Derivatives Flashcards
OTC vs exchange-traded derivatives
OTC derivatives are traded directly between two parties with no centralized exchange or intermediary involved: customizable; many require to have central clearing house;
Exchange-traded derivatives are traded on a formal exchange is facilitated by the exchange;
standardaized contracts on date, quantity, delivery obligations; liquid; transparent; lower cost; central clearing=min counterparty risk by taking initial margin and MTM one day P&L for each side
forward contracts
customized, no active secondary market, specified asset and date;
FP set @inception, but no one pays until delivery date;
long, buy contract and buy the asset, pay forward price, received underlying (gain if underlying price increase); short, sell the contract and sell the asset, recieve forward price, pay underlying(underlying price fall, gain);
deliverable contract=seller give shares, buyer buy at FP;
cash settle=loser pay winner in the contract; buyer receive/pay (-FP+underlying spot price)unit; seller receive/pay (-underlying spot price+FP)unit;
owner of underlying can hedge with a derivatives; nonowners can speculate with derivatives
futures
like forward but standardized, exchange traded, active secondary market, require margin deposit, no counterparty risk, regulated, clearing house
vs
forwards otc, customized, default risk, no margin, less regulation unless central clearing;
specify quality, tick size, daily price limit;
tick size
min price fluctuation the contract will move;
contract size=$index points;
tick size=0.25index points
tick value=tick size*contract size
futures margin
initial margin deposited before trade made;
maintenance margin=min margin that triggers margin call;
if margin<maintenance margin, must deposit variation margin to restore initial margin or else position closed;
price=(1-initial margin)/(1-maintenance margin)
settlement price
closing price=avg price of trades at closing (last 30 seconds) to calc margin
spot price
current price of underlying for immediate delivery;
futures price converges to spot price over time as contract expires;
forward/future price>spot price=contango; <spot=backwarddated
@expiration, settlement price=spot price
price limits
exchange put limits on how much price can change each day e.g.+-20 cents vs previous close;
or even circuit breaks=if limit breached, triggers x minutes of trading halt or closes market (+-20%)
mark to market
process of adjusting margin balance in a futures each day for the change in futures price
swaps
2 legs, 1 pays floating payment, another pays fixed payment based on interest rate, index, bond or commodity;
payments netted, may or may not require margin, multiple settlement dates; customized, like series of forwards
credit default swap
protection buyer (short credit risk, interest rate risk through IR swap) make periodic coupons to protection seller (long credit risk), seller only pays if bond default;
greater PoD or LGD increase swap fixed payment and spread;
hedge credit risk by buy CDS to short credit risk;
if expect credit risk worsen, buy CDS now (less risk, protection cheaper) and sell CDS later, speculate credit quality;
options
long, option buyer pay premium to get the right to excercise strike price at future trade; buy call=right to buy (strike<stock price) v short call=obligation to sell;
short, option seller receive premium for obligation to sell if owner excercise; buy put=right to sell vs sell put=obligation to buy;
europuean excercise only @expiration;
american option excercise any time so worth at least as european options;
call option value
St>X=stock price>strike, excercise;
St<X, lapse;
call option intrinsic value prior to maturity/payoff=MAX (0, St − Strike);
[price that it need to be at] call option breakeven=(strike price+premium)
intrinsic value=value or payoff of option at expiration;
graph=x is stock price, y is payment (tick for call)
max profit=unlimited; max loss=premium;
exercise value put option is positive if the underlying price<exercise price;
exercise value=0 if the underlying asset price>=exercise price;
exercise value of an option cannot be negative;
put option value
[price that it need to be at] breakeven=(strike price-preimum); max loss=premium;
put option intrinsic value prior to maturity or payoff=MAX (0, Strike-St);
option premium=intrinsic value+time value=(X-St)+(volitility&time); always positive cuz its reward for risk;
time value>0 and decays overtime to 0 at expiry; out-of-the-money option has an exercise value of zero, its price is its time value
forward commitments vs contingent claims
futures, forward, swaps are forward commitments;
options and credit derivatives are contingent claims only one party can excise depending on event
benefits of derivatives
easier to get into short postions (vs short sale where have to borrow first then sell and buy back), lower transaction cost vs buying underlying directly on cash market (operaetional advanrage), smaller initial capital requirements higher leverage, greater liquidity, manages risk, improve efficiency of market prices;
price discovery function=use market data from deriv market to get insight about future market price:
expected implied volitility (derived from premium, t, spot, strike, rf), esimate of forward price vs spot merging, expected future interst rate changes
risks of derivatives
transparency=hard to understand risk exposures;
basis risk=the underlying of a derivative might not fully match a position being hedged in risk, expected value, expiration date and delivery date;
liquidity risk=mismatch of derivative CSAH FLOW with those of existing risk to be hedged e.g. if margin call, where to get cash?;
counterparty credit risk=OTC if not covered by clearing house or depend on derivative position e.g. option seller could default;
systemic risk=excess speculation (LEVERAGED) can have negative impact on financial institutions e.g. sold too much OTC CDS protection and cant pay back when recession caused ripple effect, price limit violation
short futures means you dont have underlying and will buy in future so airline short fuel and long futures contract to hedge risk; basis risk of maturity date difference
gold miner need gold (long gold), short gold futures to hedge risk;
buy futures, pay cash receive gold; short futures, get cash, pay gold
swap reduce duration risk of fixed rate debt with floating rate payer swap (pay floating, receive fixed and pay off fixed rate debt so left with floating payment only, 0 duration)
issuer pay floating receive fixed and pay off fixed debt, reduce duration risk (duration is sensitivity of bond to change in yield), floating rate bond price always the same cuz coupon changes with interest rate rest the par so duration risk is reduced
hedge accounting
use g&l on derivatives to offset effects of change in A&L values; Cash flow hedges protect against fluctuations in future cash flows, while fair value hedges protect against changes in the fair value of an asset or liability
cash flow hedges=Protects against variability in future cash flows in a recognized asset, liability, or a forecasted transaction.
e.g. Hedging the risk of interest rate fluctuations on a floating rate loan or hedging the risk of FX fluctuations on a forecasted foreign currency transaction.
(issuer have FRN and enter into contract fixed rate payer swap, issuer pay fix, receive floating; fixed rate so higher duration cuz price wont adjust to par);
fair value hedge=Protects against changes in the fair value of a recognized asset or liability, INVENTORY, or an unrecognized firm commitment FIXED-RATE BOND LIABILITY.
e.g. Hedging the risk of a decline in the fair value inventory or hedging the risk of a rise in the fair value of a liability.
(issuer receive fixed coupon and trade with floating rate payer swap, issuer pay floating and receive fixed so bond price can adjust to par with floating to offset changes in balance sheet value of fixed rate bond liability);
net value hedge=protect from changes in the value of foreign OPERATIONS, hedge foreign subsidary’s equity on parent’s balance sheet with currency forwards, if exchange rate changes, any gain or loss on net asset on foreign subsidray is offet by change on derivative on parents b/s;
fixed payer swap vs fixed receiver swap (not fixed rate payer)?????????
pay floating, receive fixed
vs
pay fix receive floating
bond hedge by swap out fixed rate for floating rate if credit quality constant, can reset coupon rate at par (cp is floating rate payer); cash flow and fx hedge by swap out floating rate for fixed rate
Floating-rate bonds have near-zero modified duration because coupons reset frequently, so prices are less sensitive to changes in interest rates.
Fixed-rate bonds have positive modified duration because it measures their sensitivity to interest rate changes
Duration of Zero coupon bond is equal to the life of the bond
long forward, price rise, gain, long risk;
sell forward, price rise, loss, sell risk;
long put to limit downside exposure=protective put (long stock long put);
buy call for leveraged long exposure, if loss, loss premium
- Fixed rate bond has a positive modified duration.
- Floating rate bond has almost 0 modified duration.
- Paying fixed meaning you pay out duration, and receive 0 duration so
pay fixed swap is negative duration because it is hedge against rising interest rates.
pay floating rate receive fixed rate, give 0 duration , receive positive duration=positive duration
arbitrage
two assets that have same payoffs but different prices; buy lower priced asset and sell higher priced asset gives riskless arbitrage profit;
action of arbitrage will close the gap, price converge to no-arbitrage price
swap is equivalent to a series of forward contracts
Ease of taking short positions with derivatives compared to underlying assets, and implied volatility is revealed by option prices, are two of the advantages of derivative instruments
but basis risk is the most accurate thing about derivatives
Using an interest rate swap to hedge changes in the value of a balance sheet liability is considered a fair value hedge. If the interest rate swap is used to convert the floating-rate payments on a bond liability to fixed-rate payments, it would be considered a cash flow hedge.
replicate
replicate a deriv by creating a portfolio with same payoffs
no-arbitrage 1yr forward price means
forward price at which payoff is the same at St-FP
F0(T)=S0(1+Rf)^T
no arbitrage price; fwd price=spot price at inception
F0(T)=forward price at time 0 expiring at T;
FP is equal to spot price compounded at risk-free rate;
sell asset forward for F0(T) at time 0 and get paid later equals buy asset now and hold till time T;
F0(T)/S0=(1+Rf)^T buy asset now sell forward at F0(T) will earn Rf at time T
if fwd price>arbitrage-free price, asset is overvalued, sell the asset and receive higher fwd price at time T
time 0: short fwd contract today at fwd priace, borrow at spot price @Rf and buy underlying asset;
time t: deliver asset and earn fwd price, repay spot price*(1+Rf)^T, difference is profit;
this arbitrage is called cash+carry arbitrage beceause buying asset in cash and hold it till delivery date;
short fwd contract price push down and buy underlying asset will push price up until arbitrage gone
if fwd price<arbitrage-free price, asset is undervalued, buy the asset at time T and short asset (no arb, fwd price=spot price with Rf=arbitrage price)
time 0, long forward and short underlying at spot price and invest at Rf rate;
time T, investment return is spot price*(1+Rf)^T, pay and receive underlying at fwd price;
reverse cash and carry=dont hold inventory, short
no arbitrage price with cost and benefit
F0(T)=[S0-PV0(benefit)+PV0(cost)]*(1+Rf)^T
or S0*(1+Rf)^T-FV(benefit)+FV(cost);
S0+cost-benefit=cost of buying the underlying;
PV0(cost)=PV of storage/insurance costs (monetary cost); PV0(benefit)=PV of cash flows (div, monetary benefits) and convinience yield (holding asset in supply shortage=non-monetary);
e^x plug in time and get growth.
ln(x) plug in growth and get the time it would take.
For example:
e^3 is 20.08. After 3 units of time, we end up with 20.08 times what we started with.
ln(20.08) is about 3. If we want growth of 20.08, we’d wait 3 units of time (again, assuming a 100% continuous growth rate).
continuous compounding PV FV
FV=Se^rT
PV=Se^(-rT)
continuous compounding with cost and benefit
F0(T)=S0e^(Rf+c-i)
no-arbitrage forward exchange rate (discrete) covered interest parity
forward(A/B)=[(1+RfA)/(1+RfB)]*spot(A/B)
A=price currency;B=base currency;
spot USD/EUR=1.1;
base currency (foreign) is trading at forward discount (depreciation) if interest rate is higher than price currency (domestic)
no-arbitrage forward exchange rate (continuous)
F0(1)=spot(A/B)*e^(RfA-RfB);
if B has higher Rf rate, due to higher inflation rate so B is depreciating
replicate is different from arbitrage e.g. investor can replicate a long forward on a stock that pays no dividends by
borrowing at the risk-free rate to buy the underlying
Borrowing S0 at Rf to buy the underlying asset at S0 has a zero cost and pays the spot price of the underlying asset minus the loan repayment of at time = T of S0(1 + Rf)T, which is the same payoff as a long forward at F0 = S0(1 + Rf)T, the no-arbitrage forward price.
Convenience yield refers to nonmonetary benefits from holding an asset. One example of convenience yield is the advantage of owning an asset that is difficult to sell short when it is perceived to be overvalued. Interest and dividends are monetary benefits. Storage and insurance are carrying costs.
vs PV of cash flow like div
Derivatives pricing models use the risk-free rate to discount future cash flows (risk-neutral pricing) because
they are based on constructing arbitrage relationships that are theoretically riskless=are based on portfolios with certain payoffs.
Risk-neutral pricing
=investors are indifferent to risk and determines the theoretical price of derivatives based on the expected future payoff discounted at the risk-free rate.
It is possible to profit from arbitrage when there are no costs or benefits to holding the underlying asset and the forward contract price is:
less than the future value of the spot price.
no arbitrage price is the forward price that forward has a zero value at initiation
forward contract value long
initial contract value=0: F0(T)=S0(1+Rf)^T; fwd price=future value of the current spot price, compounded at the risk-free rate over the term of the forward contract; contract value= Vt(0)=0;
value of forward during t contract life:
Vt(T)=St-F0(T)/[(1+R)^(T-t)] spot price of the asset minus the present value of the forward contract price;
value of forward at settlement T:
VT(T)=ST-F0(T) [long forward, spot price-present value of the forward price];
value of fwd contract at initiation=0 and will change as time goes because no MTM;
discount at remaining life is 3month not since inception, (T-t)=(1-0.75);
long party gain when underlying price rise;
short counterparty value is equal and opposite
long floating rate agreement (a forward)=pay fixed rate (rate agreed in contract), receive floating rate (forward rate=markets expectations of the floating rate)
at settlement:
if MRR>fixed, long receives [MRR-fixed]notional amount;
if MRR<fixed, long pays [fixed-MRR]NA
replicate FRA borrow for 90 days starting at 30 days from now
replicat in cash market: borrow for 120 days where you lend for 30 days
Floating rate agreenment (FRA) calc 360 days
lock in a fixed interest rate for a future period, for a payment; 2 legs in long: pay fixed rate in contract, receive floating rate in market foward rates; forward rate increase, value of FRA increase;
F3,6 price at time 0 means 6 month of FRA forward rate starting at month 3=3 is expiry of FRA, 6 is length of borrowing and lending;
(1+spot rate9)=(1+spot rate3)(1+F3,6), annualize (F3,6)/612;
price of FRA at time 0 is the price of interest rate on a 6 months loan starting at month 3 calced from 3month and 9 month floating rate;
dont know about market MRR until expiry of FRA at month 3 vs F3,6;
[MRR-F3,6]/126notional=payoff at time 9;
discount to month 3 using MRR because Fwd contract rate is not available
long FRA pay fixed (concern IR rise), short FRA pay floating (concern of IR fall); fixed leg tells the concern
forwards vs futures price & value
at initiation, foward price=futures price, both contract value at 0;
as time goes, fwd price does not change but underlying price changes so fwd contract value changes from 0 at initiation to (ST-F0(T)) cuz there are no MTM cash flows;
but future price will reset to settlement price daily, after MTM cash flow netted to margin (one day of profit loss settled), value of futures contract remains at 0 daily
forwards vs futures price and interest rates
if interest rates are positively correlated with underlying, asset value rise as well, due to cost of carrying model, futures price goes up too.
prefer long futures than a forward because if IR increase underlying price increase, futures price increase, more margin added to account, can reinvest margin profit at higher IR; if IR decrease underlying price fall, futures price fall, loss posted to margin, if need variation margin to prevent margin call, can borrow at lower IR;
higher rate when lending, lower rate at borrowing so prefer futures over forwards;
short term interest rate future (STIR)
based on depositing from the expiry date of future;
implied fwd rate (forward MRR) computed same way as FRA but priced differently:
FRA price=annualized implied fwd rate;
STIR price using IMM index convention=(100-annualized forward rate aka implied interest rate or MRR*100) (which is the fwd interest rate from expiry to end of the borrowing and end period F3,6);
if forward rate increase, STIR price goes down, long STIR loss; if fwd rate decrease, STIR price increase, long STIR gain;
if long IR futures, decrease in annualized fwd rate, value of futures increase;
long FRA (pay fixed, receive floatiing), increase fwd rate increase MRR [MRR>fixed] so increase value of FRA increase;
to hedge a borrowing, concern of rising IR, long FRA (pay fixed) or short IR future (value increase);
to hedge an investment, concern of falling IR, short FRA (receive fixed) or long IR future
an interest rate swap is not the same as a floating rate agreement (FRA). While both involve interest rates, swaps are a derivative contract to exchange interest payments, while FRAs are agreements to lock in a future interest rate for a specific period
FRA is discounted back
futures calc
calc payoff on each bps change: 0.0001notional/2 if semiannual;
new deposit price=annualized ratenotional/2;
profit in future from change in rates=payoff*change in bps;
add them together equal notoinal*old rate, guaranteed to make this amount regardless to what happened to interest rate at delivery date;
fwd contract payoff is nonlinear (FRA) but futures contract is.
convexity of forward using FRA as e.g.
fwd contract payoff is nonlinear (FRA) but futures contract is.
FRA value=[(MRR-fixed rate)*notional]/(1+MRR)
when MRR (interest rate, forward rate) increase, gains in FRA value but discount rate (MRR) also increase, dampening the gain;
when MRR decrease, losses in FRA value, smaller discount rate amplify the loss;
gain to long<loss to long;
long FRA is losing because of positive convexity because long party is long interest rate not long bonds; if long bond, convexity is good to have;
hence, loss from rate decrease are larger than gains from MRR increase due to discounting;
for short FRA, gains from interest rate decrease is greater than losses from interest rate increase (IR increase, loss, less less, IR decrease, gain, more gain);
convexity is small for short-dated FRA but larger for longd-dated
centrally cleared OTC deriv contract most concerned with
not interest rate not counterparty credit risk but systemic risk
novation
clearing house taking the other side of the futures contract as intermediary
long call option payoff & profit &breakeven
payoff=max(0,ST-X)
profit=payoff-premium=max(0,ST-X)-C0
breakeven=C0+X
upside no limit; downside=C0
long put option payoff & profit & breakeven
payoff=max(0, X-ST)
profit=max(0, X-ST)-P0
breakeven=X-P0
upside X-P0; downside=P0
CDS gains
if underlying bond have credit migration, 100bps spread increase to 120bps spread, portection buyer gain: change in CDS spreadsnotionaleffective duration;
if defaulted, payment to buyer=LGD%*notional (amt lost is paid)
Effective duration is a measure of a bond’s price sensitivity to changes in interest rates, specifically designed for bonds with embedded options
intereat rate decrease, higher duration, higher the price appreciation;
to increase duration, enter fixed receiver swap (buy fixed rate swap); decrease duration, enter fixed payer swap (sell fixed rate swap)
rates
- MRR simple compounding: FV=PV(1+MRR/360)
- periodic compounding: FV=PV(1+Rf)^T/365
- continuous compounding: FV=PVe^RfT/365
convinience yield
reflects the preference that market participants exhibit for buying in the spot market for NON-CASH reasons, including low inventories in the underlying cash market
Because payments on forward rate agreements are discounted at MRR, they exhibit convexity, whereas payments on interest rate futures are linear (no convexity).
bootstrapping spot rates
1yr spot rate=ytm on one year;
ytm is weighted average of undelying spot rate; last spot rate should be closest and above ytm;
bootstraping means using previous spot rate to calc later spot rate
The mark to market adjustment to futures contracts resets the price of the futures contract to the new settlement price, which returns the value of the contract to zero each day
all forward contract, long gains when underlying goes up; STIR futures is inverse
if underlying is interest rate, long gain if interest rate go up
implied fwd rate
derived from spot rates
If the price of a forward contract is greater than the price of an identical futures contract, the most likely explanation is that:
the futures contract requires daily settlement and the forward contract does not.
The reason there may be a difference in price between a forward contract and an identical futures contract is that a futures position has daily settlement and so makes or requires cash flows during its life.
interest rate swap
fixed rate swap payer pay fixed receive MRR*notional; a series of FRA with each FRA priced at swap fixed rate, value of each FRA does not need to be 0 but should sum to zero at initiation because swap value at initiation is 0, swap price is fixed throughout;
swap value at initiation=0 but individual FRAs at fixed rate may have different values;
only know first MRR, rest unknown;
can replicate a one year interest rate swap with 3 FRA of 180, 270 and 360 days. dont need 90 days because already know MRR at time 0; fixed rate for each zero arbitrage FRA is different for each FRA: Fzero=(1+180 day MRR)/(1+90 day MRR)-1; but if sum value of all FRAs up, is 0;
use spots to find fwd rates; use fwd rates discount by spot rate to find PV of fixed rate; use PV of fixed rate and annuity (discount using spot) to find fixed rate(coupon);
swap price is the fixed rate;
at initiation, swap value is 0;
if issuer has fixed rate debt, buy floating swap (pay floating, receive fixed) so converted to a floating rate debt; pay floating swap lose value when forward rate curve shifts up: IR increase, market fwd rates increase, floating pmt increase, swap value decrease;
payer swap (pay fixed, receive floating) interest rate increase, floating rate increase will increase value of swap
pricing a swap
at t=0, use spot rates to derive forward rate to get floating payment;
PV of floating rate payment=discount forward rate (numerator) using spot rates (denominator);
PV of floating payment=PV of fixed payment=use spots (denominator aka annuity factor) to derive coupon;
at t=1, fixed rate PV not equal to floating rate PV because market interest rate will change; value of swap=PV of remaining floating-PV of remaining fixed (fixed payer); increase in MRR increases the value of fixed rate payer because fixed rate on a new swap with same remaining life would increase;
interest rate swap2
price of a fixed-for-floating interest-rate swap is defined as the fixed rate specified in the swap contract.
Typically a swap will be priced such that it has a value of zero at initiation and neither party pays the other to enter the swap.
value of swap may increase or decrease during the life of swap as floating rate change but price (initiate fixed rate) is the same.
When replicating a swap with a series of forward contracts, each forward contract is likely to have a non-zero value at initiation, but they can replicate a swap with a value of zero at initiation if the values of the forward contracts sum to zero at swap initiation.
investor could best replicate the position of the floating rate payer in a swap by:
borrowing at a floating rate and buying a fixed-rate bond or borrowing at a floating rate and entering a series of FRAs, but these would not necessarily be zero-value FRAs as long as sum to 0 at initiation; zero-value FRAs would typically not all have the same fixed rate as swap payments do.
forward commitment vs contingent claims
fwd commitments have: 0 value at initiation, symmetric payoffs, no price paid upfront, unlimited g&l unless St is 0;
contingent claims have:
positive value at issuance=premium=PV of expected option payoff at expiry, asymmetric payoffs, max loss=option price (premium) for long; max gain=option price for short;
value of premiums/value of option
american options premium is at least of european options; if excercise american before expiry, lose the time value of premium;
call: max value=current value of St because would buy the stock directly; min value=max(0, St-X*(1+Rf)^-(T-t))
put: max value=PV of X=X(1+Rf)^-(T-t) ; min value=max(0, X*(1+Rf)^-(T-t)-St); because would never more than strike to sell the underlying and receive that strike;
higher volaility incresae call and put values;
longer time to expiry incresae call and put values unless european option deep in the money, long time to expiration and high Rf (e.g. long call X=20 St=1, would rather excercise now and reinvest 19 at rf rate but cant, so european option value decrease);
higher Rf rate: increase call value (delay borrow to purchase) decresae put value (delay sell);
benefits of holding (paying div): decrease call value increase put value;
carrying cost: increase call value decrease put value
put call parity: protective put and fiduciary call (SIP COKE=LONG STOCK LONG PUT LONG CALL LONG Riskfree BOND or replace stock with fwd)
long stock long put: P0+S0=C0+X(1+Rf)^-T: long call+PV of strike (zero coupon bond);
at expiry, should have same value;
if ST>=X,
call option: ST-X, bond: X, payoff=ST;
put option: 0, stock:ST, payoff=ST;
if ST<=X,
call option: 0, bond: X, payoff=X;
put option: X-ST, stock:ST, payoff=X;
cuz identical payoff at expiry and european, so payoff must be idential prior to expiry to prevent arbitrage;
rearrange the equation to replicate, produce synthetic option:
P=C-S+PV(X): long call short stock long zero coupon bond equals synthetic long put;
C=P+S+PV: long call long stock long zero coupon bond equals synthetic long call;
S=C+PV(X)-P: long call long bond short put equals synthetic stock
equity is a synthetic call option
A=L+E
assume A is marekt value of firms asset which firm value V0,
assume L is zero coupon bond PV(D), will not default unless maturity;
firm value V0=E0+PV(D);
solvency: VT>D payoff debt,
equity value: VT-D (like call ST-X, residual claim after debt)
Debt: D;
insolvency: VT<D will not pay equity,
equity value: 0
Debt: D-VT=loss (X-ST negative so short put, debt holder get whats remaining in firm value) hence
=>owning equity is equivalent to a long call on firm’s asset, get upside if firm value exceed debt otherwise value is 0: max(0,ST-X); can be created by long asset long put which is protective put aka synthetic call
=>debt either gets riskfree D or firm value<D, so risky debt is like a long Rf bond (X)+short put (X-ST)=X*(1+Rf)-P; equivalent to short put option on company assets;
C0+PV(X)=P0+S0
C0+PV(D) risk free zero coupon bond=P0+V0 firm value
V0=C0+PV(D)-P0
because equity equals synthetic call
firm value=equity value+risk free debt-short put
firm value=equity+debt
put call forward parity
F0(T)=S0*(1+Rf)^T [cost of carrying fwd contract]
long call has the right to excercise a long foward, short call has the obligation to excercise long forward;
long put has the right to excericise a short forward, short put has the obligation to short forward;
S0=F0(T)(1+Rf)^-T
replace S0 in put call parity:
F0(T)(1+Rf)^-T+P=C+X(1+Rf)^-T
P=C+PV(X & F0(T))
option value (premium)
option value=PV of expected payoffs at option expiry
one period binomial model (risk neutral probability)
the risk neutral probability (ESTIMATE) needed to be consistent witht the risk neutral portfolio to give a risk free rate;
U=up-move factor=1.5=>if stock rises, go up by 15% driven by assumed volatility;
D=down-move factor=0.87=>if stock down, goes down by 13%;
πU=risk neutral prob of up-move=(1+Rf-D)/(U-D);
πD=risk neutral prob of down-dove=1-πU;
use PV option value to derive prob;
work out hedge ratio first,
plug in h to calc P1,
P1*(1+Rf)^-1=P0 made of long stock and short call,
P0=ST-C0 to get C0;
value of option C0=PV of expected payoff=[(payoff upπU)(payoff down(1-πU)]/(1+Rf)=>
payoff down(1-πU)=0 =>
C0(1+Rf)/payoff up=πU
one period binomial model (risk neutral portfolio)
assume volatility constant: change in price is due to volatility;
valuing call option:
risk neutral portfolio: long stock short call, need to find stock/call ratio or hedge ratio:
Method1:
Value up=hS up-C up=hS down-C down=Value down=>
h(60)-5=h(42)-0=>hedge ratio=(C+ - C-)/(S+ - S-);
h=hedge ratio, need h shares per short call so have no risk, value of option is fixed so rate of return is Rf rate;
plug in h, to calc V1 payoff [long stock short put for up or down stock price],
V1(1+Rf)=V0 to get portfolio value at time 0;
P0 consists of long stock and short call so hS0-C0=V0
Method2:
diff expected payoff/diff expected stock price;
valuing put option:
risk neutral portfolio: long stock long put:
Value up=hS up+P up=hS down+P down=Value down=>
hS up+P up=V1,
V1(1+Rf)=V0,
h*S0+P0=V0, P0 is the no arbitrage price of put at time 0;
put option hedge ratio is always negative so convert short put to long put to make it positive;
hedge ratio=(P+ - P-)/(S+ - S-)
put call parity arbitrage
market value of put>synthetic put (long call long bond short stock), buy lower value sell higher value, buy call short put and short stock position would cancel out the exercise of either the put or call option (P+S=C+X);
market value of put<synthetic put (long call long bond short stock), buy lower value sell higher value, buy put short call and borrow? the PV of the exercise price (X), and buy the stock (P+S=C+X);
A one-period binomial model for option pricing uses risk-neutral probabilities because:
the model is based on a no-arbitrage relationship.
Because a one-period binomial model is based on a no-arbitrage relationship (investors cannot make risk-free profit), we can discount the expected payoff at the risk-free rate.
In a one-period binomial model based on risk-neutral probabilities, the value of an option is the present value of a probability-weighted average of two possible option payoffs at the end of a single period, during which the price of the underlying asset is assumed to move either up or down to specific values.
A one-period binomial model can be constructed based on replication and no-arbitrage pricing, without regard to the probabilities of an up-move or a down-move.
We can use the risk-free rate to value an option with a one-period binomial model because:
combining options with the underlying asset in a specific ratio will produce a risk-free future payment.
A portfolio of an option position and a position in the underlying asset can be constructed so that the portfolio value at option expiration is certain, the same for an up-move and for a down-move.
One method of valuing a call option with a one-period binomial model involves:
A portfolio combining the call option with the underlying asset can be constructed that will have the same value at option expiration whether there is an up-move or a down move in the asset price. The present value of this portfolio is the discounted present value of the certain future payment, which can then be used to value the option.
An option valuation model based on risk neutrality uses risk-neutral PSEUDO-probabilities of an up-move and a down-move, NOT ACTUAL probabilities.
The average call value is NOT a certain future payment.
The risk-free rate, the volatility of the price of the underlying, and the current asset price are three of the required variables needed to value an option with a one-period binomial model. The risk-adjusted rate of return and (actual) probability of an up-move are not required.
diff in payoff/diff in price discount to PV
long option is add, short option is minus
Reasons why the no-arbitrage approach to pricing options is different from the no-arbitrage approach to pricing forward commitments least likely include that: Many forward commitments are exchange-traded derivatives, while most options are over-the-counter derivatives.
Arbitrage and replication concepts apply to both exchange-traded and over-the-counter derivatives.
The differences between the models for forward commitments and options arise from the facts that initial values are positive for options and typically zero for forward commitments,
gains and losses are one-sided (nonlinear, asymmetry: can have limited loss with the potential for unlimited gain) for options and two-sided for forward commitments
swap price vs value
price of a swap contract is the par swap rate, which is the fixed rate that gives the swap a value of zero at initiation. The value of a swap contract to the fixed-rate payer is the present value of the expected future floating-rate payments minus the present value of the future fixed-rate payments.
Convexity bias can cause interest rate forward and futures prices to differ significantly for contracts with LONG maturities.
Forward and futures prices may differ if a forward contract does not require daily cash settlement of mark-to-market gains and losses (as futures contracts require); futures price=settlement price as it changes daily
and interest rates are positively or negatively correlated with futures prices over time.
long asset and a short forward contract on the asset replicates investing at the risk-free rate because the future payoff is certain.
Cash flows related to futures margin least likely include:
interest on the margin loan but may post interest-bearing securities as collateral and earn interest (collateral yield) on these securities and deposits to meet margin calls.
option moneyness (in and out of money) does not account for premium
An investor could best replicate the position of the floating rate payer in a swap by:
borrowing at a floating rate and buying a fixed-rate bond.
The payments could also be replicated by taking a floating-rate loan (or issuing a floating-rate bond) and entering a series of FRAs, but these would not necessarily (or likely) be zero-value FRAs; zero-value FRAs would typically not all have the same fixed rate as swap payments do.
swap OTC, options exchange traded or OTC, fwd OTC, futures exchange
When the spot curve is upward sloping, the forward curve will be above it, and the par curve will be below it.
When the spot curve is downward sloping, the forward curve will be below it, and the par curve will be above the spot curve
FRA expiry date marks the start of the loan or deposit that the FRA is meant to hedge. The notional period is the duration of the underlying loan or deposit that the FRA covers.
Floating rate agreement contracts settle at FRA expiry, not the end of the notional period, because the FRA’s purpose is to hedge against interest rate risk for a specific future period. The settlement at the FRA’s expiry date ensures that the interest rate differential between the fixed and floating rates is resolved when the loan or deposit starts
if interest rate is positively correlated with underlying asset value, long futures is preferred to a forward without MTM cuz higher rate when lending than when borrowing
STIR is exchange traded FRA, standardized and liquid
FRA price=annualized imlied fwd rate; vice versa
STIR price=100-(100implied fwd rate)=(1-annauized MRR)100; long gain when price increas and implied fwd rate fall
fwd=0 value at inception, symmetric payoff, unlimited g&l unless at 0 price;
contingent claims: positive value at issuance - premium, asymmeetric payoff, max loss =option price for long, and max gain=option price for short
- hedge ratio
- plug hedge ratio to get payoff
- discount payoff to get options value
- calc portfolio value long stock=shot call price and put call parity to get put price
if margin falls below maintenance margin, need to get back to initial margin
moneiness and payoff, intrinsic value does not include premium
lend a bond=invest in a bond=buy bond
hedge ratio=(payoff when price is up-payoff when price down)/(stock price up-stock price down)
price an option with one period binomial model will need S0, X, T, rf, volaility (asset price up & down)
if actual, market or empirial probaility is used to price an option in one period binomial model, the increase in probability will not change option price because its risk neutral model
