Formulario Derivative Securities Pricing

St-Stra

R

Discretely compounded returns = St 1

-

Continuously returns In (st)

R In (St 1)

= - -

el

St St-1

= r)

SEE (1

St + I

Variance -

for At

5

s At .

= =

Tiny Bearly St

* : 152

5 E

Saily

yoly = 50

= .

252 f

It

Port Mp S

+ Ps

: t

= , , Upto

RiSkLess DAT ht

0 +

=

=

AS

↑ p

,

-

=

st

Ito’s Lemma + Taylor series

Process

FOLLOW AN 1TO E[5St]

= At

5 :

1

0 .

ITO'S LEMMA

Forward Price Process: Ito’s Lemma

TO's Price

1 Processes

STORk

ketiz) 1 o

Black-Scholes: European call

on non-dividend paying stocks the Option

DELTA of

PROBABILITY TO

ITM

BE

Value of a Forward contract at time 0: Formula

B S

early

American call NDP never .

IF

American put NDP Numerical Procedure

LWAYS

A ITM

EARLY DEEP

European DP BS Pr(D)

+

ret

=

PV(Dir) - Dive

= it dividend be the

American call DP large to before

exercise

I especially

and

divident date

to

if close T

BOUND

LOWER

Put-Call parity (dividend)

Early exercise convenient if

.

· ↑ DIVIDEND

1

MORE THAN

WITH :

~ I

I B

a

-- O

MATURITY/ til

NEXT DIVIDEND til is

Index options as portfolio insurance ·

J

↓ ↓

portfolio INDEX -M

VALUE VALUE

V INDEX

NO OPTIONS

PUT MULTIPLIER

1

=

Insurance

To Have EXPLICIT)

SIF NOT CAPM (Rm 2f)

-p 28 B

+

= . -

BS to stocks which pay a dividend yield D P PARITy

C

-

↳ Sot

Cake-Ep +

European index options:

X PARITY

PUT-CA -

ke Foe

p

c +

+ p) er +

Fo (c

k +

= -

Relationship Forward-Spot (Investment + conversion or conversion +investment)

O

Pricing European currency options like

VI is a

yield

Dividend Forward

PARITY

P C

- -T -

-

Soe + ke

c

p + =

Futures and Forward on currencies

Generalisation binomial tree—> Ex: long Delta futures and short 1 derivatives

=> &

·

2

Valuing European future options BLACK'S

MODEL

European future options and spot

F ST options are equivalent when future

T

T = contracts mature at the same time of the

, option

Futures style option

=

- Delta hedging S

As c

~ = :

- dividend

NJoe) call

Delta european non

a

on

=

Delta portfolio

Eq Si

Ap :

= 2 gioi Ap

=

ns = =

=

elta neutral Portfolio

* Ita If hedging

a

Gamma hedging Gamma

Neutral

(1 )

neutral

For A portf =

.

Vega neutral portfolio

Delta real—> volatility not constant

·

Ars A real because

> Io when

Timp SI

CDS spread Spread

CDS

YTM RF S

=

- bond

Short REPO Long

cas =

PD)

St

St =

= -

- =

Street

St =>

= Present value on 1 bp paid on the premium Change in price of a CDS for a 1 bp

leg until default or maturity increase in spread

Mark to market CDS

BINARY

CDS P(t T)

Price ZGB , (T t)

= -

Disc FACT. : e

. Black’s Model for European bond

options CURRENT

ACCRUED Bona

included

i Dirty

-

Pu coupons to

be paid during

the life of the

Forward

Forward bond and forward yield

⑧ Yield to

Vols

From

-Fon kand Price Vols

FORWARD yield .

val mobility

prica yield

Today Fonw .