G12 Lecture 4

Introduction to Financial Engineering

Financial Engineering• FE is concerned with the design and valuation of

“derivative securities”• A derivative security is a contract whose payoff is tied

to (derived from) the value of another variable, called the underlying– Buy now a fixed amount of oil for a fixed price per barrel to

be delivered in eight weeks• Value depends on the oil price in eight weeks

– Option (i.e. right but not obligation) to sell 100 shares of Oracle stock for $12 per share at any time over the next three months

• Value depends on the share price over next three months

What are these financial instruments used for?

• Hedge against risk– energy prices– raw material prices– stock prices (e.g. possibility of merger)– exchange rates

• Speculation – Very dangerous (e.g. Nick Leason of Berings

Bank)

Characteristics of FE Contracts• Contract specifies

– an exchange of one set of assets (e.g. a fixed amount of money, cash flow from a project) against another set of assets (e.g. a fixed number of shares, a fixed amount of material, another cash flow stream)

– at a specific time or at some time during a specific time interval, to be determined by one of the contract parties

• Contract may specify, for one of the parties, – a right but not an obligation to the exchange (option)

• In general the monetary values of the assets change randomly over time

• Pricing problem: what is the “value” of such a contract?

Dynamics of the value of money

• Time value of money: receiving £1 today is worth more than receiving £1 in the future

• Compounding at period interest rate r: • Receiving £1 today is worth the same as receiving £ (1+r) after

one period or receiving £ (1+r)n after n periods • Investing £1 today costs the same as investing £ (1+r) after

one period or £ (1+r)n after n periods

• Discounting at period interest rate r:• Receiving £1 in period n is worth the same as receiving

£1/(1+r)n today• Investing £1 in periods costs the same as investing £ 1/(1+r)n

today

Continuous compounding• To specify the time value of money we need

– annual interest rate r – and number n of compounding intervals in a year

• Convention: – add interest of r/n for each £ in the account at the end of each of n

equal length periods over the year

• If there are n compounding intervals of equal length in a year then the interest rate at the end of the year is (1+r/n)n which tends to exp(r ) as n tends to infinity

(1+0.1/12)12=1.10506.., exp(0.1)=1.10517...

• Continuous compounding at an annual rate r turns £1 into £ exp(r ) after one year

Why “continuous” compounding?

• Cont. comp. allows us to compute the value of money at any time t (not just at the end of periods)

• Value of £1 at some time t=n/m is £(1+r/m)n=£(1+tr/n)n

• (1+tr/n)n tends to Exp(tr) for large n– Can choose n as large as we wish if we choose number of

compounding periods m sufficiently large

• £X compounded continuously at rate r turn into £exp(tr)*X over the interval [0,t]

Net present value of cash flow

• What is the value of a cash flow x=(x0,x1,…xn) over the next n periods?– Negative xi: invest £ xi,, positive xi: receive £ xi

• Net present value NPV(x)=x0+x1/(1+r)+…+xn/(1+r)n

• Discount all payments/investments back to time t=0 and add the discounted values up

• If cash flow is uncertain then NPV is often replaced by expected NPV (risk-neutral valuation)

• Benefits and limitations of NPV valuations and risk-neutral pricing can be found in finance textbook under the topic “investment appraisal”

• Let’s now turn to asset dynamics…

A simple model of stock prices

• Stock price St at time t is a stochastic process– Discrete time: Look at stock price S at the end of

periods of fixed length (e.g. every day), t=0,1,2,…

• Binomial model: If St=S then • St+1=uSt with probability

• St+1=dSt with probability (1-p)

• Model parameters: u,d,p

• Initial condition S0

The binomial lattice model

S

uS

dSd2S

udS

d4S

ud3S

d3S

ud2S

u2dSu2d2S

u3dS

u4S

u3S

u2S

t=0 1 2 3 4 5

State

Time

Binomial distribution

• Stock price at time t St can achieve values

utS,ut-1dS, ut-2d2S,…, u2dt-2S,udt-1S, dtS

• P(St=ukdt-kS)=(nCk)*pk*(1-p)t-k

– Here (nCk):=n!/((n-k)!k!)

A more realistic modelSt+1=utSt, t=0,1,2,…

• where ut are random variables

– Assume ut, t=0,1,2,… to be independent

– Notice that ut=St+1/St is independent of the units of measurement of stock price

– Call ut the return of the stock

• What is a realistic distribution for returns?

An additive model• Passing to logarithms gives

ln St+1= ln St +ln ut

• Let wt = ln ut

• wt is the sum of many small random changes between t and t+1

• Central limit theorem: The sum of (many) random variables is (approximately) normally distributed (under typically satisfied technical conditions)

– Most important result in probability theory– Explains the importance and prevalence of the normal distribution

Log-normal random variables

• Assume that ln ut is normal– Central limit theorem is theoretical argument for this

assumption – Empirical evidence shows that this is a reasonably

realistic assumption for stock prices • however, real return distributions have often fatter tails

• If the distribution of ln u is normal then u is called log-normal– Notice that log-normal variables u are positive since

u=elnu and with normally distributed ln u

Distribution of return• Assume that the distribution of ut is independent of t• Under log-normal assumption the distribution is defined by

mean and standard deviation of the normal variable ln ut

Growth rate =E(ln ut), Volatility =Std(ln ut)

• Typical values are=12%, =15% if the length of the periods is one year =1%, =1.25% if the length of the periods is one month

• Recall 95% rule: 95% of the realisations of a normal variable are within 2 Stds of the mean

• Careful: if ln u is normal with mean and variance 2 then the mean of the log-normal variable u is NOT exp() but E(u)=exp(+2/2) and Var(u)=exp(2 + 2)(exp(2)-1)

Model of stock prices

St+1=utSt, t=0,1,2,…• ut`s are independent identically log-normal

random variable with E(u) = exp(+2/2) Var(u)= exp(2 + 2)(exp(2)-1)

• Model is determined by growth rate and volatility , which are the mean and std of ln ut

• Values for and 2 can be found empirically by fitting a normal distribution to the logarithms of stock returns

Simulation• Find and for a basic time interval (e.g. =14%, =30%

over a year)• Divide the basic time interval (e.g. a year) into m intervals of

length t=1/m (e.g. m=52 weeks)– Time domain T={0,1,…,m}

• Use model ln St+ 1= ln St +wt

• Know ln Sm= ln S0 +w1+…+wm • w1+…+wm is N(,2) • Assume all wi are independent N(’,’2),

=E(w1+…+wm)=m’, hence ’ = /m

2=V(w1+…+wm)=m ’2, hence ’2 =2/m

Simulation

• Hence ln St+t= ln St +wt,• wt is normal with mean t and variance

2t • If Z is a standard normal variable (mean=0,

var=1) then

ln St+t= ln St + t + Zsqrt(t)• Such a process is called a Random Walk• Can use this to simulate process St

Simulation• Inputs:

– current price S0, – growth rate (over a base period, e.g. one year)– volatility (over the same base period)– Number of m time steps per base period (t=1/m is the length of

a time step)– Total number M of time steps

• Iteration St+1= exp(t + Zsqrt(t))St

Z is standard normal (mean=0, std =1)

Options• Call option: Right but not the obligation to buy a

particular stock at a particular price (strike price) – European Call Option: can be exercised only on a particular

date (expiration date)– American Call Option: can be exercised on or before the

expiration date

• Put option: Right but not the obligation to sell a particular stock for the strike price– European: exercise on expiration date– American exercise on or before expiration date

• Will focus on European call in the sequel…

Payoff

Payoff of European call option at expiration time T:

Max{ST-K,0}

– If ST>K: purchase stock for price K (exercise the option) and sell for market price ST, resulting in payoff ST-K

– If ST<=K: don’t exercise the option (if you want the stock, buy it on the market)

Pricing an option • What’s a “fair” price for an option today? • Economics: the fair price of an option is the expected

NPV of its “risk-neutral” payoff • Risk-neutral payoff is obtained by replacing stock price

process St by so-called “risk-neutral” equivalent Rt

St+1= exp(t + Zsqrt(t))St

Rt+1= exp((r- 2/2)t + Zsqrt(t))Rt

– Recall that the expected annual return of the stock is =+2/2; expected annual return of the risk-neutral equivalent is r

– Volatility of both processes is the same

Option pricing by simulation

• Model: – Generate a sample RT of the risk-neutral equivalent

using the formula

RT= exp((r- 2/2)T + Zsqrt(T))S0

– Compute discounted payoff

exp(-rT)*max{RT-K,0}

• Replication: – Replicate the model and take the average over all

discounted payoffs

The Black-Scholes formula

• Risk-neutral pricing for a European option has a closed form solution

• The value of a European call option with strike price K, expiration time T and current stock price S is

SN(d1)-Ke-rTN(d2),

where

xy dyexxN

Td

TTrsSd

2/

2

21

2

2

1)(Normsdist)(

)/())2/()/(ln(

Key learning points• Stochastic dynamic programming is the discipline that

studies sequential decision making under uncertainty

• Can compute optimal stationary decisions in Markov decision processes

• Have seen how stock price dynamics can be modelled by assuming log-normal returns

• Risk-neutral pricing is a way to assign a value to a stock price derivatives

• European options can be valued using simulation (also for more complicated underlying assets)