G 12 Lecture 4 Introduction to Financial Engineering

G 12 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=(x 0, x 1, …xn) over the next n periods? – Negative xi: invest £ xi, , positive xi: receive £ xi • Net present value NPV(x)=x 0+x 1/(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=u. St with probability • St+1=d. St with probability (1 -p) • Model parameters: u, d, p • Initial condition S 0

The binomial lattice model u 4 S u 3 S u 2 S u. S ud. S S d 2 S u 2 d. S ud 2 S d 3 S State u 3 d. S u 2 d 2 S ud 3 S d 4 S Time t=0 1 2 3 4 5

Binomial distribution • Stock price at time t St can achieve values ut. S, ut-1 d. S, ut-2 d 2 S, …, u 2 dt-2 S, udt-1 S, dt. S • P(St=ukdt-k. S)=(n. Ck)*pk*(1 -p)t-k – Here (n. Ck): =n!/((n-k)!k!)

A more realistic model St+1=ut. St, 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=ut. St, 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 S 0 +w 1+…+wm is N( , 2) Assume all wi are independent N( ’, ’ 2), =E(w 1+…+wm)=m ’, hence ’ = /m 2=V(w 1+…+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 + Zsqrt( t) • Such a process is called a Random Walk • Can use this to simulate process St

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 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 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))S 0 – Compute discounted payoff exp(-r. T)*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(d 1)-Ke-r. TN(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)