Lec 10 Discover Option Prices Lec 10: How to Discover Option Prices (Hull, Ch. 10) Suppose S0 = $50 and r = 25% . Q: What might be reasonable prices for C0E, C0A, or P0E, P0A (given K=40, T=1 year)?. Intuition, or what questions to think about. ▸ Is the stock price expected to ↑ or ↓? ▸ If call is American, I would pay at least $10. Why? ▸ If call is European, why pay anything? (Exercise ONLY on the Expiration Day!.) ▸ Is it ever possible for C0E = C0A, or P0E = P0A ? The purpose of this Lecture is to help you develop “good intuition” about option pricing. Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices European Call, Stock pays no dividends: C0E (p.2) Do these prices make sense? S0 = $50, C0E(K = $40, T=1yr) = $5, and r = 25%(simple interest) Intuition. There are two ways to buy stock: A: Buy the stock right now, CF0 = -50 Or B: Buy the call and a bond and wait until expiration {+C, +B(FV=$40, T=1yr)} ➟ CF0 = -5-32 = -$37 At Expiration, for the synthetic stock: if call is in the money (ST > 40) ➟ CT + 40 = ST if call is out of the money (ST < 40) ➟ CT + 40 = 40. Which is the better investment A or B ? Is it possible to make some “free money”? Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices Yes, try the following strategy: {-S, +C, +B(FV=40, T=1yr)} {Short the stock for $50, Buy the call for $5, Buy a bond for $32 =40/1.25} CF0 = +50-5-32 = +$13 At Expiration, if ST ≥ 40, call is in the money. Bond matures for $40, use $40 plus call to buy stock. Use stock to cover short position. CFT=0. if ST < 40, call is worthless. Bond matures for $40. Use some of $40, buy stock and cover the short position. CFT = 40 - ST > 0. Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices This strategy is great! Think about it: Receive $13 now. If stock price ↓, make more money (40 - ST). If ST ↑, lose nothing! This is known as an ARBITRAGE OPPORTUNITY. The “Arbitrage Profit” = $13. Clearly, Call is mis-priced. To preclude this arbitrage C0E must be at least 5+13 = $18. In sum, If S0 = $50, and r = 25%, then C0E(K = $40, T=1yr) > $18 (Compare this answer with initial intuition: “European Call has little value” ). Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices European Call Price C0E, Stock pays a Dividend (p. 3) Assume stock pays a $5 Dividend (for sure) in 3 months. How will this affect the Call value? Do these prices make sense? S0 = $50, C0E(K = $40, T=1yr)=$6, r=25%, and Div=$5 in 3 months. There are two ways to buy stock: A: Buy the stock right now: CF0 = -50, Or B: Buy the call. Buy a bond to mimic the dividend, and another bond to cover the $40. Wait until expiration. {+C, +B(FV=5, t=3 months), +B(FV=40, T=1yr) } CF0 = -6 -4.71 -32 = -$42.71 Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices Cash Flows for the synthetic stock: In 3 months, 1st bond matures for $5, just like the $5 Dividend from the stock. At Expiration, if call is in the money (ST > 40) CT + 40 = ST if call is out of the money (ST < 40) CT + 40 = 40 Which is the better investment A or B ? (Synthetic is better: it costs less and has better future outcomes) Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices Arbitrage Strategy: {-S, +C, +B(FV=5, T=3 months), +B(FV=40, T=1yr) } CF0 = +50 - 6 - 32 - 4.71 = +$7.29 In 3 months, use $5 from the first bond to cover Dividend on short position. At Expiration, if ST ≥ 40, Bond matures, receive $40. Call is in the money; use $40 (from the bond) plus the call to buy stock. Use stock to cover short position. CFT= 0. if ST < 40, Bond matures, receive $40. Call is worthless. Use some of the $40 from the bond to buy stock and cover the short. CFT = 40 - ST > 0. Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices Thus, we have an arbitrage opportunity. Receive a CF0 = $7.29 now. If S↑, lose nothing! If S ↓, make even more money (40 - ST). To preclude the arbitrage C0E must be at least 6 + 7.29 = $13.29. (Exercise: Assume a $10 Div. in 3 months. Show that C0E > $8.59). In sum: if S0 = $50, C0E($40, T=1yr), r = 25%, plus a dividend in 3 months No Div $5 Div $10 Div C0E ≥ $18 $13.29 $8.59 Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices American Call Price C0A, no Dividends (p. 4) Do these prices make sense? S0 = $50, C0A(K = $40, T=1yr) = $5 There are two ways to buy the stock: Pay $50 and buy the stock immediately. Or Buy the Call for $5, exercise immediately, Pay only $45 Smell Arbitrage? Buy the Call for $5, pay $40 to exercise call, sell the stock for + $50, CF0 = -5 - 40 + 50 =+5. ➟ Arb. profit = $5 To preclude arbitrage we must have: C0A + 40 > 50; i.e., C0A > $10; Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices American Call C0A, Stock Pays a SMALL Dividend (p. 5) If the stock paid a dividend, you would want to exercise in order to collect the dividend. Yes or No? Suppose: S0 = $50, C0A(K = $40, T=1yr) = $11 Div = $5 (for sure) in 3 months(t*), r = 25%/yr ⇒ r for 3 months = 25%/4 = 6¼%. Must consider: 1) exercise before dividend is paid out 2) forgo dividend, wait till expiration Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices 1. Create a Synthetic Stock Position for 3 months. (Exercise just before dividend is paid) Strategy: {+C, +B(FV = $40, t* = 3 months), -S}. CF0 = - C0 - PV(K) + S0 = -11 - 40/1.0625 + 50 = $1.35 > 0 ☺ At t* = 3 months, just before ex-dividend day if S3 > $40 Call is in the money. Use $40 from bond to exercise call, receive stock, use it to cover short position before dividend is paid. CF3= 0. If S3 < $40 Call out money ∴ do not exercise. Bond matures for $40; use some of it to buy stock and cover short . CF3=40-S3 > 0, and you still own the call! Clearly, this is an Arb. opp. ⇒ C0 must be > $11. C0A must be > S0 - PV(K,t*) = $12.35 = 11 + 1.35. Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices 2. Create a Synthetic Stock Position for 1 year. (Do not Exercise, give up the dividend) Synthetic Stock={+C, +B (FV =$5, t*=3 months), +B(FV =$40, T =1yr)}, Synthetic Stock Price = { C0 + D/(1+r/4) + 40/1.25 } = 11 + 4.71 + 32 = $47.71 Actual Stock Price = S0 = $50 ⇒ Arb. opp. Set up an arbitrage: {Buy Synthetic Stock, Short the actual (i.e., physical) stock} {-S, +C, +Bond(FV=5, t*=3 months) , +Bond(FV=40, T=1) }. ➟ Net CF0 = +$50 - (11 + 4.71 +32) = $2.29 Will it work? Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices In 3 months, use the 1st bond (=$5) to cover dividend on short stock. At T = 1 yr (= Expiration) if ST > 40 Call in the money, receive $40 from 2nd bond, use it to exercise call, receive stock, cover short. Net CFT = 0. if ST < 40 Call out of the money, throw it away. Receive $40 from bond, use some of it to buy back stock and cover short. CFT = 40 - ST > 0! To preclude Arb. C0A > $13.29 (=11+2.29) Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices What do we learn? Go back to original question: “Is it a good idea to exercise just to receive the dividend?” If you exercise right before the dividend payment, C0A = $12.35, If you DO NOT plan to exercise in 3 months, C0A = $13.29, It seems that the option is worth more if we forgo the dividend. Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices American Call C0A, Stock Pays a LARGE Dividend (p. 6) Stock pays a $10 Dividend (for sure) in 3 months (time t*). Again, consider: 1) Exercise before dividend is paid out, or 2) Wait till expiration 1. Create a Synthetic Stock Position for 3 months. (Exercise before dividend is paid) The synthetic position in the stock for 3 months consists of: {+C, +B (FV = $40, t = 3months)} = { C0 + 40/(1.0625) } = C0A + $37.65 The real stock costs $50. ➟ C0A > $12.35 ( = 50-37.65) Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices 2) Create a Synthetic Stock Position for 1 year. Synthetic stock for 1 yr consists of: {+C, +B(FV=$10, t=3months), +B(FV=$40, T=1)} = C0 + $10/(1+r/4) + 40/1.25 = C0 + $41.41 ➟ C0A > 50 - 41.41 = $8.59 What is the math telling us? If you plan to exercise in 3 months, C0A = $12.35, If you plan to hold call for 1 yr, C0A = $8.59. Implication: If the dividend is large, then we should Exercise right before dividend is paid Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices The right to exercise at any time: How much is it worth? (p. 7) Asume S0 = $50, K=$40, T=1 year, Dividend in 3 months, and r = 25%. No Div $5 Div $10 Div C0E > $18 $13.29 $ 8.59 C0A > $18 $13.29 $12.35 $0 $0 $ 3.76 Right to Early exercise Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices Put Option Prices European Puts on stocks that pay NO dividends. (p. 7) Do these prices make sense? S0 = $75, P0E(K = $100, T=1yr) = $4, and r = 25% There are two ways to buy a bond: A: {Buy the stock and the put} and wait until expiration or B: {Buy the bond right now}, {+S, +P} ➟ CF0 = -75-4 = -$79 {+B(FV=$100, T=1yr)} ➟ CF0 = -$80 Arb Strategy: {+S, +P, -B(FV=100, T=1yr)}. CF0 = -75-4+80 = +$1 Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices At Expiration, if ST ≥ 100 Put is worthless. Sell stock, use some of this cash to pay loan. CFT = ST - K > 0. ST < 100 Put is in the money, exercise it. Hand over stock; receive $100, cover loan. CFT =0 To preclude arbitrage PE0 > $5 (=4+1). In general, PE0 > max(0, PV(K) - S0) Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices European PUT prices for stocks that pay dividends (p. 7) Assume a $5 Dividend in 3 months. Do these prices make sense? S0 = $75, $5 Div, P0E(K = $100, T=1yr) = $6, and r = 25% A synthetic position in a 1-year bond consists of: {+S, +P, -B(FV=$5, t=3 months) } ➟ Synthetic bond costs: $76.29 (= -6 - 75 + 5/1.0625) The actual bond costs: 100/1.25 = $80 Is this possible? ➟ There must be an arb. opp. Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices Buy cheap: synthetic bond , and Sell the expensive one, (actual or physical). Arb Portfolio ST < 100 ST > 100 CF0 + Put -(ST -100) 0 -6 + Stock + ST + ST -75 - B for Div* -5* -5* +4.71 - B for K -100 -100 +80 0 + $3.71 Arbitrage-free Price: PE0 > 6 + 3.71 = $9.71 *in 3 months, receive $5 div, pay off first bond. Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices If the Dividend is $10 Dividend in 3 months, then PE0 ≥ -75 + 10/1.0625 + 100/1.25 = $14.41 Summary: If S0 = $75, K = $100, T=1 year, and r = 25% and Dividend in 3 months. Then, No Div $5 Div $10 Div PE0 ≥ $5 $9.71 $14.41 P0A ≥ $25 $25 $28.53 $20 $15.29 $14.12 Right to Early exercise ▸ For an American PUT, the right to early exercise is worth quite a bit. ▸ For an American CALL if S0 = $50, K=$40, T=1 year, r = 25%. No Div $5 Div $10 Div C0E ≥ $18 $13.29 $8.59 C0A ≥ $18 $13.29 $12.35 $0 $0 $3.76 Right to Early Exercise Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices Put-Call Parity (p. 11) European Options on stocks that pay no dividends Proposition: For European Options on a stock that pays no dividends (Call and Put with same K and T), +S, +P = +C, +B(FV=K,T) And By the law of one price: +C0 = + S0 + P0 - B(FV=K,T) - C0 = - S0 - P0 + B(FV=K,T) +P0 = - S0 + C0 + B(FV=K,T) - P0 = + S0 - C0 - B(FV=K,T), etc. Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices Put-Call Parity (p. 11) European Options on stocks that pay dividends +C0 + B(FV=K,T) + B(FV=Dividend, t*) = +S0 + P0 American Options on stocks with/without Dividends +C0 + B0(FV=K,T) ≤ +S0 + P0 Lec 10 Discover Option Prices dfdf

Lec 10 Discover Option Prices Thank You (A Favara) Lec 10 Discover Option Prices dfdf