A&J Flashcards for Exam MFE/3F Spring Alvin Soh

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1 A&J Flashcards for Exam MFE/3F Spring 2010 Alvin Soh

2 Outline DM chapter 9 DM chapter 10&11 DM chapter 12 DM chapter 13 DM chapter 14&22 DM chapter 18 DM chapter 19 DM chapter 20&21 DM chapter 24 Parity and Other Option Relationship Binomial Option Pricing The Black-Scholes Formula Market-Making and Delta-Hedging Exotic Options The Lognormal Distribution Monte Carlo Valuation Brownian Motion and Itô s Lemma Interest rate models

3 DM Chapter 9- Parity and Other Option Relationship Put-call parity for European Options

4 DM Chapter 9- Parity and Other Option Relationship rt,, 0, T C K T P K T e F K or,, 0 T rt C K T P K T S e Ke or,, C K T P K T S PV Div Ke 0 0, T rt

5 DM Chapter 9- Parity and Other Option Relationship Put call parity for American Options

6 DM Chapter 9- Parity and Other Option Relationship C K PV Div S P C PV K S or P S PV K C P S PV Div K

7 DM Chapter 9- Parity and Other Option Relationship Create synthetic stock by applying put-call parity when the stock pays discrete dividends.

8 DM Chapter 9- Parity and Other Option Relationship,, S C K T P K T PV Div Ke 0 0, T This means that a stock is equivalent to: 1. Purchasing a $K-strike call option; 2. Selling a $K-strike put option; rt PV Div Ke at risk-free rate. 3. Lend 0, T rt

9 DM Chapter 9- Parity and Other Option Relationship Create synthetic stock by applying put-call parity when the stock pays continuous dividend.

10 DM Chapter 9- Parity and Other Option Relationship T S0 e C K, T P K, T Ke This means that a stock is equivalent to: T e 1. Purchasing unit of $K-strike call option; T 2. Selling e unit of $K-strike put option; rt 3. Lend Ke at risk-free rate. rt

11 DM Chapter 9- Parity and Other Option Relationship Given that K1 K2 K3, C K C K C K C K K K K K P K P K P K P K K K K K To exploit the mispricing, 1. Sell n units of CK2 or 2 K3 K2 2. Buy n K3 K1 K2 K1 3. Buy n K K 3 1 P K ; units of 1 units of 3 C K or P K ; C K or 1 P K 3.

12 DM Chapter 10 & 11- Binomial Option Pricing The amount of money lent, B to replicate an European option under binomial pricing model

13 DM Chapter 10 & 11- Binomial Option Pricing B uc dc u d rh d u e

14 DM Chapter 10 & 11- Binomial Option Pricing The risk-neutral probability that the underlying stock price will move to on the date of expiry of the option Su

15 DM Chapter 10 & 11- Binomial Option Pricing p * e d u d r h

16 DM Chapter 10 & 11- Binomial Option Pricing 1. One plus the rate of capital gain on the stock if it goes up in binomial pricing model, u ; 2. One plus the rate of capital loss on the stock if it goes down in binomial pricing model, d.

17 DM Chapter 10 & 11- Binomial Option Pricing u e r h h d e r h h

18 DM Chapter 10 & 11- Binomial Option Pricing The value of the option at a node for: 1. An American call; 2. An American put.

19 DM Chapter 10 & 11- Binomial Option Pricing 1. * * Call S, K, t max K S, Cup Cd 1 p 2. * * Put S, K, t max K S, Pup Pd 1 p

20 DM Chapter 12- The Black Scholes Formula The assumptions of Black-Scholes formula

21 DM Chapter 12- The Black Scholes Formula 1. The continuously compounded returns on the stock are normally distributed and independent over time; 2. The volatility of continuously compounded return is known and constant; 3. The future dividends are known, either as a dollar amount or as a fixed dividend yield; 4. The risk-free rate is known and constant; 5. There are no transaction costs or taxes; 6. It is possible to short-sell costlessly and borrow at the risk-free rate.

22 DM Chapter 12- The Black Scholes Formula Call option premium under the assumption of Black-Scholes Framework, given that the stock pays dividend as a fixed dividend yield

23 DM Chapter 12- The Black Scholes Formula where 1. d 1 T Se ln rt Ke 2. d2 d1 T rt T C Se N d1 Ke N d T

24 DM Chapter 14 & 22- Exotic Options CallOnCall PutOnCall BSCall xe rt 1 CallOnPut PutOnPut BSPut xe rt 1

25 DM Chapter 14 & 22- Exotic Options Black Scholes pricing formula for Gap options: 1. Call option with strike price K 1 and trigger price K 2 ; 2. Put option with strike price K 1 and trigger price K 2.

26 DM Chapter 14 & 22- Exotic Options where 1. T rt 1. Call Se N d1 K1e N d2 rt T 2. Put K e N d Se N d d S 2 ln r 0.5 K2 T d d T T

27 DM Chapter 19- Monte Carlo Valuation 1 2 h hzn 2 1 or S S e ST nh n h S e 0 n T h Zi 2 n i1

28 DM Chapter 19- Monte Carlo Valuation Monte Carlo Valuation of plain vanilla options: 1. Call option; 2. Put option.

29 DM Chapter 19- Monte Carlo Valuation N 2 max r rt C e S h hz i 0e K,0 N i N 2 max r rt P e K S h hz i 0e,0 N i1

30 DM Chapter 20 & 21- Brownian Motion, Itô s Lemma Definition of Brownian Motion

31 DM Chapter 20 & 21- Brownian Motion, Itô s Lemma 1. Z 0 0; 2. Z t s Z t N 0, s ; 3. Z t s1 Z t is independent of Z t Z t s 2 4. Zt is continuous; 5. A martingale: E Z t s Z t Z t. ;

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