storm) independent

source

sink

SYSTEM FAILURETIME =

max(min( 1 , T2 , T7 ), min(T1 , T6 , T5 , T8 ), min(T1 , T2 , T3 , T4 , T8 ), min(T1 , T6 , T5 , T4 , T3 , T7 ))

T

Simulation of Montreal power

grid

• function L=lifepowr(T)

• % input vector T of length 8,

lifetimes of components of montreal

• % power grid. Outputs lifetime L of

the system.

• L1=min(T([1 2 7]));

• L2=min(T([1 5 6 8]));

• L3=min(T([1 2 3 4 8]))

• L4=min(T([1 3 4 5 6 7]));

• L=max([L1,L2,L3,L4]);

Run Montreal power grid

simulation.

• Assume exponential lifetimes, mean=20

years. (Matlab code)

• L=[];

• for i=1:10000

• L=[L lifepowr(exprnd(20,1,8))];

• end

• hist(L,50)

(or try L=[L lifepowr(unifrnd(0,40,1,8))]; )

Results:

min(L)

ans =

0.0185

» mean(L)

ans =

9.1255

Number of simulations for given

accuracy.

95 % CI for parameter is mean ± 2 S / n

2

where S is sample variance obtained from pilot simulation .

To ensure estimator within δ ,

(with confidence around 95%)

set δ = 2 S / n

2

⎛ 2S ⎞

Solve for n . n ≥ ⎜ ⎟

⎝δ ⎠

95% confidence interval for

mean system life

• M=mean(L)

• S=sqrt(var(L))

• M+[-1 1]*2*S/sqrt(length(L))

• to achieve accuracy to 2 decimals, need

delta=.01

• sample size=(2*S/.01)^2

• TOO LARGE-- we need VARIANCE

REDUCTION (better simulations)

The Brownian Motion (Wiener)

Process

Brownian Motion Path

Stochastic integral and

approximating sums

•Denote by dW a small

increment

•Then dW distributed

N(0,dt).

•dW has standard

deviation

Taylor™s expansion

•Define a new process

(e.g. derivative price)

•By Taylor™s Theorem

¦..

Meaning of differentials

Ito™s lemma

Example of Ito™s lemma

Martingale property of

stochastic integrals

The Black-Scholes model for

Stock Prices (discrete time)

• Assume that the price of a stock on day m

is

St = S 0 exp{‘ j =1 Z j }

m

where S 0 is the stock price at time 0 and the random

variables Z i are independent normal N ( µ , σ / N )

2

rσ 2

where µ = ’ ,

N 2N

r is the annual interest rate (e.g. 0.05)

and N is the number of (trading) days in a year (e.g. 252).

The model (for pricing financial

derivatives)

• Notice that under this model, S has a

m

lognormal distribution

rm / N

• with expected value S 0 e

• annual volatility= σ

• Expected value of future stock price is the

same as that of bank deposit of equal

amount. (this assumption is forced on us by no-

arbitrage conditions whenever we are pricing a

financial derivative)

Financial derivatives

• Financial instruments that derive their value from

an associated asset (e.g. stock, index)

• Used to speculate on a rise (call option) or fall

(put option) in asset price.

• used 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, etc.

• Financial equivalent of insurance company: they

allow for transfer of RISK

What is a Call Option?

• A call option is a right, but • Where stock price at

maturity = ST

not an obligation, to

purchase an underlying

• If interest rates are

stock for a fixed price K

constant r

(exercise price) at a fixed

compounded

time T (maturity).