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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