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Options. . call options put options writer. . K : exercise price C : P : S : F : . Long : - PowerPoint PPT Presentation

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• Options

• call options put options writer

• K : exercise priceC : P : S : F :

• Long : Long :

• St K at-the-money Stin-the-money Stout-of-the-money

• () () (premium) (Cert Pert )

• - Put-call paritya long call + a short putlong callshort putcall put -, Put-call parity

• Put-call parity

• Put-call parity: r*Put-call parity:

• (1999 FRM Exam Q.35)Put-call parity: A. B. C. D.

• (2002 FRM Exam Q.47)\$50\$140\$1002%5%\$140: A. \$77 B. \$10 C. \$90 D. \$81

• (2002 FRM Exam Q.25)6%\$20\$18\$4\$1.47()? A. \$2.00 B. \$2.53 C. \$14.00 D.

• Short Covered call: + short a callLong protective put: + long a putLong straddle(): long a call + long a putShort straddle: short a call + short a putStrangle: ( strangleout-of-money, straddle )

• spread bull spread: : 1. K1 2. K2 : C(S,K1) - C(S,K2) >0 ST > K2:

• (2001 FRM Exam Q.90)? A. B. C. D.

• (1999 FRM Exam Q.33) bull spread? A. =50 =55 B. =55 =50 C. =5=7 D. =50 =55

• (2000 FRM Exam Q.5)bullish spread: \$3\$30\$1.50\$40\$42bullish spread: A. \$8.50 B. \$9.00 C. \$9.50 D. \$12.50

• (2002 FRM Exam 42):\$7\$50\$4\$42\$2\$37\$33: A. \$1B. \$2 C. \$3D. \$4

• Put-call parity:

• () : St K :

• : : K St : ,

• (1999 FRM Exam 34)805%90: A. 14.61 B. 13.90 C. 10.00 D. 5.90

• (risk-neutral)Binomial process: r = 25% S1 = 150, C1 = 50 S0 =100 S2 = 50, C2 = 0

• p:

• Black-Scholes :1. 2. 3. 4. ( )

• geometric Brownian motion (GBM): (dt) =dt=2dt dz=0=dt

• : N(0,1)

• : N(d)

• put-call parity: : S=1005%K=100=20%?

• Call N(d1) = 59.77% c = \$59.77 - \$52.88 =\$6.89

• K(S > K)

• B-SMerton(1973) (q)B-S in-the-money (SK)K-S d1d2 N(d1)N(d2) K-S

• in-the-moneyBack(1976)

• F S: B-S(volatility)volatility (implied standard deviation, ISD)

• B-S KISDISD (volatility smile) 1987/10

• (2001 FRM Exam Q.91) = 100 = 110 = 10% = 0.5N(d1) = 0.457185N(d2) = 0.374163B-S: A. \$10.90 B. \$9.51 C. \$6.57 D. \$7.92

• (1998 FRM Exam Q.2)B-S: A. d1 B. d2 C. N(d1) D. N(d2)

• (binary optionsdigital options) Q I(x) (indicator variable)

• in-the-moneyN(d2) (STK STKQ)

• (barrier options) HS: (knock-out): SH

• Down-and-out call: S < HDown-and-in call: S < HUp-and-out call: S > HUp-and-in call: S > H

• Down-and-out calldown-and-in call: C = CDO + CDICDO CDI CH

• (Asian options): ST: volatility(/3)

• (1997 FRM Exam Q.10)Knock-out: A. Knock-out options B. Knock-out options C. Knock-out options D. Knock-out optionsgamma

• (2002 FRM Exam Q.19) =100? A. down-and-out :=90,=110 B. down-and-in :=90,=110 C. up-and-in: =110,=100 D. up-and-in: =110,=100

• : f : (S)

• f Taylor :

• delta gamma delta delta gammaTaylor : 1. : 2. ( exotic options) 3.

• (1999 FRM Exam Q.65)(vanilla)delta-gamma? A. Taylor deltagamma B. deltagamma C. deltagamma D. (A)(C)(B)

• (1999 FRM Exam Q.88) delta(portfolio) A. - B. C. D. B-S

• (2001 FRM Exam Q.79)100,000300,000 =50 = 49=20%=5%delta?() A. 65,000 B. 100,000 C. 21,000 D. 100,000

• (sensitivity) deltadelta

• delta: Gamma ()gamma(pdf)

• Call delta: at-the-money 0.5 in-the-money 1 out-of-the-money 0Put delta: at-the-money -0.5 in-the-money 1 out-of-the-money 0at-the-money

• gammaconvexity.convexitygammagammaconvexity:

• gamma: long call: > 0 > 0 long put: < 0 > 0 short call: < 0 < 0 short put: > 0 < 0

• (volatility) lambda( vegakappa) : (long) lambdaAt-the-mony lambdaLambda

• rho : :

• N(d2)=1

• : : :

• (passage of time) theta () (time decay)

• :

• Gamma () at-the-moneytheta () at-the-moneytheta

• GBM f(S ,t) Itos lemma (!) Taylor

• Greeks : f S dz :

• (GBM df), dz (immunized) B-S

• (y)

• greeks dd: Black-Scholes(partial differential equation, PDE) ( )

• PDEdeltaPDE=0 rf

• deltagamma ()theta( (time decay))straddle() deltagamma S

• Delta gammatheta

• (2001 FRM Exam Q.123) in-the-money Greeks ? A. Lambda (vega) B. Rho C. Gamma D. Delta

• (1998 FRM Exam Q.43) Greeks (long): A. deltavegarho B. vegarho C. deltavegagammarho D. deltavegagammathetarho

• (1998 FRM Exam Q.44) Greeks (short): A. deltavegarho B. vegarho C. deltavegagammarho D. deltavegagammathetarho

• (1998 FRM Exam Q.45) Greeks (long) straddle(): A. deltavegarho B. vegarho C. deltavegagammarho D. deltavegagammathetarho

• (1999 FRM Exam Q.39)(time decay)? A. in-the-money B. out-of-the-money C. at-the-money D.

• (1999 FRM Exam Q.38)(time value)? A. out-of-the-moneyat-the-money B. in-the-moneyat-the-money C. at-the-money out-of-the-moneyin- the-money D. at-the-money

• (1999 FRM Exam Q.56): (1) Gamma(2) Vega(3) theta(4) rho A. (2) B. (1)(2) C. D. (3)(4)

• (1998 FRM Exam Q.36)at-the-moneystraddle? 1. delta2. gamma3. vega4. theta5. A. (1)(2) B. (1) (2) (3) C. (1) (3) (4) (5) D. (1) (2) (3) (4) (5)

• (1998 FRM Exam Q.37)at-the-moneystraddle? 1. delta2. gamma3. vega 4. theta5. A. (1)(2) B. (1) (2) (3) C. (1) (3) (4) (5) D. (1) (2) (3) (4) (5)

• (2000 FRM Exam Q.76) vega gama? A. B. C. D.

• Dynamic hedgingB-S

• Delta (Delta)delta

• ()() Short

• : 1.( ) ! 2. ( 1987 long in put) 3. (loss-limit)

• GBM

• (asymmetric) (, symmetric)

• Long option: long gamma long right tailShort option: short gamma long left tail

• VaR VaR: (95% =1.64)

• VaR VaR

• (2001 FRM Exam Q.80)? A. gammadelta B. gammadelta C. gammadelta D.gamma delta

• (1997 FRM Exam Q.28):VaR=7.8% at-the-moneyVaR: A. \$39,000 B. \$39,000 C. \$78,000 D. \$78,000

• (1998 FRM Exam Q.27) deltagamma 100,000-50,000 \$2delta-gamma VaR: A. \$100,000 B. \$200,000 C. \$300,000 D. \$400,000