1. FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche Lecture Advanced Derivatives

2. Equity Collar Equity Swaps Exotics a) Asian options (Monte Carlo) b) Barrier Options (BOPM) Other Exotics TOPICS

3. EQUITY COLLAR

4. 1) You already hold stocks but you want to limit downside (buy a put) but you are also willing to limit the upside if you can earn some cash today (by selling an option - a call) COLLAR = long stock + long put (K 1 ) + short call (K 2 ) {0,+1,0} = {+1,+1,+1} + {-1,0,0} + {0,0,-1} EQUITY COLLAR

5. Equity Collar: Payoff Profile +1 00 Long Stock Long Put Short Call 00 0 0 +1 Equity Collar plus equals

6. S T K 2 Long SharesS T S T S T Long Put (K 1 )K 1 - S T 00 Short Call (K 2 )00 -(S T - K 2 ) Gross PayoffK 1 S T K 2 Net Profit (1) K 1 - (P -C) S T - (P - C) K 2 - (P - C) Note : 1. Net Profit = Gross Payoff – (P-C) Table 16.1 : Equity Collar Payoffs

7. EQUITY SWAPS

8. EQUITY SWAP (Excel T16.2) Pension fund already holds $Q in FRN’s with payout based on LIBOR (every 90 days) Fancies a ‘punt’ on the S&P500, for a while. Should she sell FRN’s and invest in the stocks ? Maybe cheaper to: Agree to receive the % return on the S&P500 = R 90 (minus a spread), every 90 days and payout LIBOR - THIS IS AN EQUITY SWAP Net receipts = $Q ( R 90 - spread ) - LIBOR (90/360) Pension fund now effectively has an investment in the stock market.

9. OTHER EQUITY SWAPS 1)You hold 100% US portfolio of $Q Agree in SWAP to pay out on S&P500 and receive return on Nikkei 225 (in USD, at fixed known exchange rate). On notional principal of say $Q/2. 2) You hold 100% US Corporate bonds of $Q Agree 2 separate swaps a) receive S&P500 and pay LIBOR on $Q/4 b) receive Nikkei 225 and pay LIBOR on $Q/4 You have changed the composition of your portfolio a) + b) from one swap dealer = ‘ structured finance’

10. EXOTICS

11. EXOTICS (often path dependent) a) Average price ASIAN CALL payoff = max { 0, S av - K } - used by corporates to hedge risk of series of foreign currency receipts or payments in the future (over the life of the option) ~ cheaper than an ‘ordinary’ f.c. option b) Barrier Options (e.g. up and out put) - pension fund holds stocks and is worried about fall in price but does not think price will rise by a very large amount Ordinary put? - expensive Up and out put - cheaper

12. Pricing an Asian Option (BOPM)~ Excel T16.3 Average price ASIAN CALL(T=3)  Calculate stock price at each node of tree calculate the average stock price S av,i at expiry, for each of the 8 possible paths (i = 1, 2, …, 8).  Calculate the option payoff for each path, that is max[S av,i – K, 0] (for i = 1, 2, …, 8).  The risk neutral probability for a particular path is q i *=q k (1-q) n-k, q= risk neutral probability of an ‘up’ move k= number of ‘up’ moves (n – k) = the number of ‘down’ moves

13. Pricing an Asian Option (BOPM)~ Excel T16.3 Average price ASIAN CALL(T=3) Weight each of the 8 outcomes for the call payoff max[S av,i – K, 0] by the q i * to give the expected payoff: ES* =  The call premium is then the PV of ES*, discounted at the risk free rate, hence:  C Asian = e -rT (ES*)

14. Pricing an Asian Option (MCS)~ Excel T16.4 Average price ASIAN CALL (MCS) Simulate path for underlying, S and calculate S av,i - K for each run of the MCS C = exp(-rT)x Average of max { 0, S av,i - K }

15. Pricing Barrier Options (BOPM) Down-and-out call S 0 = 100. Choose K = 100 and H = 90 (barrier) Construct lattice for S Payoff at T is max {0, S T -K } Follow every ‘path’ (ie DUU is different from UUD) If on say path DUU we have any value of S 0). Use BOPM risk neutral probabilities for each path and each payoff at T

16. Example: Down-and-out call (Excel Table 16.5): S 0 =100, K= 100, q = 0.857, (1-q) = 0.143 H = 90 UUU ={115, 132.25, 152.09} Payoff = 52.09 (q * = 0.629) DUU ={80, 92,105.8} Payoff = 0 NOT 5.08 (q * = 0.629) C = exp(-rT ) x ‘ Sum of [ q * x payoffs at T ] ’ where q i *=q k (1-q) n-k,

17. Other Exotics Lookback call the strike price is set at expiration at the lowest price S min of the underlying stock during the life of the option (ie. the payoff is S T – S min ). Lookback put sets the strike price at expiry, equal to the highest price reached by the stock over the option’s life (ie. the payoff is S max – S T ). These options are also referred to as no-regrets options Shout options allow the holder to lock in a minimum payoff S t – K, at time t>0 but which will not be received until expiration. The payoff is max[S t – K, S T – K]

18. Other Exotics Barrier options knockout options If the option is terminated when the stock price falls to the barrier, then they are referred to as down-and-out options, while if they are terminated when the price rises to the barrier, they are up-and-out-options. Up-and-in-option, whereby the option’s ‘life’ does not begin until the stock price hits an upper barrier. The option premium is paid up front but the option cannot be exercised until after the barrier has been hit. Down-and-in-option is not activated until the stock price hits the designated lower barrier.

19. Other Exotics Compound options~ options on options. For example an investor might want the right to buy an option at a later date, at a price (premium) fixed today. This is a ‘call on a call’ and acts as a hedge against a future increase in the options price Rainbow options Options can also be structured to have a payoff based on the better or worse of two underlying assets and are referred to as min-max or rainbow options or alternative options. For example, a call may payoff according to which of 2 stocks has the higher price (or return) at expiry.

20. LECTURE ENDS HERE