Exam MFE
Raise Your Odds® with Adapt
PUT-‐CALL PPUT-CALL ARITY (PCP) PARITY (PCP) Prepaid Forward & Forward ' 𝑆𝑆 ⋅ 𝑒𝑒 *($,") 𝐹𝐹",$ 𝑆𝑆 = 𝐹𝐹",$ ' 𝐹𝐹",$ (𝑆𝑆) Dividend Structure No 𝑆𝑆" Discrete 𝑆𝑆" − PV(Divs) Continuous 𝑆𝑆" 𝑒𝑒 ,5($,") 𝐹𝐹",$ (𝑆𝑆) Dividend Structure No 𝑆𝑆" 𝑒𝑒 *($,") Discrete 𝑆𝑆" 𝑒𝑒 *($,") − AV(Divs) Continuous 𝑆𝑆" 𝑒𝑒 (*,5)($,") PCP for Stock ' 𝐶𝐶 − 𝑃𝑃 = 𝐹𝐹",$ 𝑆𝑆 − 𝐾𝐾𝑒𝑒 ,*($,") PCP for Exchange Option 𝐶𝐶 𝐴𝐴, 𝐵𝐵 𝑃𝑃 𝐴𝐴, 𝐵𝐵 receive 𝐴𝐴, give up 𝐵𝐵 give up 𝐴𝐴, receive 𝐵𝐵 ' ' 𝐶𝐶(𝐴𝐴, 𝐵𝐵) − 𝑃𝑃(𝐴𝐴, 𝐵𝐵) = 𝐹𝐹",$ 𝐴𝐴 − 𝐹𝐹",$ 𝐵𝐵 𝐶𝐶 𝐴𝐴, 𝐵𝐵 = 𝑃𝑃 𝐵𝐵, 𝐴𝐴 PCP for Currency Exchange 𝑆𝑆C → 𝑥𝑥C 𝑟𝑟 → 𝑟𝑟G 𝛿𝛿 → 𝑟𝑟I 𝐶𝐶 𝑥𝑥C , 𝐾𝐾 − 𝑃𝑃 𝑥𝑥C , 𝐾𝐾 = 𝑥𝑥C 𝑒𝑒 ,*J $ − 𝐾𝐾𝑒𝑒 ,*K $ L L , where 𝑥𝑥C is in 𝑑𝑑 /𝑓𝑓 𝐶𝐶G 𝑥𝑥C , 𝐾𝐾 = 𝑥𝑥C ⋅ 𝐾𝐾 ⋅ 𝑃𝑃I MN O
PCP for Bonds 𝐶𝐶 − 𝑃𝑃 = 𝐵𝐵" − 𝑃𝑃𝑉𝑉",$ 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 − 𝐾𝐾𝑒𝑒 ,*($,") 𝐵𝐵" = Bond price at time 𝑡𝑡 COMPARING OPTIONS COMPARING OPTIONS Bounds for Option Prices Call and Put 𝑆𝑆 ≥ 𝐶𝐶bcd* ≥ 𝐶𝐶ef* ≥ max 0, 𝐹𝐹 ' 𝑆𝑆 − 𝐾𝐾𝑒𝑒 ,*$ 𝐾𝐾 ≥ 𝑃𝑃bcd* ≥ 𝑃𝑃ef* ≥ max 0, 𝐾𝐾𝑒𝑒 ,*$ − 𝐹𝐹 ' 𝑆𝑆 European vs. American Call 𝐹𝐹 ' 𝑆𝑆 ≥ 𝐶𝐶ef* ≥ max 0, 𝐹𝐹 ' 𝑆𝑆 − 𝐾𝐾𝑒𝑒 ,*$ 𝑆𝑆 ≥ 𝐶𝐶bcd* ≥ max (0, 𝑆𝑆 − 𝐾𝐾) European vs. American Put 𝐾𝐾𝑒𝑒 ,*$ ≥ 𝑃𝑃ef* ≥ max 0, 𝐾𝐾𝑒𝑒 ,*$ − 𝐹𝐹 ' 𝑆𝑆 𝐾𝐾 ≥ 𝑃𝑃bcd* ≥ max (0, 𝐾𝐾 − 𝑆𝑆) Early Exercise of American Option American Call • Nondividend-‐‑paying stock o Early exercise is never optimal. o 𝐶𝐶bcd* = 𝐶𝐶ef* • Dividend-‐‑paying stock o Early exercise is not optimal if 𝑃𝑃𝑃𝑃 Dividends < 𝑃𝑃𝑃𝑃 Interest on the strike + Implicit Put American Put Early exercise is not optimal if 𝑃𝑃𝑃𝑃 Interest on the strike < 𝑃𝑃𝑃𝑃 Dividends + Implicit Call Different Strike Prices For 𝐾𝐾L < 𝐾𝐾p < 𝐾𝐾q : Call • 𝐶𝐶 𝐾𝐾L > 𝐶𝐶 𝐾𝐾p > 𝐶𝐶 𝐾𝐾q • 𝐶𝐶 𝐾𝐾L − 𝐶𝐶 𝐾𝐾p < 𝐾𝐾p − 𝐾𝐾L European: 𝐶𝐶 𝐾𝐾L − 𝐶𝐶 𝐾𝐾p < 𝑃𝑃𝑃𝑃 𝐾𝐾p − 𝐾𝐾L s Ot ,s Ou s O ,s Ov • > u Ou ,Ot
Ov ,Ou
www.coachingactuaries.com
Put • 𝑃𝑃 𝐾𝐾L < 𝑃𝑃 𝐾𝐾p < 𝑃𝑃(𝐾𝐾q ) • 𝑃𝑃 𝐾𝐾p − 𝑃𝑃 𝐾𝐾L < 𝐾𝐾p − 𝐾𝐾L European: 𝑃𝑃 𝐾𝐾p − 𝑃𝑃 𝐾𝐾L < 𝑃𝑃𝑃𝑃 𝐾𝐾p − 𝐾𝐾L ' Ou ,' Ot ' O ,'(Ou ) < v Ou ,Ot
Ov ,Ou
BINOMIAL MODEL BINOMIAL MODEL Replicating Portfolio An option can be replicated by buying 𝛥𝛥 shares of the underlying stock and lending 𝐵𝐵 at the risk-‐‑free rate. 𝑉𝑉f − 𝑉𝑉G 𝑢𝑢𝑉𝑉G − 𝑑𝑑𝑉𝑉f 𝛥𝛥 = 𝑒𝑒 ,5x 𝐵𝐵 = 𝑒𝑒 ,*x 𝑆𝑆 𝑢𝑢 − 𝑑𝑑 𝑢𝑢 − 𝑑𝑑 𝑉𝑉 = 𝛥𝛥𝛥𝛥 + 𝐵𝐵 Call Put 𝛥𝛥 + − 𝐵𝐵 − + Risk-‐‑neutral Probability Pricing 𝑒𝑒 *,5 x − 𝑑𝑑 𝑝𝑝∗ = 𝑢𝑢 − 𝑑𝑑 𝑉𝑉 = 𝑒𝑒 ,*x 𝑝𝑝∗ 𝑉𝑉f + 1 − 𝑝𝑝∗ 𝑉𝑉G 𝑆𝑆C 𝑒𝑒 *,5 x = 𝑝𝑝∗ 𝑆𝑆f + 1 − 𝑝𝑝∗ 𝑆𝑆G Realistic Probability Pricing 𝑒𝑒 {,5 x − 𝑑𝑑 𝑝𝑝 = 𝑢𝑢 − 𝑑𝑑 ,|x 𝑝𝑝 𝑉𝑉f + 1 − 𝑝𝑝 𝑉𝑉G 𝑉𝑉 = 𝑒𝑒 𝑆𝑆C 𝑒𝑒 {,5 x = 𝑝𝑝 𝑆𝑆f + 1 − 𝑝𝑝 𝑆𝑆G 𝛥𝛥𝛥𝛥 {x 𝐵𝐵 *x 𝑒𝑒 |x = 𝑒𝑒 + 𝑒𝑒 𝑉𝑉 𝑉𝑉 Standard Binomial Tree (Forward Tree) 𝑢𝑢 = 𝑒𝑒 *,5 x}~ x 𝑑𝑑 = 𝑒𝑒 *,5 x,~ x 𝑒𝑒 *,5 x − 𝑑𝑑 1 𝑝𝑝∗ = = 𝑢𝑢 − 𝑑𝑑 1 + 𝑒𝑒 ~ x Cox-‐‑Ross-‐‑Rubinstein Tree 𝑢𝑢 = 𝑒𝑒 ~ x 𝑑𝑑 = 𝑒𝑒 ,~ x Lognormal Tree (Jarrow-‐‑Rudd Tree) u u 𝑢𝑢 = 𝑒𝑒 *,5,C.Ä~ x}~ x 𝑑𝑑 = 𝑒𝑒 *,5,C.Ä~ x,~ x No-‐‑Arbitrage Condition Arbitrage is possible if the following inequality is not satisfied: 𝑑𝑑 < 𝑒𝑒 *,5 x < 𝑢𝑢 Option on Currencies 𝑆𝑆C → 𝑥𝑥C 𝑟𝑟 → 𝑟𝑟G 𝛿𝛿 → 𝑟𝑟I
𝑢𝑢 = 𝑒𝑒 *K ,*J x}~ x 𝑑𝑑 = 𝑒𝑒 *K ,*J x,~ x 𝑒𝑒 *K ,*J x − 𝑑𝑑 𝑝𝑝∗ = 𝑢𝑢 − 𝑑𝑑 Option on Futures Contracts 𝐹𝐹",$Å = 𝑆𝑆" 𝑒𝑒 (*,5)($Å ,") 𝑇𝑇 = Expiration date of the option 𝑇𝑇Ö = Expiration date of the futures contract 𝑇𝑇 ≤ 𝑇𝑇Ö 𝑆𝑆" → 𝐹𝐹",$Å 𝛿𝛿 → 𝑟𝑟 1 − 𝑑𝑑Ö 𝑉𝑉f − 𝑉𝑉G 𝑝𝑝∗ = 𝛥𝛥 = 𝑢𝑢Ö − 𝑑𝑑Ö 𝐹𝐹 𝑢𝑢Ö − 𝑑𝑑Ö ,*x ∗ ∗ 𝑝𝑝 𝑉𝑉f + 1 − 𝑝𝑝 𝑉𝑉G 𝐵𝐵 = 𝑒𝑒
Utility Values and State Prices 𝑈𝑈f : Utility value per dollar in the up state 𝑈𝑈G : Utility value per dollar in the down state 𝑄𝑄f = 𝑝𝑝×𝑈𝑈f = 𝑝𝑝∗ ×𝑒𝑒 ,*x 𝑄𝑄G = 1 − 𝑝𝑝 ×𝑈𝑈G = 1 − 𝑝𝑝∗ ×𝑒𝑒 ,*x 𝑒𝑒 ,*x = 𝑄𝑄f + 𝑄𝑄G 𝑆𝑆 = 𝑄𝑄f 𝑆𝑆f 𝑒𝑒 5x + 𝑄𝑄G 𝑆𝑆G 𝑒𝑒 5x 𝑉𝑉 = 𝑄𝑄f 𝑉𝑉f + 𝑄𝑄G 𝑉𝑉G 𝑄𝑄f 𝑝𝑝∗ = 𝑄𝑄f + 𝑄𝑄G 𝑟𝑟, 𝛼𝛼, 𝛾𝛾sêëë , 𝛾𝛾'f" 𝛾𝛾'f" ≤ 𝑟𝑟 ≤ 𝛼𝛼 ≤ 𝛾𝛾sêëë MODEL LOGNORMAL LOGNORMAL MODEL Lognormal Model for Stock Prices 𝑋𝑋~𝑁𝑁 𝑚𝑚, 𝑣𝑣 p ⟺ 𝑌𝑌 = 𝑒𝑒 ô ~𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝑚𝑚, 𝑣𝑣 p u • 𝐸𝐸 𝑌𝑌 = 𝑒𝑒 c}C.Äù u • 𝑉𝑉𝑉𝑉𝑉𝑉 𝑌𝑌 = 𝐸𝐸 𝑌𝑌 p 𝑒𝑒 ù − 1 𝑆𝑆$ For 𝑇𝑇 > 𝑡𝑡, ln ~𝑁𝑁 𝑚𝑚, 𝑣𝑣 p 𝑆𝑆" • 𝑚𝑚 = 𝛼𝛼 − 𝛿𝛿 − 0.5𝜎𝜎 p 𝑇𝑇 − 𝑡𝑡 • 𝑣𝑣 p = 𝜎𝜎 p 𝑇𝑇 − 𝑡𝑡 𝐸𝐸 𝑆𝑆$ |𝑆𝑆" = 𝑆𝑆" 𝑒𝑒 ({,5)($,") u 𝑉𝑉𝑉𝑉𝑉𝑉 𝑆𝑆$ |𝑆𝑆" = 𝐸𝐸 𝑆𝑆$ |𝑆𝑆" p 𝑒𝑒 ù − 1 u
𝑆𝑆$ = 𝑆𝑆" 𝑒𝑒 {,5,C.Ä~ $," }~ $,"⋅£ , 𝑍𝑍~𝑁𝑁(0,1) u Median = 𝑆𝑆" 𝑒𝑒 {,5,C.Ä~ $," Covariance 𝑆𝑆$ ⋅ 𝑉𝑉𝑉𝑉𝑉𝑉 𝑆𝑆" 𝑆𝑆C 𝐶𝐶𝐶𝐶𝐶𝐶 𝑆𝑆" , 𝑆𝑆$ = 𝐸𝐸 𝑆𝑆" Probability Pr 𝑆𝑆$ < 𝐾𝐾 = 𝑁𝑁 −𝑑𝑑p Pr 𝑆𝑆$ > 𝐾𝐾 = 𝑁𝑁 +𝑑𝑑p 𝑆𝑆 ln " + (𝛼𝛼 − 𝛿𝛿 − 0.5𝜎𝜎 p )(𝑇𝑇 − 𝑡𝑡) 𝐾𝐾 𝑑𝑑p = 𝜎𝜎 𝑇𝑇 − 𝑡𝑡 Prediction Interval (Confidence Interval) The (1 − 𝑝𝑝) prediction interval is given by 𝑆𝑆$¶ and 𝑆𝑆$ß such that Pr 𝑆𝑆$¶ < 𝑆𝑆$ < 𝑆𝑆$ß = 1 − 𝑝𝑝. u © 𝑆𝑆$¶ = 𝑆𝑆" 𝑒𝑒 {,5,C.Ä~ $," }~ $,"⋅® ß {,5,C.Ä~ u $," }~ $,"⋅® ™ 𝑆𝑆$ = 𝑆𝑆" 𝑒𝑒 𝑝𝑝 𝑝𝑝 Pr 𝑍𝑍 < 𝑧𝑧 ¶ = ⇒ 𝑧𝑧 ¶ = 𝑁𝑁 ,L 2 2 𝑝𝑝 𝑧𝑧 ß = −𝑧𝑧 ¶ = −𝑁𝑁 ,L 2 Conditional and Partial Expectation 𝑃𝑃𝑃𝑃 𝑆𝑆$ 𝑆𝑆$ < 𝐾𝐾 𝐸𝐸 𝑆𝑆$ 𝑆𝑆$ < 𝐾𝐾 = Pr 𝑆𝑆$ < 𝐾𝐾 {,5 $," 𝑁𝑁 −𝑑𝑑L 𝑆𝑆" 𝑒𝑒 = 𝑁𝑁 −𝑑𝑑p 𝑃𝑃𝑃𝑃 𝑆𝑆$ 𝑆𝑆$ > 𝐾𝐾 𝐸𝐸 𝑆𝑆$ 𝑆𝑆$ > 𝐾𝐾 = Pr 𝑆𝑆$ > 𝐾𝐾 {,5 $," 𝑁𝑁 +𝑑𝑑L 𝑆𝑆" 𝑒𝑒 = 𝑁𝑁 +𝑑𝑑p 𝑆𝑆 ln " + 𝛼𝛼 − 𝛿𝛿 + 0.5𝜎𝜎 p 𝑇𝑇 − 𝑡𝑡 𝐾𝐾 𝑑𝑑L = 𝜎𝜎 𝑇𝑇 − 𝑡𝑡 𝑑𝑑p = 𝑑𝑑L − 𝜎𝜎 𝑇𝑇 − 𝑡𝑡 Expected Option Payoffs 𝐸𝐸 Call Payoff = 𝑆𝑆" 𝑒𝑒 {,5 $," 𝑁𝑁 𝑑𝑑L − 𝐾𝐾𝐾𝐾 𝑑𝑑p 𝐸𝐸 Put Payoff = 𝐾𝐾𝐾𝐾 −𝑑𝑑p − 𝑆𝑆" 𝑒𝑒
{,5 $,"
𝑁𝑁 −𝑑𝑑L
Copyright © 2016 Coaching Actuaries. All Rights Reserved. 1
BLACK-SCHOLES MODEL BLACK-‐SCHOLES PRICING MPRICING ODEL Generalized B-‐‑S Formula 𝐶𝐶 = 𝐹𝐹 ' 𝑆𝑆 ⋅ 𝑁𝑁 𝑑𝑑L − 𝐹𝐹 ' 𝐾𝐾 ⋅ 𝑁𝑁 𝑑𝑑p 𝑃𝑃 = 𝐹𝐹 ' 𝐾𝐾 ⋅ 𝑁𝑁 −𝑑𝑑p − 𝐹𝐹 ' 𝑆𝑆 ⋅ 𝑁𝑁 −𝑑𝑑L 1 𝐹𝐹 ' 𝑆𝑆 + 𝜎𝜎 p 𝑇𝑇 − 𝑡𝑡 ln ' 2 𝐹𝐹 𝐾𝐾 𝑑𝑑L = 𝜎𝜎 𝑇𝑇 − 𝑡𝑡 1 𝐹𝐹 ' 𝑆𝑆 − 𝜎𝜎 p 𝑇𝑇 − 𝑡𝑡 ln ' 2 𝐹𝐹 𝐾𝐾 𝑑𝑑p = = 𝑑𝑑L − 𝜎𝜎 𝑇𝑇 − 𝑡𝑡 𝜎𝜎 𝑇𝑇 − 𝑡𝑡 B-‐‑S Formula for Stock 𝐶𝐶 = 𝑆𝑆" 𝑒𝑒 ,5 $," ⋅ 𝑁𝑁 𝑑𝑑L − 𝐾𝐾𝑒𝑒 ,* $," ⋅ 𝑁𝑁 𝑑𝑑p 𝑃𝑃 = 𝐾𝐾𝑒𝑒 ,* $," ⋅ 𝑁𝑁 −𝑑𝑑p − 𝑆𝑆" 𝑒𝑒 ,5 $," ⋅ 𝑁𝑁 −𝑑𝑑L 𝑆𝑆 1 ln " + 𝑟𝑟 − 𝛿𝛿 + 𝜎𝜎 p 𝑇𝑇 − 𝑡𝑡 2 𝐾𝐾 𝑑𝑑L = 𝜎𝜎 𝑇𝑇 − 𝑡𝑡 𝑆𝑆 1 ln " + 𝑟𝑟 − 𝛿𝛿 − 𝜎𝜎 p 𝑇𝑇 − 𝑡𝑡 2 𝐾𝐾 𝑑𝑑p = 𝜎𝜎 𝑇𝑇 − 𝑡𝑡
= 𝑑𝑑L − 𝜎𝜎 𝑇𝑇 − 𝑡𝑡 B-‐‑S Formula for Currency 𝑆𝑆C → 𝑥𝑥C 𝑟𝑟 → 𝑟𝑟G 𝛿𝛿 → 𝑟𝑟I 𝐶𝐶 = 𝑥𝑥C 𝑒𝑒 ,*J $," ⋅ 𝑁𝑁 𝑑𝑑L − 𝐾𝐾𝑒𝑒 ,*K $," ⋅ 𝑁𝑁 𝑑𝑑p 𝑃𝑃 = 𝐾𝐾𝑒𝑒 ,*K $," ⋅ 𝑁𝑁 −𝑑𝑑p − 𝑥𝑥C 𝑒𝑒 ,*J $," ⋅ 𝑁𝑁 −𝑑𝑑L 𝑥𝑥 1 ln C + 𝑟𝑟G − 𝑟𝑟I + 𝜎𝜎 p 𝑇𝑇 2 𝐾𝐾 𝑑𝑑L = 𝜎𝜎 𝑇𝑇 𝑥𝑥 1 ln C + 𝑟𝑟G − 𝑟𝑟I − 𝜎𝜎 p 𝑇𝑇 2 𝐾𝐾 𝑑𝑑p = 𝜎𝜎 𝑇𝑇
= 𝑑𝑑L − 𝜎𝜎 𝑇𝑇 B-‐‑S Formula for Futures 𝐹𝐹",$Å = 𝑆𝑆" 𝑒𝑒 (*,5)($Å ,") 𝑇𝑇 = Expiration date of the option 𝑇𝑇Ö = Expiration date of the futures contract 𝑇𝑇 ≤ 𝑇𝑇Ö 𝑆𝑆C → 𝐹𝐹C,$Å 𝛿𝛿 → 𝑟𝑟 𝐶𝐶 = 𝐹𝐹C,$Å 𝑒𝑒 ,*$ ⋅ 𝑁𝑁 𝑑𝑑L − 𝐾𝐾𝑒𝑒 ,*$ ⋅ 𝑁𝑁 𝑑𝑑p 𝑃𝑃 = 𝐾𝐾𝑒𝑒 ,*$ ⋅ 𝑁𝑁 −𝑑𝑑p − 𝐹𝐹C,$Å 𝑒𝑒 ,*$ ⋅ 𝑁𝑁 −𝑑𝑑L 𝐹𝐹C,$Å 1 ln + 𝜎𝜎 p 𝑇𝑇 2 𝐾𝐾 𝑑𝑑L = 𝜎𝜎 𝑇𝑇 𝐹𝐹C,$Å 1 ln − 𝜎𝜎 p 𝑇𝑇 2 𝐾𝐾 𝑑𝑑p = = 𝑑𝑑L − 𝜎𝜎 𝑇𝑇 𝜎𝜎 𝑇𝑇 Greeks Delta ÆØ∞±≤≥ ¥± µ ∂∑¥∏± π∫¥ª≥ æø • 𝛥𝛥 = = ÆØ∞±≤≥ ¥± º∑∏ªΩ π∫¥ª≥
æ¿
• 𝛥𝛥s = 𝑒𝑒 ,5$ 𝑁𝑁 𝑑𝑑L 𝛥𝛥' = −𝑒𝑒 ,5$ 𝑁𝑁 −𝑑𝑑L • 0 ≤ 𝛥𝛥s ≤ 1 −1 ≤ 𝛥𝛥' ≤ 0 • 𝛥𝛥¡ − 𝛥𝛥' = 𝑒𝑒 ,5$ • Delta increases as the stock price increases. Gamma • 𝛤𝛤 =
ÆØ∞±≤≥ ¥± √ ≥ƒ∑∞
ÆØ∞±≤≥ ¥± º∑∏ªΩ π∫¥ª≥
=
æ≈ æ¿
=
æuø æ¿
u
• 𝛤𝛤s ≥ 0 𝛤𝛤' ≥ 0 • 𝛤𝛤s = 𝛤𝛤' Theta • 𝜃𝜃 = Change in the option price as time advances 𝜕𝜕𝜕𝜕 = 𝜕𝜕𝜕𝜕 • 𝜃𝜃 is usually negative. Vega ÆØ∞±≤≥ ¥± µ ∂∑¥∏± π∫¥ª≥ æø • 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = = ÆØ∞±≤≥ ¥± »∏ƒ∞∑¥ƒ¥∑…
• 𝑉𝑉𝑉𝑉𝑉𝑉𝑎𝑎s ≥ 0 𝑉𝑉𝑉𝑉𝑉𝑉𝑎𝑎' ≥ 0 • 𝑉𝑉𝑉𝑉𝑉𝑉𝑎𝑎s = 𝑉𝑉𝑉𝑉𝑉𝑉𝑎𝑎'
æ~
www.coachingactuaries.com
Rho • 𝜌𝜌 =
ÆØ∞±≤≥ ¥± µ ∂∑¥∏± π∫¥ª≥
=
ÆØ∞±≤≥ ¥± À¥ÃΩ,Õ∫≥≥ ∫∞∑≥
æø æ*
• 𝜌𝜌s ≥ 0 𝜌𝜌' ≤ 0 Psi ÆØ∞±≤≥ ¥± µ ∂∑¥∏± π∫¥ª≥ æø • 𝜓𝜓 = = ÆØ∞±≤≥ ¥± √ ¥œ¥–≥±– —¥≥ƒ–
æ5
• 𝜓𝜓s ≤ 0 𝜓𝜓' ≥ 0
Greek '”*"I”ë‘” =
’
‘÷L
𝑁𝑁‘ ⋅ Greek ‘
Elasticity % change in option price 𝛥𝛥𝛥𝛥 = 𝛺𝛺 = 𝑉𝑉 % change in stock price 𝜎𝜎Ÿ⁄"‘”’ = 𝛺𝛺 𝜎𝜎¿"”¡€ 𝛾𝛾 − 𝑟𝑟 = 𝛺𝛺(𝛼𝛼 − 𝑟𝑟) 𝛺𝛺sêëë ≥ 1 𝛺𝛺'f" ≤ 0 ’ 𝛥𝛥'”*"I”ë‘” ⋅ 𝑆𝑆 = 𝜔𝜔‘ 𝛺𝛺‘ 𝛺𝛺'”*"I”ë‘” = 𝑉𝑉'”*"I”ë‘” ‘÷L
𝛾𝛾'”*"I”ë‘” − 𝑟𝑟 = 𝛺𝛺'”*"I”ë‘” (𝛼𝛼 − 𝑟𝑟)
DELTA HEDGING DELTA HEDGING Overnight Profit 3 components in overnight profit: • Gain on stocks • Gain on options • Interest on borrowed/lent money Breakeven The price movement with no gain or loss to delta-‐‑ hedger is: ±𝑆𝑆𝑆𝑆 ℎ Delta-‐‑Gamma-‐‑Theta Approximation 1 𝑉𝑉"}x = 𝑉𝑉" + 𝛥𝛥" 𝜖𝜖 + 𝛤𝛤" 𝜖𝜖 p + 𝜃𝜃" ℎ 2 Boyle-‐‑Emanuel Formula Boyle-‐‑Emanuel periodic variance of return when rehedging every ℎ in period 𝑖𝑖 : 1 𝑉𝑉𝑉𝑉𝑉𝑉 𝑅𝑅x,‘ = 𝑆𝑆 p 𝜎𝜎 p 𝛤𝛤ℎ p 2 Boyle-‐‑Emanuel annual variance of return when rehedging every ℎ in period 𝑖𝑖 : 1 𝑉𝑉𝑉𝑉𝑉𝑉 𝑅𝑅x,‘ = 𝑆𝑆 p 𝜎𝜎 p 𝛤𝛤 p ℎ 2 Greeks for Binomial Trees 𝑉𝑉f − 𝑉𝑉G 𝛥𝛥 𝑆𝑆, 0 = 𝑒𝑒 ,5x 𝑆𝑆 𝑢𝑢 − 𝑑𝑑 𝛥𝛥 𝑆𝑆𝑆𝑆, ℎ − 𝛥𝛥(𝑆𝑆𝑆𝑆, ℎ) 𝛤𝛤 𝑆𝑆, 0 ≈ 𝛤𝛤 𝑆𝑆, ℎ = 𝑆𝑆(𝑢𝑢 − 𝑑𝑑) 𝜃𝜃 𝑆𝑆, 0 1 𝑉𝑉 𝑆𝑆𝑆𝑆𝑆𝑆, 2ℎ − 𝑉𝑉 𝑆𝑆, 0 − 𝛥𝛥 𝑆𝑆, 0 𝜖𝜖 − 𝛤𝛤 𝑆𝑆, 0 𝜖𝜖 p 2 = 2ℎ
EXOTIC OPTIONS EXOTIC OPTIONS Asian Option 𝐴𝐴 𝑆𝑆 arithmetic average 𝑆𝑆 = 𝐺𝐺 𝑆𝑆 geometric average
𝐴𝐴 𝑆𝑆 =
‰ "÷L 𝑆𝑆"
𝑁𝑁
𝐺𝐺 𝑆𝑆 =
𝐺𝐺 𝑆𝑆 ≤ 𝐴𝐴 𝑆𝑆 Average Price PayoffÆ∞ƒƒ PayoffπÂ∑
max 0, 𝑆𝑆 − 𝐾𝐾 max 0, 𝐾𝐾 − 𝑆𝑆
‰
"÷L
𝑆𝑆"
L ‰
Average Strike max 0, 𝑆𝑆 − 𝑆𝑆 max 0, 𝑆𝑆 − 𝑆𝑆
The value of an Asian option is less than or equal to the value of an otherwise equivalent ordinary option. As 𝑁𝑁 increases: • Value of average price option decreases • Value of average strike option increases
Barrier Option Three types: • Knock-‐‑in Goes into existence if barrier is reached. • Knock-‐‑out Goes out of existence if barrier is reached. • Rebate Pays fixed amount if barrier is reached. Down vs. Up: • If 𝑆𝑆C < 𝐵𝐵: Up-‐‑and-‐‑in, up-‐‑and-‐‑out, up rebate • If 𝑆𝑆C > 𝐵𝐵: Down-‐‑and-‐‑in, down-‐‑and-‐‑out, down rebate Knock-‐‑in + Knock-‐‑out = Ordinary Option Barrier option ≤ Ordinary Option Special relationships: • If barrier ≤ strike: up-‐‑and-‐‑in call = ordinary call • If barrier ≥ strike: down-‐‑and-‐‑in put = ordinary put Compound Option The value of the underlying option at time 𝑡𝑡L = 𝑉𝑉 𝑆𝑆"t , 𝐾𝐾, 𝑇𝑇 − 𝑡𝑡L The value of the compound call at time 𝑡𝑡L = max 0, 𝑉𝑉 𝑆𝑆"t , 𝐾𝐾, 𝑇𝑇 − 𝑡𝑡L − 𝑥𝑥 The value of the compound put at time 𝑡𝑡L = max 0, 𝑥𝑥 − 𝑉𝑉 𝑆𝑆"t , 𝐾𝐾, 𝑇𝑇 − 𝑡𝑡L Put-‐‑call parity for compound option: • CallonCall − PutonCall = 𝐶𝐶ef* − 𝑥𝑥𝑒𝑒 ,*"t • CallonPut − PutonPut = 𝑃𝑃ef* − 𝑥𝑥𝑒𝑒 ,*"t
Gap Option 𝐾𝐾L : Strike Price 𝐾𝐾p : Trigger Price 𝐾𝐾L determines the amount of the payoff. 𝐾𝐾p determines whether the option will have a payoff. 0, 𝑆𝑆$ ≤ 𝐾𝐾p PayoffË∞∂ Æ∞ƒƒ = 𝑆𝑆$ − 𝐾𝐾L , 𝑆𝑆$ > 𝐾𝐾p 𝐾𝐾L − 𝑆𝑆$ , 𝑆𝑆$ ≤ 𝐾𝐾p PayoffË∞∂ πÂ∑ = 0, 𝑆𝑆$ > 𝐾𝐾p ,5$ ,*$ GapCall = 𝑆𝑆C 𝑒𝑒 𝑁𝑁 𝑑𝑑L − 𝐾𝐾L 𝑒𝑒 𝑁𝑁 𝑑𝑑p GapPut = 𝐾𝐾L 𝑒𝑒 ,*$ 𝑁𝑁 −𝑑𝑑p − 𝑆𝑆C 𝑒𝑒 ,5$ 𝑁𝑁 −𝑑𝑑L where 𝑑𝑑L and 𝑑𝑑p are based on 𝐾𝐾p GapCall − GapPut = 𝑆𝑆C 𝑒𝑒 ,5$ − 𝐾𝐾L 𝑒𝑒 ,*$ Exchange Option 𝐶𝐶(𝐴𝐴, 𝐵𝐵) = 𝐹𝐹 ' 𝐴𝐴 ⋅ 𝑁𝑁 𝑑𝑑L − 𝐹𝐹 ' 𝐵𝐵 ⋅ 𝑁𝑁 𝑑𝑑p 𝑃𝑃(𝐴𝐴, 𝐵𝐵) = 𝐹𝐹 ' 𝐵𝐵 ⋅ 𝑁𝑁 −𝑑𝑑p − 𝐹𝐹 ' 𝐴𝐴 ⋅ 𝑁𝑁 −𝑑𝑑L 1 𝐹𝐹 ' 𝐴𝐴 + 𝜎𝜎 p 𝑇𝑇 − 𝑡𝑡 ln ' 2 𝐹𝐹 𝐵𝐵 𝑑𝑑L = 𝜎𝜎 𝑇𝑇 − 𝑡𝑡 𝑑𝑑p = 𝑑𝑑L − 𝜎𝜎 𝑇𝑇 − 𝑡𝑡 𝜎𝜎 =
𝜎𝜎bp + 𝜎𝜎Èp − 2𝜌𝜌𝜎𝜎b 𝜎𝜎È
All-‐‑or-‐‑nothing Option Option Payoff Time-‐‑t Price 0, 𝑆𝑆$ < 𝐾𝐾 Asset 𝑆𝑆" 𝑒𝑒 ,5 $," 𝑁𝑁 𝑑𝑑L 𝑆𝑆$ , 𝑆𝑆$ > 𝐾𝐾 Call 𝑆𝑆$ , 𝑆𝑆$ < 𝐾𝐾 Asset 𝑆𝑆" 𝑒𝑒 ,5 $," 𝑁𝑁 −𝑑𝑑L 0, 𝑆𝑆$ > 𝐾𝐾 Put 0, 𝑆𝑆$ < 𝐾𝐾 Cash 𝑒𝑒 ,* $," 𝑁𝑁 𝑑𝑑p $1, 𝑆𝑆$ > 𝐾𝐾 Call $1, 𝑆𝑆$ < 𝐾𝐾 Cash 𝑒𝑒 ,* $," 𝑁𝑁 −𝑑𝑑p 0, 𝑆𝑆$ > 𝐾𝐾 Put Maxima and Minima • max 𝐴𝐴, 𝐵𝐵 = max 0, 𝐵𝐵 − 𝐴𝐴 + 𝐴𝐴 max 𝐴𝐴, 𝐵𝐵 = max 𝐴𝐴 − 𝐵𝐵, 0 + 𝐵𝐵 • max 𝑐𝑐𝑐𝑐, 𝑐𝑐𝑐𝑐 = 𝑐𝑐 ⋅ max 𝐴𝐴, 𝐵𝐵 𝑐𝑐 > 0 max 𝑐𝑐𝑐𝑐, 𝑐𝑐𝑐𝑐 = 𝑐𝑐 ⋅ min 𝐴𝐴, 𝐵𝐵 𝑐𝑐 < 0 • max 𝐴𝐴, 𝐵𝐵 + min 𝐴𝐴, 𝐵𝐵 = 𝐴𝐴 + 𝐵𝐵 ⇒ min 𝐴𝐴, 𝐵𝐵 = − max 𝐴𝐴, 𝐵𝐵 + 𝐴𝐴 + 𝐵𝐵
Copyright © 2016 Coaching Actuaries. All Rights Reserved. 2
Forward Start Option For a call option expiring at time 𝑇𝑇 whose strike is set on future date 𝑡𝑡 to be 𝑋𝑋 𝑆𝑆" : 𝐶𝐶 𝑆𝑆" , 𝑋𝑋𝑆𝑆" , 𝑇𝑇 − 𝑡𝑡 = 𝑆𝑆" 𝑒𝑒 ,5 $," 𝑁𝑁 𝑑𝑑L − 𝑋𝑋𝑆𝑆" 𝑒𝑒 ,* $," 𝑁𝑁 𝑑𝑑p = 𝑆𝑆" 𝑒𝑒 ,5 $," 𝑁𝑁 𝑑𝑑L − 𝑋𝑋𝑒𝑒 ,* $," 𝑁𝑁 𝑑𝑑p 𝑆𝑆 ln " + 𝑟𝑟 − 𝛿𝛿 + 0.5𝜎𝜎 p 𝑇𝑇 − 𝑡𝑡 𝑋𝑋𝑆𝑆" 𝑑𝑑L = 𝜎𝜎 𝑇𝑇 − 𝑡𝑡 1 ln + (𝑟𝑟 − 𝛿𝛿 + 0.5𝜎𝜎 p )(𝑇𝑇 − 𝑡𝑡) 𝑋𝑋 = 𝜎𝜎 𝑇𝑇 − 𝑡𝑡 𝑑𝑑p = 𝑑𝑑L − 𝜎𝜎 𝑇𝑇 − 𝑡𝑡 The time-‐‑0 value of the forward start option is: ' 𝑆𝑆 × 𝑒𝑒 ,5 $," 𝑁𝑁 𝑑𝑑L − 𝑋𝑋𝑒𝑒 ,* $," 𝑁𝑁 𝑑𝑑p 𝑉𝑉C = 𝐹𝐹C,"
Chooser Option For an option that allows the owner to choose at time 𝑡𝑡 whether the option will become a European call or put with strike 𝐾𝐾 expiring at time 𝑇𝑇: 𝑉𝑉" = max 𝐶𝐶 𝑆𝑆" , 𝐾𝐾, 𝑇𝑇 − 𝑡𝑡 , 𝑃𝑃 𝑆𝑆" , 𝐾𝐾, 𝑇𝑇 − 𝑡𝑡 = 𝑒𝑒 ,5
$,"
𝑉𝑉C = 𝑒𝑒 ,5
max 0, 𝐾𝐾𝑒𝑒 ,
$,"
*,5 $,"
− 𝑆𝑆"
+ 𝐶𝐶 𝑆𝑆" , 𝐾𝐾, 𝑇𝑇 − 𝑡𝑡
⋅ 𝑃𝑃 𝑆𝑆C , 𝐾𝐾𝑒𝑒 ,
*,5 $,"
, 𝑡𝑡 + 𝐶𝐶 𝑆𝑆C , 𝐾𝐾, 𝑇𝑇
CARLO VALUATION MONTE CMONTE ARLO VALUATION Simulating Standard Normal Variables 𝑧𝑧 =
Lp
‘÷L
𝑢𝑢‘ − 6 𝑧𝑧‘ = 𝑁𝑁 ,L 𝑢𝑢‘
Simulating Lognormal Stock Prices • Not interested in the intermediate prices: u 𝑆𝑆$ = 𝑆𝑆" 𝑒𝑒 {,5,C.Ä~ $," }~ $,"⋅£ • Interested in the intermediate prices: u 𝑆𝑆"}x = 𝑆𝑆" 𝑒𝑒 {,5,C.Ä~ x}~ x⋅£t {,5,C.Ä~ u x}~ x⋅£u 𝑆𝑆"}px = 𝑆𝑆"}x 𝑒𝑒 . . . u 𝑆𝑆$,x = 𝑆𝑆$,px 𝑒𝑒 {,5,C.Ä~ x}~ x⋅£ÌÓt {,5,C.Ä~ u x}~ x⋅£Ì 𝑆𝑆$ = 𝑆𝑆$,x 𝑒𝑒 Risk-‐‑neutral vs. True • Use the risk-‐‑neutral distribution only when discounting is needed. • Use the true distribution when discounting is not needed. Control Variate Method 𝑌𝑌 ∗ = 𝑌𝑌 + 𝛽𝛽 𝑋𝑋 − 𝑋𝑋 where 𝑌𝑌 ∗ = Control variate estimate for Option 𝑌𝑌 𝑌𝑌 = Monte Carlo estimate for Option 𝑌𝑌 𝑋𝑋 = Exact/True price of Option 𝑋𝑋 𝑋𝑋 = Monte Carlo estimate for Option 𝑋𝑋 𝑉𝑉𝑉𝑉𝑉𝑉 𝑌𝑌 ∗ = 𝑉𝑉𝑉𝑉𝑉𝑉 𝑌𝑌 + 𝛽𝛽 p 𝑉𝑉𝑉𝑉𝑉𝑉 𝑋𝑋 − 2𝛽𝛽𝛽𝛽𝛽𝛽𝛽𝛽 𝑌𝑌, 𝑋𝑋 𝑉𝑉𝑉𝑉𝑉𝑉 𝑌𝑌 ∗ is minimized when: ’ 𝐶𝐶𝐶𝐶𝐶𝐶 𝑌𝑌, 𝑋𝑋 ‘÷L 𝑌𝑌‘ − 𝑌𝑌 𝑋𝑋‘ − 𝑋𝑋 = 𝛽𝛽 = ’ p 𝑉𝑉𝑉𝑉𝑉𝑉 𝑋𝑋 ‘÷L 𝑋𝑋‘ − 𝑋𝑋 When 𝛽𝛽 is set to minimize 𝑉𝑉𝑉𝑉𝑉𝑉 𝑌𝑌 ∗ : 𝑉𝑉𝑉𝑉𝑉𝑉 𝑌𝑌 ∗ = 𝑉𝑉𝑉𝑉𝑉𝑉 𝑌𝑌 1 − 𝜌𝜌ô,Ò p
Antithetic Variate Method For every 𝑢𝑢‘ , also simulate using 1 − 𝑢𝑢‘ . For every 𝑧𝑧‘ , also simulate using – 𝑧𝑧‘ . Stratified Sampling Break the sampling space into equal size spaces. Then, scale the uniform numbers into the equal size spaces.
www.coachingactuaries.com
MOTION BROWNIAN MBROWNIAN OTION Basics of Brownian Motion 𝑍𝑍 𝑡𝑡 : Pure/Standard Brownian Motion Characteristics: 1. 𝑍𝑍 0 = 0 2. 𝑍𝑍 𝑡𝑡 + 𝑠𝑠 − 𝑍𝑍 𝑡𝑡 ~𝑁𝑁 0, 𝑠𝑠 𝑍𝑍 𝑡𝑡 − 𝑍𝑍 0 ~𝑁𝑁(0, 𝑡𝑡) 3. 𝑍𝑍 𝑡𝑡 + ℎ − 𝑍𝑍(𝑡𝑡) is independent of 𝑍𝑍 𝑡𝑡 − 𝑍𝑍(𝑡𝑡 − 𝑠𝑠) 4. 𝑍𝑍(𝑡𝑡) is continuous 𝑍𝑍(𝑡𝑡) is a martingale if • 𝐸𝐸 𝑍𝑍 𝑡𝑡 + ℎ − 𝑍𝑍 𝑡𝑡 = 0 • 𝐸𝐸 𝑍𝑍 𝑡𝑡 + ℎ 𝑍𝑍 𝑡𝑡 = 𝑍𝑍(𝑡𝑡) Properties: 1. Quadratic variation = 𝑇𝑇 2. Cubic or higher order variation = 0 3. Total variation = ∞ Arithmetic Brownian Motion 𝑑𝑑𝑑𝑑 𝑡𝑡 = 𝑎𝑎 𝑑𝑑𝑑𝑑 + 𝑏𝑏 𝑑𝑑𝑑𝑑 𝑡𝑡 𝑋𝑋 𝑇𝑇 − 𝑋𝑋 0 = 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑏𝑏(𝑇𝑇) 𝑋𝑋 𝑡𝑡 − 𝑋𝑋 0 ~𝑁𝑁 𝑎𝑎𝑎𝑎, 𝑏𝑏 p 𝑡𝑡 Ornstein-‐‑Uhlenbeck Process 𝑑𝑑𝑑𝑑 𝑡𝑡 = 𝜆𝜆 𝛼𝛼 − 𝑋𝑋 𝑡𝑡 𝑑𝑑𝑑𝑑 + 𝜎𝜎𝜎𝜎𝜎𝜎 𝑡𝑡 Geometric Brownian Motion 𝑑𝑑𝑑𝑑 𝑡𝑡 = 𝑎𝑎𝑎𝑎 𝑡𝑡 𝑑𝑑𝑑𝑑 + 𝑏𝑏𝑏𝑏 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑡𝑡 = 𝑎𝑎 𝑑𝑑𝑑𝑑 + 𝑏𝑏 𝑑𝑑𝑑𝑑 𝑡𝑡 𝑋𝑋 𝑡𝑡 1 𝑑𝑑 ln 𝑋𝑋 𝑡𝑡 = 𝑎𝑎 − 𝑏𝑏 p 𝑑𝑑𝑑𝑑 + 𝑏𝑏 𝑑𝑑𝑑𝑑(𝑡𝑡) 2 L u
𝑋𝑋 𝑡𝑡 = 𝑋𝑋 0 𝑒𝑒 ê,p˜ "}˜⋅£(") 1 𝑋𝑋 𝑡𝑡 ~𝑁𝑁 𝑚𝑚 = 𝑎𝑎 − 𝑏𝑏 p 𝑡𝑡, 𝑣𝑣 p = 𝑏𝑏 p 𝑡𝑡 ln 𝑋𝑋 0 2 The followings are equivalent: • The Black-‐‑Scholes framework applies. • 𝑑𝑑𝑑𝑑 𝑡𝑡 = 𝛼𝛼 − 𝛿𝛿 𝑆𝑆 𝑡𝑡 𝑑𝑑𝑑𝑑 + 𝜎𝜎𝜎𝜎 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑡𝑡 G¿ " = (𝛼𝛼 − 𝛿𝛿 ) 𝑑𝑑𝑑𝑑 + 𝜎𝜎 𝑑𝑑𝑑𝑑 𝑡𝑡 • ¿ "
• 𝑑𝑑 ln 𝑆𝑆 𝑡𝑡
L
= 𝛼𝛼 − 𝛿𝛿 − 𝜎𝜎 p 𝑑𝑑𝑑𝑑 + 𝜎𝜎 𝑑𝑑𝑑𝑑 𝑡𝑡 t {,5 , ~ u u
p
"}~£ " • 𝑆𝑆 𝑡𝑡 = 𝑆𝑆 0 𝑒𝑒 ¿ " L ~𝑁𝑁 𝑚𝑚 = 𝛼𝛼 − 𝛿𝛿 − 𝜎𝜎 p 𝑡𝑡, 𝑣𝑣 p = 𝜎𝜎 p 𝑡𝑡 • ln ¿ C
Ito’s Lemma
p
1 𝑑𝑑𝑑𝑑 = 𝑉𝑉¿ 𝑑𝑑𝑑𝑑 + 𝑉𝑉¿¿ 𝑑𝑑𝑑𝑑 p + 𝑉𝑉" 𝑑𝑑𝑑𝑑 2 Multiplication Rules 𝑑𝑑𝑑𝑑 × 𝑑𝑑𝑑𝑑 = 0 𝑑𝑑𝑑𝑑 × 𝑑𝑑𝑑𝑑 = 0 𝑑𝑑𝑑𝑑 × 𝑑𝑑𝑑𝑑 = 0 𝑑𝑑𝑑𝑑 × 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑
Sharpe Ratio 𝛼𝛼 − 𝑟𝑟 𝜙𝜙 = 𝜎𝜎 Two Ito’s processes depending on the same 𝑑𝑑𝑑𝑑(𝑡𝑡) will have equal Sharpe ratios. Risk-‐‑free Portfolio For a portfolio consisting of Asset 1 and Asset 2: Return on Asset 1 = 𝑑𝑑𝑆𝑆L + 𝛿𝛿L 𝑆𝑆L 𝑑𝑑𝑑𝑑 Return on Asset 2 = 𝑑𝑑𝑆𝑆p + 𝛿𝛿p 𝑆𝑆p 𝑑𝑑𝑑𝑑 Total return on the portfolio = 𝑁𝑁L 𝑑𝑑𝑆𝑆L + 𝛿𝛿L 𝑆𝑆L 𝑑𝑑𝑑𝑑 + 𝑁𝑁p 𝑑𝑑𝑆𝑆p + 𝛿𝛿p 𝑆𝑆p 𝑑𝑑𝑑𝑑 The coefficient of 𝑑𝑑𝑑𝑑 = 𝑁𝑁L 𝜎𝜎L 𝑆𝑆L + 𝑁𝑁p 𝜎𝜎p 𝑆𝑆p Risk-‐‑free ⇒ The coefficient of 𝑑𝑑𝑑𝑑 = 0 −𝑁𝑁p 𝜎𝜎p 𝑆𝑆p −𝑁𝑁L 𝜎𝜎L 𝑆𝑆L 𝑁𝑁p = 𝑁𝑁L = 𝜎𝜎L 𝑆𝑆L 𝜎𝜎p 𝑆𝑆p Risk-‐‑neutral Pricing 𝑑𝑑𝑑𝑑 𝑡𝑡 = 𝛼𝛼 − 𝛿𝛿 𝑑𝑑𝑑𝑑 + 𝜎𝜎 𝑑𝑑𝑑𝑑(𝑡𝑡) 𝑆𝑆 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑡𝑡 = 𝑟𝑟 − 𝛿𝛿 𝑑𝑑𝑑𝑑 + 𝜎𝜎 𝑑𝑑 𝑍𝑍(𝑡𝑡) 𝑆𝑆 𝑡𝑡 𝑑𝑑𝑍𝑍 𝑡𝑡 = 𝑑𝑑𝑑𝑑 𝑡𝑡 + 𝜙𝜙𝜙𝜙𝜙𝜙 𝑍𝑍 𝑡𝑡 = 𝑍𝑍 𝑡𝑡 + 𝜙𝜙𝜙𝜙 True Measure Risk-‐‑neutral Measure 𝑍𝑍 𝑡𝑡 ~𝑁𝑁 0, 𝑡𝑡 𝑍𝑍 𝑡𝑡 ~𝑁𝑁 0, 𝑡𝑡 𝑍𝑍 𝑡𝑡 ~𝑁𝑁 −𝜙𝜙𝜙𝜙, 𝑡𝑡 𝑍𝑍 𝑡𝑡 ~𝑁𝑁 𝜙𝜙𝜙𝜙, 𝑡𝑡
Proportional Portfolio 𝑥𝑥 ∶ percentage invested in Asset 𝐴𝐴 1 − 𝑥𝑥 ∶ percentage invested in Asset 𝐵𝐵 𝑊𝑊 𝑡𝑡 ∶ value of the portfolio at time 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑡𝑡 + 𝛿𝛿˚ 𝑑𝑑𝑑𝑑 𝑊𝑊 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑡𝑡 + 𝛿𝛿b 𝑑𝑑𝑑𝑑 + 1 − 𝑥𝑥 + 𝛿𝛿È 𝑑𝑑𝑑𝑑 = 𝑥𝑥 𝐴𝐴 𝑡𝑡 𝐵𝐵 𝑡𝑡 If Asset 𝐴𝐴 is a risk-‐‑fee asset, then: 𝑑𝑑𝑑𝑑 𝑡𝑡 + 𝛿𝛿b 𝑑𝑑𝑑𝑑 = 𝑟𝑟 𝑑𝑑𝑑𝑑 𝐴𝐴 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑡𝑡 + 𝛿𝛿˚ 𝑑𝑑𝑑𝑑 𝑊𝑊 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑡𝑡 = 𝑥𝑥 𝑟𝑟 𝑑𝑑𝑑𝑑 + 1 − 𝑥𝑥 + 𝛿𝛿È 𝑑𝑑𝑑𝑑 𝐵𝐵 𝑡𝑡 The Black-‐‑Scholes Equation 𝑟𝑟 − 𝛿𝛿 𝑆𝑆𝑉𝑉¿ + 0.5𝜎𝜎 p 𝑆𝑆 p 𝑉𝑉¿¿ + 𝑉𝑉" = 𝑟𝑟 − 𝛿𝛿 ∗ 𝑉𝑉 𝑟𝑟 − 𝛿𝛿 𝑆𝑆𝑆𝑆 + 0.5𝜎𝜎 p 𝑆𝑆 p 𝛤𝛤 + 𝜃𝜃 = 𝑟𝑟 − 𝛿𝛿 ∗ 𝑉𝑉 where 𝛿𝛿: dividend yield on stock 𝛿𝛿 ∗ : dividend yield on derivative S^a u 𝐸𝐸 𝑆𝑆 𝑇𝑇 ê = 𝑆𝑆 𝑡𝑡 ê 𝑒𝑒 ê {,5 }C.Äê ê,L ~ ê 𝐹𝐹",$ 𝑆𝑆 𝑇𝑇 = 𝐸𝐸 ∗ 𝑆𝑆 𝑇𝑇 ê u = 𝑆𝑆 𝑡𝑡 ê 𝑒𝑒 ê *,5 }C.Äê ê,L ~ $," 𝛿𝛿 ∗ = 𝑟𝑟 − 𝑎𝑎 𝑟𝑟 − 𝛿𝛿 − 0.5𝑎𝑎 𝑎𝑎 − 1 𝜎𝜎 p 𝛾𝛾 = 𝑎𝑎 𝛼𝛼 − 𝑟𝑟 + 𝑟𝑟
$,"
𝑑𝑑𝑆𝑆 ê = 𝑎𝑎 𝛼𝛼 − 𝛿𝛿 + 0.5𝑎𝑎 𝑎𝑎 − 1 𝜎𝜎 p 𝑑𝑑𝑑𝑑 + 𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑 𝑡𝑡 𝑆𝑆 ê
INTEREST RINTEREST ATE MODELS RATE MODELS General Ito’s Process for 𝒓𝒓 𝒕𝒕 and 𝑷𝑷 𝒓𝒓, 𝒕𝒕, 𝑻𝑻 𝑑𝑑𝑑𝑑 𝑡𝑡 = 𝑎𝑎 𝑟𝑟 𝑑𝑑𝑑𝑑 + 𝜎𝜎 𝑟𝑟 𝑑𝑑𝑑𝑑 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑟𝑟, 𝑡𝑡, 𝑇𝑇 = 𝛼𝛼 𝑟𝑟, 𝑡𝑡, 𝑇𝑇 𝑑𝑑𝑑𝑑 − 𝑞𝑞 𝑟𝑟, 𝑡𝑡, 𝑇𝑇 𝑑𝑑𝑑𝑑 𝑃𝑃 𝑟𝑟, 𝑡𝑡, 𝑇𝑇 where 1 1 𝛼𝛼 𝑟𝑟, 𝑡𝑡, 𝑇𝑇 = 𝑎𝑎 𝑟𝑟 ⋅ 𝑃𝑃* + 𝜎𝜎 𝑟𝑟 p ⋅ 𝑃𝑃** + 𝑃𝑃" 𝑃𝑃 2 𝑃𝑃* 𝑞𝑞 𝑟𝑟, 𝑡𝑡, 𝑇𝑇 = − 𝜎𝜎 𝑟𝑟 𝑃𝑃 Sharpe Ratio 𝛼𝛼 𝑟𝑟, 𝑡𝑡, 𝑇𝑇 − 𝑟𝑟 𝜙𝜙 𝑟𝑟, 𝑡𝑡 = 𝑞𝑞 𝑟𝑟, 𝑡𝑡, 𝑇𝑇 Partial PDE for Bond 1 𝑟𝑟𝑟𝑟 = 𝜎𝜎 𝑟𝑟 p 𝑃𝑃** + 𝑎𝑎 𝑟𝑟 + 𝜎𝜎 𝑟𝑟 𝜙𝜙 𝑟𝑟, 𝑡𝑡 𝑃𝑃* + 𝑃𝑃" 2 Risk-‐‑neutral Process 𝑑𝑑𝑑𝑑 𝑡𝑡 = 𝑎𝑎 𝑟𝑟 + 𝜎𝜎 𝑟𝑟 ⋅ 𝜙𝜙 𝑟𝑟, 𝑡𝑡 𝑑𝑑𝑑𝑑 + 𝜎𝜎 𝑟𝑟 𝑑𝑑 𝑍𝑍(𝑡𝑡) 𝑑𝑑𝑑𝑑 𝑟𝑟, 𝑡𝑡, 𝑇𝑇 = 𝑟𝑟 𝑑𝑑𝑑𝑑 − 𝑞𝑞 𝑟𝑟, 𝑡𝑡, 𝑇𝑇 𝑑𝑑 𝑍𝑍(𝑡𝑡) 𝑃𝑃 𝑟𝑟, 𝑡𝑡, 𝑇𝑇 𝑑𝑑𝑍𝑍 𝑡𝑡 = 𝑑𝑑𝑑𝑑 𝑡𝑡 − 𝜙𝜙 𝑟𝑟, 𝑡𝑡 𝑑𝑑𝑑𝑑
𝑍𝑍 𝑡𝑡 = 𝑍𝑍 𝑡𝑡 −
"
C
𝜙𝜙 𝑟𝑟, 𝑠𝑠 𝑑𝑑𝑑𝑑
Delta-‐‑Gamma-‐‑Theta Approximation 1 𝑃𝑃 𝑡𝑡 + ℎ, 𝑇𝑇 − 𝑃𝑃(𝑡𝑡, 𝑇𝑇) = 𝛥𝛥𝛥𝛥 + 𝛤𝛤𝜖𝜖 p + 𝜃𝜃ℎ 2 𝜖𝜖 = 𝑟𝑟 𝑡𝑡 + ℎ − 𝑟𝑟(𝑡𝑡) 𝜕𝜕𝜕𝜕 𝜕𝜕 p 𝑃𝑃 𝜕𝜕𝜕𝜕 𝛥𝛥 = 𝛤𝛤 = p 𝜃𝜃 = 𝜕𝜕𝑟𝑟 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
Copyright © 2016 Coaching Actuaries. All Rights Reserved. 3
Rendlemen-‐‑Bartter Model 𝑑𝑑𝑑𝑑 𝑡𝑡 = 𝑎𝑎 ⋅ 𝑟𝑟 𝑡𝑡 𝑑𝑑𝑑𝑑 + 𝜎𝜎 ⋅ 𝑟𝑟 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑡𝑡 1 𝑟𝑟 𝑡𝑡 ~𝑁𝑁 𝑚𝑚 = 𝑎𝑎 − 𝜎𝜎 p 𝑡𝑡, 𝑣𝑣 p = 𝜎𝜎 p 𝑡𝑡 ln 𝑟𝑟 0 2 L u
𝑟𝑟 𝑡𝑡 = 𝑟𝑟 0 𝑒𝑒 ê,p~ "}~£ " Characteristics: • No mean-‐‑reverting • 𝑟𝑟 cannot go negative • Volatility varies with 𝑟𝑟 Vasicek Model 𝑑𝑑𝑑𝑑 𝑡𝑡 = 𝑎𝑎 𝑏𝑏 − 𝑟𝑟 𝑡𝑡 𝑑𝑑𝑑𝑑 + 𝜎𝜎 𝑑𝑑𝑑𝑑 𝑡𝑡 𝑃𝑃 𝑟𝑟, 𝑡𝑡, 𝑇𝑇 = 𝐴𝐴 𝑡𝑡, 𝑇𝑇 ⋅ 𝑒𝑒 ,È ",$ ⋅* 𝐴𝐴 𝑡𝑡, 𝑇𝑇 = Don" t bother 1 − 𝑒𝑒 ,ê($,") 𝐵𝐵 𝑡𝑡, 𝑇𝑇 = 𝑎𝑎 $,"|ê = 𝑎𝑎 𝑞𝑞 𝑟𝑟, 𝑡𝑡, 𝑇𝑇 = 𝐵𝐵 𝑡𝑡, 𝑇𝑇 ⋅ 𝜎𝜎 Yield on infinite bond: 𝜎𝜎𝜎𝜎 1 𝜎𝜎 p 𝑟𝑟 = 𝑏𝑏 + − 𝑎𝑎 2 𝑎𝑎 p Useful facts: • 𝐴𝐴 𝑡𝑡 + 𝑐𝑐, 𝑇𝑇 + 𝑐𝑐 = 𝐴𝐴 𝑡𝑡, 𝑇𝑇 𝐵𝐵 𝑡𝑡 + 𝑐𝑐, 𝑇𝑇 + 𝑐𝑐 = 𝐵𝐵 𝑡𝑡, 𝑇𝑇 • 𝜙𝜙 𝑟𝑟, 𝑡𝑡 is a constant. Characteristics: • Mean-‐‑reverting • 𝑟𝑟 can go negative • Volatility does not vary with 𝑟𝑟
Cox-‐‑Ingersoll-‐‑Ross Model 𝑑𝑑𝑑𝑑 𝑡𝑡 = 𝑎𝑎 𝑏𝑏 − 𝑟𝑟 𝑡𝑡 𝑑𝑑𝑑𝑑 + 𝜎𝜎 𝑟𝑟 𝑑𝑑𝑑𝑑 𝑡𝑡 𝑃𝑃 𝑟𝑟, 𝑡𝑡, 𝑇𝑇 = 𝐴𝐴 𝑡𝑡, 𝑇𝑇 ⋅ 𝑒𝑒 ,È ",$ ⋅* 𝐴𝐴 𝑡𝑡, 𝑇𝑇 = Don" t bother 𝐵𝐵 𝑡𝑡, 𝑇𝑇 = Don" t bother 𝑞𝑞 𝑟𝑟, 𝑡𝑡, 𝑇𝑇 = 𝐵𝐵 𝑡𝑡, 𝑇𝑇 ⋅ 𝜎𝜎 𝑟𝑟 𝜙𝜙 𝜙𝜙 𝑟𝑟, 𝑡𝑡 = ⋅ 𝑟𝑟 𝜎𝜎 Useful facts: • 𝐴𝐴 𝑡𝑡 + 𝑐𝑐, 𝑇𝑇 + 𝑐𝑐 = 𝐴𝐴 𝑡𝑡, 𝑇𝑇 𝐵𝐵 𝑡𝑡 + 𝑐𝑐, 𝑇𝑇 + 𝑐𝑐 = 𝐵𝐵 𝑡𝑡, 𝑇𝑇 • 𝜎𝜎 𝑟𝑟 ∝ 𝑟𝑟 • 𝜙𝜙(𝑟𝑟, 𝑡𝑡) ∝ 𝑟𝑟 Characteristics: • Mean-‐‑reverting • 𝑟𝑟 cannot go negative • Volatility varies with 𝑟𝑟
Duration-‐‑hedging To duration-‐‑hedge a 𝑇𝑇p -‐‑year bond with 𝑇𝑇L -‐‑year bond: 𝑇𝑇p − 𝑡𝑡 ⋅ 𝑃𝑃 𝑡𝑡, 𝑇𝑇p 𝑁𝑁 = − 𝑇𝑇L − 𝑡𝑡 ⋅ 𝑃𝑃 𝑡𝑡, 𝑇𝑇L Delta-‐‑hedging To delta-‐‑hedge a 𝑇𝑇p -‐‑year bond with 𝑇𝑇L -‐‑year bond: 𝛥𝛥' ",$u 𝑁𝑁 = − 𝛥𝛥' ",$t
Black-‐‑Derman-‐‑Toy 1. Effective interest rates 2. 𝑝𝑝∗ = 0.5 3. The ratio between two consecutive nodes is 𝑒𝑒 p~$ x 𝜎𝜎" : short-‐‑term volatility 𝑦𝑦 1, 𝑇𝑇, 𝑟𝑟f 1 Yield volatility $ = ⋅ ln 𝑦𝑦 1, 𝑇𝑇, 𝑟𝑟G 2 ℎ Forward Price 𝑃𝑃" 𝑇𝑇, 𝑇𝑇 + 𝑠𝑠 = 𝐹𝐹",$ 𝑃𝑃 𝑇𝑇, 𝑇𝑇 + 𝑠𝑠 𝑃𝑃" 𝑡𝑡, 𝑇𝑇 + 𝑠𝑠 𝑃𝑃" 𝑡𝑡, 𝑇𝑇 Black’s Formula 𝐶𝐶 = 𝑃𝑃 0, 𝑇𝑇 𝐹𝐹 ⋅ 𝑁𝑁 𝑑𝑑L − 𝐾𝐾 ⋅ 𝑁𝑁 𝑑𝑑p 𝑃𝑃 = 𝑃𝑃 0, 𝑇𝑇 𝐾𝐾 ⋅ 𝑁𝑁 −𝑑𝑑p − 𝐹𝐹 ⋅ 𝑁𝑁 −𝑑𝑑L 𝐹𝐹 + 0.5𝜎𝜎 p 𝑇𝑇 ln 𝐾𝐾 𝑑𝑑L = 𝜎𝜎 𝑇𝑇 𝑑𝑑p = 𝑑𝑑L − 𝜎𝜎 𝑇𝑇 𝑃𝑃C (0, 𝑇𝑇 + 𝑠𝑠) 𝐹𝐹 = 𝐹𝐹C,$ 𝑇𝑇, 𝑇𝑇 + 𝑠𝑠 = 𝑃𝑃C (0, 𝑇𝑇)
=
𝜎𝜎 =
0 < 𝑡𝑡 ≤ 𝑇𝑇
Caplet At time 𝑇𝑇, the value of 𝑇𝑇 + 1 -‐‑year caplet max 0, 𝑅𝑅$ − 𝐾𝐾' = × Notional 1 + 𝑅𝑅$ A caplet is equivalent to 1 + 𝐾𝐾' puts with strike L . price
www.coachingactuaries.com
𝑉𝑉𝑉𝑉𝑉𝑉 ln 𝐹𝐹C,$ 𝑇𝑇, 𝑇𝑇 + 𝑠𝑠 𝑡𝑡
L}O )
Copyright © 2016 Coaching Actuaries. All Rights Reserved. 4 Copyright © 2016 Coaching Actuaries. All Rights Reserved. 4 Personal copies permitted. Resale or distribution is prohibited.
