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Options II

# Options II - Econ 174 FINANCIAL RISK MANAGEMENT LECTURE...

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Econ 174 – FINANCIAL RISK MANAGEMENT LECTURE NOTES Foster, UCSD October 16, 2011 OPTIONS II – OPTION PRICING THEORY A. Simple Patterns in Option Prices 1. Preliminaries and Notation: a) The value of an option at time t is the premium C t or P t at which it can be bought or sold. b) Because option holders can’t be forced to exercise at a loss, C t and P t are never negative: Both C and P ≥ \$0 c) The premium value has two components. 1) Intrinsic value (“exercise value”) = max {immediate payoff, 0}. Intrinsic value of call S t - K > 0 or 0 Intrinsic value of put K - S t > 0 or 0 2) Time value (“speculative value”) = total value (P t or C t ) − intrinsic value. d) Examples. [Table 1] 1) July \$21 call with C t = \$1.68. Immediate payoff = S t - K = \$21.24 - \$21 = \$0.24 > 0, so this option is in the money intrinsic value = \$0.24; time value = C t - \$0.24 = \$1.44 2) April \$19 put with P t = \$0.50. immediate payoff = K - S t = \$19 - 21.24 < \$0, so this option is out of the money intrinsic value = \$0; time value = P t - \$0 = \$0.50 3) Note that, ceteris paribus : Both P and C are higher for options with longer-term T. For example, the \$23 puts all have the same intrinsic value K – S 0 = 23 - 21.24 = \$1.76, so the greater time value must be due to the longer-term. C is lower and P is higher for options with bigger K. 2. Principles of Portfolio Replication: Stock Option Notation (for 0 t T) t = 0…τ… T Now, early exercise date, expiration date T-t Time remaining to maturity at time t (yrs) S t Spot (stock market) price of stock (\$/share) K Strike (exercise) price (\$/share) C, P American put/call option premium (\$/share) c, p European put/call option premium (\$/share) C t (T), etc. Premium at time t of option expiring at T D PV of known dividends (discount at r f ) Table 1. INTEL STOCK OPTIONS (Jan 14, 2010) S 0 = \$21.24 Calls Puts K FE B APR JUL FEB APR JUL \$19 \$21 \$23 \$2.3 1 0.90 0.21 \$2.5 7 1.28 0.49 \$2.9 6 1.68 0.82 \$0.1 8 0.81 2.15 \$0.5 0 1.20 2.85 \$0.9 2 1.66 2.90

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Ec 174 OPTIONS II p. 2 of 17 a) These variations on the “Law of One Price” are used heavily in the arbitrage arguments underlying most option pricing theory. b) Consider portfolios A and B and future states of the world i = 1…m: Portfolio A costs A 0 to obtain today and will be worth {A(i) T , i = 1…m} at time T Portfolio B costs B 0 to obtain today and will be worth {B(i) T , i = 1…m} at time T c) From these, we can deduce the following principles: #1 If the distribution of values {A(i) T } {B(i) T }, then the current values A 0 = B 0 , or else an arbitrage opportunity exists. #2 If all values in the distribution in {A(i) T } ≥ values in the distribution in {B(i) T }, then A 0 B 0 , or else an arbitrage opportunity exists. #3 If the value of portfolio A = A T in all states i = 1…m, then A 0 = A T e −rT with certainty, or else an arbitrage opportunity exists. 1 3. Factors Affecting Option Value: [Table 2] a) The effect of stock price (S t ) and strike price (K) on intrinsic value.
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