University of Minnesota, Jan. 21, 2011

1. Equity Derivatives Dave Engebretson Quantitative Analyst Citigroup Derivative Markets, Inc. January 21, 2011 University of Minnesota, Jan. 21, 2011

2. Contents • Vanilla Options • Terminology • Pricing Methods • Risk and Hedging • Spreads • Exotic Options • Questions/Discussion University of Minnesota, Jan. 21, 2011

3. Options Terminology Call/put: a call permits the holder to buy a share of stock for the strike price; a put permits the holder to sell a share of stock for the strike price Spot price (S): the price of the underlying stock Strike price (K): the price for which a share of stock may be bought/sold Expiration date: the final date on which an option may be exercised American/european: european-style options may only be exercised on their expiration dates; american-style options may be exercised on any date through (and including) their expiration dates Example: an IBM $100 american call expiring on 20-Jan-2012 permits its holder to buy a share of IBM for $100 on any business day up to, and including, 20-Jan-2012 University of Minnesota, Jan. 21, 2011

4. Volatility Volatility (s) is a measure of how random a product is, usually defining a one-year standard deviation The left picture shows low volatility - the path is very predictable The right picture shows high volatility - the path cannot be well predicted University of Minnesota, Jan. 21, 2011

5. T = 0 T > 0 Options 101 Call payoff Put payoff Spot price Spot price Put payoff Call payoff Spot price Spot price University of Minnesota, Jan. 21, 2011

6. s ~ 0 Options 102 Convolving the probability distribution with the final payoff gives today's fair price for the option. Higher volatility gives higher value because, while it samples more lower spots, it also samples more higher spots. Different strikes correspond to shifting the red payoff curve horizontally; different spot prices correspond to shifting the blue probability distribution s >> 0 Probability Probability Spot price Spot price University of Minnesota, Jan. 21, 2011

7. T = 0 Limiting Values Call payoff Put payoff Spot price Spot price American call >= 0 >= S – K (intrinsic) American put >= 0 >= K – S (intrinsic) Why? European options can be worth less than intrinsic value University of Minnesota, Jan. 21, 2011

8. Limiting Values If S << K and s = 0 then a european put will be worth K – S at expiration Consider the following scenario: Buy the european put for K e-rt - S, buy a share of stock for S, pay interest on the borrowed difference of K e-rt At expiration exercise the put, receiving K and closing my position, and use the K to repay the loan of K e-rt Net profit: 0 European call >= 0 >= S – K e-rt (intrinsic) European put >= 0 >= K e-rt – S (intrinsic) University of Minnesota, Jan. 21, 2011

9. Limiting Values Why must american calls be worth at least S – K? If an american call is worth less than S – K, I could do the following: Buy the call for C < S – K Sell a share simultaneously for S Immediately exercise the call (american), paying K to receive a share I then have no net shares (sold one, exercised into one) and my total cash intake is -C + S – K Is this advantageous? -C + S – K > 0 S – K > C This was our initial assumption, so we have an arbitrage ? ? University of Minnesota, Jan. 21, 2011

10. Pricing Methods Closed-form solutions for option prices apply only in certain cases (european options without dividends, etc.) Iterative solutions can handle far more types of derivatives, but cost more in calculation time Monte-Carlo pricing for some very exotic derivatives – this converges very slowly and introduces randomness into pricing Closed-form Iterative Monte-Carlo Speed Modern computing and parallel processing mean fewer resources devoted to building faster iterative or closed-form solutions Capability University of Minnesota, Jan. 21, 2011

11. Black-Scholes European options without dividends can be priced in closed form using this model University of Minnesota, Jan. 21, 2011

12. (spot2a, time2) (spot2b, time2) val2a val2b pu pd val1 (spot1, time1) Binomial Trees Link the value at one unknown point (spot1, time1) with values at two known points (spot2a, time2) and (spot2b, time2) Several choices of pu, pd, Su=spot2b/spot1, Sd=spot2a/spot1 exist, each with advantages and disadvantages University of Minnesota, Jan. 21, 2011

13. T = 0 Pricing an Option with a Binomial Tree 1. Discretize the payoff at expiration, choose normal vs. log-normal evolution 2. Evolve the first timestep 3. Repeat step 2 to cover the entire lifetime of the option 1 Call payoff Spot price T = Dt T = 2Dt 3 2 Call payoff Call payoff Spot price Spot price University of Minnesota, Jan. 21, 2011

14. Monte-Carlo Generate a multitude of paths consistent with desired distribution and dynamics For each path, compute the value of the option Appropriately average values for all the paths Greeks: best to compute with perturbations to existing paths. Why? Slow convergence, but able to handle just about any type of option; may obtain slightly different results when recalculating the same option University of Minnesota, Jan. 21, 2011

15. Risk Greeks for call, plotted vs. K / S Delta Gamma, Vega Theta Rho University of Minnesota, Jan. 21, 2011

16. Risk ATM greeks, plotted vs. time University of Minnesota, Jan. 21, 2011

17. Hedging Delta - shares of stock Rho - interest rate futures Gamma, Vega, Theta - other options Gamma, Theta ~ Vega ~ Gamma, Vega Theta University of Minnesota, Jan. 21, 2011

18. Spreads A spread is a group of trades done together Netting of risk Often a cheaper way to take specific positions Some spreads are listed on exchanges, many are OTC All spreads have at least two legs, but can have many Payoff Spot price University of Minnesota, Jan. 21, 2011

19. Combo A combo is a long call with a short put at the same strike The payoff replicates a forward Combo payoff Payoff Spot price Spot price University of Minnesota, Jan. 21, 2011

20. Call Spread A call spread is a long call of one strike with a short call of another strike These can be bullish or bearish depending which strike is bought Call spread payoff Payoff Spot price Spot price University of Minnesota, Jan. 21, 2011

21. Straddle A straddle is a long call with a long put at the same strike The payoff is a bet on volatility Payoff Straddle payoff Spot price Spot price University of Minnesota, Jan. 21, 2011

22. Butterfly A butterfly is a combination of three equally spaced strikes in 1/-2/1 ratios Butterflies pay off when the stock ends near the middle strike, price is probability Butterfly payoff Payoff Spot price Spot price University of Minnesota, Jan. 21, 2011

23. Put-Call Parity Compare a combo’s payoff with the payoff of a share of stock minus a bond Call – Put = S – K e-rt Combo payoff Payoff Spot price Spot price University of Minnesota, Jan. 21, 2011

24. Exotic Option Types American - not solvable in closed form, so are they exotic? Asian – payoff depends not on terminal spot, but on average spot over defined time period Bermudan – can only be exercised on predetermined dates, so something between european and american Binary (digital) – all-or-nothing depending on a condition being met Cliquet (compound) – an option to deliver an option. Call on call, call on put, etc. Knock-in/knock-out (barrier) – options that come into/go out of existence when a condition is met, such as spot reaching a predetermined value Variance/volatility/dividend swap – an agreement to exchange money based on realized variance, volatility, or dividends University of Minnesota, Jan. 21, 2011

25. American Options Use a binomial tree, raise values to intrinsic at each time step If an option is raised to intrinsic, exercise it Non-dividend calls don’t get exercised Bermudan – same, but only raise to intrinsic at exercise dates Raise this point to intrinsic and exercise! Call payoff Call payoff Spot price Spot price University of Minnesota, Jan. 21, 2011

26. Binary Options Start with a call spread, bring the strikes closer together, and increase the number of units of call spread Binary option payoff Call spread payoff Spot price Spot price University of Minnesota, Jan. 21, 2011

27. Questions? University of Minnesota, Jan. 21, 2011