1. Derivatives Financial products that depend on another, generally more basic, product such as a stock

2. Examples Forward contracts Futures Options Swaps

3. Forward contracts A agrees to buy and B agrees to sell an asset –at specific price, K (forward price) –On specific date, T (delivery date) A takes ‘long position’ B takes ‘short position’ Agricultural crops and commodities

4. K Y(T) But price fluctuates in the future time! Actual price at time T:Y(T) and K can differ Payoff can be both positive and negative –Whatever is gained by A is lost by B –And vice versa K Y(T) Short position of seller Long position of buyer

5. Options Gives holder right to exercise a given action –Buy (or sell) underlying asset –At time T exercise date; expiration date; date of maturity –for price K Strike price; exercise price European option –can only be exercised at maturity (t= T) American option –can be exercised anytime between t=0 & t=T Exotic options –Even more complicated boundary conditions!

6. European Calls and Puts –Call option A has right (but not obligation) to buy underlying asset at strike price at maturity B is obliged to sell Secured by paying fee C(Y,t) to B (via option dealer) –Unlike future: No symmetry between buyer and seller Buyer pays C at time t=0 and acquires right to buy at t=T Seller receives cash but faces potential liabilities at t=T

7. Payoff for European call options K Y(T) C(Y,0) Payoff for buyer of call option K Y(T) Payoff for seller of call option

8. Put options K Y(T) C(Y,0) K Y(T) C(Y,0) Payoff for buyer of put option Payoff for seller of put option Buyer has right to sell underlying asset at strike price, K at maturity time, T

9. ? How much would one pay for the option? How can the seller minimise the risk associated with his obligation?

10. Simple example Suppose in 3 months time using a call option, may purchase 1 share in Acme Ltd for 2.50 Scenario 1 –In 3 months Acme Ltd trades at 2.70 Exercise option: buy for 2.50; sell at 2.70; profit is.20 Scenario 2 –In 3 months Acme Ltd trades at 2.30 Let option lapse

11. Assume 2 equal probability scenarios Expected profit is –½*0 +½*20 = 10 –Ignore interest rate effects reasonable to assume value of option is 10 Scenario 1 –Profit on exercise:20 –cost of option: (10) –Net profit:10 gain100% Scenario 2 –Cost of option: (10) –Net loss (10) Loss 100%

12. But suppose buy shares At T = 0 share price 250 –Buy 1 share Scenario 1 –Sell at 270; –profit 20; Gain 20/250*100 = +8% Scenario –Sell at 230; –loss 20; Loss -8% Options respond in more exaggerated way More highly geared Used for speculating/ gambling and insurance

13. Put options Allows holder to sell asset at prescribed price strike or exercise price Holder of ‘calls’ hopes asset price will rise Holder of ‘puts’ hopes price to fall –Can also use as insurance against fall of prices in portfolio

14. Hedging - a form of insurance Say UK company must pay S = 10000 euro to Irish firm in 180 days 1 can write forward contract at present exchange rate for S 2 Buy call option for given strike price at 180 days maturity –Eliminates risk associated with exchange rate fluctuations –Risk is exposure to losses in forward contract Or cost of option contract

15. Equity options (Financial Times 22 November 2003)

16. Extent of trade in calls and puts (vanilla options) ~ $10,000 billion worldwide In late 1992 Citicorp alone had contracts totalling ~ $1426 billion May in some markets have a value greater than the underlying asset In some cases, options are more liquid than actual asset

17. Y(21 November)=4319

18. Y(T)=4325 Option values at expiry (FTSE = 4319 at 21 Nov 2003 - 3 rd Friday in month) At expiry C = -(K-4325) K<Y(T) C = 0 K>Y(T)

19. ‘Risk-less’ portfolio ( Binomial model) Y changes with time, hence  h must also be changed to maximise hedging process and minimise risk time t=0 t=t 1 Y; C Y  ; C  Y  ; C 

20. Rational and fair price Holder of shares selling a derivative of stock at time, t Change must equal gain obtained by investing in riskless security (eg cash)

21. Differential v stochastic calculus

22. Assumptions of Black Scholes 1973

23. Black Scholes SPDE Valid for all types of options Choice of solution determined by boundary conditions

24. Boundary conditions: Call option C(Y, K, T) = Y(T)-K if Y>K C(Y, K, T) = 0 if Y<K C = max{Y-K, 0} C(Y,K,T) K Y(T)

25. Relation to heat transfer equation

27. Solution for puts

28. Problems Interest rates, r may vary Volatility, σ is not constant –Fat tails, not a Gaussian Historical volatility –Over what time period? Implied volatility –Use Black Scholes formula in inverse sense to compute volatility given set of C values (For BS it would be constant –Gives indication of level of volatility expected by market traders