (t) By Itˆo 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)

.

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Geometric Brownian Motion (Cont.)

0 20 40 60 80 100 120

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

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The Stratified Sampling

Definition

Stratified sampling refers broadly to any sampling mechanism that constrains the fraction of observations drawn from specific subsets of the sample space.

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

E [Y ] =

K

X

i =1

P (Y ∈ Ai) E [Y |Y ∈ Ai] =

K

X

i =1

piE [Y |Y ∈ Ai]

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

An unbiased estimator of E [Y ] is provided by Y =ˆ

K

X

i =1

pi

1 ni

n_i

X

j =1

Yij= 1 n

K

X

i =1 n_i

X

j =1

Yij

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Variance Reduction Techniques Quasi Monte Carlo Method

Stratified Sampling (Cont.)

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

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Antithetic Variates

Definition

The method of antithetic variates attempts to reduce variance by introducing 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 mutually antithetic.

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 ˆY =_2n¹ Pn

i =1

Yi+ ˜Yi

Var ˆY

=1 nVar

n

X

i =1

Yi+ ˜Yi

2

=1 2

Var (Y1) + Cov Y1, ˜Y1

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Variance Reduction Techniques Quasi Monte Carlo Method

Importance Sampling

Definition

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

EQ[X ] = EQ˜

XdQ

d ˜Q

.

define Radon-Nikodym derivative L = ^dQ

d ˜Q and measure ˜Q is called an importance measure which give more weight to important outcomes.

Theorem

Let Q^∗be define by

dQ^∗ dQ = |X |

E |X |

the importance sampling estimator ZL^∗under Q^∗ has a smaller variance than the estimator ZL under any other ˜Q.

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Quasi Monte Carlo Method

Definition

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

Properties of Quasi Monte Carlo Method

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

Accelerate convergence of ordinary Monte Carlo method from O

√1 n

to Quasi Monte Carlo method O ¹_n.

Example: Suppose the objective is to calculate

E [f (U1, · · · , Ud)] = ˆ

[0,1)^d

f (x ) dx ≈ 1 n

n

X

i =1

f (xi)

for carefully and deterministically chosen points x1, · · · , xnin [0, 1)^d.

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Black-Scholes Equations

Monte Carlo Simulations for Option Pricing

European Option

Definition

An option is a derivative financial instrument that specifies a contract between two 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 in the transaction, while the seller incurs the corresponding obligation to fulfill the transaction.

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

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

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

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

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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 constant drift and volatility.

It is possible to borrow and lend cash at a known constant risk free interest rate.

It is possible to but and sell any amount, even fractional of stock.

The transactions do not incur any fees or costs and underlying security does not pay a dividend.

Definition

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

∂V

∂t +1

2σ²S²∂²V

∂S² + rS∂V

∂S − rV = 0.

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Black-Scholes Solution

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

C (S, t) = N (d1) S − N (d2) Ke^{−r (T −t)}

d1= ln

S K

+

r +^σ₂²

(T − t) σ√

T − t

d2= ln

S K

+

r −^σ₂²

(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 normal distribution.

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.

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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.

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No-Arbitrage Pricing Formula

Non-Arbitrage Pricing Formula

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

Vt = e^{−r (T −t)}E_Q[Π(T )|Ft] .

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

C (St, t) = e^{−r (T −t)}E_Q(S_T− K )⁺|Ft

and the stock price follows geometric Brownian motion dSt

St

= µdt + σdWt.

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Monte Carlo Simulation

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

Sk= Sk−1exp

µ −1 2σ²

4t + σW4t

, W4t ∼ N (0, 4t)

Start from S_i⁰= St, i = 1, · · · , N we have N paths ended with S

T −t 4t

i ,

then the simulated Monte Carlo result of call option can be approximated as

C (St, t) ≈ 1 N

N

X

i =1

S

T −t 4t

i − K

+

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Monte Carlo Simulation (Cont.)

0 0.2 0.4 0.6 0.8 1

1 1.2 1.4 1.6 1.8 2 0

0.2 0.4 0.6 0.8

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Monte Carlo Methods in Finance

Monte Carlo Methods in Finance

Author: Yiyang Yang

Advisor: Pr. Xiaolin Li, Pr. Zari Rachev

October 2, 2012

Outline

Introduction

Implement of Monte Carlo Method Generating Random Numbers Generating Random Variables Generating Sample Path

Techniques for Elaborate Simulation Variance Reduction Techniques Quasi Monte Carlo Method

Example of Pricing European Options Black-Scholes Equations

Monte Carlo Simulations for Option Pricing

Introduction

Background History:

Monte Carlo methods:

Introduction (Cont.)

Direct simulation of a naturally random system

Adding artificial randomness to a system, then simulating the new system

Example

Principles of Derivative Pricing

Principles of theory for Monte Carlo

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

Discounted asset prices are martingales under a probability measure associated with the choice of discount factor. Prices are expectation of discounted payoffs under such martingale measure.

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

Random Number Generator

A generator of genuinely random numbers has the mechanism for producing random variables U

, U

, · · · such that

A random number generator produces a finite sequence of numbers u

, u

, · · · , u

in the unit interval

Linear Congruential Gnerators

Definition

The general linear congruential generator proposed by Lehmer takes the form

x

= (ax

+ c) mod m, u

= x

m .

a, c, m are integer constants that determine the value generated.

Initial value x

is called seed.

Inverse Transform Method

Definition

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 from Unif [0, 1]. If U ≤

/

, this is the expected X ; else, repeat above steps.

Brownian Motion

Definition

Simulation of Brownian Motion

Brownian Motion (Cont.)

Geometric Brownian Motion

Definition

Geometric Brownian Motion (Cont.)

The Stratified Sampling

Definition

Stratified sampling refers broadly to any sampling mechanism that constrains the fraction of observations drawn from specific subsets of the sample space.

Stratified Sampling (Cont.)

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

Antithetic Variates

Definition

Importance Sampling

Definition

Theorem

Quasi Monte Carlo Method

Definition

European Option

Definition

Black-Scholes Equations

Definition

Black-Scholes Solution

Numerical Scheme

No-Arbitrage Pricing Formula

Non-Arbitrage Pricing Formula

Monte Carlo Simulation

Monte Carlo Simulation (Cont.)

References

Paul Glasserman, Monte Carlo Methods in Financial Engineering, Springer, ISBN-10: 0387004513, ISBN-13: