G12 Lecture 4

1. G12 Lecture 4 Introduction to Financial Engineering

2. 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

3. 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)

4. 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?

5. 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

6. 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

7. 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]

8. 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…

9. 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

10. The binomial lattice model u4S u3S State u3dS u2S u2dS uS u2d2S udS ud2S dS ud3S S d2S d3S d4S Time t=0 1 2 3 4 5

11. 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!)

12. 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?

13. 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

14. 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

15. 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)

16. 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

17. 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

18. 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

19. 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)

20. 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…

21. 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)

22. 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

23. 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

24. 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

25. 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)