Options - 2

Option Strategies CBOE lists 12 basic option strategies for investors. These are: buying or selling naked calls, covered calls, buying or selling naked puts, protective puts, bear spread, bull spread, straddle, butterfly spread, strangle, synthetic stock   The CBOE has a nice online tutorial that supplements most of what appears in my notes. They even have questions to test yourself. The link to that site was in the previous set of notes. This section pertains to the advanced strategies portion of their online course.

COVERED CALLS Buy stock @ $ 75 Write 80 CALL for $ 1.5. Net Cost = 75 - 1.50 = 73.50 (recover to break-even).   Stock Stock Call Total Price Gain/Loss G/L G/L 65.00 -10.00 1.50 -8.50 70.00 - 5.00 1.50 -3.50 73.50 - 1.50 1.50 0.00 75.00 0.00 1.50 1.50 76.50 + 1.50 1.50 3.00 78.50 + 3.50 1.50 5.00 80.00 + 5.00 1.50 6.50 81.50 + 6.50 0 6.50 85.00 +10.00 -3.50 6.50 90.00 +15.00 -8.50 6.50 Gains limited on the upside, essentially gains above 80 are given up for current income. 6.50 -8.50

COLLARS ADD 70 PUT (@ 3.125 TO THE ABOVE COVERED CALL, New net Cost = 73.50 + 3.125 = 76.625 This 76.625 must be recovered to break even   Stock Stock Call Put Total Price Gain/Loss G/L G/L G/L 55.00 -20.00 1.50 11.875 -6.625 60.00 -15.00 1.50 6.875 -6.625 65.00 -10.00 1.50 1.875 -6.625 70.00 - 5.00 1.50 -3.125 -6.625 75.00 0.00 1.50 -3.125 -1.625 76.63 +1.625 1.50 -3.125 0 80.00 + 5.00 1.50 -3.125 3.375 85.00 +10.00 -3.50 -3.125 3.375 90.00 +15.00 -8.50 -3.125 3.375 3.375 -6.625

Bull Spread Stock = 100 Strike Price Call Price Costs Buy 1 call at 95 7.00 -7.00 Write 1call at 100 4.00 +4.00 Graph turns at the strikes Total Cost = -3.00 Expiration Cash flows S 90 95 97 98 100 102 103 105 95 call (B) 0 0 2 3 5 7 8 10 100 call (W) 0 0 0 0 0 - 2 - 3 - 5 Total 0 0 2 3 5 5 5 5 - cost -3 -3 -3 -3 -3 -3 -3 -3 Net CF -3 -3 -1 0 2 2 2 2   +2 95 100 -3

BEAR SPREAD Buy 1 call at 105 3.00 -3.00   Stock = 100 Strike Price Call Price Costs Buy 1 call at 105 3.00 -3.00 Write 1 calls at 100 4.00 +4.00 Total Cost = +1.00 Expiration Cash flows S 95 97 98 100 102 103 105 110 105 call (B) 0 0 0 0 0 0 0 5 100 call (W) 0 0 0 0 - 2 - 3 - 5 -10 Total 0 0 0 0 - 2 - 3 - 5 - 5 -Cost 1 1 1 1 1 1 1 1 Net C 1 1 1 1 - 1 - 2 - 4 - 4

BUY STRADDLE:  Stock Price = $ 60, Strike = $60, Call Price = $ 3, Put Price = $ 2, same exp. Buy Put = - 2 Buy Call = - 3 Initial Cost = - 5   At expiration, the gains (G) or losses (L) are: 60 65 55

WHAT IF EXPECTATION IS: PRICE MORE LIKELY TO ?   Load up on the CALL side: BUY STRAP, i.e BUY 2 CALLS and BUY 1 PUT, FOR COST OF $ 8. Now BREAK-EVEN => $ 8 movement on 1 put OR $ 4 movement in each call of 2. OR $8 movement on 1 put So the prices are 52 AND 64. Graph similar to straddle but skewed more to the left (break-even of 52 vs 55 for straddle) WHAT IF EXPECTATION IS: PRICE MORE LIKELY TO ?   Load up on the PUT side: BUY STRIP, i.e BUY 1 CALL and BUY 2 PUTS, FOR COST OF $ 7. Now BREAK-EVEN => $ 7.00 movement on 1 CALL, OR 3.50 movement on each PUT (of 2). Break even prices are 56.5 or $ 67 WITH BOTH STRIPS & STRAPS, LOSSES IF THE STOCK PRICE DOES NOT MOVE.

STRANGLE: (changes the strike price). Buy 60 Call = -3.00 Buy 55 put = -0.50 (versus 60 put for $ 2.00) Initial cost = -3.50   Break-evens: Above 60 call gains, S = 63.50 Below 55 put gains, S = 51.50 Between 55 and 60, neither gains, and max loss = 3.50 51.5 63.5 -3.50 55 60

LONG BUTTERFLY   Stock = 100 Strike Price Call Price Costs Buy 1 call at 105 3.00 -3.00 Write 2 calls at 100 4.00 +8.00 Buy 1 call at 95 7.00 -7.00 Total Cost = -2.00 Expiration Cash flows S 95 97 98 100 102 103 105 110 105 call (B) 0 0 0 0 0 0 0 5 100 call (2W 0 0 0 0 - 4 - 6 -10 -20 95 call (1B) 0 +2 +3 +5 +7 +8 +10 +15 Total 0 2 3 5 3 2 0 0 - cost -2 -2 -2 -2 -2 -2 -2 -2 Net CF -2 0 1 3 1 0 -2 -2

Comments on STRATEGIES There is no reason to keep the number of options on each leg the same, they can be in a ratio. There is no reason to keep the expirations on each leg the same, they can be along a calendar. There are many more, think of them as a tool kit, from which you extract what seems appropriate for a given market situations. Most traders don’t seem to do that- they learn one or two and implement them regularly. Brokerage houses offer strategy finders for option beginners.

PUT-CALL-PARITY IMPLIES THAT S + P = C + PV(X) Recall from earlier strategy discussion that a) buying stock and a put looks like a call; b) buying stock and selling calls => writing a put. So clearly the three are related. As proof: Using S = 75, X = 75, P = 4, C = 4.74, rates are 1% to exp, think of two strategies. I. a) Buy Put + Buy Stock; b) Buy Call AND Invest PV (X) in T-Bills   a) Cost of (a) is 4.00 + 75 = $79 ; Cost of (b) is 4.74 + 75/(1.01) = $79 COST EQUAL II. Evaluate at expiration. S= 60 S= 90 a) Stock 15 0 Put 60 90 75 90 PAYOFFS THE SAME! b) Call 0 15 Bill 75 75 75 90 PUT-CALL-PARITY IMPLIES THAT S + P = C + PV(X)

This holds exactly for European options since the payoff above is at expiration. For American options, because of early exercise,put-call parity holds approximately. The main message is really of linkages between stock, option and interest rate markets. Think in terms of arbitrage. Suppose the call is priced at $4. One side of PCP is greater than the other, but both payoff the same at expiration. Buy Put + Buy Stock = 79 > Buy Call AND Invest PV (X) in T-Bills = 78.26 So, sell the expensive side, buy the cheap side, arbitrage profit is the difference. \   Sell the put + 4.00 Short the stock + 75.00 Buy the call - 4.00 Invest in bills - 74.26   Keep 0.74 in cash per unit. At expiration, everything washes out !

Once again, the strategy was SELL PUT SHORT STOCK BUY CALL INVEST IN BILLS Suppose at expiration, the stock is below 75 (the strike price). Call is worthless Put gets exercised, you buy the stock for 75, using the proceeds from the T-bill, whose FV is 75 And deliver against the short.   Suppose the stock closes above strike (say at 80). Put expires worthless You exercise the call, buying the stock at 75 using the proceeds from the T-Bill and delivering it against the short.

MOTIVATING BINOMIAL PRICING   1. START WITH COVERED CALLS, S = 100, C= 7, X = 100 Say stock price increases to 110 or drops to 95 (crude volatility) a) Buy 1 Share + write 1 Call Cost S = 110 S = 95 S -100 110 95 C + 7 - 10 0 - 93 100 95 b) Buy 1 Share + write 2 Calls Cost Exp S = 110 S = 95 S -100 110 95 C + 14 - 20 0 86 90 95

c) Buy 2 Shares + write 3 Calls Cost S = 110 S = 95 S -200 220 190 C + 21 - 30 0 -179 190 190   What happened here! In (a) cash flow varied with stock price change (100 or 95) In (b) cash flow varied with stock price change (90 or 95) In (c) cash flow SAME with stock price change (190) (IS RISKFREE). So investment of 179 should earn risk-free rate. Thisis the notion of a risk-free hedge created from purchasing two shares and writing 3 calls. DELTA OR HEDGE RATIO

A: BINOMIAL PRICING MODEL (FORMULA VERSION)   S = 100, X = 100, risk-free rate r = 6% (over the period). Suppose price increases to 110 or decreases to 90 (volatility). Stock Call 110=Su u=1.1 10 Cu S=100 C u, d = up, down 90=Sd d= 0.9 0 Cd Step 1: Find Hedge ratio (h) = Cu - Cd = 10 - 0 = 1 Su - Sd 110 – 90 2 pCu + (1-p) Cd 1+r -d Value of call C = ---------------------, p = ---------- (1+r) u - d p = (1+ 0.06 – 0.9)/(1.1-0.9) = 0.8 C = [0.8 *10 + 0.2 * 0]/1.06 = 7.55.

REVISIT NOTION OF SYNTHETICS BY REARRANGING P-C-P DYNAMIC HEDGING ISLAMIC FINANCE ANYONE.