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Model-Independent Option Valuation Techniques
- 1. Model-Independent Option Valuation Dr. Kurt Smith Overview Introduction Boundaries Vertical Spread Butterfly Spread Calendar Spread Conclusion Model-Independent Option Valuation; Dr. Kurt Smith Finance (Derivative Securities) Finance (Derivative Securities) Introduction Model-Independent Option Valuation; Dr. Kurt Smith 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
- 2. 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. • Finance (Derivative Securities) Introduction Model-Independent Option Valuation; Dr. Kurt Smith Expiry Payoff ST K European Call Expiry payoff = MAX(ST – K, 0)=(ST – K)+ 100 120 70 ST=120, K=100; then (ST – K)+=20 ST=70, K=100; then (ST – K)+=0 Examples: Expiry Payoff ST K 55 110 30 European Put Expiry payoff = MAX(K – ST, 0)=(K – ST)+ ST=110, K=55; then (K – ST)+=0 ST=30, K=55; then (K – ST )+=25 Examples:
- 3. Finance (Derivative Securities) Introduction Model-Independent Option Valuation; Dr. Kurt Smith Long Call Long Put Short Put Short Call Finance (Derivative Securities) Boundaries Model-Independent Option Valuation; Dr. Kurt Smith Finance (Derivative Securities) Boundaries Model-Independent Option Valuation; Dr. Kurt Smith Contract maturity (Tj). Strike price (Ki). Spot price (S0). American exercise. For European Call options Ci,j and European Put options Pi,j. Finance (Derivative Securities)
- 4. Boundaries Model-Independent Option Valuation; Dr. Kurt Smith Contract maturity (Tj): At expiry: Before expiry: • • At the limit: • t=0 T S0 Ki T=0 S0,Ki t=0 T=∞ S0 Ki Call Price S0 Ki Finance (Derivative Securities)
- 5. Boundaries Model-Independent Option Valuation; Dr. Kurt Smith Strike price (Ki): Zero: At the limit: • • Call Price S0 Ki Finance (Derivative Securities) Boundaries Model-Independent Option Valuation; Dr. Kurt Smith Spot price (S0): Zero: • Call Price S0 Ki • At the limit:
- 6. 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. Finance (Derivative Securities) Boundaries Model-Independent Option Valuation; Dr. Kurt Smith Why is the value of a Call option non-negative (Ci,j ≥ 0) whereas the value of a forward contract Ft can be negative? A Call option expiry payoff (ST - Ki)+ ≥ 0. Since there is no possibility of loss at T, the option value at t Ci,j ≥ 0. In contrast, the expiry payoff of a forward contract (ST - ft;S,T) is positive, negative, or zero. Expiry Payoff ST Ki Call Option Expiry Payoff ST Forward Contract 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
- 7. forward contract created by options must have the same value. Expiry Payoff ST f(t;S,T)=K Buy at K thru long Call if ST > K Buy at K thru short Put if ST < K Finance (Derivative Securities) Boundaries Model-Independent Option Valuation; Dr. Kurt Smith Put options: Maturity: Strike: • • Spot: •
- 8. American: • Put Price S0 Ki Ki Finance (Derivative Securities) Vertical Spread Model-Independent Option Valuation; Dr. Kurt Smith Finance (Derivative Securities) Vertical Spread Model-Independent Option Valuation; Dr. Kurt Smith Bull spread: different strikes (Ki), same maturity (Tj). Also r=0 & div=0. Expiry Payoff Expiry Payoff ST ST K1 K2 In general: Example: That is, cannot pay a negative amount today for a future payoff that at worst is zero. Finance (Derivative Securities)
- 9. Vertical Spread Model-Independent Option Valuation; Dr. Kurt SmithHow?Now tPayoff at Expiry TPortfoliotST < 5050 ≤ ST ≤ 55ST > 55Buy K2 = 551200ST – 55Sell K1 = 50-180-(ST – 50)-(ST – 50)Sub- Total-6050 – ST -5Lend Cash6≥ 6≥ 6≥ 6Total0> 0> 0> 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(K1=50)=$18 and C(K2=55)=$12. Is there an arbitrage? If so, how would you exploit it? Sell the bull spread. Why? Finance (Derivative Securities) Vertical Spread Model-Independent Option Valuation; Dr. Kurt Smith Bear spread: different strikes (Ki), same maturity (Tj). Also r=0 & div=0. Expiry Payoff Expiry Payoff ST ST K1 K2 In general: Example: That is, cannot pay a negative amount today for a future payoff that at worst is zero.
- 10. Finance (Derivative Securities) Butterfly Spread Model-Independent Option Valuation; Dr. Kurt Smith Finance (Derivative Securities) Butterfly Spread Model-Independent Option Valuation; Dr. Kurt Smith Expiry Payoff Expiry Payoff ST ST K1 K2 K3 If K2 - K1 = K3 - K2 then C(K1) – 2C(K2) + C(K3) must have a value greater than zero. In general: Example: Butterfly spread: different strikes (Ki), same maturity (Tj). Also r=0 & div=0. That is, cannot pay a negative amount today for a future payoff that at worst is zero. Finance (Derivative Securities) Butterfly Spread Model-Independent Option Valuation; Dr. Kurt Smith
- 11. 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) 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(K1=70), sell 2 Call(K2=80), buy Call(K3=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). Finance (Derivative Securities) Calendar Spread Model-Independent Option Valuation; Dr. Kurt Smith Finance (Derivative Securities) Calendar Spread Model-Independent Option Valuation; Dr. Kurt Smith Expiry Payoff ST Call Price S0
- 12. Calendar spread: same strikes (Ki), different maturities (Tj). 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. Finance (Derivative Securities) Calendar Spread Model-Independent Option Valuation; Dr. Kurt Smith Example: the price of a Call option expiring at T1 is $5 and T2 is $4, where T1 < T2. Is there an arbitrage? If so, how would you exploit it? Expiry Payoff at T2ST2 < KST2 > KNowExpiry Payoff at T1PortfoliotST1 < KST1 > KST1 < KST1 > KSell C(T1)-50-(ST2-K)0-(ST2-K)Buy C(T2)400ST2-KST2-KTotal- 10K-ST2ST2-K0 The trader receives $1 today (t) for non-negative expiry payoffs at T2 . This is a Type 2 arbitrage violation. Sell near (T1) and buy far (T2) maturity to extract the arbitrage profit. Finance (Derivative Securities) Conclusion Model-Independent Option Valuation; Dr. Kurt Smith Finance (Derivative Securities) Conclusion Model-independent means value relationships between different
- 13. 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: Finance (Derivative Securities) Conclusion Model-Independent Option Valuation; Dr. Kurt Smith Expiry Payoff Expiry Payoff ST ST K1 K2 Expiry Payoff Expiry Payoff ST ST
- 14. K1 K2 Vertical Spread Bull Spread Bear Spread Finance (Derivative Securities) Conclusion Model-Independent Option Valuation; Dr. Kurt Smith Expiry Payoff Expiry Payoff ST ST K1 K2 K3 Butterfly Spread Finance (Derivative Securities) Conclusion Model-Independent Option Valuation; Dr. Kurt Smith Call Price S0
- 24. -- -+³ -- ( ) ( ) ( ) 31 21 123 3232 0 KK KK CKCKCK KKKK - - -+³ -- 00.511.522.5507090110130150Expiry PayoffExpiry Spot K=70, 72, 90 024681012507090110130150Expiry PayoffExpiry Spot K=70, 80, 90 02468101214161820507090110130150Expiry PayoffExpiry Spot K=70, 88, 90 ( ) ( ) 90708070 7641 90809080
- 27. ASSIGNMENT 2 Note: a variant of these questions was in a previous final exam paper. Students are referred to the unit outline which provides details on the assessments, including the due date and time which will be strictly enforced. Students are reminded that their assignment should be their own work. Both questions must be answered. Answers are mathematical and hence a marking rubric is not required. Questions Let ( )C K denote a European vanilla Call option with strike price K . Assume that all options are identical except for strike price, and strike prices satisfy 1 2 3K K K< < and 2 1 32K K K= + . Assume also that interest rates are zero. Question 1 [5 marks] What are the no-arbitrage lower bound, and the no-arbitrage upper bound, of the vertical spread ( ) ( )1 2C K C K− ? Question 2 [10 marks] What is the functional relationship between the no-arbitrage values of the two vertical spreads, ( ) ( )1 2C K C K− and ( ) ( )2 3C K C K− ?