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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
u4S
u3S
u2S
uS
udS
S
dS
d2S
u2dS
ud2S
d3S
State
u3dS
u2d2S
ud3S
d4S
Time
t=0
1
2
3
4
5
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 model
St+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
2t
• 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
d1  (ln( S / s )  (r   2 / 2)T ) /( T )
d2     T
1
N ( x)  Normsdist ( x) 
2
x
y /2
e
dy

2

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)