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Options Introduction Finance 30233, Fall 2011 Advanced Investments S. Mann The Neeley School at TCU Call and put option contracts Notation Definitions Graphical representations (payoff diagrams)

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Options Right, but not the obligation, to either buy or sell at a fixed price over a time period (t,T) Call option - right to buy at fixed price Put option - right to sell at fixed price fixed price (K) : strike price, exercise price (K = X in BKM) selling an option: write the option Notation: call value (stock price, time remaining, strike price) = c ( S(t), T-t, K) at expiration (T): c (S(T),0,K) = 0if S(T) < K S(T) - K if S(T) K or:c(S(T),0,K) = max (0,S(T) - K)

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"Moneyness" Kasset price (S) Call “moneyness” Out of the money in the money (S K) 0 Call value Put “moneyness” 0 in the moneyout of the money (S K) Kasset price (S) Put value

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Call value at maturity Value 5 0 K(K+5)S(T) Call value = max (0, S(T) - K) c (S(T),0,K) = 0 ; S(T) < K S(T) - K ; S(T) K

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Short position in Call: value at maturity Value 0 -5 K(K+5)S(T) Short call value = min (0, K -S(T)) c (S(T),0,K) = 0 ; S(T) < K S(T) - K ; S(T) K short is opposite: -c(S(T),0,K) = 0; S(T) < K -[S(T)-K]; S(T) K

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Call profit at maturity Value 0 KS(T) Profit = c(S(T),0,K) - c(S(t),T-t,K) Call value at T: c(S(T),0,K) = max(0,S(T)-K) Profit is value at maturity less initial price paid. Breakeven point Call profit

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Put value at maturity Value 5 0 (K-5) K S(T) Put value = max (0, K - S(T)) p(S(T),0,K) = K - S(T) ; S(T) K 0 ; S(T) > K

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Short put position: value at maturity Value 0 -5 (K-5) K S(T) Short put value = min (0, S(T)-K) p(S(T),0,K) = K - S(T) ; S(T) K 0 ; S(T) > K short is opposite: -p(S(T),0,K) = S(T) - K; S(T) K 0 ; S(T) > K

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Put profit at maturity Value 0 K S(T) Put value at T: p(S(T),0,K) = max(0,K-S(T)) put profit Profit = p(S(T),0,K) - p(S(t),T-t,K) Breakeven point Profit is value at maturity less initial price paid.

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Option values at maturity (payoffs) 0 K K K K 00 0 long call short put long put short call

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European Put-Call parity: Asset plus Put Asset K K Put Asset plus European put: S(0) + p[S(0),T;K] K KS(T) K

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European Put-Call parity: Bond plus Call Bond K K Call K Bond + European Call: c[S(0),T;K] + KB(0,T) K S(T) K 0

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European Put-Call parity European Put-Call parity: S(0) + p[S(0),T;K] = c[S(0),T;K] + KB(0,T) Value at expiration Positioncost nowS(T) KS(T) > K Portfolio A: Stock S(0) S(T)S(T) putp[S(0),T;K]K - S(T)0 total A:S + PKS (T) Portfolio B: Callc[S(0),T;K]0S(T) - K BillKB(0,T)KK total B:C + KB(0,T)KS(T)

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Bull Spread: value at maturity S(0) = $50value at maturity position:S(T) 45 45 S(T) 50 S(T) > 50 Long call with strike at $450S(T) - 45 S(T) -45 Short call w/ strike at $5000- [ S(T) - 50] net:0S(T) -45 5 4045505560 S(T) Position value at T 10 5 0

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Bear Spread: value at maturity S(0) = $30value at maturity position:S(T) 25 25 S(T) 35 S(T) >35 Long call with strike at $3500 S(T) -35 Short call w/ strike at $250-[S(T) - 25]- [ S(T) -25] net:025 - S(T) -10 2025303540 S(T) Position value at T 0 - 5 -10

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Butterfly Spread: value at maturity S(0) = $50value at maturity position: S(T) 45 45 S(T 50 50 S(T) 55 S(T) > 55 Long call, K= $450 S(T) - 45 S(T) - 45 S(T) - 45 Short 2 calls, K= $500 0 -2 [S(T) - 50]-2[S(T) - 50] Long call, K = $5500 0 S(T) - 55 net:0 S(T) -45 55 - S(T) 0 4045505560 S(T) Position value at T 10 5 0

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Straddle value at maturity S(0) = $25 value at maturity position:S(T) 25 S(T) > 25 Long call, K= $25 0 S(T) - 45 Long put, K= $2525 - S(T) 0 net:25 - S(T)S(T) - 25 1520253035 S(T) Position value at T 10 5 0 Bottom straddle: call strike > put strike: put K = 23; call K = 27 straddle Bottom straddle

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