a&j flashcards for soa exam mfe cas exam 3f

A&J Flashcards for

Exam MFE/3F Spring 2010

Alvin Soh

Outline

A&J © 2010

DM chapter 9 Parity and Other Option Relationship

DM chapter 10&11 Binomial Option Pricing

DM chapter 12 The Black-Scholes Formula

DM chapter 13 Market-Making and Delta-Hedging

DM chapter 14&22 Exotic Options

DM chapter 18 The Lognormal Distribution

DM chapter 19 Monte Carlo Valuation

DM chapter 20&21 Brownian Motion and Itô’s Lemma

DM chapter 24 Interest rate models

DM Chapter 9- Parity and Other Option Relationship

A&J © 2010

Put-call parity for European Options


A&J © 2010

0,, , rT

TC K T P K T e F K or

0, , T rTC K T P K T S e Ke or

0 0,, , rT

TC K T P K T S PV Div Ke


A&J © 2010

Put –call parity for American Options


A&J © 2010

C K PV Div S P C PV K S or

P S PV K C P S PV Div K


A&J © 2010

Create synthetic stock by applying put-call parity when the stock pays discrete dividends.


A&J © 2010

0 0,, , rT

TS C K T P K T PV Div Ke

This means that a stock is equivalent to:

1. Purchasing a $K-strike call option; 2. Selling a $K-strike put option;

3. Lend 0,

rT

TPV Div Ke at risk-free rate.


A&J © 2010

Create synthetic stock by applying put-call parity when the stock pays continuous dividend.


A&J © 2010

0 , ,T rTS e C K T P K T Ke

This means that a stock is equivalent to:

1. Purchasing Te unit of $K-strike call option;

2. Selling Te unit of $K-strike put option;

3. Lend r T

Ke

at risk-free rate.


A&J © 2010

Given that 1 2 3K K K ,

1. 1 2 2 3

2 1 3 2

C K C K C K C K

K K K K

2. 1 2 2 3

2 1 3 2

P K P K P K P K

K K K K

To exploit the mispricing,

1. Sell n units of 2C K or 2P K ;

2. Buy 3 2

3 1

K Kn

K K

units of 1C K or 1P K ;

3. Buy 2 1

3 1

K Kn

K K

units of 3C K or 3P K

.

DM Chapter 10 & 11- Binomial Option Pricing

A&J © 2010

The amount of money lent, B to replicate an European option under binomial pricing model


A&J © 2010

rh d uuC dC

B eu d


A&J © 2010

The risk-neutral probability that the underlying stock price will move to uS

on the date of expiry of the option


A&J © 2010

*

r he d

pu d


A&J © 2010

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 .


A&J © 2010

1. r h h

u e

2. r h h

d e


A&J © 2010

The value of the option at a node for:

1. An American call; 2. An American put.


A&J © 2010

1. * *, , max , 1u dCall S K t K S C p C p

2. * *, , max , 1u dPut S K t K S P p P p

DM Chapter 12- The Black Scholes Formula

A&J © 2010

The assumptions of Black-Scholes formula


A&J © 2010

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.


A&J © 2010

Call option premium under the assumption of Black-Scholes Framework, given that the stock pays dividend as a fixed dividend yield


A&J © 2010

Black Scholes pricing formula for Gap options:

1. Call option with strike price 1K and trigger price 2K ;

2. Put option with strike price 1K and trigger price 2K .


A&J © 2010

1. 1 1 2

T rTCall Se N d K e N d

2. 1 2 1

rT TPut K e N d Se N d

where

1.

2

21

ln 0.5S

r TK

dT

2. 2 1d d T

DM Chapter 19- Monte Carlo Valuation

A&J © 2010

21

2

1

h hZ n

nh n hS S e

or

2

1

1 1

2

0

n

i

T h Z in

TS S e

DM Chapter 20 & 21- Brownian Motion, Itô’s Lemma

A&J © 2010

1. 0 0Z ;

2. 0,Z t s Z t N s ;

3. 1Z t s Z t is independent of 2Z t Z t s ;

4. Z t is continuous;

5. A martingale: |E Z t s Z t Z t .

a&j flashcards for soa exam mfe cas exam 3f

Documents