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

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

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 University of Minnesota, Jan. 21, 2011

Volatility Volatility (  ) 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

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

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 Probability Spot price  ~ 0 Probability Spot price  >> 0 University of Minnesota, Jan. 21, 2011

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

Limiting Values University of Minnesota, Jan. 21, 2011 If S << K and  = 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)

Limiting Values University of Minnesota, Jan. 21, 2011 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: 1.Buy the call for C < S – K 2.Sell a share simultaneously for S 3.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 ? ?

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 University of Minnesota, Jan. 21, 2011 Speed Capability Closed-formIterativeMonte-Carlo Modern computing and parallel processing mean fewer resources devoted to building faster iterative or closed-form solutions

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

Binomial Trees Link the value at one unknown point (spot 1, time 1 ) with values at two known points (spot 2a, time 2 ) and (spot 2b, time 2 ) Several choices of p u, p d, S u =spot 2b /spot 1, S d =spot 2a /spot 1 exist, each with advantages and disadvantages val 1 (spot 1, time 1 ) val 2b (spot 2b, time 2 ) val 2a (spot 2a, time 2 ) pupu pdpd University of Minnesota, Jan. 21, 2011

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 Call payoff Spot price T = 0 1 Call payoff Spot price T =  t 2 Call payoff Spot price T = 2  t 3 University of Minnesota, Jan. 21, 2011

Monte-Carlo University of Minnesota, Jan. 21, 2011 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

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

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

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

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 University of Minnesota, Jan. 21, 2011 Payoff Spot price

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

Call Spread University of Minnesota, Jan. 21, 2011 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 Payoff Spot price Call spread payoff Spot price

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

Butterfly University of Minnesota, Jan. 21, 2011 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 Payoff Spot price Butterfly payoff Spot price

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 University of Minnesota, Jan. 21, 2011 Combo payoff Spot price Payoff Spot price

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

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 University of Minnesota, Jan. 21, 2011 Call payoff Spot price Call payoff Spot price Raise this point to intrinsic and exercise!

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

Questions? University of Minnesota, Jan. 21, 2011