stochastic process and their applications.pdf

7/28/2019 Stochastic process and their applications.pdf

1/31

Stochastic Processes and their Applications to

Mathematical Finance

Sean Fanning Jay Parekh

August 17, 2004


2/31

Contents

1 Introduction 3

2 Financial Background 4

2.1 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 Stocks: the Underlying Asset . . . . . . . . . . . . . . . . 42.1.3 Payoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.4 Financial Leverage . . . . . . . . . . . . . . . . . . . . . . 62.1.5 Trading Strategies: Option Spreads . . . . . . . . . . . . . 7

2.2 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Stochastic Processes 10

3.1 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . 11

3.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.2 A Better Model . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Stochastic Calculus 13

4.1 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 134.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 134.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2 Itos Formula: A Stochastic Chain Rule . . . . . . . . . . . . . . . 164.2.1 Motivation by Taylor Expansion . . . . . . . . . . . . . . . 16

4.2.2 Formal Itos Formula . . . . . . . . . . . . . . . . . . . . . 16

5 The Black-Scholes Formula 17

5.1 Deriving the Black-Scholes Formula . . . . . . . . . . . . . . . . . 175.2 Feynman-Kac Theorem . . . . . . . . . . . . . . . . . . . . . . . . 185.3 Exact Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1


3/31

6 Numerical Solutions 20

6.1 Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . 206.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6.2 Euler-Maruyama Method . . . . . . . . . . . . . . . . . . . . . . . 21

7 Application: 3M Company 23

7.1 Estimating the Parameters . . . . . . . . . . . . . . . . . . . . . . 247.2 The Application: The Black-Scholes Price vs. The True Market

Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 247.2.2 Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . 25

8 Appendix: Code 26

8.1 Payoff Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268.2 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . 278.4 Euler-Maruyama Payoff . . . . . . . . . . . . . . . . . . . . . . . 278.5 Pricing European Call . . . . . . . . . . . . . . . . . . . . . . . . 28

9 References 30

2


4/31

Chapter 1

Introduction

Worldwide, there is an estimated $10,000 billion gross in the derivative market.This paper provides a brief overview of options and the stochastic processes usedto model them. We assume the reader has a general understanding of probability,calculus, and differential equations. We do not assume the reader has priorknowledge about options, stochastic processes, or stochastic calculus.

We begin with a brief background on basic financial concepts, with an em-phasis on options. We then cover Brownian Motion and Geometric BrownianMotion, the latter of which we will use as a model for stock prices. It then be-comes necessary to introduce stochastic calculus concepts. Using the stochasticcalculus, we procede to the Black-Scholes Formula, which is used to price options.Then using the Feynman-Kac Theorem, we relate the Black-Scholes partial dif-

ferential equation to a stochastic diffential equation. Basic numerical methodsfor solving the stochastic differential equation are introduced. MATLAB code forthe numerical methods is supplemented. We conclude with an application to thereal-world stock data of 3M Company.

We would like to thank Professors Jeffrey Cooper and Kyoung-Sook Moon fortheir instruction, guidance, and time as well as the University of Maryland MathDepartment for sponsoring this project.

3


5/31

Chapter 2

Financial Background

2.1 Options

2.1.1 Basics

An option is a contract that gives the purchaser the right to buy or sell a specifiednumber of shares of an underlying asset at a fixed price on a specified future date.There is no obligation to exercise the option. In our case, we will discuss optionswhose underlying asset is a publicly traded stock.

The price at which the buyer of an option agrees to buy or sell an option, ifhe or she so chooses, is called the strike price. It is denoted as K. The time atwhich the option expires is called expiry, denoted as T.

Options can be classified as either call or put options. A call option givesthe purchaser the right to buy a security. A put option gives the purchaser theright to sell a security. Generally, one option corresponds to the purchase or saleof 100 shares of stock.

There are many types of options. Two specific types are European andAmerican options. A European option gives the purchaser the right to buy orsell stock only upon expiry of the option. Alternatively, an American option givesthe purchaser the right to buy or sell stock at any time between purchase andexpiry. In this paper, we will only consider European options.

2.1.2 Stocks: the Underlying Asset

A stock represents ownership in a company. By purchasing stock, an investor isputting a claim on a companys assets and earnings. A shareholder purchasesstock because he or she expects to receive compensation in the form of eitherdividends1 or capital gains2.

1We will not take into account dividends when considering the pricing of options.2A capital gain is the profit resulting in an increase in stock price, between the time of

purchase and the time of sale.

4


6/31

Stocks are traded on an exchange or market, where buyers and sellers arelinked together. Stock prices are determined by supply and demand. If investorsfeel strongly about a stock, they will demand more shares, causing the stock priceto increase.

Company earnings play a major role in investor demand. Many other factors,like speculation, news events, and dividend payouts, also drive stock prices. How-ever, there is no single parameter responsible for changes in stock price. Often,it is the case that investors buy or sell stock based on feelings. Because ofthis, there is an inherent randomness to the stock market. Mathematically, thisrandomness can be addressed with stochastic processes.

2.1.3 Payoffs

0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

16

18

20Payoff for a European Call Option

Stock Price

Payoff

Figure 2.1: Payoff for a European call option with strike price $20

The payoff3 from purchasing a European call option can be represented bythe following function:

C = max(S(T) K, 0)That is, if the stock price at time T is greater than the strike price, the optionwill be exercised. The purchaser can buy a stock for K < S(T) and immediatelysell it for S(T), thus earning a payoff of S(T) K. If, however, the stock priceupon expiry is less than the strike price, the option will not be exercised. If it

3It should be noted that there is a difference between payoff and profit. Profit takes into

account the transaction costs associated with buying an option. Payoff treats these transaction

costs as sunk costs.

5


7/31

were, the payoff would be negative; the investor chooses a zero payoff rather thana negative payoff.

The payoff from a European put can be represented by the following function:

P = max(K S(T), 0)

If the stock price at time T is less than the strike price, the owner of a put willexercise the option, agreeing to sell the specified shares at the price K. Thepayoff from this transaction will be the difference of K S(T). Stock can bebought for S(T) and sold for K, where S(T) < K. If S(T) > K, then the putwill not be exercised; doing so would result in a negative payoff.

0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

16

18

20Payoff for a European Put Option

Stock Price

Payoff

Figure 2.2: Payoff for a Eurpean put option with strike price $20

2.1.4 Financial Leverage

By purchasing stock through the use of options, an investor is taking advantage offinancial leverage. By purchasing a stocks option, there is more potential profit

and loss.Consider the following example:An investor purchases 100 call options with strike price $5. This gives the

investor the right to buy 10,000 shares of stock. Upon expiry, the price of thestock is $10. The market value of the stock is $100,000. If the options areexercised, they will cost the investor $50,000. Therefore, the investor will earn a$50,000 profit.

Now suppose the stock price increases by 50% to $15. The market value ofthe stock is now $150,000, but the cost of exercising the options is the same,

6


8/31

$50,000. The profit is now $100,000; a 50% increase in stock price leads to a100% increase in profit.

2.1.5 Trading Strategies: Option Spreads

An option spread involves buying or selling two or more options on the samestock. Spreads allow the investor to limit risk and profit under certain expectedconditions. We will describe a straddle, a strangle, and a butterfly spread.

Straddle

A long straddle involves buying a call and put option with the same strike priceon the same stock. A long straddle is ideal for a volatile stock; the investor profits

when the stock goes up or down drastically.A short straddle involves selling a call and put with the same strike on thesame stock. It gives a profit only when the stock price does not change greatly.

A long straddle has a limited loss and an unlimited profit. A short straddlehas an ulimited loss and a limited profit.

0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

16

18

20Payoff for a Long Straddle

Stock Price

Pa

yoff

Figure 2.3: Payoff for a long straddle

Strangle

A strangle involves the same strategy as a straddle, except that the strike pricesfor the call and put are not equal. The strike for the call is greater than the strikefor the put. For a long strangle, a drastic change (either positive or negative) instock price results in a profit. However, the change in stock price must be greaterfor a long strangle than for a long straddle.

7


9/31

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10Payoff for a Long Strangle

Stock Price

Payoff

Figure 2.4: Payoff for a long strangle

Butterfly Spread

A butterfly spread (using call options) involves purchasing four calls. One call ispurchased with a low strike price and another with a higher strike price. Thentwo call options are sold with a strike price inbetween the long options.

The butterfly spread is profitable when the stock price does not change dras-tically or remains close to the strike prices of the two short calls. In addition,there is a limited loss to this strategy. The butterfly spread can also be formedwith put options (profitable with a volatile stock).

0 5 10 15 20 25 30 35 400

1

2

3

4

5

6

7

8

9

10Payoff for a Butterfly Spread

Stock Price

Payoff

Figure 2.5: Payoff for a Butterfly Spread of Call Options

8


10/31

2.2 Arbitrage

In deriving a model for the pricing of options, an important assumption is maderegarding arbitrage. Arbitrage is the ability to buy an asset in one market andinstantaneously sell it in another market for a profit.

Arbitrage is possible when one of the following three conditions is not met:

1. The Law of One Price: any given asset must trade at the same price on anymarket

2. Two assets with identical cash flows must trade at the same price

3. An asset with a known future price must trade at its risk-free discountedprice today

9


11/31

Chapter 3

Stochastic Processes

3.1 Brownian Motion

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

0

0.5

1

t

W(t)

Figure 3.1: A random Brownian Motion along the time interval [0, 1]

3.1.1 Definition

Definition: Brownian Motion. A Brownian Motion W(t, ) is a functionW : [0,] R, where represents the set of outcomes in a probabilityspace, satisfying the following three conditions:

1. With probabilty 1, W(0) = 0 and the mapping t W(t, ) is almost surelycontinuous

10


12/31

2. For 0 s < t T, W(t) W(s) N(0, t s) where N is the normalGaussian distribution, with mean 0 and variance t s.

3. For0 s < t < u < v T, the increments W(t)W(s) and W(v)W(u)are independent for all s,t,u,v [0, T]

3.1.2 History

The long studied model known as Brownian Motion, also known as a WienerProcess, is named after the English botanist Robert Brown. In 1827, Browndescribed the unusual motion exhibited by a small particle totally immersed in aliquid or a gas.

In 1900, the French mathemetician Bachelier independently introduced Brow-

nian motion to model the price movements of stocks and commodities.In 1905, Albert Einstein was able to explain this motion mathematically. Heassumed that the immersed particle was continuously bombarded by moleculesof the surrounding medium.

In a series of papers originating in 1918, Norbert Wiener provided a mathe-matically concise definition and other mathematical properties of Brownian Mo-tion.

3.2 Geometric Brownian Motion

3.2.1 DefinitionDefinition: Geometric Brownian Motion. A Geometric Brownian Motion is

an almost surely continuous time stochastic process S(t) that solves the stochasticdifferential equation

dS(t) = Sdt + SdW(t)

where dW(t) is a Brownian Motion and the constants and represent drift andvolatility, respectively.

The equation has solution

S(t) = S(0)e(1

22)t+W(t)

A variant of Brownian Motion, Geometric Brownian Motion (GBM) is log-normally distributed; it takes on only nonnegative values. The random variableln( S(t)S(0)) is normally distributed with mean ( 122)t and variance 2t. Futurechanges in value are independent of past changes in value.

11


13/31

3.2.2 A Better Model

When using a Brownian Motion to describe stock prices, two major flaws arise.

First, a Brownian Motion could become negative. Stock prices, however, are nevernegative. Second, a Brownian Motion assumes the price difference, regardless ofthe initial price, follows the same normal distribution. In the case of stocks, theprobability that a stock would drop from say $100 to $90 (a 10% change) is notthe same as if the stock were to drop from $50 to $40 (a 20% change). TheBrownian Motion model assigns these two events equal probability.

Because a Geometric Brownian Motion is nonnegative, it provides for a morerealistic model of stock prices. Also, the GBM model considers the ratio of stockprices to have the same normal distribution. Therefore, the percentage changein price as opposed to the absolute change in price is modelled.

12


14/31

Chapter 4

Stochastic Calculus

4.1 Stochastic Integrals

4.1.1 Introduction

A bounded, smooth function f : [a, b] R is said to be integrable provided thatthere is exactly one number A such that L(f, P) A U(f, P), where P is apartition of the interval [a, b], L(f, P) is the lower Darboux sum, and U(f, P) isthe upper Darboux sum.

This defines the integralba

f A, which can be approximated by usingRiemann sums.

N1n=0

f(tn)(tn+1 tn) A =ba

f (4.1)

N1n=0

f(tn+1 + tn

2)(tn+1 tn) A =

ba

f (4.2)

The sum (4.1) is the lefthand sum and the sum (4.2) is the midpoint sum. As

the number of partitions goes to infinity, both sums approximateba

f as A.

Now consider the case of a stochastic function W(t) and its integral T

0W(t)dW(t).

This stochastic integral can be approximated in the same manner ba f was ap-proximated by (4.1) and (4.2). However, the approximations resulting from alefthand sum and a midpoint sum on a stochastic function are not equivalent;the lefthand sum results in the Ito Integral while the midpoint sum results inthe Stratonovich Integral. We will concern ourselves with the Ito Integral.

13


15/31

4.1.2 Example

Consider the stochastic integral

T0

W(t)dW(t). (4.3)

Also, recall from the definition of a Brownian Motion that

E[W(t)W(s)] = 0E[(W(t)W(s))2] = t s

for all t > s [0, T].

Ito Integral

The Ito Integral of (4.3) is the limiting case of the lefthand sum

Sn =N1n=0

W(tn)(W(tn+1)W(tn)). (4.4)

The Riemann sum (4.4) can be rewritten as

Sn =1

2

N1

n=0(W(tn+1)

2 W(tn)2) 12

N1

n=0(W(tn+1)W(tn))2. (4.5)

The first sum telescopes to W(T)2 W(0)2 = W(T)2, since W(0) = 0 fromthe definition of Brownian Motion. Now (4.5) becomes

Sn =1

2W(T)2 1

2

N1n=0

(W(tn+1) W(tn))2. (4.6)

Let us now focus on the second sum, the quadratic variation:

QV =N1

n=0(W(tn+1)W(tn))2.

For any partition, the expected value

E[QV] =N1n=0

E[(W(tn+1)W(tn))2] =N1n=0

(tn+1 tn)

= T

14


16/31

and

E[Sn] =1

2

(E[W(T)2]

T) =

1

2

(T

T)

= 0.

It can be shown that the sum QV tends to T, in the L()2 sense. If t W(t, ) were a smooth function, we would have

QV =N1n=0

W(n)2(tn+1 tn) CtT,

which would tend to zero as t 0.In conclusion, in L()2, the Ito Integral

T0

W(t)dW(t) = limt0

Sn = 12

(W(T)2 T)

and

E[

T0

W(t)dW(t)] = limt0

E[Sn] = 0.

OPTIONAL: The Stratonovich Integral

Let us now see what happens when the integral (4.3) is approximated by themidpoint sum

N1

n=0

(W(t

n+1) + W(t

n)

2 )(W(tn+1)W(tn)). (4.7)

(4.7) can be rewritten as

1

2

N1n=0

(W(tn+1)2 W(tn)2) = 1

2W(T)2 (4.8)

The expected value of (4.8) is

1

2E[W(T)2] =

T

2.

Therefore, in L2(), T0

W(t)dW(t) =1

2W(T)2

with

E[

T0

W(t)dW(t)] =T

2

when using the midpoint sum. This isT0

W(t)dW(t) in the Stratonovich sense.

15


17/31

4.2 Itos Formula: A Stochastic Chain Rule

4.2.1 Motivation by Taylor ExpansionGiven a function f(t, S(t)), where S(t) solves the stochastic differential equationdS = Sdt + SdW, we must find a formula for determining df.

Let us apply a Taylor Expansion 1 to f(t, S(t)), disregarding the fact that S(t)is stochastic:

df = ftdt + fSdS+1

2fttdt

2 +1

2fSSdS

2 + ftSdtdS+ . . .

where . . . represents higher order terms.

Proposition 1. dS

dt as dt 0 in theL2

senseMaking use of this proposition results in

df = ftdt + fS

dt +1

2fttdt

2 +1

2fSSdt + ftSdt

3/2 + . . .

Because dt 0, dt2 and dt3/2 can be eliminated: a number less than the absolutevalue of one raised to an exponent greater than one is smaller than the originalnumber (n < n, for n < |1| and > 1). This also means that the higher orderterms can be eliminated; they all contain a dt term. Therefore,

df = ftdt + fSdS+ 12fSSdt

= ftdt + fS(Sdt + SdW) +1

2fSSdt

= (ft + SfS +1

2fSS)dt + SfSdW (4.9)

(4.9) is Itos Formula applied to f(t, S(t)).

4.2.2 Formal Itos Formula

Itos Formula. Let X(t) solve the stochastic differential equation

dX(t) = a(X, t)dt + b(X, t)dW(t)

and let f(X(t), t) be C2.Then

df(X(t), t) = (a(X, t)fX + ft +1

2b2(X, t)fXX)dt + b(X, t)fXdW(t)

1NOTATION: ft ft,

2ft2

ftt, 2f

tS ftS, etc . . .

16


18/31

Chapter 5

The Black-Scholes Formula

5.1 Deriving the Black-Scholes Formula

Let f(t, S(t)) represent the value of a European call option on a stock S.Let S follow the stochastic differential equation

dS = Sdt + SdW

with constants equal to the drift and equal to the volatility. dW follows aBrownian Motion.

Our goal is to construct a portfolio that replicates a European call option. Ifwe can price the replicating portfolio, then we can also price the option.

To construct the replicating portfolio , consider a short position in the calloption, f, (t) shares of stock S, and (t) shares of a risk-free bond B. There-fore,

= f + (t)S+ (t)B (5.1)ASSUMPTIONS:

f(t, S(t)) represents the value of a European call option. S follows a Geometric Brownian Motion. That is, dS = Sdt + SdW,

where W is a Brownian Motion and the constants and represent driftand volatility, respectively.

The risk-free bond B grows at the rate dB = rBdt, where r is the risk-freeinterest rate.

There is no arbitrage. The portfolio is self-financing.

17


19/31

Now differentiating (5.1) with respect to t gives

d =

df + (t)dS+ Sd(t) + (t)dB + Bd(t) + ddS (5.2)

The self-financing assumption says that changes in the value of the portfolio are not caused by changes in the number of shares or bonds. Rather, changesin the value of are caused only by changes in the value of S, r, and f. Thismeans that (5.2) can be re-written as

d = df + (t)dS+ (t)dB (5.3)Now applying Itos Formula to f(t, S(t)) gives

df = (ft + SfS +1

22S2fSS)dt + S fSdW

Substituting this result and dS and dB into (5.3) gives

d = (ftSfS12

2S2fSS+(t)S+(t)rB)dt+((t)SSfS)dW (5.4)

Choose (t) = fS. This eliminates the stochastic element, dW. Now (5.4)becomes

d = (ft SfS 12

2S2fSS + SfS + (t)rB)dt (5.5)

Since there is no arbitrage, the portfolio must grow by the rate of d =rdt = r(f + fS + (t)B)dt. Substituting into (5.5) yields

ft + 12

2S2fSS + rSfS rf = 0 (5.6)

(5.6) is the BLACK-SCHOLES PDE, with final condition f(T, S) = (SK)+

5.2 Feynman-Kac Theorem

Feynman-Kac Theorem. Let a,b, and g be smooth, bounded functions. Let X

solve the stochastic differential equation

dX(t) = a(t, X(t))dt + b(t, X(t))dW(t)

and let

u(x, t) = E[g(X(T))|X(t) = x]Then u is a solution of

ut + aux +1

2b2uxx = 0

u(x, T) = g(x)

for t < T.

18


20/31

The Feynman-Kac Theorem provides a way of relating the Black-Scholes par-tial differential equation with a stochastic differential equation. The SDE isindependent of the drift parameter.

The PDE

ft +1

22S2fSS + rSfS rf = 0 (5.7)

with final condition f(T, S) = max(S(T)K, 0) and the SDE

f(t, S(t)) = E[er(Tt)max(S(T)K, 0)] (5.8)with

dS = rSdt + SdWare equivalent. Finite difference or finite element methods can be used with

(5.7), while (5.8) is ideal for the Monte Carlo Simulation.

5.3 Exact Solution

Exact Solution. The Black-Scholes PDE (5.7) has exact solution

C(S, t) = S(1)Ker(Tt)(2)

where

1 =

ln( SK) + (r +12

2)(T

t)

T t2 =

ln( SK) + (r 122)(T t)

T tand represents the Gaussian CDF.

5.4 History

In 1973, Myron Scholes and Fisher Black derived the Black-Scholes formula.Their work was built upon the earlier research of Paul Samuelson and Robert

Merton. The Black-Scholes formula revolutionized the trading of options. It givesinvestors a mathematical approach to pricing options, as opposed to guessing.

It assumes that the stock or underlying asset follows a random walk withconstant drift and volatility, there are no arbitrage opportunities, stock tradingis continuous, there are no dividends, there are no transaction costs, stocks areperfectly divisible (can buy a fraction of a stock), and the stock can be sold short.Not all of these assumptions are realistic, especially the assumption that a stockhas constant volatility. However, Black-Scholes is still useful.

19


21/31

Chapter 6

Numerical Solutions

6.1 Monte Carlo Method

6.1.1 Overview

In mathematics, generally speaking, a problem is given, a formulation of theproblem is made, then solved analytically or numerically. In a Monte Carlosimulation the opposite occurs. A mathematical problem is given and then solvedby constructing a game of chance that in some way leads to an approximatesolution to the given problem. We have previously stated that the expectedvalue of the solution of the SDE stock model is equivalent to the solution of theBlack-Scholes PDE, by the Feynman-Kac Theorem. A Monte Carlo Simulation

can be used with this SDE model.In many of the problems where the Monte Carlo simulation is applicable, there

is already an element of chance built into the system. In our case, the element ofchance is the volatility of the movement of the stock price. The various possibilitesof Monte Carlo simulations began to be studied in the 1940s.

20


22/31

6.1.2 Example

The following figure is an example, using MATLAB, of a Monte Carlo Simulation:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

1

1.5

2

2.5

3

3.5

4

t

U(t)

mean of 1000 paths5 individual paths

Figure 6.1: Monte Carlo Simulation along 1000 discretized Brownian paths ofGBM

In this example we evaluate the stock model function U(t, S(t)) along 1000discretized Brownian paths. The expected value of this solution can be seen as

the center line with a smooth appearence. Notice that although U(t, S(t)) is non-smooth along the individual paths, the expected value of the solution appears tobe smooth. This can be established by noting that the properties of the Brownianmotion require the expected value of S(t) to be zero. Therefore, the expectedvalue of U(t, S(t)) is solely dependent on the drift and not the volatility. In thisexample, the expected value turns out to be e9/8t.

6.2 Euler-Maruyama Method

The Euler-Maruyama Method is an extension of the Euler Method. The Euler

Method, also known as the tangent line method, originated in 1758. It computesan approximation to a deterministic differential equation along a set of timevalues. The Euler-Maruyama Method computes an approximation to stochasticdifferential equations. It can be stated as follows:

X(t) = X0 +

t0

f(X(S))dS+

t0

g(X(S))dW(S)

for 0 t T.

21


23/31

Let us take a look at Figure 6.2. The black line represents the true solutionand the x-line represents the Euler-Maruyama approximated solution. The truesolution and Euler-Maruyama approximated solution were computed along 1000discretized Brownian paths for different initial values, ranging from 70 to 95. Thestrike price (K) was set at 82.96, the interest rate (r) at 10.8%, the time (T) at.2 years, and volatility () at .1195.

70 75 80 85 90 950

1

2

3

4

5

6

7

8

9Price of European Call Option

Initial Value S(0)

Expected

Value

Figure 6.2: Euler-Maruyama Approximated Payoff Diagram for a European Call

Notice that the graph in Figure 6.2 resembles the call payoff graph shown inFigure 6.2. The computed graphs curve begins to rise slightly before the strikeprice of 82.96, which differs from the previously shown payoff graph. This differ-ence can be explained by the fact that our computed graph is for the expectedvalue of the payoff at the initial time, while the previously shown payoff graphwas drawn at the expiration date. Given any set of initial values (time, interestrate, strike price and volatility), our code can compute the expected value orprice of the option under the set of Black-Scholes assumptions.

22


24/31

Chapter 7

Application: 3M Company

0 200 400 600 800 1000 1200 140030

40

50

60

70

80

90

100

3M COMPANY

Figure 7.1: 3M Comany: 5 Year Chart of Daily Closing Prices

As a real world application, we have taken five years of historical data fromthe 3M Company (NYSE symbol MMM) to estimate the parameters for thestochastic Black-Scholes model. Using numerical methods, we wanted to see how

close our model predicted the price of an option with respect to the real option.The model used is the result of the Feynman-Kac Theorem (5.8), as appliedto the Black-Scholes deterministic PDE.

f(t, S(t)) = E[er(Tt)max(S(T)K, 0)]dS = Srdt + SdW

23


25/31

7.1 Estimating the Parameters

For our model, which is independent of drift , it was only necessary to estimatethe volatility . However, even though it was unnecessary, we estimated .

To estimate the parameters, five years of historical daily closing prices werecollected for MMM. Next, the daily percentage change in stock price was com-puted. This was done in the following manner:

i = ln(Si

Si1)

where Si is the closing price on day i and i is the daily change in closing price.Next, the sample standard deviation of the i terms was calculated:

Sdaily = 1N 1

Ni=1

(i)2

This gives an estimate for the daily volatility, but we are interested in the yearlyvolatility. Since there are 252 trading days in a year, multiplying Sdaily by

252

will give us our yearly volatility estimate:

= 252

N 1Ni=1

(i)2

To estimate , we can calculate the sample mean and multiply by 252:

1

N

N1

i

The risk-free rate r can be approximated by the interest rate for 3 monthTreasury Bonds.

Using our historical data, our calculations resulted in the following estimates:

r = .108 = 0.007528

= .1195

7.2 The Application: The Black-Scholes Price

vs. The True Market Price

7.2.1 Introduction

In reality, the Black-Scholes model is based upon several unrealistic assumptions.These assumptions include continuity of stock price, no transaction costs, and

24


26/31

most importantly, constant volatility.In the real market, stocks often change in their volatilities. For example,

the airline industry is more volatile when gas prices increase. A well-knowndiscrepency arises between the market price of options and the computed Black-Scholes price. This discrepency can be better accounted for when using a stochas-tic volatility model. It is important to note that there is no generally acceptedstochastic volatility model.

7.2.2 Implied Volatility

Implied volatility is the volatility that when substituted into the Black-ScholesFormula produces the actual market price of an option. In Figure 7.2, the com-puted Black-Scholes price can be seen as the straight black line, the true market

price as the x-line, and the implied volatility as the dashed-line.

70 75 80 85 90 950

2

4

6

8

10

12

14

16

18Price and Implied Volatility of a European Call Option

Strike Price

PriceorImpliedVolatility

Figure 7.2: Implied Volatility

Notice that the implied volatility is not constant; it increases the further awayfrom the strike price (82.96). This smile-effect gives rise to what is known as

the Smile Curve. Intuitively, a stock with a higher volatility is harder to predict.Therefore, the stocks option requires a higher premium due to the increasedstock volatility.

25


27/31

Chapter 8

Appendix: Code

8.1 Payoff Diagramsclear all;

close all; syms k k_0 d k_01; k = 20; k_0=10; k_2=30; d=.01;

%EURO CALL

s=0:d:k; y=0; AXIS([0 40 -5 40]); AXIS manual; plot(s,y, k*) hold

on; s=k:d:2*k; y=s-k; plot(s,y,k*) title(Payoff for a European

Call Option); hold off; pause;

%EURO PUT

s=0:d:k; y=k-s; plot(s,y,k*); hold on; s=k:d:2*k; y=0;

plot(s,y,k*); title(Payoff for a European Put Option); hold off;

pause;

%EURO STRADDLE

s=0:d:k; y=k-s; plot(s,y,k*); hold on; s=k:d:2*k; y=s-k;

plot(s,y,k*); title(Payoff for a Long Straddle); hold off;

pause;

%EURO STRANGLE

s=0:d:k_0; y=k_0-s; plot(s,y, k*); hold on; s=k_0:d:k; y=0;

plot(s,y, k*); s=k:d:k+k_0; y=s-k; plot(s,y,k*); hold off;

title(Payoff for a Long Strangle); pause;

%BUTTERFLY SPREADs=0:d:k_0; y=0; plot(s,y,k*); hold on; s=k_0:d:k; y=s-k_0;

plot(s,y,k*); s=k:d:k_2; y=s-k_0-2*s+2*k; plot(s,y,k*);

s=k_2:d:k_2+k_0; y=0; plot(s,y, k*); title(Payoff for a Butterfly

Spread);

26


28/31

8.2 Brownian Motion

%BPATH2 Brownian path simulation: vectorized

randn(state,500) % set the state of randn

T = 1; N = 500; dt = T/N;

dW = sqrt(dt)*randn(1,N); % increments

W = cumsum(dW); % cumulative sum

plot([0:dt:T],[0,W], k-) % plot W against t

xlabel(t,FontSize,16) ylabel(W(t),FontSize,16,Rotation,0)

8.3 Monte Carlo Simulation

%BPATH3 Function along a Brownian path

randn(state,0) % set the state of randn

T = 1; N = 500; dt = T/N; t = [dt:dt:1];

M = 1000; % M paths simultaneously

dW = sqrt(dt)*randn(M,N); % increments

W = cumsum(dW,2); % cumulative sum

U = exp(repmat(t,[M 1]) + 0.5*W); Umean = mean(U);

plot([0,t],[1,Umean],k x), hold on % plot mean over M paths

plot([0,t],[ones(5,1),U(1:5,:)],k-), hold off % plot 5 individual paths

xlabel(t,FontSize,16)

ylabel(U(t),FontSize,16,Rotation,0,HorizontalAlignment,right)

legend(mean of 1000 paths,5 individual paths,2)

averr = norm((Umean - exp(9*t/8)),inf) % sample error

8.4 Euler-Maruyama Payoff

% Sean and Jay

% Expected Value of a European Call Option

% makes the graph that looks like the call payoff graph

randn(state,100) % set seed for psuedorandom number generator

% problem parameters

vol = 0.1195; r = 0.108; XzeroStart = 65; K = 82.96; M = 1000;

T = .2; N = 2^8; dt = T/N; t = [dt:dt:T]; R = 4; Dt = R*dt; L =

N/R;

for i = 1:11 %loop runs throough different initial values

Xzero = XzeroStart + 2.5*(i-1); %increment inital value

dW = sqrt(dt)*randn(M,N); % Brownian increment

W = cumsum(dW,2); % discretized Brownian path

27


29/31

Xtrue = Xzero*exp((r-0.5*vol 2)*repmat(t, [M 1])+vol*W);

XendTrue = Xtrue(1:M,N)-K; % true solution based off Black-Scholes

Gtrue = exp(-r*T)*max(XendTrue,0); % defines funtion max(X-K, 0)MeanGtrue(i) = mean(Gtrue); % expected value of G

Xem = zeros(M,L);

Xtemp = repmat(Xzero, [M 1]);

for j = 1:L

Winc = sum(dW(1:M,R*(j-1)+1:R*j),2);

Xtemp = Xtemp + r*repmat(Dt, [M 1]).*Xtemp + vol*Winc.*Xtemp;

Xem(1:M,j) = Xtemp;

end

XendEm = Xem(1:M,L)-K;

GEm = exp(-r*T)*max(XendEm, 0);

MeanGEm(i) = mean(GEm);

end

plot([70:2.5:95],MeanGtrue, k-), hold on plot([70:2.5:95],MeanGEm,

k--*), hold off title(Price of European Call Option);

xlabel(Initial Value S(0)); ylabel(Expected Value);

8.5 Pricing European Call

% Sean and Jay

% Expected Value of a European Call Option

randn(state,300) % set seed for psuedorandom number generator

% problem parameters

vol = 0.1195; r = 0.108; Xzero = 82.96; KStart = 70; M = 10^4; N =

2^8;

T = .2; %8/3/04 thru 10/15/04 = 73/365

dt = T/N; t = [dt:dt:T];

dW = sqrt(dt)*randn(M,N); % Brownian increment

W = cumsum(dW,2); % discretized Brownian path

%true market 3M option data taken 8/03/04

Cobs = [12.5 9 4.6 2 .55 .2]; plot([70:5:95],Cobs, k--*), hold on

%initialized variables

Ivol = [0 0 0 0 0 0]; min = [100 100 100 100 100 100]; ivolatility =

[0 0 0 0 0 0]; MeanGtrue = [0 0 0 0 0 0]; diff = [0 0 0 0 0 0];

for i = 1:6 %loop runs through different strike values

K = KStart + 5*(i-1); %increment strike value

28


30/31

Xtrue = Xzero*exp((r-0.5*vol^2)*repmat(t, [M 1])+vol*W);

XendTrue = Xtrue(1:M,N)-K; % true solution based off Black-Scholes

Gtrue = exp(-r*T)*max(XendTrue,0); % defines funtion max(X-K, 0)

MeanGtrue(i) = mean(Gtrue); % expected value of G

diff(i) = MeanGtrue(i) - Cobs(i);

count = 0;

value = MeanGtrue(i);

done = 1;

temp = 100;

while done == 1

count = count + 1;

Ivol(i) = Ivol(i) + .01;

Itrue = Xzero*exp((r-0.5*Ivol(i) 2)*repmat(t, [M 1])+Ivol(i)*W);

IendTrue = Itrue(1:M,N)-K; % true solution based off Black-Scholes

Ihold = exp(-r*T)*max(IendTrue,0); % defines funtion max(X-K, 0)value = mean(Ihold); % expected value of G

temp = abs(value - Cobs(i));

if temp < min(i)

min(i) = temp;

ivolatility(i) = count;

else

done = 1; %as vol increases so does the mean

%therefore if not closer to true value heading further from true

end

if count == 100

done = 0;

end

end

end

diff min ivolatility

plot([70:5:95],ivolatility,k-.) plot([70:5:95],MeanGtrue, k),

hold off title(Price and Implied Volatility of a European Call

Option); xlabel(Strike Price); ylabel(Price or Implied

Volatility);

29


31/31

Chapter 9

References

1. Fitzpatrick, Patrick M., Advanced Calculus: A Course in MathematicalAnalysis, PWS Publishing Company, 1996.

2. Goodman, J., Moon, K.S., Szepessy, A., Tempone, R., Zouraris, G., Stochas-tic and Partial Differential Equations with Adapted Numerics, 2004.

3. Higham, Desmond J., An Algorithmic Introduction to Numerical Simula-tion of Stochastic Differential Equations, Society for Industrial and AppliedMathematics, 2003.

4. Hull, John C., Options, Futures, and Other Derivatives: Fifth Edition,Pearson Education, Inc., 2003.

5. Ross, Sheldon M., An Elementary Introduction to Mathematical Finance:Options and Other Topics Second Edition, Cambridge University Press,2003.

stochastic process and their applications.pdf

Documents