Notation
c: European call option price
p: European put option price
S0: Stock price today K: Strike price T: Life of option s: Volatility of stock
price
C: American call option price
ST > K ST − K
K ST 0 ST ST
ST < K 0 K K
K− ST ST K
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The Put-Call Parity Result (Equation
P: American put option price
ST: Stock price at option maturity
D: PV of dividends paid during life of option
r Risk-free rate for maturity T with cont. comp.
(Equation 10.5, page 221)
p Ke -rT–S0
Options, Futures, and Other Derivatives, 8th Edition,
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Put-Call Parity: No Dividends
Cc Pp
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Calls: An Arbitrage Opportunity?
Suppose that
c= 3 T=1 K = 18
S0 = 20 r = 10% D= 0
What are the arbitrage possibilities when
p = 2.25 ? p= 1 ?
Options, Futures, and Other Derivatives, 8th Edition,
c S0 –Ke -rT
Options, Futures, and Other Derivatives, 8th Edition,
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Puts: An Arbitrage Opportunity?
Suppose that
p= 1 T = 0.5 K = 40
S0 = 37 r =5% D =0
Is there an arbitrage opportunity?
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Lower Bound for European Put Prices; No Dividends
Options, Futures, and Other Derivatives, 8th Edition,
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Arbitrage Opportunities
Suppose that
c= 3 T = 0.25 K =30
S0= 31 r = 10% D= 0
10.6, page 222)
Both are worth max(ST , K ) at the maturity of the options They must therefore be worth the same today. This means that
c + Ke -rT = p + S0
Consider the following 2 portfolios: Portfolio A: European call on a stock + zerocoupon bond that pays K at time T Portfolio C: European put on the stock + the stock
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Effect of Variables on Option Pricing (Table 10.1, page 215)
Variable
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Values of Portfolios
Portfolio A Portfolio C
Call option Zero-coupon bond Total Put Option Share Total
Is there an arbitrage opportunity?
Options, Futures, and Other Derivatives, 8th Edition,
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Lower Bound for European Call Option Prices; No Dividends (Equation 10.4, page 220)
c
p
C
P
S0
+
−
+
−
K
−
+
−
+
T
?
?
+
+
s
+
+
+
+
r
+
−
+
−
D
−
+
−
+
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American vs European Options
An American option is worth at least as much as the corresponding European option