Lecture 12. Stock Option boundary conditions. Agenda:

Transcription

1 Lecture 12 Stock Option boundary conditions Agenda: I. Option boundary conditions: ~ Option boundary conditions based on arbitrage force ~ American call options without dividend ~ American put options without dividend II. Effects of dividends, market frictions, and exercise price

2 I. Option boundary conditions ~ Option boundary conditions (based on arbitrage force): Two portfolios with the same payoff pattern must have the same price. Assumptions: Market is frictionless Borrow or lend at risk free rate Tax rate are fixed Call and put options have the same K and T, and traded at the same time Notation: S 0, K, T, S T, r, C, P, c, p Upper bounds for option premiums: ~ Upper bounds for call options: c C S 0 Intuition: Stock is like an option with infinite time to maturity. Arbitrage force: buy stock sell options. ~ Upper bounds for put options: p P K Intuition: a put maximum value at maturity is K, p Ke -rt Arbitrage: Sell put and lend Ke -rt at the risk-free rate. Lower bound for European calls without dividends: c S 0 Ke -rt c max(s 0 Ke -rt, 0) Intuition: c + Ke -rt max (S T, K) at maturity Portfolio 1 S 0 S T at maturity Portfolio 2 Arbitrage: buy a call, lend Ke -rt, short a stock S 0 =51, K=50, T=0.5, r=12% then c

3 Lower bound for European puts without dividends: p Ke -rt S 0 p max(ke -rt S 0, 0) Intuition: p+s 0 max (S T, K) at maturity Portfolio 3 Ke -rt K at maturity Portfolio 4 Arbitrage: buy a put, long a stock, borrow Ke -rt S 0 =51, K=50, T=0.5, r=12% then p?? Put-call parity for European options: c- p =S 0 - Ke -rt Intuition: Portfolio 1 = max (S T, K) = Portfolio 3 c + Ke -rt = p+s 0 Arbitrage: Buy low value portfolio and sell high value portfolio Put-call inequality for American options: S 0 K C P S 0 Ke -rt Intuition: S 0 K + P C C + K S 0 + P P C (S 0 K) C S 0 Ke -rt + P C (S 0 Ke -rt ) P C + Ke -rt S 0 + P Arbitrage: Buy low value portfolio and sell high value portfolio 2

4 ~ Boundary for American call options without dividend: The American call option should not be exercised early if there is no dividend. C c S 0 Ke -rt > S 0 K Intuition: American call premium must be greater than the intrinsic value because of time value and insurance from the option. S0=$50 K=$40 Holding period = 1 month A. Hold the American call for 1 month and exercise it one month later B. Exercise the American call right now and hold the stock for one month. A B S T >K Ke rt + S T -40 S T S T < K Ke rt S T If stock price is expected to drop, investors are better off selling the option than exercising it: Sell the American call at market price: (50-40) + Time value Exercise the option: C > S 0 K Call K S 0 r, T, and volatility affect the shape. ATM has the largest time value, DITM or DOTM has the lowest time value. 3

5 ~ Boundary American put options without dividend: The American option may be exercised early anytime before maturity. A put option should be exercised early if it is sufficiently deep in the money. K=10, S 0 =0 Maximum gain is $10. If investors exercise a put early, they can earn interest on $10. The early exercise of a put becomes more attractive if S drops deeper and deeper r increases volatility decreases P K S 0 (In comparison with p: p Ke -rt S 0 ) Intuition: American put premiums must be no less than its intrinsic value. Sometimes American put is equal to its intrinsic value. Therefore, European put could be lower than its intrinsic value. Put European put American put A Ke -rt K S 0 B American put = intrinsic value (if S is sufficiently low) Ke -rt S 0 European put < intrinsic value (if S is sufficiently low) B > A 4

6 II. Effects of dividends, market frictions, and exercise price ~ Effect of dividends (present value of expected future dividends): Boundary conditions: c S 0 d Ke -rt p d + Ke -rt - S 0 ~ Replace S 0 by (S 0 d). d is the present value of the dividends during the life of options. Early exercise for American call options: Sometimes it is optimal to exercise an American call immediately prior to an exdividend date. Put-call parity for European options: c + Ke -rt = p+s 0 d Put-call inequality for American options: S 0 d K C P S 0 Ke -rt 5

7 ~ Effect of market frictions (transaction costs, bid-ask spread in option and underlying asset markets): Cash Flow time=0 The Boundary Condition for the European Call Option With Market Frictions Cash Flow time=t Portfolio S a X-T X,P X-T X,P S a X+T X,C X+T X,C S b 1: Buy Call -(C a +T C) 0 0 S b -X- T X,C 2: Buy Put -(P a +T P) X- S a -T X,P 0 0 Lend one unit of -S 0 a e -rf, b t S b foreign currency S b S b Borrow domestic X e -rd, a t T X,P -T S X+(T X,P +T S) e rd, a t X+(T X,P +T S) e rd, a t X+(T X,P +T S) e rd, a t currency Value of portfolio 2 -(P a +T P) -S 0 a e -rf, b t S b - S a + T S e rd, a t S b X+(T X,P +T S) e rd, a t S b X+(T X,P +T S) e rd, a t + X e -rd, a t T X,P -T S +T X,P(e rd, a t -1) 0 0 S b -X- T X,C Because the payoff of portfolio 2 is no less than that of a call option in all future states, the value of portfolio 2 shall be no less than a call. This upper boundary for a European call option is specified as equation: (P a + S a 0 e -rf, b t - X e -rd, a t ) + (T X,P +T S +T P ) C a + T C C a - T C C b - T C. The Boundary Condition for the European Put Option With Market Frictions Portfolio Cash Flow Cash Flow time=0 time=t S a X-T X,P X-T X,P S a X+T X,C X+T X,C S b 1: Buy Put -(P a +T P) X - S a - T X,P 0 0 2: Buy Call -(C a +T C) 0 0 S b -X - T X,C Borrow one unit of S 0 b e -rf, a t -S a foreign currency -S a -S a Lend domestic -(X e -rd, b t +T X,C +T S) X+(T X,C +T S) e rd, b t X+(T X,C +T S) e rd, b t X+(T X,C +T S) e rd, b t currency Value of portfolio 2: -(C a +T P)+ S 0 b e -rf, a t X+(T X,C +T S) e rd, b t -S a X+(T X,C +T S) e rd, b t -S a S b +(T X,C +T S) e rd, b t -S a -(X e -rd, b t +T X,C +T S) X - S a - T X,P 0 0 Again, because the payoff of portfolio 2 is no less than that of a put option in all future states, the value of portfolio 2 shall be no less than a put. This upper boundary for European put options is written as equation: (C a S b 0 e -rf, a t + X e -rd, b t ) + (T X,c +T S +T C ) P a + T P P a - T P P b - T P. 6

8 Cash Flow time=0 The Boundary Condition for the American Call Option With Market Frictions Cash Flow time =, 0< T Portfolio S a X-T X,P X-T X,P S a X+T X,C X+T X,C S b 1: Buy Call -(C a +T C) 0 0 S b -X- T X,C 2: Buy Put -(P a +T P) X- S a -T X,P Lend one unit of -S a foreign currency S b e rf, b S b e rf, b S b e rf, b Borrow domestic X e -rd, a T T X,P -T S X e -rd, a (T- ) X e -rd, a (T- ) -rd, a (T- ) X e Currency +(T X,P +T S) e rd, a +(T X,P +T S) e rd, a +(T X,P +T S) e rd, a Value of portfolio 2 -(P a 0 +T P) -S a X- S a -T X,P + S b e rf, b S b e rf, b S b e rf, b +X e -rd, a T T X,P -T S +(T X,P +T S) e rd, a X e -rd, a (T- ) X e -rd, a (T- ) -rd, a (T- ) X e 0 +(T X,P +T S) e rd, a 0 +(T X,P +T S) e rd, a S b -X- T X,C Since the cash flow generated from portfolio 2 is no less than a call option in all future states, the value of portfolio 2 shall be no less than the value of a call. This boundary is: (P a + S a 0 - X e -rd, a T )+ (T X,P +T S +T P ) C a + T C C a - T C C b - T C The Boundary Condition for the American Put Option With Market Frictions Portfolio Cash Flow Cash Flow time=0 time= 0< T S a X-T X,P X-T X,P S a X+T X,C X+T X,C S b 1: Buy Put -(P a +T P) X - S a - T X,P 0 0 2: Buy Call -(C a +T C) 0 0 S b -X - T X,C Borrow one unit of S 0 b e -rf, a T -S a e -rf, a (T- ) -S a e -rf, a (T- ) -S -rf, a (T- ) a e foreign currency Lend domestic -(X+T X,C +T S) (X+T X,C +T S) e rd, b (X+T X,C +T S) e rd, b (X+T X,C +T S) e rd, b currency Value of portfolio 2: -(C a +T P) - S 0 b e -rf, a T (X+T X,C +T S) e rd, b (X+T X,C +T S) e rd, b rd, b (X+T X,C +T S) e +X +T X,C +T S -S a e -rf, a (T- ) X-S a -T X,P -S a e -rf, a (T- ) 0 + S b -X - T X,C -S a e -rf, a (T- ) 0 Because the payoff of portfolio 2 is no less than a put option in all future states, the value of portfolio 2 shall be no less than the value of a put. This upper boundary for the American put is specified as: (C a S b 0 e -rf, a T + X) + (T X,c +T S +T C ) P a + T P P a - T P P b - T P. 7

9 ~ Effect of exercise price: if K 2 > K 1 (K 2 -K 1 )e -rt c(k 1 ) c(k 2 ) (K 2 -K 1 ) c(k 1 ) c(k 2 ) Intuition: The advantage of buying an option with a lower exercise price over the one with a higher exercise price will not be more than the difference in the exercise prices. Switch an option with K=$50 to an option with K=40, the payoff change is $10. (K 2 -K 1 )e -rt p(k 2 ) p(k 1 ) (K 2 -K 1 ) p(k 2 ) p(k 1 ) 8