STAT 906: Comp Intensive Methods In Finance

•Instructor: Don McLeish office: MC6138email:dlmcleis@uwaterloo.ca

•Text: available at

http://www.stats.uwaterloo.ca/~dlmcleis/s906/

Other References• The Mathematics of Financial Derivatives• (Wilmott, Howison, Dewynne) (bookstore)• Monte Carlo: Concepts, Algorithms and Applications.

George Fishman QA298.F57(one day reserve)• Arbitrage Theory in Continuous Time. Tomas Bjork• Mathematics of Financial Markets. R.J. Elliott and P.E.

Kopp (Springer) HG4515.3.E37(one day reserve)• Glasserman, P. (2003) Monte Carlo Methods in

Financial Engineering (Applications of Mathematics, 53) Springer, New York

• Jaeckel, P. (2003) Monte Carlo Methods in Finance, Wiley, New York

Marking

•Marks DISTRIBUTION

– Midterm Test: 20– Project&presentation 40 – Class presentations 40

•Tentative Calendar: Week of

– Sept 15: McLeish=M– Sep 22 : M,A,B– Sep 29 : M,C,D– Oct. 6: M,E,F– Oct 13: M,G,H– Oct 20: M,I,J– Oct 27: M,A,B– Nov 3 : C,D,midterm– Nov 10 : M,E,F– Nov 17: M,C,D– Nov 24: M,A,B– Dec 1: projects

Contents •A general discussion of simulation.•Example: 2 person game. •Some basic theory of finance (Chapter 2)•How to generate random numbers (Uniform and non-uniform)(Chapter 3)

•Variance Reduction•Quasi Monte Carlo•Simulation and Option valuation•Estimation and Calibration•Sensitivity Analysis•Numerical solutions to PDE’s•..\book\contents.pdf

Course Prerequisites

• Knowledge of (or tolerance of) Stochastic Calculus.

• Some finance knowledge/interest• Computing: Some computer language

(e.g. MATLAB, R, C++)• “Tolerance of”= have taken or willing to

study on side

Simulation: Monte Carlo Methods

•Simulation: Imitation of a real-world process or system. •Model: set of assumptions concerning the operation of the system•Monte Carlo:simulation of random phenomena

•Parameters: numerical values associated with the model•Why use simulation? •Permits changing model and parameters, examining interactions, changes to system,

Option

• Gives the holder the right (no obligation) to buy or sell an asset at a prescribed price,time determined by the contract.

• e.g. a (European) call option on IBM stock. exercise price $120. expiry date Oct 20, 2003. If S=stock price on Oct 20 the value of option on expiry is max(S-120,0)

Uses of financial derivatives

• Speculate on a rise (call option) or fall (put option) in asset price.

• Hedge a portfolio already held-e.g. a promise to deliver IBM stock at point in future. Insurance against disadvantageous moves in asset prices, currency exchange rates, interest rates, credit changes, weather, electricity demand, …...

Asset prices as random variables

• Random models often applied to complex systems which are essentially causal with many factors influencing result in a complex way. e.g. Brownian motion, toss of coin, currency exchange rates, …

• What are the constants---- a dollar and if so whose…..? Choice of “numeraire”.

Why use simulation?

•Permits experimenting with the controllable system parameters to identify optimal settings•Permits examining effect of environmental or exogenous changes.

•Identify which of several systems is most efficient•Determine which variables are most important.•Verify and check robustness of analytic solutions.

Advantages of Simulation

•investigate effects of changes, new designs, new models, etc. without costly implementation.•Stress testing: test systems under different scenarios (e.g. higher interest rates, different exchange rates)

•“What if” questions.•NON-FINANCIAL •Identify bottlenecks in systems and rectify•Increase experience with complex system at lower cost.

Disadvantages

•Two models for same process may differ. •Simulation output is random so hard to interpret results. •Building models and running simulation is time consuming

•ANALYTIC SOLUTIONS, IF AVAILABLE, SHOULD BE USED! (analytic solutions to a similar (simpler) model can be used to improve a simulation)

Non Finance Applications

•Manufacturing systems: e.g. material handling, inventory, assembly plants, scheduling, •Public Systems: health care- hospital management, emergency room, Military

•Natural resource management, transportation, traffic systems, airport (e.g. Motorway)•Construction systems: project planning, scheduling, •entertainment: restaurants, movies etc.

Types of Systems

•State variables: determine the state of the system in order to simulate future events.•Discrete: state variables change only at discrete time points. (discrete event simulation)

•Continuous: state variables change continuously over time •Stochastic, deterministic

Example (Discrete): Bank queue

•Interested in required number of tellers.•State variables: number of tellers available, number of customers in bank, (possibly also the time in service of customers being served)

EXAMPLE: QUEUE

•Calling population•queue (waiting line)•server(s)

One-server queue

Examples of queueing systems

•Repair facility(garage, mechanics, etc.)•hospital, airport,production line, job shop, computer, telephone, ticket office, grocery store, taxi, mass transit, warehouse

•Identify the customers and the server(s) in each.

Example (continuous-time model):TSE300 index.

•X(t) = TSE300 index at time t. Changes (almost) continuously in time•On what present values does the future depend? (state variables) : Current value of index

•possibly other factors depending on model (e.g. interest rates, exchange rates, balance of trade, agricultural prices, forest product prices, kitchen sink …)

Steps in contructing a simulation

•Formulate Problem (why do a simulation?)•Set objectives•suggest candidate models.•Collect real data-identify most appropriate model

•Computer implementation of model (code or program)•Verify model (i.e. debug)•validate model (i.e. accurately represents system?)

Designing Simulation

•Design the simulation-what alternatives are to be simulated? How long a run- what data is collected etc.•Run simulation and analyse output.

•More runs? •Change model? •Change parameters? •Document•report on results Conclusion

Prisoner’s dilemna• non-zero-sum game called the "Prisoner's Dilemma" (Axelrod, 1984). The two players choose between two

moves, either "cooperate" or "defect". The idea is that each player gains when both cooperate, but if only one of them cooperates, the other one, who defects, will gain more. If both defect, both lose (or gain very little) but not as much as the "cheated" cooperator whose cooperation is not returned. The whole game situation and its different outcomes can be summarized below. Points=payoff to A

• The game got its name from the following hypothetical situation: imagine two criminals arrested under the suspicion of having committed a crime together. However, the police does not have sufficient proof in order to have them convicted. The two prisoners are isolated from each other, and the police visit each of them and offer a deal: the one who offers evidence against the other one will be freed. If none of them accepts the offer, they are in fact cooperating against the police, and both of them will get only a small punishment because of lack of proof. They both gain. However, if one of them betrays the other one, by confessing to the police, the defector will gain more, since he is freed; the one who remained silent, on the other hand, will receive the full punishment, since he did not help the police, and there is sufficient proof. If both betray, both will be punished, but less severely than if they had refused to talk. The dilemma resides in the fact that each prisoner has a choice between only two options, but cannot make a good decision without knowing what the other one will do.

Example: Simulate the Prisoner’s Dilemma.

•Payoff matrix•Prisoners Alf and Butch either confess or not. Payoffs taken from table: A chooses row (don’t confess/ confess) •B chooses column (don’t confess/confess)

•Payoff (to Alf)•cooperate: 5 -10•Defect 10 0•(to B)•cooperate defect• 5 10•-10 0

•

Summary of game& simulation results.

[p,q,val]=nonzerosum(A,B,100)A =

5 -1010 0

B =5 10

-10 0

>> [p,q,val]=nonzerosum(A,B,100)p = 0 1q = 0

1val = 0 0

TWO PERSON GAME-MATLAB CODE

• [p,q]=nonzerosum(A,B,nsim)• n=size(A); p=ones(1,n(1)); % q=ones(n(2),1);

• for i=1:nsim• [m,s]=max(A*q); p(s)=p(s)+1;• [m,s]=max(p*B); q(s)=q(s)+1;• end• p=p-ones(1,n(1)); p=p/sum(p);• q=q-ones(n(2),1); q=q/sum(q);

Another two-person game.

•Payoff matrix•Two companies A and B each choose bid . Payoffs taken from table:

•Payoff to A•A’s Bid Payoffs•12 3 2 -2•13 1 -4 4•14 0 -5 6•Payoff to B•B’s bid B1 B2 B3• -1 -2 3• 0 4 -6• 0 5 -5

OUTPUT

• [p,q]=nonzerosum(A,B,50000)• p = 0.6667 0 0.3333• q = 0 0.53334 0.46666

• and the value of the game• 0 to A and 0.3333 to B.

Generating random bid

• Generate a uniform [0,1] random variable.• If U<2/3, then bid $12,• otherwise bid $14.

Random Numbers

•Almost all random numbers are generated from uniform [0,1] random numbers•Uniform [0,1] random numbers X are generated from discrete uniform{0,1,….m-1} random numbers, m large.

•We then convert discrete uniform to continuous U[0,1] using (U=X+0.5)/m

Example: Monte Carlo Integration

•Notice that

4/13/1

1

0

)1()(

ref(x)dx whe

xxxf

Find

−=

∫.t independenfor )(

n1

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1

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i

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ii UUf

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=

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Graph of function

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

One-Dimensional Monte-Carlo Integral-Variance of error

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Monte Carlo integration in Excel

• Put =RAND() in cell A1• Copy to cells A2:A20• put =A1^(1/3)*(1-A1)^(1/4) in cell B1• copy to cells B2:B20• in any cell you wish put

=AVERAGE(B1:B20) for mean of values in column B.

• Similarly =STDEV(B1:B20)• and =STDEV(B1:B20)/SQRT(20)

Worksheet for integral0.727331 0.64986 0.587992 0.103846 0.0232210.332666 0.6262610.325482 0.6233860.766509 0.636170.097599 0.4487440.735855 0.6472290.038771 0.3351270.982999 0.3590330.246993 0.5844690.931513 0.499610.603536 0.6705820.54235 0.670749

0.627847 0.6688030.293298 0.609180.688517 0.6596750.132834 0.4923740.77347 0.63328

0.460555 0.6618310.323538 0.6225910.682537 0.660894

Matlab Code for integral

• u=rand(1,1000000);• mean(u.^(1/3).*(1-u).^(1/4))• (Answer=.57325, se=.00011) • true value approx .57326)• How much error is there? • Can we decrease the error? • script2

Standard Error of Estimator

• The standard error is the square root of the variance of the estimator

• se=sqrt(var(fn(U))/length(U));• This gives standard error of estimator. • An approximate 95% confidence interval is estimate +-2

se

))((n

1Error Standard

))((1))(1(1

i

n

iii

UfVar

UfVarn

Ufn

Var

=

=∑=

Changing Variables to obtain integral over [0,1]

U[0,1].is where

]1)11([1)11(

becomes integral above The

1,11, 1

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fEduuu

f

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−=−

−=−=+

=

∫

∫∞

Example: A double integral

dww

dydzz

dx

yw

xz

yxyxg

g(x,y)dxdy

22

33

0 0

1 and 11

1 and 1

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−=−=

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g( g(x,y)dxdy

−+−+=

=

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−−=

∫ ∫

∫∫∫ ∫∞ ∞

Matlab Code for integral

• w=rand(1,1000000);• z=rand(1,1000000);• f=(z.*w)./(w.^3.*z.^3+w.^3.*(1-z).^3+(1-w).^3.*z.^3);• mean(f)• sqrt(var(f)/1000000)• (gives 2.1399 and SE=.0019)

Example: estimating pi

• Estimate pi by randomly generating points in square and counting how many fall in unit circle.

• Matlab code:• u=rand(1000000,2);• 4*mean(u(:,1).^2+u(:,2).^2 <1)

4]1P[U

then U[-1,1]indep are , If

22

2 1

21

π=<+U

UU

Estimating Pi (picture)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Example: 3 Monte Carlo estimators of pi.

Stat 906 Class Random Number Generator

• Write down five digits “randomly selected” from {0,1,2,…,9}.

• e.g. 12345

Some elementary tests:

•N=number of digits equal to 7 is binomial(5n,1/10) where n=number in class. •

•Probability at least one digit repeated=1-probability all different =

nSD

n

n

209

209)

1011)(

101(5n)(Variance

2n1)

101)(5( valueExpected

=

=−=

== 6976.010

6*7*8*9*101 5 =−

Basic Theory of Finance: The No Arbitrage Principle

• Can I Make money with positive probability with net investment of $0??( this is called an arbitrage)

One stock, One bond, one period

• Assume stock presently worth $s will be worth su or sd after one month where d<1<u.

• A bond, presently worth $1 goes to $(1+r)over the next month

• Buy x bonds and y shares of stock.• Value in 1 month =x(1+r)+ysu• or x(1+r)+ysd depending on whether

stock goes up or down.

Possible stock movements.

Find an arbitrage

• Can you find values of (x,y) so that the net cost is $0 (e.g. sell short bonds and buy stocks) so that you are certain you will not lose money and you make a positiveamount with positive prob. i.e. both x(1+r)+ysu>=0 and

• x(1+r)+ysd >=0 with one >0.• For net investment =0, need sy=-x.

Condition for no arbitrage

• Arbitrage IS possible UNLESS• d<1+r<u• i.e. $1 in stock either returns more (u) or

less (d) than bond. In this case, the bond return is a CONVEX linear combination of the stock returns.

• THAT IS 1+r=qu+(1-q)d for some 0<q<1.

RISK NEUTRAL MEASURE

• Solve for q: • q=(1+r-d)/(u-d)• The distribution which assigns probability

q to the stock rising to su and probability • 1-q to the stock falling to sd is the risk

neutral measure, denoted Q.

Expected value of stock price under Q

If there is to be no arbitrage, there exists a probability measureQ such that the expected price of future value of the stock S1 discountedto the present using the return from a risk-free bond is exactly the present valueof the stock.

Denote by Q the probability distribution which puts probabilities q and 1 − qon these points su, sd.

Risk neutral measure

• consensus investors' attitude to risk avoidance. It is not necessarily true that the expected value of S1 under the actual probability distribution P describing the future probabilities of the stock is equal to s, only the expected value under Q.

DerivativesDerivatives are investments which derive their value from that of a corresponding asset, such as a stock. A European call option is an option which permits you (but does not compel you) to purchase the stock at a fixed future date ( the maturity date) or for a given predetermined price, the exercise price of the option). For example a call option with exercise price $10 on a stock whose future value is denoted S1, is worth on expiry S1 − 10 if S1 > 10 but nothing at all if S1 < 10. The difference S1 − 10 between the value of the stock on expiry and the exercise price of the option is your profit if you exercises the option, purchasing the stock for $10 and sell it on the open market at $S1.However, if S1 < 10, there is no point in exercising your option as you are not compelled to do so and your return is $0. In

Replicating a Contingent ClaimA function of the stock price V (S1) which may represent the return from aportfolio of stocks and derivatives is called a contingent claim. V (S1) represents the payoff to an investor from a certain financial instrument or derivative when the stock price at the end of the period is S1.In a complete market, there is an investment solely in the stock and bond which reproduces these values V (su) and V (sd) exactly.

Discounted Expected value under Q

Recap

• Under the Q measure, the expected discounted-to-present value of any marketed asset (stock or option) is its present price.

• The Q measure is uniquely determined by this property in a complete market.

Multiperiod Models

• When an asset price evolves over time, decisions about an investment are made at various periods during the life of investment.

• Decisions made with the benefit of information including the price of the asset and any related assets at all previous time periods, beginning at some time t=0 when we started observing the process.

The “history” sequence. • When an asset price evolves over time, the investor normally makes

decisions about the investment at various periods during its life. Such decisions are made with the benefit of current information, and this information, whether used or not, includes the price of the asset and any related assets at all previous time periods, beginning at some time t = 0 when we began observation of the process. We denote this information available for use at time t as Ht. Ht is what is called a sigma-field generated by the past. There are two fundamental properties of this sigma-field

1. the sigma-fields increase over time. In other words, our informationabout this and related processes increases over time because we have

observed more of the relevant history. In the mathematical model, we do not “forget” relevant information

• Ht includes the value of the asset price Sτ , τ · t at all times τ · t. In measure-theoretic language, St is adapted to or measurable with respect to Ht.

In general, under Q, the discounted stock price is a

martingale

Martingales

Recap II

• Prices of derivatives (in a complete market) determined, not by actual distribution of stock, but by risk neutral measure Q.

• Under Q, the discounted price of any marketed security (stocks and options) are martingales.

• This martingale measure Q is unique in a complete market.

Determining Q from derivative prices.

Possible stock movements

• At each node, assume probability stock

• so discounted price is a martingale.• Leaves us 4 parameters, the probability it

moves up at each of four nodes.

.11

wheredown movesy probabilit and is up moves

duc

cpp ii

−−=

=

Equating option prices with discounted expectation

• In this case we can solve for all four unknowns ( the p’s)

• What if there are only two options sold on this stock--say the first two?

• This is an incomplete market since market prices do not uniquely determine Q

• Can use principle of maximum entropy(minimum cross-entropy)

Entropy

• For a discrete distribution, such that P[X = x] = p(x), the entropy may be defined to be

Discrete analogue to Normal

Sum of values on two dice compared with max entropy

distribution

Monte Carlo Price

Generating S from a discrete uniform

Congruence modulo mReduction Modulo m

For positive integers a and m, the value a (mod m) is the remainder (between 0 and m-1) obtained when a is divided by m.

So 7( mod 3)=1 .

A Linear congruential pseudo-random number generator

• Start with seed• and certain constants a,c,m• Repeat• Note that

0x

0,1,...n ),mod()(1 =+=+ mcaxx nn

}1,...,2,1,0{ −∈ mxn

Example.Example: Linear (or mixed) congruential generator

Matlab Code for linear congruentialgenerator (m small)

• function x=lcg(x0,a,c,m,)• y=x0; x=[]; • for i=1:n • y=rem(a*y+c,m); x=[x y];• end

Better linear congruentialgenerators

• Period =length of sequence before it repeats• =8 for the last generator ( in which m=8).

• Better to choose m large, near word-length e.g.

• Want period large and random-looking behaviour

machine.bit 32 afor 12 31 −

Example (script3)

• Consider (x0,a,c,m,n)=(1,33,1,m,2^10).

• Plot the values of x(i-1) versus x(i) for m=50,500,1100. Do they appear random? What if c=113 or a=113?

• x=lcg(1,33,1,m,2^10);• n=length(x);• plot(x(1:n-1)),x(2:n),’.’);

Multiplicative CongruentialGenerators

script3b

example

• Consider (x0,a,c,m,n)=(1,33,1,m,2^10).

• Plot the values of x(i-1) versus x(i) for m=50,500,1100. Do they appear random? What if c=113 or a=113?

• x=lcg(1,33,1,m,2^10);• n=length(x);• plot(x(1:n-1)),x(2:n),’.’);

More random if we use only a fraction of the period.

• For example a=397, m=1024, c=1 if we plot only the first 200 or so, they look more or less randomly placed.

Full period Linear CongruentialGenerators (script3b)

.1 divides alsoit divides 4 If )(1 divides also offactor primeEach (b)

prime relatively are and (a) ifonly and

if period thishasIt . period possible maximum has) )(mod(

generator alcongruentilinear The .

1

−−

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amcam

mc

mmcaxx

THEOREM

nn

Various linear congruentialgenerators

Recent Generators: MersenneTwister

From Matsumoto and Nishimura (1998). A twisted Generalized Feedback Shift Register with period 2^19937 - 1 and equidistribution in 623 consecutive dimensions (over the whole period). The "seed" is a 624-dimensional set of 32-bit integers plus a current position in that set.

This is the default in R although there are other options.

See their web page:math.keio.ac.jp/~matumoto/emt.html

Recent Generators: Marsaglia-Multicarry

• A multiply-with-carry RNG recommended by George Marsaglia. It has a period of more than 2^60 and has passed all tests (according to Marsaglia). The seed is two integers (all values allowed).

Testing uniform generators: runs(up) test

Runs Test...

t!significan So

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==−

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Runs test on Mersenne Twister

• u<-runif(100000)• n<-length(u)• v<-u[1:n-1]<u[2:n]• r<-sum(v[1:99998]!=v[2:99999])• er<-(2*n-1)/3• s<-sqrt((3*n-5)/18)• z=(r-er)/s

The chi-squared Test.

Chi-squared test. Example.

Example (cont)

•N1=7, N2=5, N3=2, N4=2•All P(Ai)=1/4

The poker test.

•E.g. .123 .112 .344 .977 .882 .234 .788 .835 .854 .444•Are there too many repeated digits?????•Probability all digits different=

Poker test (cont)

Tests for serial correlation

•Estimate the autocorrelations for the sequence e.g. cor(x(i),x(i+m))•These theoretical values all 0. Determines whether observed values significantly non-zero.

Tests of lattice structure:e.g. a=23,c=0,m=97.

Spectral test.

•e1 is shortest distance between two distinct points on lattice•e2 is shortest vector not spanned by e1. Try to ensure that relative size e2/e1 not too great. Since all generators with common period have common

•area in parallelogram spanned by e1 and e2, we can look at the length of the vector e1 (spectral test statistic). Repeat in 3,4,5,6,7 dimensions (i.e. 3-7 spanning vectors ei)

In 3 dim. e.g. a=383,c=263,m=10,000

RANDU in three dimensions

MATLAB.

x=lcg(17,65539,0,2^(31),10000);

plot3(x(1:9998),x(2:9999),x(3:10000),'.')

0,65539,231 === cam

Spectral Test results

1. toclose allwith generator Chooses)hyperplane parallelbetween distancemax /()(StatisticTest mck=