AI

Question

The forward price of $1,200 is worse for us if we want to buy a forward contract. To understand this, suppose the index after one year is $1,150. While we have already made money in part a) with a forward price of $1,100, we are still losing $50 with the new price of $1,200. As there was no advantage in buying either the stock or forward at a price of $1,100, we now need to be "bribed" to enter into the forward contract. We somehow need to find an equation that makes the two strategies comparable again. Suppose that we lend some money initially together with entering into the forward contract so that we will receive $100 after one year. Then, the payoff from our modified forward strategy is: $ST − $1,200 + $100 = $ST − $1,100, which equals the payoff of the "borrow to buy index" strategy. We have found the future value of the premium somebody needs us to pay. We still need to find out what the premium we will receive in one year is worth today. We need to discount it: $100/(1 + 0.10) = $90.91.

Put0call pariiy

The net cost of cost of buying the idnex using options must equal the cost of buying the index using a forard contracts

Buy

Two ends

Short selling

Which entails borriwng a secruity, selling it, making deividen payments to the secruity lender, and then returing it

Example: Buy and sell 100 shares of XYZ

XYZ: bid = $49.75, offer = $50, commission = $15 Buy: (100 x $50) + $15 = $5,015 Sell: (100 x $49.75) -$15 = $4,960 Transaction cost: $5015 -$4,960 = $55

Arbituage

a situation in which one can generate positive cash flow by simultaneously buying and selling related assets, with no net investment and with no risk free money!!!

Exicse

act of paying the tike price

Buy

add commison since its what you paid

buy

add commission since its buy total is what you lsot from buying

Credit risk

after you sell the stock the lender can hold the money you recieved form selling

QWitten

an oblaition to buyt the underlying asset a t fixed price if it ists advatnge to the option buyer to sell

Non-diversifiable risks

are reallocated

F-

ask

Pjsical

avoided due to sginfiacnt cots

puttut

bad if strike price is less than asset price since you can sell it for more in the money

Forward why itnerest

because the seller is missing out on itnerst, if it was preparid he could reisnnest so euqals prepaid fprward plus interset If preparid could have eanred that interet

Prepaid why no interest

because the seller is not missing out on any interest

Why is the forward price equal to that

because when the seller gets the money at the end could invsted buy ,missed out so charges prepadi *ERTF so he can get the interest he could have gotten with prepaid

market-makers

bid-ask spread

Markers

bid-offer

Forwatd

bidning agrrement to buy/selll an underlying asseta

1 step of short selling

borrow and sell an asset

Creating a sysnethic forwad without pruchasing options

borrowing and prushae shares

straddle and strangle both

both betting on high vaolity buying put and call

2

buy back and return the assets

transciton

buy-sell $4,125.00 − $4,075.00 = $50

straddle

buying a call and a put with the same stirke price and time to expaitions

Synymical long foeward

buying a call otpion and sell with the same underlying wit he each otpion have the same stikr price

insured

buying an option

For every one 1020

buyx950 and buy 1-x 1060

which is more epxenive

c) The same logic as in question 2.4 (c) applies. If we compare the two contracts, we see that the put option has a protection for increases in the price of the asset: If the spot price is above $50, the buyer of the put option can walk away and need not incur a loss. The buyer of the short forward incurs a loss and must meet her obligations. However, she has the same payoff as the buyer of the put option if the spot price is below $50. Therefore, the put option should be more expensive. It is this attractive option to walk away if things are not as we want that we have to pay for.

Buy 100 stocks and buy put option

call option unlimit loss at zero

out of the

calll wih a strike price greatr than the asset price and a put with pricess than

Strangle

can be used to reduce the highe premium cost

A bull spread or a bear spread

can never have an initial premium of zero b e we are buying the same number of calls (or puts) that we are selling and the two legs of the bull and bear spreads have different strikes. A zero initial premium would mean that two calls (or puts) with different strikes have the same price—and we know by now that two instruments that have different payoff structures and the same underlying risk cannot have the same price without creating an arbitrage opportunity

Symettic

cannot have a premium of zero

A symmetric butterfly spread

cannot have a premium of zero because it would violate the convexity condition of options.

insuring a short position

caps

Types of settlements

cash and physical

SP 500

cash setltled

Remember payoff with call and option

check if payoff postivie if negative then no

regualtory

circumaenav taxes and account ruels defer taxes on tyhe sale of stock

The peark of asymettric peak is

clsoer to the thigh stieke price than low

Credit risk in shorting selling

colletral and haircut

Credit risk in short selling

colletral and harit

Brokers

commison

you pay

commison twice

Brokers and market makers

commisons bid-ask spread

Spread

constiting of aonly calls or onlu puts in which options are pcurhcased and some writtenn

over the counter contracts

credit check, collateral, bank letter of credit

details

credit risk and secuity

Price of the prepaid forward contract is the same the

current stock price

undefed

dffered

spread

different stirke price

everything is in future so need to

discount

but you have to atht ebbging o

discount to presnt

The lender must be comepsnated for

dividends recieved

Paprale

doesnt meant cnaclceing out

if payoff is negative

dont go thoru

Markets make risk-sharing more

effiency

Buy

ends

forware and cash index are

euwalivanet, differe only

With the over-the-counter

even more

new secruites can be designed by using

exiting securites

the payoff of a short is

extactly the opposite of the l0ng

haircut

extra amount to protect the lender agianst your fialure to return the wi

fa<f-

fa<f-

F-

fb t- small will increase

fb>f+

fb>f+

index

find

nsuring a flong psition

floors

If we can price the fP then we can calute the price of a forward

for a forward cotnru

x axis shrt forward

foreard price

stock=

forward and bond

Why does stock=

forward becuase in a forward you need to pay forard price so to make up profit from abond

Difference between tock and forward

forward price

X axi long forward

forward price

shorttt forward

forward price-pot price sell it, buy it lower

short

forward-spot

Maximum profit and loss for options rember include

future ofvlaue of option premoum

General asset allocation:

futures overlay

why is excange rate risk

gartunees tranctions, requires cooleterla

rmeer maximma nd loss incldes

gave to icnkdye ci in clacualtuetion

put option

gives the the right but not the obligation

Call options

givs the owner the righ but not the obligation to buy

remnber maxm9me and loss

have to include cost

1200

have to offer 100 because if price is 3000 2000-1100=900 2000-1200=800 so lose

Insurance companies and individual communities/families

have traditionally helped each other to share risks

The distavange of stradle

high premiy

Straddle

high volaity

Different profitcc rors sment

imperect corrlwetion Cross-hedging with imperfect correlation

rEASONS WHY exchange traded CONTRACTS BECOME POPUALR

increased voality in oil prices, $,Euro

floors

insured long psotions

Butterfly sprads

insured witten straddle

short rebate is less than

interest rate

borrow need topay back

interest rates

The payoff of a short

is extactly h opposte mirrow image across the axmi

In thr absecne of didivneds the timing of delivery

is irrevlanet

The seller of a put option

is obligated to buy if asked

the seller of a put option

is obligated to buy if asked

For the case of continuous dividends, the forward premium

is simply the difference between the risk-free rate and the dividend yield:

calcualting commison for buy

its total what you dd for buy so add but for sell susbtract since its hwta you gained from selling

sell

just sell

cancel out

just stop and do straight ine

contously compoundeed without anauual or monthly

just that

weird y

k3-k2/k3-k1

Premium equal ewacuiton if

k=F Buying a call and selling a put on the same underlying asset, with each option having the same strike price and time to expiration

A butterfly sptead insures against

large losses on a covered

Whuch neds to be larger

larger

what

lease-rate

Shortsllers typically

leave the short sale proceeds on despoits and long with a hair

Cash

less costly and more protiectioned

increase striek price call

less expensive

long forward loss

limited

short forward proift

limited

how to find maximum profit

look at profit for less than strike price and more than the stirke price combined profit lowest loss

high demand for borrowed wine

low interet

Writtne

low voality

reducued transciton

lower const way to take on a aptticulat for example expensive to buy stock and sell bonds but with detiaves can acheveie same economic reglatory artbituarge

Intermdiates

market makers trade

Interndaites

market-makers traders

Open itnerst

measure the numbe of contacts outstanding

Sell

middle

Selldsf what

middle

sell

middle

sell most in ASB

middle

sell what

middle

sell what in asmyritv

middle

With aytesmtic callauciton

middle stirke is what you want point to bppoint

maximium profit for a long option is

minus future value ofpt

Maximum profit for a long option

minus future vlaue of option premoun

stike price higher

more diffcult for spot price> strike price

why

more likely you can sell it for more than its worth in the market

Put option

more value when it depricates

Fully leveraged why interset

need to pay back interst

short forward

negative line through forward price

short

negative line, through forward price

Out of the money

negative payoff if exericed immedaility

Put Call prraity

net cost of buying the index using optins must equal the net of buying the diex using a forward contract

Pull cost

net cost of buying the index using options must equal the net cost of buying the index using a forward contract

Put call

net cost of buying uising options must equal the net cost o buying the index using index

synemtirc

no

When no dividend

no advantge in investign tohe

A part of commisosns and bid-as psread

no intial payments

Arbitutage

no risk free money

bull or spread

no zero premim

Call option

non binding right but not an obligTION TO BUY ANA SSET IN THE FUTRUE

iniitital cot

not future value

off market forward

not zoerm

Long forward

oblgiation

Forward

obligation

the rate of interest pays on the clleoratios depnded

on how many people want to borrow wine from the particulare and how willing

maximima dn loss check to see if you include

opion premoum

rember payoff neagvie

option

Expriation

option must be exercised or become worthless

Types of

options, future, and sawps

Intitla interest

our intital sum we gain is 49 but we gain 2%

iinterst is 2

our intitl what wegain si 2%, but we gain 50 at time T since its 50

stranglef

out of call lower preoium cost

Which has more of credit risk

over the counter contracts

fund

paid in full

1000

pay 100, more profit no matter the rice

remberb

payoff only iif ti would be in payoff

intreest rate implied

payoff/intital cost

ijterest rate

payoff/intital cost not future value

remmebr check

payoffRe

buy com

plus commission

Remeber when calcauting what you paid for but

posititve need to add commsison

forward

positive line, through forward price

In the money

postitve payoff if exercised immedaility

Long forward

pot price at expiration-forward rpcie

at a

predetermiend price

during a

predetermined time period

strike price increases put

premoum increases since its is more liekly ypu can seel it for more than its worth in the market

Forwatd price

prepaird froward *ERTF

call0-pull

present value

Dont care about

price

offer

price at which you cna buy

Call options

price set today

put

price to sell at sell hugh, then buy low want price to go down

remember wit maximum profit and loss chek to see if you invlufr

prmoum

SInce this sort of

profits are traded away quickly, and cannot persist, at equilibrium we can expect

If there no cost then it is

psotive

Collar

purhcase of a put option and the sale of call option higher higher stike price

Option premium incfrease with stirk esprice for

put option

Cot to carry

r-sS

cost ot carry

r-sdidivendT

antoher cost

rate and market rate what you could have earned, what you did earn

another cost

rate and rate of itnerest

stock

rebate

non diveagele

recolalced

F o,T

regulat

bond

repo

rate on collerbatl bond

repo rate

always need more of

right

maximum proit and loss

right note saying did you include option premium calculation write note to include option in maximum profit or loss

Puchcaed call

right to buy

Call

right, but not obligation

call

rightto buy want price to go go up so can buy and sell high want

how many uses

risk management, sepcualtion, redce transciton costs, regualtory arbitage

Uses

risk management, speculation, reduce transction costs, regualtory arbitarge

forwrd premum

risk-didiveinded annaulized-risk-dvidn

Perfect corrleation

same profit no matter

Natuonal value

scale valye od underyling

FT

sell

bid

sell

If prices fell int the meteamn

sell-buy

Synethic

selling a put otpion. buying a call

covered

selling an option

short selling

selling it, making didivend, and then respltieu

at a price

set today

when is the price

set today

rate paid on collebratio

short rebate

stock market

short rebate

short

should be higher for box

low demand

small free offer a rate clsoe to the market

everything is future

so discount

profit

sold-buy

strike price above

someone needs to pay us

why do you short sell

specualtion fiancning hedging

why short selling

specualtion, fiancing, hedging

forward payoff

spo

Prepaid ewuals

stock

Can be used to enter to infure

stock price

underlying

stock, itnerest rate, foreign exhcnage, commodity, other heating and cooling degree days, credit, real estate

Difference between straddle and steanlge

straddle-same strike price Strangle-out of call call and put strangle can be used to reduce high eprrmeum cost

Long forward

straight positve line, across forward price

buy 100 stock and write contract maximum profit

strike pirce-future value of stock price low profit at zero

call out of the pimone

strike price greater than asset price can buy it cheaper in market

put out of money

strike price less than asset, can sell it for more in the makret

Selling

subtract commison since its what gained from selling

One can offest eht risk of cofward of a rofward creating

synethif forward to offset a psotion int eha ctualy foward contraect

Strike price

the amount paid by the otpion buyer for the asset if he/she deciedes

Np because

the holder of the forward will not ereicve ediviend

stike

the price paid by the otpion buyer for asset if eh deicdes to exercise

to sell

the underling asset at a predertemined price during a predetermined time period

buy call

the underlyng asset a predtermiend price during a pretrmeineed period

white at the same elinatiing

the unpleasent downside for the buyer

Call options preserve

the upside potential

trinalge same ends

then regular6s

down

then staright

where it was change

then stragith ine after evens again

where it goes down

then straight

borrowing to buy the stock

thetic minices the effects of entinter into the stock

sell what

this example buy three 950-strike and seven 1,050-strike calls, and sell ten 1,020-strike calls. The profit diagram looks as follows:

Aysmteitc

three

Butterfly you buy assymetc

three

sell ten 1,020-strike calls. The profit diagram looks as follows:

three 950-strike and seven 1,050-strike calls, and

asysmeitc

title ed right

the eller of a call options is oblaigtated

to deliver

Option buyer decies wherether ot not

toexciers

buy

two ends

Does the forward price predict future price

udnerremations

froward

unfroear

long forward profit

unlimited

short forward loss

unlimited

payoff

value at experiation

Diversable risk

vanish

strike price below

we need tp pay someone

In order to construct the asymmetric butterfly, for every 1,020-strike call

we write, we buy λ 950-strike calls and 1 − λ 1,050-strike calls. Since we can only buy whole units of calls, we will in

tasnction cost difference

what uou pay and recieve

Short selling

when the price of an asset is expected to fall

Call option becomes more profitable

when the underlying asset apperciates in value

zero

which part gives you a profit of equation use to calculate break even price

higher strike price call

worse for us since harder for aet ot above

Insured written staddle

write a straddle add a strangle

buyer

written staddle strnalge

asysmeite

x35+(1-x)45=43

asset and option

x<strike price x>strike price payoff for each for each cost subtract for written, long -(intital cost-premium) for call, short -(premium-intitla cost)

Calcualting shares for peak

xk2+(1-x)k3

Ratio spread

yes can have a zero intitla premiu

Ratio spred

yes zero premiun

ratio spread

zero

At the money option

zero payoff if excesied immediality

zero cost collar has

zero width

Three methods to price prepaid forward

Pricing by analogy Pricing by discounted present value Pricing by arbitrage

sell bid

($40.95 × 100) − $20 = $4,075.00

buy ask

($41.05 × 100) + $20 = $4,125.00

Buy:

(100 x $50) + $15 = $5,015 buy total of what you paid so need to add commission so you calculate what you paid for buy sell need to subtract since tis what you gained from selling but buy is what you paid so plus commission

F+= what

(Sa0+2k)er^bT

f-=

(Sb-2k(eTT

written put option

-(p-s)+premium=0 -(p-s)

written call option

-(s-c)+premium=0 -(s-c) -(sT-c)_premium profit call price you buy at want to sell hierhger than in person

payoff written put

-Max[0,strike-spot]

to decrease price of a call option,

. By increasing the strike price of the call, the buyer of the call must wait for larger increases in the underlying index before the option pays off. This makes the call option less attractive, and the buyer of the option is only willing to pay a smaller premium. We receive less money, thus pushing the net option premium towards zero.

how to micic thelogn forward

1000 and the same time entering a long forward micices the effect of buying index

remmebr if it

20.212

Forward-

? plu interst

Fb, FP

A

risk premoum

A bull spread or a bear spread can never have an initial premium of zero because we are buying the same number of calls (or puts) that we are selling and the two legs of the bull and bear spreads have different strikes. A zero initial premium would mean that two calls (or puts) with different strikes have the same price—and we know by now that two instruments that have different payoff structures and the same underlying risk cannot have the same price without creating an arbitrage opportunity. A symmetric butterfly spread cannot have a premium of zero because it would violate the convexity condition of options.

Deriative

An agreement between two parties which has a value determined by the price of something else

Fa,F-

B

F-

BUY

F+

Bid

Strangles

Biying an out call and put with the same time to expairtoin

Basic tranctions

Brokers, market-makers

IF FP>S0

Buy stock, sell prepaid

End user

Corproations Investment managers Invesotrs

Forward premiun

Difference between current forward price and stock price

Buy most of

End

Buy what

Ends

What if there is diviend

FP=So0

Fiancial engineering principles

Facilitate hedging of existing positions Enable understanding of complex positions Allow for creation of customized products Render regulation less effective redner reuglation csutomezlied compelx exisitng

F+

Fb to large will decrease

Forward contract

Features and QUALITY DELVIERY CONJRERCIATES PRICE THE BUYER WILL PAY

If didvends

Fo<ST

Stock

Forward and bond

Basic deriatves contracts

Forward contracts call option put options

Short

Forward price-spot at expriation

Short forward

Forward price-spot price at expriation sell it, buy it a lower price

Stock payoff

Forward+bond

Forward=

Future value of prepaid foatd

Forward price=

Future value of prepaid forward

National value SP 500

Futures 250*index

Marign and market

Initial margin Maintenance margin (70 - 80% of initial margin) Margin call Daily mark-to-market

why

Intuitively, whenever the 35-strike put option pays off (i.e., has a payoff bigger than zero), the 40-strike and the 35-strike options also pay off. However, there are some instances in which the 40-strike option pays off and the 35-strike option does not. Similarly, there are some instances in which the 45-strike option pays off but neither the 40-strike nor the 35-strike pay off. Therefore, the 45-strike offers more potential than the 40- and 35-strike, and the 40-strike offers more potential than the 35-strike. We pay for these additional payoff possibilities by initially paying a higher premium. It makes sense that the premium is increasing in the strike price.

Cross fedl

Investing in index and doing opposite for foward

nONDITCT

Investors who do not care whether the stock goes up or down,

Ype of postions

Long psoiston short positons

Why bue an index contracy of ssynethigiing it susing inded

Lower tteanctions

Change

Mutltipleber by price change

Quanto

Nikie no exhcngreate

Financial futures on furuoward inuclude

On stocks and indexes On currencies on interset rates

Ceeating a ynethif foward

One can offset the risk of a forward by creating a synthetic forward to offset a position in the actual forward contract

Open inters

Open interest: total number of buy/sell pairs

Scualting on voality

Options can be used to create psotions that are nondirction with responect to udnerlying asset

Want to invest in sp 500 and want to invest in bonds

Own stock Short foward

Graphical resprents

Payoff diagrams profit diagrams

It needs to be

Premium and call is present but forward is future since you pay in the future so you need to discount

Equation

Premium-premium=PV(FV-K)

Foward=

Prepare future value

Cross geldougn

Risk management for stock-pickers

Prepaid=

S0, not ST

FP)T=

S0_PV

Prepaid=

S0_PV all didivends

Prepaid==

SO-stockE-RT

What payoff are your trying to replication

ST-F0,T

Interest recieved from lender on collateral

Scarcity decreases the interest rate Repo rate in bond markets Short rebate in the stock market

Wrriten stadle

Selling a call and put with samestke price

Fiancing

Short wale to borrow money

Forward+Bond

Spot price at expiration -$1,020 + $1,020= Spot price at expiration

Forward price

Spot price+Interst to carry the asset--asset lease rate

Examples of viality

Straddles, stangles, buyyer glyi

Fiancnail engineering

The construction of a financial product from other products

Futures prices vs. forwards price SP 500

The difference negligible especially for short-lived contracts Can be significant for long-lived contracts and/or when interest rates are correlated with the price of the underlying asset

Forward contracts

The feautes and quanity of the asset to be leivered Delivery logstics, uch as time, date, and place the price the buyer will pay the time of delivery

difference betwen

The forward contract has a zero premium, while the synthetic forward requires that we pay the net option premium With the forward contract, we pay the forward price, while with the synthetic forward we pay the strike price no otouim, sysnti