STAT 906: Computer
Intensive Methods In Finance
• Don McLeish: MC6138
email:dlmcleis@uwaterloo.ca
•Text: Monte Carlo Simulation and Finance. Don L.
McLeish (Wiley, 2005) Available from me @60% of
$119. Early version of book and slides available at
•www.mcleish.ca (follow links)
Other References
• The Mathematics of Financial Derivatives
• (Wilmott, Howison, Dewynne)
• Monte Carlo: Concepts, Algorithms and Applications.
George Fishman QA298.F57
• Arbitrage Theory in Continuous Time. Tomas Bjork
• Mathematics of Financial Markets. R.J. Elliott and P.E.
Kopp (Springer) HG4515.3.E37
• 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
•Tentative Calendar: Week of
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Sept 13+15:
Sep 20+22 :
Sep 27+29 :
Oct. 4+6:
Oct 11+13:
Oct 18+20:
Oct 25: Midterm Test 1
Oct 27:
Nov 1 :
Nov 10 :
Nov 17:
Nov 24:
Nov 26: Midterm Test 2
– Dec 1+3+5: Projects
•Marks distribution
– Midterm Tests (2):
35
– Project&presentation (15 minutes per
presentation)
50
– Class problem presentations
15
Contents
•A general discussion of
simulation.
•Example: 2 person
game.
•Some basic theory of
finance (Chapter 2)
•How to generate
random numbers
(Uniform and nonuniform)(Chapter 3)
•Variance Reduction
•Quasi Monte Carlo
•Simulation and Option
valuation
•Estimation and
Calibration
•Sensitivity Analysis
•Numerical solutions to
PDE’s
Course Prerequisites
• Knowledge of (or tolerance of) Stochastic
Calculus.
• Some finance knowledge/interest
• Computing: Some computer language
(e.g. MATLAB, R, C++)
• “Tolerance of” means have taken course,
seen before, or willing to read 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?
•A cheap source of
experience. E.g. Airbus
pilots
• 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?
•Experience
•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-FINANCE
applications
– 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 dataidentify 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 simulationwhat 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
•
•
•
•
A non-zero-sum game called the "Prisoner's Dilemma"
The two players choose between two moves, either "cooperate" or
"defect".
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
Mathematical model of “plea-bargain”. The dilemma resides in the fact that each
prisoner has a choice between only two options, but cannot make an optimal
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.
A=
5 -10
10 0
B=
5 10
-10 0
>> [p,q,val]=nonzerosum(A,B,10000)
p= 0 1
q=0
1
val = 0 0
“Defect”=second strategy dominates for both players
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.
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 riskless bond, presently worth $1 is
valued $(1+r) 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 positive
amount with positive prob. i.e. both
x(1+r)+ysu>=0 and
• x(1+r)+ysd >=0 with either or both is >0.
• For net investment =0, need sy=-x.
No point on 0-investment line in
yellow region…no arbitrage.
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
Denote by Q the probability distribution which puts
probabilities q and 1 − q on these points su, sd.
If there is to be no arbitrage, there exists a probability
measure Q such that the expected price of future value of
the stock S1 discounted to the present using the return from
a risk-free bond is exactly the present value of the stock.
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.
Derivatives
Financial Derivatives: 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).
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, nothing 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
(you are not compelled to do so) .. return = $0.
Replicating a Contingent Claim
A function of the stock price V (S1) which may represent the return from a
portfolio 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.
Solve for x and y
•x(1+r)+ysu=V(su)
•x(1+r)+ysd=V(sd)
Solving we obtain
V ( su )  V ( sd )
*
y 
su  sd
*
V ( su )  y su
*
x 
1 r
Investment in stock
• Replicating portfolio requires that
investment in stock is slope of value
function between (sd,su)
• A complete market is one in which every
contingent claim (every function V(S1) of a
stock price) can be replicated with using
(marketable) stocks and risk-free bonds.
Discounted Expected value of
V(S1) under Q
Recap
• Under the Q measure, the expected
discounted-to-present value of any
marketed asset (stock or contingent claim)
is its present price.
• In a complete market, there is only one
measure Q measure with this
``martingale’’ property.
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, including the price
of the asset and any related assets at all previous time
periods, beginning at some time t = 0.
• Denote this information available for use at time t as Ht.
• Ht is what is sigma-field generated by the past.
• two fundamental properties of Ht:
1. Ht increases in t. (our information about this and related
processes increases over time).
2. Ht includes the value of the asset price Su , for all u
less than or equal to t. (St is adapted to or measurable
with respect to Ht).
In general, under Q, the
discounted stock price is a
martingale
CAPM AND PORTFOLIO
SELECTION
Consider a column vec tor of (random) returns
R  ( R1 ,..., Rn )' a vector of weights
w  ( w1 ,..., wn )'. We assume that
E ( R )   (a vector) and
cov(R)   ( a matrix).
Then E ( w' R )  w' 
Var ( w' R )  w' w
Consider t he set of all possible values of
(standard deviation, mean)
 ( w' w , w'  ) for weight s that add to 1.
Forms the interior of a region like
this:
The boundary of this region is the
“efficient frontier”. The point with
smallest standard deviation is the
most conservative investment
The conservative portfolio
min  2  w' w (min portfolio variance)
subject to w'1  1
and
w'    (fixed mean)
Solution :  2 satisfies  2  A22 2  2 A12  A11 where


A  [1  ]'  [1  ]
-1
-1
Minimum value of  satisfying this is
1
 g  1 and correpondi ng weights are
1'  1
w g   g  11
(proportio nal to row sums of inverse covariance matrix)
Add a risk-free bond paying r

(p, p)
Everything below this line L is now feasible
Any point on L can be achieved with a weighted
average of the market (point p) and the risk-free
bond
The Market portfolio
has weights
wm proportion al to
 (  - r1)
weight  sd proportion al to
-1
R  standardiz ed excess return
-1
i  r
standardiz ed excess return 
i
Entropy
• Consider ascertaining the value of a random variable X
with distribution
On average it will take 1+1(.5)=1.5 queries with
yes/no answers to determine its value
For a discrete distribution, such that P[X = x] =
p(x), the entropy may be defined to be
Entropy depends on the
probabilities p(x), not the values x.
So for example if g(.) is a one-one function, X and g(X) have
the same entropy. The maximum entropy principle asserts
that given whatever constraints on a distribution are present,
distributions tend to maximize entropy. Examples:
•Distribution on [a,b] with maximum entropy is uniform.
•Distribution on [0, 1) with maximum entropy subject to
E(X)=1 is exponential(1)
•Distribution on (-1,1) with maximum entropy subject to
constraints E(X)=, var(X)=2
is Normal( , 2)
Discrete analogue to Normal
The Lagrangian becomes
Settting the derivative with respect to p(x) equal to zero and
solving we obtain a normal p.d.f.
Maximum entropy dice
• Maximum entropy distribution on integers
{2,3,…,12} subject to having mean 7,
variance 2 £ 35/12 is given below
Sum of values on two dice
compared with max entropy
distribution