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Model-Independent Option Valuation Dr. Kurt Smith

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Overview Introduction Boundaries Vertical Spread Butterfly Spread Calendar Spread Conclusion Model-Independent Option Valuation; Dr. Kurt Smith Finance (Derivative Securities)

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Introduction Model-Independent Option Valuation; Dr. Kurt Smith

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Finance (Derivative Securities) Introduction Model-independent means value relationships between different options on the same underlier that must hold to prevent arbitrage. These relationships must hold for every option pricing model. The absence of vertical spread, butterfly spread and calendar spread arbitrages is sufficient to exclude all static arbitrages from a set of option price quotes across strikes and maturities on a single underlier. Option buyers have the right, not the obligation, to exercise the option at expiry (European) or anytime up to and including expiry (American). Expiry payoff diagrams for options can be obtained via simple rotations about the x- and y-axis. Model-Independent Option Valuation; Dr. Kurt Smith The focus in this lecture is on a single underlier with zero intermediate cash flows (e.g., no dividends). For simplicity, interest rates are assumed to be zero unless stated otherwise.

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Finance (Derivative Securities) Introduction Model-Independent Option Valuation; Dr. Kurt Smith Expiry Payoff STST K European Call Expiry payoff = MAX(S T – K, 0)=(S T – K) + 10012070 S T =120, K=100; then (S T – K) + =20 S T =70, K=100; then (S T – K) + =0 Examples: Expiry Payoff STST K 5511030 European Put Expiry payoff = MAX(K – S T, 0)=(K – S T ) + S T =110, K=55; then (K – S T ) + =0 S T =30, K=55; then (K – S T ) + =25 Examples:

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Finance (Derivative Securities) Introduction Model-Independent Option Valuation; Dr. Kurt Smith Long CallLong Put Short PutShort Call

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Finance (Derivative Securities) Boundaries Model-Independent Option Valuation; Dr. Kurt Smith

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Finance (Derivative Securities) Boundaries Model-Independent Option Valuation; Dr. Kurt Smith Contract maturity (T j ). Strike price (K i ). Spot price (S 0 ). American exercise. For European Call options C i,j and European Put options P i,j.

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Finance (Derivative Securities) Boundaries Model-Independent Option Valuation; Dr. Kurt Smith Contract maturity (T j ): At expiry: Before expiry: At the limit: t=0T S0S0 KiKi T=0 S 0,K i t=0T=∞ S0S0 KiKi Call Price S0S0 KiKi

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Finance (Derivative Securities) Boundaries Model-Independent Option Valuation; Dr. Kurt Smith Strike price (K i ): Zero: At the limit: Call Price S0S0 KiKi

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Finance (Derivative Securities) Boundaries Model-Independent Option Valuation; Dr. Kurt Smith Spot price (S 0 ): Zero: Call Price S0S0 KiKi At the limit:

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Finance (Derivative Securities) Boundaries Model-Independent Option Valuation; Dr. Kurt Smith American exercise (Amex.): An American option has all of the features of a European option PLUS the ability to exercise early if it is in the buyer’s interest. Therefore, an American option cannot be worth less than a European option.

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Finance (Derivative Securities) Boundaries Model-Independent Option Valuation; Dr. Kurt Smith Why is the value of a Call option non-negative (C i,j ≥ 0) whereas the value of a forward contract F t can be negative? A Call option expiry payoff (S T - K i ) + ≥ 0. Since there is no possibility of loss at T, the option value at t C i,j ≥ 0. In contrast, the expiry payoff of a forward contract (S T - f t;S,T ) is positive, negative, or zero. Expiry Payoff STST KiKi Call Option Expiry Payoff STST Forward Contract

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Finance (Derivative Securities) Boundaries Model-Independent Option Valuation; Dr. Kurt Smith European Put-Call parity for an underlier with no interim cash flows (e.g., no dividends): a forward contract and a synthetic forward contract created by options must have the same value. Expiry Payoff STST f(t;S,T)=K Buy at K thru long Call if S T > K Buy at K thru short Put if S T < K

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Finance (Derivative Securities) Boundaries Model-Independent Option Valuation; Dr. Kurt Smith Put options: Maturity: Strike: Spot: American: Put Price S0S0 KiKi KiKi

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Finance (Derivative Securities) Vertical Spread Model-Independent Option Valuation; Dr. Kurt Smith

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Finance (Derivative Securities) Vertical Spread Model-Independent Option Valuation; Dr. Kurt Smith Bull spread: different strikes (K i ), same maturity (T j ). Also r=0 & div=0. Expiry Payoff STST STST K1K1 K2K2 In general: Example: That is, cannot pay a negative amount today for a future payoff that at worst is zero.

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Finance (Derivative Securities) Vertical Spread Model-Independent Option Valuation; Dr. Kurt Smith How?Now tPayoff at Expiry T PortfoliotS T < 5050 ≤ S T ≤ 55S T > 55 Buy K 2 = 551200S T – 55 Sell K 1 = 50-180-(S T – 50) Sub-Total-6050 – S T -5 Lend Cash6≥ 6 Total0> 0 Therefore, pay zero today to get a guaranteed positive payoff in the future (Type 3 arbitrage violation). The trader will do this as many times as possible to pay a multiple of zero today to earn a multiple of a positive amount in the future. Example: Let C(K 1 =50)=$18 and C(K 2 =55)=$12. Is there an arbitrage? If so, how would you exploit it? Sell the bull spread. Why?

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Finance (Derivative Securities) Vertical Spread Model-Independent Option Valuation; Dr. Kurt Smith Bear spread: different strikes (K i ), same maturity (T j ). Also r=0 & div=0. Expiry Payoff STST STST K1K1 K2K2 In general: Example: That is, cannot pay a negative amount today for a future payoff that at worst is zero.

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Finance (Derivative Securities) Butterfly Spread Model-Independent Option Valuation; Dr. Kurt Smith

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Finance (Derivative Securities) Butterfly Spread Model-Independent Option Valuation; Dr. Kurt Smith Expiry Payoff STST STST K1K1 K2K2 K3K3 If K 2 - K 1 = K 3 - K 2 then C(K 1 ) – 2C(K 2 ) + C(K 3 ) must have a value greater than zero. In general: Example: Butterfly spread: different strikes (K i ), same maturity (T j ). Also r=0 & div=0. That is, cannot pay a negative amount today for a future payoff that at worst is zero.

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Finance (Derivative Securities) Butterfly Spread Model-Independent Option Valuation; Dr. Kurt Smith Asymmetric butterflies Symmetric butterfly C(K=70)-1.11C(K=72)+0.11C(K=90)C(K=70)-10C(K=88)+9C(K=90) C(K=70)-2C(K=80)+C(K=90)

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Finance (Derivative Securities) Butterfly Spread Model-Independent Option Valuation; Dr. Kurt Smith Example: Let C(K=70)=$7, C(K=80)=$6 and C(K=90)=$4. Is there an arbitrage? If so, how would you exploit it? Hence, yes there is an arbitrage. Buy Call(K 1 =70), sell 2 Call(K 2 =80), buy Call(K 3 =90). The trader will receive $1 now [i.e., at t=0 will pay 7-2(6)+4=-$1 ]; and will have zero probability of loss in the future (refer to expiry payoff figure).

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Finance (Derivative Securities) Calendar Spread Model-Independent Option Valuation; Dr. Kurt Smith

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Finance (Derivative Securities) Calendar Spread Model-Independent Option Valuation; Dr. Kurt Smith Expiry PayoffSTST Call PriceS0S0 Calendar spread: same strikes (K i ), different maturities (T j ). Also r=0 & div=0. In general: Example: That is, cannot pay a negative amount today for a future payoff that at worst is zero.

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Finance (Derivative Securities) Calendar Spread Model-Independent Option Valuation; Dr. Kurt Smith Example: the price of a Call option expiring at T 1 is $5 and T 2 is $4, where T 1 < T 2. Is there an arbitrage? If so, how would you exploit it? Expiry Payoff at T 2 S T 2 < KS T 2 > K NowExpiry Payoff at T 1 PortfoliotS T 1 < KS T 1 > KS T 1 < KS T 1 > K Sell C(T 1 )-50-(S T 2 -K)0 Buy C(T 2 )400S T 2 -K Total0K-S T 2 S T 2 -K0 The trader receives $1 today (t) for non-negative expiry payoffs at T 2. This is a Type 2 arbitrage violation. Sell near (T 1 ) and buy far (T 2 ) maturity to extract the arbitrage profit.

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Finance (Derivative Securities) Conclusion Model-Independent Option Valuation; Dr. Kurt Smith

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Finance (Derivative Securities) Conclusion Model-independent means value relationships between different options on the same underlier that must hold to prevent arbitrage. These relationships must hold for every option pricing model. The absence of vertical spread, butterfly spread and calendar spread arbitrages is sufficient to exclude all static arbitrages from a set of option price quotes across strikes and maturities on a single underlier. Option buyers have the right, not the obligation, to exercise the option at expiry (European) or anytime up to and including expiry (American). Expiry payoff diagrams for options can be obtained via simple rotations about the x- and y-axis. Model-Independent Option Valuation; Dr. Kurt Smith Vertical Spread:Butterfly Spread:Calendar Spread:

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Finance (Derivative Securities) Conclusion Model-Independent Option Valuation; Dr. Kurt Smith Expiry Payoff STST STST K1K1 K2K2 STST STST K1K1 K2K2 Vertical Spread Bull SpreadBear Spread

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Finance (Derivative Securities) Conclusion Model-Independent Option Valuation; Dr. Kurt Smith Expiry Payoff STST STST K1K1 K2K2 K3K3 Butterfly Spread

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Finance (Derivative Securities) Conclusion Model-Independent Option Valuation; Dr. Kurt Smith Call PriceS0S0 Calendar Spread

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