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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)