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= , exp(0.1)= 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=(x 0,x 1,…x n ) over the next n periods? –Negative x i : invest £ x i,, positive x i : receive £ x i Net present value NPV(x)=x 0 +x 1 /(1+r)+…+x n /(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 S t 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 S t =S then S t+1 =uS t with probability S t+1 =dS t with probability (1-p) Model parameters: u,d,p Initial condition S 0

The binomial lattice model S uS dS d2Sd2S udS d4Sd4S ud 3 S d3Sd3S ud 2 S u 2 dS u2d2Su2d2S u 3 dS u4Su4S u3Su3S u2Su2S t= State Time

Binomial distribution Stock price at time t S t can achieve values u t S,u t-1 dS, u t-2 d 2 S,…, u 2 d t-2 S,ud t-1 S, d t S P(S t =u k d t-k S)=(nCk)*p k *(1-p) t-k –Here (nCk):=n!/((n-k)!k!)

A more realistic model S t+1 =u t S t, t=0,1,2,… where u t are random variables –Assume u t, t=0,1,2,… to be independent –Notice that u t =S t+1 /S t is independent of the units of measurement of stock price –Call u t the return of the stock What is a realistic distribution for returns?

An additive model Passing to logarithms gives ln S t+1 = ln S t +ln u t Let w t = ln u t w t 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 u t 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=e lnu and with normally distributed ln u

Distribution of return Assume that the distribution of u t is independent of t Under log-normal assumption the distribution is defined by mean and standard deviation of the normal variable ln u t Growth rate =E(ln u t ), Volatility  =Std(ln u t ) 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 S t+1 =u t S t, t=0,1,2,… u t `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 u t 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 S t+ 1 = ln S t +w t Know ln S m = ln S 0 +w 1 +…+w m w 1 +…+w m is N(,  2 ) Assume all w i are independent N( ’,  ’ 2 ), =E(w 1 +…+w m )=m ’, hence ’ = /m  2 =V(w 1 +…+w m )=m  ’ 2, hence  ’ 2 =  2 /m

Simulation Hence ln S t+  t = ln S t +w t, w t is normal with mean  t and variance  2  t If Z is a standard normal variable (mean=0, var=1) then ln S t+  t = ln S t +  t +  Zsqrt(  t) Such a process is called a Random Walk Can use this to simulate process S t

Simulation Inputs: –current price S 0, –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 S t+1 = exp(  t +  Zsqrt(  t))S t 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{S T -K,0} –If S T >K: purchase stock for price K (exercise the option) and sell for market price S T, resulting in payoff S T -K –If S T <=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 S t by so-called “risk-neutral” equivalent R t S t+1 = exp(  t +  Zsqrt(  t))S t R t+1 = exp((r-  2 /2)  t +  Zsqrt(  t))R t –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 R T of the risk-neutral equivalent using the formula R T = exp((r-  2 /2)T +  Zsqrt(T))S 0 –Compute discounted payoff exp(-rT)*max{R T -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(d 1 )-Ke -rT N(d 2 ), where

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)