soa exam mfe flash cards

7/30/2019 SOA Exam MFE Flash Cards

1/21

Pricing American Call Options with One Discrete Dividend

Stock pays dividend D at time 1t

( ))1(,, )(, 111tTrrt

tD eKDKSCallOnPutKeSllAmericanCa +=

where

)1( )( 1tTreKDx =


2/21

Gap Option

o Gap options are options where the option payoff jumps at the price where the

option comes into the money. Put-Call Parity Holds.

o Thegap callpays1KS when 2KS> . 1K is the strike price (the price the

option holder pays at expiration to acquire the stock) and 2K is the payment

trigger (the price at which payment on the option is triggered).

( ) )()(,,,,,, 21121 dNeKdNSeTrKKSCrTT =

( ) )()(,,,,,, 21121 dNeKdNSeTrKKSPrTT +=

T

TeK

Se

d

rT

T

2

2

1

2

1ln +

=

Tdd = 12


3/21

Gap Option continued

Above: A gap option mustbe exercised when 1KS> for a call or 1KS< for a put.Since the owner can lose money at exercise, the term option is a bit of a misnomer.

This possible negative payoff is reflected in a lower option price.


4/21

Exchange Option

An exchange option, also called an outperformance option, pays off only if the

underlying asset outperforms some other asset, called the benchmark.

Exchange calls maturing T in periods provide the right to obtain one unit of risky asset

1 in exchange for one unit of risky asset 2.

tS = Price of risky asset 1 at time t

tK = Price of risky asset 2 at time t

S and K are the dividend yields of the assets.

S and K are the volatilities of the assets. is the correlation between the continuously compounded returns on the two assets.

The payoff of the option is [ ]tT KS ,0max .


5/21

Exchange Option Pricing

( ) )()(,,,,,21

dNeKdNSeTrKSCTT KS =

( ) )()(,,,,, 21 dNeKdNSeTrKSPTT KS +=

T

T

Ke

Se

d

T

T

K

S

2

12

1ln +

=

Tdd = 12

KSKS 222 +=

The volatility, , is the volatility of KS

ln over the life of the option.


6/21

Brownian Motion and Its Lemma

Stochastic Differential Equations

(Will reference as equation 20.1)

)()(

)(tdZdt

tS

tdS += where

o S(t) is the stock price.o dS(t) is the instantaneous change in the stock price.

o is the continuously compounded return on the stock.

o is the continuously compounded volatility.

o Z(t) is a normally distributed random variable that follows a process called

Brownian Motion.

o A stock obeying 20.1 follows a process called Geometric Brownian Motion.


7/21

Sharp Ratio

The Sharp Ratio for any asset is the ratio of the risk premium to volatility.

Sharp Ratio for Call =

r

Sharp Ratio for Put =

)( r

for

)()(

)(tdZdttS

tdS +=


8/21

Implications of Equation 20.1

1. If the stock price follows 20.1, the distribution S(T), given the current priceS(0), is lognormal, i.e.

( ) ( )

+ TTSTS 22 ,)

2

1()0(ln~)(ln

The assumption that the stock price follows geometric Brownian motion thus

provides a foundation for our assumption that the stock price is lognormally

distributed.

2. Geometric Brownian Motion allows us to describe thepath of the stock price.This is essential for barrier options which require us to compute the probability

that the price will hit the barrier.


9/21

Arithmetic Brownian Motion

Given )()( tdZdttdX += Then )(tX is normally distributed with

mean t= variance t2=

Probability [Stock Price is Less than )(tX given )0(X ] =

t

tXtXN

)0()(


10/21

Geometric Brownian Motion

Given )()()(

)(tdZdt

tS

tdS += Then

0

lnS

St is normally distributed

with mean t)2

1( 2 = variance t2=

Probability (Stock Price is Less than tS given 0S ) =


11/21

t

tS

S

N

t

2

0 2

1ln

Definition of Brownian Motion

A stochastic process is a random process that is a function of time.

Brownian motion is a stochastic process that is a random walk occurring incontinuous time, with movements that are continuous rather than discrete.o To generate Brownian motion, we would flip a coin infinitely fast and

take infinitesimally small steps at each point.

Let Z(t) represent the value of a Brownian motion at time t. Brownian motion isa continuous stochastic process, Z(t), with the following characteristics:

o Z(0) = 0

o Z(t+s) Z(t) is normally distributed with mean 0 and variances.


12/21

o Z(t+s1) Z(t) is independent of Z(t) Z(t- s2), wheres1, s2 > 0.

This means that non-overlapping increments are

independently distributed.

o Z(t) is continuous.

Z(t) is a martingale: a stochastic process for which [ ] )()()( tZtZstZE =+ . The process Z(t) is also called a diffusion process, it is continuous yet

uncertainty increases overtime.

Its Lemma

Given S(t) follows a Geometric Brownian Motion, )()( tSdZSdtdS += .Given a process C, that is a function of S and t, the dC follows Geometric Brownian

Motion defined as follows:

dtCdSCdSCdCtSSS

++= 2

2

1

Which is VERY similar to a Delta-Gamma-Theta approximation:


13/21

hCCtttht

++=+2

2

1

Multiplication Rules

( ) dtdZ =2 ( ) 02 =dt 0* =dZdt dtdZdZ ='*

Ornstein-Uhlenbeck Mean Reversion

)()]([)( tdZdttXtdX +=


14/21

Binomial Tree Models for Interest Rates

Spot Rates: interest rates available immediately.

Forward Rates: interest rates agreed upon now in an agreement to purchase a bond at

some future date.

( ) =+sTTPt , Price of Zero Coupon BondT = Time of Paymentt = Time of Agreement


15/21

T+ s = Time of Maturity

If t=T then this is a Spot Price. If t


16/21

Black-Derman-Toy Binomial Model


17/21

BDT Assumptions and Variable Definitions

Uses Effective Annual Yields, R, not continuously compounded yields, r.

Bond Price = ( ) nR +1 where n = years to maturity Volatilities are the volatilities of yields on zero coupon bonds after one year.


18/21

( )

( )hte

jttr

jttr1

2

21

21

;,

1;, =

+ ( )

( )

( )

( )1;,

2;,

;,

1;,

21

21

21

21

+

+=

+

jttr

jttr

jttr

jttr

1t = Column 1t

h = Time Interval

t = Volatility of 1-Period Yields in Period i. This is the volatility from the BDT tree.

T = The volatility of a T year bond after one year. This is the volatility from the table.

12 =

BDT Bond Price and Yield Volatility

[ ] =)(,, hrThP Price of Zero Coupon Bondh = Time of Valuation

T = Time of Maturityr(h) = short term interest rate


19/21

Yield of this Bond = [ ] =)(,, hrThy [ ] 1)(,,1

hThrThP

Then at time h the short term rate can take on the two values ur or dr .

Yield Volatility = T =[ ]

[ ]

d

u

rThy

rThy

,,

,,ln

2

1

Implied Forward Rates

1-Period forward rate in K years = 11

1 11, +

+= ++

K

KKK

y

yF

Caplets and Caps


20/21

The value of the cap is the value of the sum of the caplets.

The value of the yearn caplet is the only the value from that year, brought back using

risk-neutral probability and BDT interest rates.

Pricing Forward Rate Agreements (FRA)

FRA pays ),( TtR - AR at time T or [ ] tTA

TtR

RTtR+

),(1

),(at time t.

),( TtR is the prevailing interest rate for a loan taken at time t and paid at timeT.


21/21

AR is calculated so that the expected value of the agreement is 0.

1),0(

),0(=

TP

tPRA

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