Arbitrage Restrictions on Option Prices. Class Objectives. Example of a mispriced call option
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1 Arbitrage Restrictions on Option Prices Lecture notes for chapter 6 by Peter Ritchken 1 Class Objectives Look at arbitrage restrictions among option contracts Identify strategies that make money above the riskless rate without any risk. Learn how arbitrageurs work. Understand why prices have to lie within certain bounds. Learn about relative pricing. This is an essential way of thinking for traders. Lecture notes for chapter 6 by Peter Ritchken 2 Example of a mispriced call option C 0 = $6.00, X= 50, T= 3months, r= 12% S 0 = $55. The call is mispriced. It is cheap. I can make money without taking on any risk. There is a riskless arbitrage profit here. Lecture notes for chapter 6 by Peter Ritchken 3 1
2 C 0 = $6.00, X= 50, T= 3months, r= 12% S 0 = $55. The call is mispriced C 0 > $6.48 S(T) <X S(T)>X A:C(0)+XB(0,T) X S(T)-X+X B: S(0) S(T) S(T) C(0) >S(0)-XB(0,T) Lecture notes for chapter 6 by Peter Ritchken 4 C 0 = $6.00, X= 50, T= 3months, r= 12% S 0 = $55. The call is mispriced C 0 > $6.48 S(T) <X S(T)>X Buy A: C(0)+XB(0,T) = = Sell B: -S(0) = -55 X S(T) S(T)-X+X S(T) Net cash flow = 0.48 S(T)-X 0 Lecture notes for chapter 6 by Peter Ritchken 5 Bounds for Calls on dividend paying stocks. 0 t 1 T C(0) > Max[0, C(1), C(2), C(3)] C(1) = S(0) - X C(2) = S(0) - XB(0,t 1 ) C(3) = S(0) - B(0,t 1 )- XB(0,T) Lecture notes for chapter 6 by Peter Ritchken 6 2
3 Show C > C(3) Initial Cost S(T)<X S(T)>X A: 1 call; bonds.., t 1 ; X bonds, T C(0) + B(0, t 1 ) + XB(0,T) 0+ G(t 1,T)+ X B: 1 stock S(0) S(T)+ d1g(t1,t) S(T)-X+ G(t 1,T)+ X S(T)+ d1g(t1,t) Lecture notes for chapter 6 by Peter Ritchken 7 S(0) = 66.25;X=50;C(0) = 16.75;r=8.3% = 0.75; t 1 =58. B(0,t 1 ) = 0.74 B(0,T)X = Cash Flow S(T)<X S(T)>X Buy 1 call; bonds.., t 1 ; X bonds, T Sell 1 stock c S(T) 50-S(T)>0 S(T) -S(T) 0 Lecture notes for chapter 6 by Peter Ritchken 8 Early exercise on a non dividend paying stock is never optimal Exercise Option to receive S(0) - X Sell Option C(0)> S(0)- XB(0,T) Early exercise is never optimal. Early exercise feature is not valuable. American vs. Bermudan vs European Options. Lecture notes for chapter 6 by Peter Ritchken 9 3
4 Early exercise feature on dividend paying stock may be valuable. Stock pays a dividend. If the dividend is very large, then exercise the option before the ex-dividend date. American options are worth more than European options. 0 t 1 T Lecture notes for chapter 6 by Peter Ritchken 10 The only possible exercise dates for a call are at expiration or just prior to an ex dividend date. Don t exercise at date t(p). We know that C(t p ) > S(t p ) - XB(t p,t 1 ) The exercise value is: Exercise Value = S(t p ) - X t p t 1 T Lecture notes for chapter 6 by Peter Ritchken 11 Dividend and Income Yield Analysis Exercise (Stop) Value = S(t 1 ) -X Ex-dividend value of stock = S(t) - Go value > [S(t 1 )- ] - XB(t 1,T). Stop value is smaller than go value if: S(t 1 )-X <[S(t 1 )- ] - XB(t 1,T). Or <X[1-XB(t 1,T)]. Dividend is smaller than interest on strike over period [t 1,T]. Lecture notes for chapter 6 by Peter Ritchken 12 4
5 Put Options P > Max[0, P a,, P b ] P a = X-S(0) P b = (X+ )B(0,t 1 ) - S(0) Lecture notes for chapter 6 by Peter Ritchken 13 Exercise after t 1 S(t1+)<X S(t1+)>X A:Buy Put P(0) X- S(t1+) --- B: Buy (d1+x) Bonds.t 1. Sell stock [ +X]B(0,t 1 )- S(0) [d1+x] [S(t1+)+] X - S(t1+) V A = V B V A > V B Lecture notes for chapter 6 by Peter Ritchken 14 Arbitrage with Puts. S(0) = 55; T= 3mth; X=60; r=12%; =$2; t 1 =1mth. P(0) >Max[P a, P b ] P b = max[0, (X+ )B(0,t 1 )- S(0)] = max[0, 62exp[-0.12(1/12)] - 55] = Lecture notes for chapter 6 by Peter Ritchken 15 5
6 American feature for puts is valuable for a stock that pays NO dividends. Assume stock price drops dramatically below the strike price. What does waiting buy you? What are the costs of waiting? Lecture notes for chapter 6 by Peter Ritchken 16 American feature may be valuable for a stock that pays dividends. If the stock pays a large dividend, you may want to wait. Exercise value = X- S(t) Wait till after the ex-div date then P(t) >P b That is P(t) >[X+ ]B(t,t 1 ) - S(t) or > X[exp[r(t 1 -t)-1] If the dividend exceeds interest generated by the strike, then DON T exercise. As t goes to t 1, this will be true. Lecture notes for chapter 6 by Peter Ritchken 17 Put Call Parityfor European contracts - No dividends P = C + XB(0,T) - S(0) How would you prove this relationship. Lecture notes for chapter 6 by Peter Ritchken 18 6
![Exercise value = X- S(t) Wait till after the ex-div date then P(t) >P b That is P(t) >[X+ ]B(t,t 1 ) - S(t) or > X[exp[r(t 1 -t)-1] If the dividend exceeds interest generated by the strike, then DON](/docs-images/44/16334568/images/page_6.jpg "T exercise. As t goes to t 1, this will be true.")
7 Put Call parity with dividends. P = C + XB(0,T) - [S(0) - B0,t 1 )] Lets try and prove this result. Lecture notes for chapter 6 by Peter Ritchken 19 Strike Price Relationships S<X1 X 1<S<X 2 X 2<S<X3 S>X3 A:2C (S-X 2 ) 2(S-X 2 ) B: C 1 +C 3 0 S-X 1 S-X 1 S-X 1 + S-X3 A=B B>A B>A B= A Lecture notes for chapter 6 by Peter Ritchken 20 7
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