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
