MFE FORMULA SHEET.pdf

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

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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    

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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~

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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      

 

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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.  

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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 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕              

 

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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    

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𝑉𝑉𝑉𝑉𝑉𝑉 ln 𝐹𝐹C,$ 𝑇𝑇, 𝑇𝑇 + 𝑠𝑠 𝑡𝑡

L}O )

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