Jorion ZusaFa

7/29/2019 Jorion ZusaFa

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4 Monte Carlo Methods

Monte Carlo simulations = simulation of random variables for the purpose of riskmanagement

- used for pricing complex financial instruments -> to build the distribution of portfoliosthat are too complex to model analytically

- are quiet flexible and are becoming easier to implement with technologicaladvancement in computing

- Drawback: result depends heavily on models assumptions = shape of distribution,parameters, pricing function

Simulations with one random variable- for creating artificial random variables with probabilities similar to risk factors of

portfolio (e.g. stock prices, exchange rates, bond yields, bond prices, commodity prices)

Simulating Markov Processes

In efficient markets prices show a random walk pattern -> they follow a Markov process = astochastic process independent of its past history = the entire distribution of future prices relyon the current price only, the past history is irrelevant

Wiener process = describes a variable z, whose change is measured over theinterval t such that its mean change is zero and variance proportional to t :

z ~ N(0, t) z = t (if is the standard normal variable N(0,1))

increments z are independent over time Generalized Wiener process = describes a variable x built up from a wiener

process, with in addition a constant trend a per unit of time and volatility b:

x= a t+ b z

particular case: martingal = a zero drift stochastic process (a = 0) which leads toE(x) = 0 -> expectation of a future value = the current value

The Ito process = describes a generalized Wiener Process, whose trend andvolatility depend on the current value of the underlying variable and time:

x= a(x,t)t+ b(x,t) z

-> a Markov process because the distribution depends only on the current value ofthe random variable x, as well as time -> in addition the innovation in this processhas a normal distribution.

The Geometric Brownian Motion (GBM)= particular example of an Ito process which describes the variable S as

S = St+ S z

expected total rate of return on the asset minus the rate of income

payment, or dividend yield (stocks)


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-> process is geometric because trend and volatility term are proportional to the current valueof S -> typical case for stock prices (rate of return appears more stationary than raw dollarreturn) -> also used for currencies because S/S represents the capital appreciation only,abstracting from dividend payments

model is important because it is the underlying process for the Black Scholes formula. ->key feature = the fact that the volatility is proportional to S = insurance that stock price stays

positive (stock price falls -> variance decreases -> unlikely to experience a large down movewhich would push price into negative values) limit of this model = a normal distribution for dS/S = dln(S) -> S follows a lognormaldistribution implies that over an interval T-t = the logarithm of the ending price isdistributed as

ln(ST) = ln(St) + ( / 2) + whether normal or lognormal distribution is seen as better depends on horizon considered

- one day only (no real difference) very unlikely to drop below zero given typicalvolatilities -> choice does not matter- measurement in years lognormal much more realistic (no negative prices)

simulation approximates this process with small steps with a normal distribution withmean und variance given by

S/S ~ N(t, t )

-> to simulate future price path for S: start with current price St and generate a sequence ofindependent standard normal variables for i = 1,2,.n. -> next price S t+1 is built asc+St(t+ t x 1) and so on until target horizon is reached. there S t+n = ST should have adistribution near to lognormal experiment can be repeated as often as needed K = number of replications shortcomings of model:

Price increments are assumed to have a normal distribution (in real world fat tails!!)

Returns may experience changing variances Jumps are not simulate able

Drawing Random Variables

most spreadsheets or statistical packages provide functions that can create uniform orstandard normal random variables -> can be easily extended to distributions that better reflectthe data (e.g. fatter tails, non-zero skewness) -> methodology involves inverse cumulative

probability distribution functionExample normal distribution: cumulative probability distribution function N(x) always is

between 0 and 1 -> analytical formula can be inverted easily1. Generate a uniform random variable u drawn from U(0,1).2. Compute x that u = N(x) or x = N -1(u)

e.g. u =0,0430 x = -1,717 because u < 0,5 -> x


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Simulating yields GBM used for stock prices and currencies -> fixed-income products are another matter bond prices display a long-term reversion to the face value, which represents therepayment of principal at maturity (assuming no default) -> process is inconsistent withGBM, which displays no such mean reversion -> volatility of bond prices also changes in a

predictable fashion as duration shrinks -> commodities also display mean reversion taken into account by modeling bond yields directly in a first step -> in a next step, bond

prices are constructed from the value of yields and a pricing function -> dynamics of interestrates rt can be modeled by (one-factor model: only one stochastic variable):

rt = K(- rt) t+ rtYzt

Features:- Mean reversion to a long run value of: K governs the speed of mean reversion ->

when interest rate high > model creates a negative drift, low current rates create a

positive drift- volatility process: model includes Vasicek model (= changes in yields are normally

distributed because r is a linear function ofz) when = 0 -> leads to closed-formsolutions for many fixed-income products -> but might lead to negative interest rateswhen initial rate starts with a low value (volatility of the change rate does not dependon the level unlike in the GBM)-> with Y = 1 this is the lognormalmodel-> with Y = 0,5 this is the Cox, Ingersoll and Ross (CIR)model

this class of models is known as equilibrium models: start with an assumption abouteconomic variables and imply a process for short term interest rate r -> generate a predicted

term structure, whose shape depends on model parameters and initial short rate are notflexible enough to provide a good fit for todays term structure

second type of models: no-arbitrage models: term structure is input for parameterestimation-> earliest model: Ho and Lee model

rt = (t)t+ zt

-> was extended to incorporate mean reversion in the Hull and Whitemodel

rt = [(t)-a rt ] t+ zt

-> Heath, Jarrow and Morton model: go one step further assume that volatility is afunction of time

Cons of no-arbitrage models: do not impose any consistency between parametersestimated over different dates -> (t) could be different from day to day which is illogical-> more sensitive to outliers or data errors in bond prices

Binominal trees

Simulations are useful to mimic uncertainty in risk factors (esp. with numerous) ->sometimes also useful to describe the uncertainty in prices with discrete trees

zt ..Wiener Process 0 < K < 1, >= 0, >= 0

t unction o time chosen so that model its the initial term structure


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tree is called binominal when the price can take one of two steps -> binominal model =discrete equivalent to GBM-> horizon T is divided in n intervals -> at each node the price is assumed to go up by a

probability of p or down by the probability of (1-p)-> u,d,p are chosen that for a small time interval the expected return and variance equal those

of the continuous process

U = e t, d = (1/u), p = etd / (u-d)This matches the mean

etd u- et et(u-d)-du+udE[S1 / S0 ] = pu + (1-p)d = * u + *d = = et

u-d u-d u-d

as the number of steps increase the discrete distribution converges to lognormal

distribution

Implementing Simulations

Simulation of VAR implementing of MC methods for risk management follows the steps

- Choose stochastic Process for risk factor price S (e.g. distribution)- Generate pseudo-random variables representing risk factor at target horizon- Calculate value of portfolio at the horizon- Repeat step 2 and 3 as often as necessary

These steps create a distribution of values , which can be sorted to derive the VAR measure the quantile and the average value VAR = average deviation from the expected value on the target date

Simulation for Derivatives focus on expected value on the target date discounted into the present -> valuation focuseson the discounted center of the distribution (cf. VAR focuses on the quantile on target date)

MC long used for pricing derivatives -> pricing can be done by assuming that underlyinggrows at risk-free rate r (assuming no income payment)e.g. pricing an option on a stock with expected return of 20% is done in assuming that stockgrows at risk-free rate of 10% and by discounting the same risk-free rate = risk-neutralapproach. contrary risk management deals with actual/physical distributions -> for measuring VAR,risk manager must simulate asset growth by using actual expected return risk management uses physical distributions, pricing uses risk-neutral distributions these methods are not applicable to all type of options -> assume that value of instrumentcan be priced as solely function of end-of-period price and perhaps of its sample path (e.g.Asian option = payoff is function of the price averaged over sample path option is path-

dependent) simulations are inadequate to price American options (early exercise) -> optimal exercisedecision too complex for model (future values) not with regular simulation methods which

Ave(FT) average valueQ(FT,,c) quantile


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consider only present and past information. -> they need backward recursion (e.g. binominaltrees) -> examines whether option should be exercised or not, starting from the end working

backward until the starting time

Accuracy

effect of sampling variability: unless K is extremely large the empirical distribution of STwill only be an approximation of the true distribution -> natural variation in statisticsmeasured from MC. MC involves independent draws -> standard error is related to square root of number ofdraws much more simulations will increase precision but only at a small rate -> greater

problem for risk management, because quintiles are estimated less precisely than the average. VAR: precision is also a function of selected confidence interval -> higher confidence =fewer observation in left tail = less precise VAR measures. -> shape of distribution alsoinfluences precision of quantile Relative to a symmetric distribution a short option has negative skewness (=long left tail) -> observations much more dispersed difficult for VAR estimation

various methods to speed up convergence:

Antithetic variable technique = uses twice the same sequence of random draws -> takesoriginal sequence and changes value (value*-1) twice the number of points in finaldistribution without running twice the number of simulations

Control variable technique = used to price options with trees when a similar option hasan analytical solution

Quasi random sequences/ Quasi Monte Carlo (QMC) = creates draws which are notindependent, but are designed to fill sample space more uniformly -> converge faster asMC (closer to the true value)

-> traditional MC also provides a standard error, therefore idea of how far the estimated valuemight be from true value important for decision on number of replications

Multiple Sources of risk more general case of simulation with many sources of risk -> if factors are independent therandomization can be performed independently for each variable (GBM model) -> in generalrisk factors are correlated, therefore the drawn independent variables need to be transformedinto correlated ones

The Cholesky Factorization generate N joint values that display the correlation structure of R -> because the matrix R

is symmetric it can be decomposed into is so-called Cholesky factors.

R=TT

in practice this decomposition yields to a number of insights: it will fail if number ofindependent factors implied in the correlation matrix is less than N

Curse of dimensionality modern risk management is about measuring risk of large portfolios, typically exposed tolarge number of risk factors. number of computations increase geometrically with N Dimension of covariance matrix = N(n+1)/2

Simplification of risk factors (use only significant ones) -> one way to do this is principal-component analyses (PCA) = finds linear combinations of risk factors that have maximalexplanatory power

T = is a lower triangular matrix with zeros on the upper right corners (above diagonal)


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5. Introduction to Derivatives

Derivate = financial contract traded private (over-the-counter OTC) or on organizedexchanges -> derive their value from some underlying index (typically price of asset) classified in two categories:

linear (forward, futures, swaps) instruments: value = linear function of underlyingindex -> are obligations to exchange payments according to specified schedule ->Forward(very simple to evaluate and price), as are futures (traded on exchanges),swaps much more complex

nonlinear instruments (options): options traded both OTC and on exchange , value =nonlinear function of underlying

Overview of derivative markets

derivative instrument = private contract whose value derives from some underlying assetprice, reference rate or index such as a stock, bond, currency or commodity contract must specify a notional amount (=principal), which is described in terms ofcurrency, shares bushels, or some other unit -> movements in value depend on notional andunderlying contracts/ private agreements between two parties -> sum of gain and losses = 0 (one sidegains, other looses same amount)-> cf. securities (bonds, stocks): are issued to raise capital Classification of derivatives markets by underlying and type of trading (=broadest level) interest rate contracts = most wide spread type -> on OTC also currency contracts (inforwards and forex swaps = combination of spot and short time forward transaction) ->exchange show interest futures and options as most common

notional amount several times the world GDP (gross domestic product) and also higher tantotal outstanding value of stocks

Forward contracts

most common transactions in financial instruments are spot transactions -> physicaldelivery as soon as practical -> beneficial to trade for delivery at some future date -> hedgeout price fluctuation forwards = private agreement to exchange a given asset against cash or other asset at afixed point in the future -> terms of contract: quantity, date and price long = position which implies buying, short = position to sell

represent contractual obligations: exchange must occur whatever happens to theintervening price (unless default) -> no choice in taking delivery or not hedge against movement in price = locking price for delivery now face/notional amount = principal value = amount to pay at maturity assumption of continuously compounded interest rates

Value of forward contract at expiration = VT = ST-F (for one unit of the underlying asset)

Value of contract at expiration is derived from the purchase and physical delivery payment of cash in exchange for actual asset

Second mode of settlement = cash settlement = measuring market value of asset uponmaturity and agreeing for the long side to receive nVT = n(ST-F) -> amount is positive=profit or negative = loss


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Valuing Forward Contracts two questions when evaluating forward contracts

1) How is current forward price determined?2) What is current value of an outstanding?

initially assumption = forward pays no income, no transaction costs = no bid-ask spread,and ability to lend and borrow at same risk-free rate.

generally forward contracts are established that initial value = 0 through setting forwardprice by a no-arbitrage relationship between cash and forward markets Arbitrage = zero-risk, zero-net investment strategy

StrategiesI) Buy one share at spot price S t and hold it to time TII) Enter forward contract to buy, sell one unit of same underlying asset at forward

price.Ft -> in order for sufficient funds at maturity = investment in an interest-bearing account

Ft = Stert

-> forward rate higher than spot rate (reflects that there is no down payment to enter theforward contract and also reflects the time value of money) -> mispriced forwards lead toarbitrage (buy cheap asset and sell the expensive one) !!!!

Valuing on Off-Market Forward contract

Use arbitrage reasoning to evaluate outstanding forward contract -> off-market: K differsfrom prevailing forward rate

Market value of outstanding long position Vt = St-Ke-rt

gains value when underlying increases in value

Valuing on Off-Market Forward contractAssets producing no income payment assumed -> but in real:

Stocks that pay regular dividend Bond that pays regular dividend Stock index that pays dividend stream approximated by continuous yield Foreign currency that pays foreign-currency denominated interest rate

Income: discrete = fixed amounts at regular points in time, continuous = accrued inproportion of time asset is held -> assumption: fixed and certain payments

prices are measured in the domestic currency

Fte-rt =St-PV(D)

Sport price of underlying asset

payoff

Payoff of profits on Long Forward Contract Payoff of profits on Short Forward Contract

Sport price of underlying asset

payoff

PV(D) present value of dividends/coupon payment -> discounted at the risk-free rateS stock rice at t


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Fte-rt =St e

-r*t for continuous payment -> known as interest rate parity when dealing withcurrencies

r* < r -> Ft > St: asset trades at a forward premium

r* > r -> Ft < St: asset trades at a forward discount

forward price differs from the spot price to reflect the time value of money and the incomeyield on the underlying asset -> is higher than the spot price if the yield on the asset is lowerthan the domestic risk-free interest rate, and vice versa. current value of an outstanding forward contract can be found by entering an offsettingforward position and discounting the net cash flow at expiration

Value of outstanding forward with income payment Vt = Ste-rt - Ke-rt Vt = (FtK)e

-rt

Futures

forward contracts allow users to take positions that are economically equivalent to those inthe underlying cash market, but do not involve substantial up-front payments forwardshave leverage futures contracts are standardized, negotiable, and exchange-traded contracts to buy or sellunderlying asset Differences from forwards:

Trading on organized exchanges Standardization: limited choice of expiration dates, fix contract sizes large

secondary market = easy to buy and sell (good liquidity) -> but also less precisely

suited to need of some hedgers (creates basis risk) Clearinghouse = standardization of counterparty (ensures the performance of the

contract, bears credit risk) Marking-to-market= settlement of gains and losses daily -> no large losses over time Margins = upfront posting of collateral that can be seized should the other party

default -> if equity below maintenance margin, customer must provide additionalfunds to cover the initial margin -> level depends on the instrument and type of

position Valuating Future contracts is similar to forward contracts, main difference is that any

profit or loss accurse during life of futures against all at once at expiration

-> correlation = 0: no difference whether payments are received earlier or later -> futuresprice = forward price-> interest-rate futures: value is negatively correlated to interest rates -> long futures pricemust be lower than the forward price (convexity effect)

Swap contracts=OTC contracts to exchange a series of cash flows according to prespecified terms -> usuallylonger periods as futures or forwards, principal = notional principal if amount is in same currency -> no need to exchange principal can be viewed as a portfolio of forward contracts -> priced using valuation formulas forforwards


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6. Options

options = non-linear derivatives

Plain vanilla options (puts, calls) Exotic options

Option payoffs

Basic options Option = instrument that gives holder the right to buy or sell an asset at a specified priceuntil a specified expiration date -> Delivery price = exercise price = strike price Options to buy = Call options, Options to sell = Put options will only be executed if generate profit -> sellers are said to write options Depending on time of exercise:

- European option: only at maturity- American Option: any time

payoff profile of long calls = CT = Max(ST-K,0),

payoff of long put = PT = Max(K-ST, 0)

Current price close to strike price = at the money Current price so that option will be exercised = in the money = intrinsic value large

- call: St > K- put: St < K

Current price so that option will not be exercised = out of the money

Payoff diagramms at exercise time (hockey stick diagrams)

Long Call Long Put

Short Call Short Put

buying options can generate only profits (or 0) at expiration -> payment is needed toacquire the contract (as it always has value >= 0) -> up-front payment = option premium option becomes more expensive as it moves in-the-money for a given spot price the sum of the profit and loss for the long and for the short = 0 long position has limited downside risk = loss of premium short position has unlimited downside risk = no upper limit on S (worse loss: S = 0)

options can also be struck on futures -> exercising a call = investor becomes long thefuture contract, exercise a put = creates short position in the future contract

K exercise priceSTasset price at T


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Put-Call ParityOption payoffs can be used to build more complex positions long Position in asset =long call (provides the equivalent of the upside risk) + short put (generates the same downsiderisk as holding the asset)

link between value of call and value of put = put-call-parity only European options are considered with the same maturity and exercise price Portfolio consists of long call and short put and investment to be sure to be able to pay

exercise price at maturity

c-p= Se-r*T-Ke-rT = (F-K)e-rt

A long position in an asset is equivalent to a long position in an European call with a shortposition in an otherwise identical put, combined with a risk-free position.

Combinating OptionsMultiply combinations possible, with each other or with underlying asset

- covered call = long position in the stock & short call (to collect the premium)

- protective put = long position in the stock & short put (to protect the downside)- collar = if long in the stock: buy long put (low strike) & short call (high strike) ->

equivalent to a short stock position -> create net payment of 0- straddle = call & put (strike price and maturity same) -> long straddle benefits from

large price moves whether up or down

- strangle = call & put (different strike prices) -> are out-of-the-money and thereforecheaper than straddles

- spreads = positions with only one type of optionso Calendar/horizontal spreads =different maturitieso Vertical spreads =different strikeso Diagonal spreads = move across maturities and strikeso Bull spread = advantage in increase of underlying asset price


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o Bear spread = represents bet on a falling priceo Butterfly spread = three types of options with same maturity -> 1 long call, two

short calls and another long call ( K1 call: MAX(St-K, 0), put: MAX(K- St, 0)- Time value = consist of remainder which reflects the possibility of future gains ->

increases with the volatility of the underlying asset options can be classified:

- at-the-money: current spot price is close to the strike price- in-the-money: intrinsic value is large- out-of-the-money: spot price is much below the strike price for calls, conversely for

puts -> intrinsic value = 0

European option: general bounds (otherwise arbitrage)- current Value of call = price of asset PV of the strike price


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- current value of a put = PV of strike priceprice of asset

American options: early exercise basic trade-off arises between value of the American option dead = exercised and alive =

not exercised -> choice between exercising or selling on the open marketAmerican call option on a non-dividend-paying stock (or asset with no income) shouldnever be exercised early -> if the asset pays income, early exercise may occur, with a

probability that increases with the size of income payment -> only exercise call early tocapture a dividend payment-> American call: exercise early = St-K -> is > St-Ke

-rt: value of the option alive must beworth more than the equivalent European call a high income payment makes holding the asset more attractive than the option ->American options on income-payment assets may be exercised early -> also applies to optionson futures (implied income stream on the underlying = risk-free rate) American puts: Pt >= K - St -> is > Ke

-rt - St (= lower bound for European puts)

-> American puts on non-income paying assets could be exercised early -> less attractive tosell the asset when lower interest rates and with higher income payments on the asset

Valuing Options

Pricing by Replication:

Philosophy of pricing models = replicate payoff on the instrument (option) by a suitableportfolio of the underlying asset plus a position (long or short) in a risk-free bill -> noarbitrage: price = price of replication portfolio

First possibility by binominal process:-> Initial price = S can only move to two different end prices (up or down)

current value of the stock S0 = [p x S1 + (1-p) x S2]/(1+r)

discounted expected payoff assuming investors are risk-neutral-> solve to calculate the risk-neutral probability p

value of the option C0= [p x c1 + (1-p)x c2]/(1+r)

concept of risk neutral pricingBlack Scholes Valuation (BS)

Provides elegant closed form solution to the pricing of European calls based on 4 assumptions:

the price of underlying asset moves in a continuous fashion interest rates are known an constant variance of underlying asset returns is constant capital markets are perfect (short-sales, no transaction costs, markets operate

continuously ) statistical process of asset price is modeled by a geometric Brownian motion (over shorttime the logarithmic return has a normal distribution)

total return dS/S = dt + dz

-> dz normally distributed with mean = 0 and variance dt-> price is lognormal distributed

dt represents the drift component

dz stochastic com onent


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position in the option can be replicated by a delta position in the underlying asset ->portfolio combining asset and option in appropriate proportions is locally risk-free risk neutrality: in risk-free world expected return on all securities = risk-free rate ofinterest, no risk premium for inducing to take risks -> artificial method to obtain correctsolution no implication that investors are risk-neutral!

limited use for risk management -> BS can be used to derive the risk-neutral probability ofexercising the option -> For risk management physical probability (= actual probability ofexercise) much more important and differs from risk-neutral probability

ExtensionsMerton expanded BS to the case of stock paying continuous dividend Garmann-Kohlhagen-model: extends formula to foreign currencies reinterpreting yield asforeign rate of interest-> call that is deep in-the-money is equivalent to a long forward contract (almost certainexercised)Black model: applies formula to options on futures

-> with an option on cash, the income is the dividend or interest on the cash instrument-> with an option on futures, the implicit income is the risk-free rate of interest-> future can be viewed as equivalent to a position in the underlying asset with the investorsetting aside an amount of cash equivalent to the present value of the future Standard options involve a choice to exchange cash for asset (special case of exchangeoption = surrender an asset in exchange for acquiring another)Margrabe has shown that valuation formula is similar to the usual model

Market versus Model Prices in practice BS is used to price options -> all parameters are observable, except volatility observing market prices: we can solve for volatility parameter that sets the model price

equal to the market price = implied standard deviation (ISD)-> Model = correct ISD should be constant for different strikes -> but this is not theobservation: plots of ISD against strike display volatility smile pattern (= ISDs increase forlow and high values of K)- Negative slope is called volatility skew

related concept is term structure of volatility: refers to observation that ISD differs acrossmaturities -> arises because option market incorporates many types of events over the lifetimeof the option (e.g. realized volatility of a stock tends to increase on the day of an earningsannouncement)

observations summarized by an implied volatility surface = three dimensional plot of ISDacross maturities and strike prices -> traders use heuristic approaches to extrapolation ofcurve over investment horizon

First scenario = sticky strike = curve does not change and ISD drops -> assumes nostructural change in volatility curve, price movement largely temporary Second scenario = sticky moneyness = curve shifts to the right and ISD stays the same ->assumes permanent shift in the volatility curve ISD of a portfolio of assets can be related to the ISD of its components through the impliedcorrelation

Usually portfolio variance is related to the individual volatilities


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-> increasing correlation = increasing total portfolio risk

Other Option Contracts

Binary/digial options = pay fixed amount if asset price ends above the strike price ->involves a sharp discontinuity around strike (just below K: value = 0, just above: value =

notional) difficult to hedge

Barrier options: payoff depends on value of asset hitting barrier during a certain period of

time Knock-out option: disappears if the price hits a certain barrier (down-and-out call:disappears when S = H < S0, up-and-out call: disappears when S = H > S0) Knock-in option: comes into existence when price hits barrier (down-and-in call: appearswhen S = H > S0, up-and-in call: appears when S = H < S0)-> down-and-out and down-and-in are perfectly complementary with identical parameters(add up to a regular call)-> sometimes the options offer a rebate if its knocked out-> similar combinations for put (up and out put: disappears when S = H > S0, down and out

put: disappears when S = H < S0) -> option is exercised at maturity if S < K-> are attractive because cheaper than equivalent European option -> less likely to be

exercised -> but difficult to hedge Asian Options/ Average rate options: generate payoffs that depend on the average value ofthe underlying spot price during life instead of the ending value -> cheaper (lower volatility),easy to hedge Chooser options: allow holder to choose whether option is call or put -> package of twooptions: regular call + option to convert to a put -> more expensive than plain-vanilla options Look back options: payoffs that depend on the extreme values of S over option lifetime ->much more expensive

Valuing options by numerical methods

Some options have analytical solutions (e.g. BS for European vanilla options) -> for moregeneral options numerical methods have to be used Basic valuation states that current value is discounted present value of expected cashflows, where asset grow at risk-free rate and discounted with same risk-free rate -> use MC togenerate sample paths, final option values and discount them to present such simulationmethods can be used for European or even path-dependent options (e.g. Asian options)Simulation cannot account for possibility of early exercise -> binominal trees for Americanoptions


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7 Fixed-Income Securities

Fixed-income securities = bonds that promise to make fixed coupon payments (primarydefinition) -> evolved to include any securities that obligates the borrower to make specific

payments to the bondholder on specific dates

Bond = security that is issued in connection with a borrowing arrangement -> in exchange forreceived cash, the borrower becomes obligated to make a series of payments to the

bondholder

Fixed-income derivates = instruments whose value derives from some bond price, interest rateor other bond market variables

Securitization = the process by which assets are pooled and securities representing interest inthe pool are issued

Overview of debt markets

fixed income markets are truly global

broad classification by borrower and currency type

domestic bonds = bonds issued by resident entities and denominated in the domesticcurrency

foreign bonds = bonds floated by a foreign issuer in the domestic currency andsubject to domestic country regulations

Eurobonds = mainly placed outside the country of the currency in they aredenominated and are sold by an international syndicate of financial institutions

Foreign bonds and Eurobonds constitute to the international bond market

Global bonds = bonds that are placed at the same time in the Eurobond and one ormore domestic markets with securities fungible between these markets

domestic bond market:

government bonds/ sovereign bonds = issued by the governments Brady bonds = sovereign bonds issued in exchange for bank loans -> hybrid securitieswhose principal is collateralized by U.S. Treasury zero-coupon bonds (no risk of defaulton the principal) Tesebonos = dollar-denominated bills issued by the Mexican government

government agency and guaranteed bonds = issued by agencies or guaranteed bythe central government which are public financial institutions

state and local bonds /mutual bonds = issued by local governments bonds issued by private financial institutions (incl. Banks, insurance companies,

issuers of asset-backed securities)

corporate bonds = issued by private nonfinancial corporations (incl. Industrials,utilities)

debt securities market

bond markets = fixed-income securities with remaining maturities beyond one year

money markets = fixed-income securities with remaining maturities below one year


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Mortgage-backed securities (MBS) = securities issued in conjunction with mortgage loans(= loans secured by the collateral of a specific real estate property) -> payments on MBSs =repackaged cash flows supported by mortgage payments made by property owners can beissued by government agencies or private financial corporations Asset-backed securities (ABS) = securities whose cash flows are supported by assets such

as credit card receivables or car loan payments

Fixed-income Securities

interest payments on a regular basis: U..S. Treasury and corporate bonds = semiannual,Eurobonds = annual, some others = quarterly

Instrument types

Fixed-coupon bonds = pay a fixed percentage of the principal every period and theprincipal as a one-time payment at maturity (balloon)

Zero-coupon bonds = pay no coupons but only the principal, return is derived fromprice appreciation only

Annuities = pay a constant amount over time which includes interest plusamortization of the principal

Perpetual bonds and consols = no set redemption date, value derives from interestpayment only

Floating-coupon bonds/ floating-rate notes (FRNs) = pay interest equal to areference rate plus a margin, reset on a regular basis

Structured notes = more complex coupon patterns inverse floaters/reverse floaters = have coupon payments that vary inversely with thelevel of interest rates (best in falling interest rate environment)

Inflation-protected notes = principal is indexed to the Consumer Price Index (CPI),provide protection against an increasing rate of inflation

additional variations:

step-up bonds = have fixed coupons that start at a low rate and increase over timebenchmark interest rates

LIBOR = London Interbank Offer Rate = reference rate based on interest rates atwhich banks borrow unsecured funds from each other in the London interbank market-> published daily by the British Bankers Association around 11:45 a.m.LT,computed from an average of the distribution rates provided by reporting banks ->calculated from 10 different currencies and various maturities (overnight to one year)

EURIBOR = Euro Interbank Offer Rate = sponsored by the European BankingFederation, published by Reuters at 11:00 a.m CET.

EONIA = Euro Overnigh Index Average = onvernight unsecured lending rate (by thesome panel of banks as the EURIBOR) -> published every day before 7:00 p.m.

Bonds with option features:

callable bonds = issuer has the right to call back the bond at fixed prices on fixeddates -> call back the bond when the cost of issuing new debt is lower than the currentcoupon paid on the bond pricing of the option feature: decompose into a long

position in a straight bond minus a call option on the bond price -> unfavourable forinvestors who require a lower price to purchase the bond, thereby increasing its yield

puttable bonds = investor has the right to put the bond back to the issuer at fixedprices on fixed dates -> dispose of the bond should its price deteriorate -> put optionmakes the bond more attractive, increasing its price and lowering its yield


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convertible bonds = bon can be converted into the common stock of the issuingcompany at a fixed price on a fixed date -> partake in the good fortunes of thecompany, allows companies to issue bonds at a lower yield than otherwise

Methods of Quotation

Clean price: without accounting for the accrued income from the last couponGross price/ dirty price: clean price + accrued interest-> computed as act/act in the U.S. Treasury market, act/360 in the LIBOR market

actual number of days since last couponAccrued Interest (AI) = Coupon *

Actual number of days between last and next coupon

Discount basis for Treasury bills: quoted in terms of an annualized discount rate

Discount rate (DR) = (FacePrice)/ Face * 360/actual number of daysPrice (P) = Face * (1DR * t /360)Face/Price = (1 + yield * t/365)

If the price is lower than the face value, the yield must be greater than the discount rate

Analysis of fixed-income Securities

The Net Present Value (NPV) Approach

yield = another way to express the price of the bond, more convenient to compare variousbonds -> is the expected rate of return on the bond. Provide all coupons are reinvested at thesame rate

Pricingfair value of the bond: discount the cash flows using the market-determined yield whichdoes not depend on t but is fixed for all payments for the bond -> ignores the shape of theterm structure of interest rates

short maturities have lower interest rates than longer maturities -> discount each cash flowat the zero-coupon rate (sport interest rate) that corresponds to each time period -> add a fixedamount/ static spread to the spot rate so that the NPV equals the current price (also possible asyield spread but less accurate) bond with a large static spread is cheaper or has a higherexpected rate of return

Ct cash flow in period t (coupon and/or principal paymentt number of periods to each payment

T number of periods to final maturityy yield to maturityP rice of thebond includin accrued interest

YT market determined yield

Rt spot interest rate for maturity t and this risk class (same currency andcredit riskSS static spread


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cash flows with different credit risks need to be discounted with different rates -> separatethe discounting of principal from the discounting of cash flows

principal is discounted into a present value using the appropriate yield present value is subtracted from the market value coupons are discounted at the stripped yield which accounts for the credit risk of the

issuer

Duration= measure of exposure/sensitivity of the bond price to movements in yields= the weighted average time to wait for each payment, but only when the bonds cash flowsare predetermined

fixed cash flows: duration is measured as the weighted maturity of each payment, wherethe weights are proportional to the present value of the cash flow

Macaulay duration

Dollar duration (D* P)= expressed in dollar value of a basis point (DVBP)

floating-rate notes:- just before the reset date the prevailing interest for setting the coupon is known ->

FRN is similar to cash/money market instrument with no interest rate risk and sellingat par with zero duration

- just after the reset date the investor is locked into a fixed coupon over the accrualperiod -> FRN is equal to a zero-coupon bond with maturity equal to the time to thenext reset date

Spot- and Forward Rates

Spot rates = zero-coupon investment rates that start at the current timeForward rates = are rates that start at a future date -> can be inferred from spot rates= the break-even future rate that equalizes the return of investment of different maturities ->can be interpreted as a measure of the slope of the term structure and can be viewed as anexpectation of the future spot rate (expectation hypothesis: assumes that there is no risk

premium embedded in forward rates and that it does not matter which maturity should beselected for investment purposes -> exchange: higher coupons capital loss)

Par yield = can be viewed as a weighted average of spot rates

D* modified duration


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when the spot rate curve is flat, the spot rate curve is identical to the par yield curve andthe forward rate curve as interest rates have to be positive, forward rates have to be positive -> otherwise therewould be arbitrage

Prepayment

Factors affecting mortgage refinancing patterns

spread between the mortgage rate and the current rates: increase in spreadincreases prepayments

age of the loan: low prepayment rates after the issuance of the loan seasoning:increasing of prepayment level until it reaches as stable level

refinancing incentives: smaller costs of refinancing makes home-owners refinancemore often

previous path of interest rates: refinancing is more likely when rates have been highin the past but recently dropped burnout: if rates are low and have been for a whilemost if the principal is already prepaid

level of mortgage rates: lower rates increase affordability and turnover economic activities: greater job turnover leads to prepayments seasonal effects: more home-buying in spring, more prepayments in fall

Public Securities Association prepayment model: industry standard to describe theschedule of projected prepayments during the remaining life if the bond

Prepayment risk mortgages are subject to market risk (e.g. fluctuations in interest risk) and credit risk(e.g. home-owner default)prepayment risk= risk that the principal will be repaid early

- contraction risk: = average life of the loan is shortened (decreasing interest rates)- extension risk = average life of the loan is lengthened (increasing interest rates)

-> mortgage investments have negative convexity which reflects the short position in anoption granted to the home-owner to repay early option feature increases the yield

option adjusted spread = most commonly used approach to model the option component

-> starting from the static spread -> running simulations of various interest rate scenarios andprepayments to establish the option cost -> with the same risk class, a security trading at a

SMM single monthly mortalityCPR conditional prepayment rate


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high OAS is preferable to others -> more stable over time than the spread (which is affectedby the option component)

Option Adjusted Spread (OAS) = Static spreadOption cost

effective duration/effective convexity-> measures are based on the estimated price of the mortgage for three yield values, makingsuitable assumptions about how changes in rates should affect prepayments

Securitization

Principles of securitization mortgage loans are not tradable

Securitization = the process by which assets are pooled and securities representing interestsin the pool are issuedMortgage-backed Securities (MBS) = tradable securities that represent claims on pools of

mortgage loans -> often with third party guarantees against credit risk (Fannie Mae:government sponsored enterprise (GSE) as mortgage insurer)

Process of securitization:1) create a new legal entity = Special-purpose vehicle (SPV)/ Special-purpose entity

(SPE)2) originator/issuer pools a group of assets and sells them to the SPV3) SPV issues tradable claims or securities that are backed by the financial assets

if clean sale to the SPV: shields the ABS investor from the credit risk of the originator pooling offers ready-made diversification across many assets

banks as financial intermediaries = raise funds (recorded as liabilities on the balance sheet)that are used for making loans (recorded as assets) securitization removes both assets and liabilities from the balance sheet requiring lessequity capital, regulatory capital relieffor the originator its an additional source of fundingmanage the banks risk profile (lowering the risk profile with securitization if collateral ismuch riskier than the rest of the assets)

Assets included in ABS: collaterals: mortgage loans, auto loans, student loans, credit card receivables, accounts

receivables, debt obligations

Servicing agent = collecting payments on the collaterals -> usually also performed by theoriginator in exchange for a servicing fee


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Pass-through = one class of bonds and all investor receive the same proportional interests inthe cash flow -> tranches = bonds, when the SPV issues different classes of securities participation certificates:= mortgage pass-through with GSE as mortgage insurer

On-balance sheet securitization = covered bonds/Pfandbriefe the bank originates the loans and issues securities secured by these loans which are kept onits books -> investors have recourse against the bank in the case of default on the mortgages,

bank provides a guarantee against credit risk

Residential mortgage-backed securities (RMBS) = collateral consists of residentialmortgage loansCommercial mortgage-backed securities (CMBS) = collateral consists of commercialmortgage loans cash flow pattern starts from an annuity, where the home-owner makes a monthly fixed

payment that covers principal and interest

subject to interest rate risk, prepayment risk and default risk

Issues with securitization less incentives to worry about the quality of the loans because its revenues depend on thevolume of issuance (moral hazard -> institutions behave differently than if fully exposed tothe risk of the activity) problem when a bank decides to securitize a loan because of negative information aboutthe borrower, rather than for diversification or capital relief reasons (adverse selection due toasymmetric information) complex securitization structures -> investors rely blindly inaccurate credit ratings easy securitization has increased the total amount of lending creating additional demandfor housing and pushing housing prices even further away from their fundamental value many originators invested in asset-backed securities by creating structured investmentvehicles (SIV) = virtual banks investing in asset-backed securities and funding themselvesusing short-term debt

Tranching prepayment is unattractive to investors (want fixed-income securities with predictable

payments)

collateralized mortgage obligations (CMOs) = securities that redirect the cash flows of an

MBS pool to various segments alternatively: collateralized bond obligations (CBOs), collateralized loan obligations(CLOs), collateralized debt obligations (CDOs)


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Tranching = rearrangement of the total cash-flows, total value and total risk of theunderlying securities -> nevertheless cash-flows and risks are fully preserved (onlyredistributedsame market value and risk profile)better tailored to investors needs

sequential-pay tranches = prioritizing the payment of principal into different tranches

planned amortization class (PAC) = to minimize prepayment risk which is transferred tosupport bonds -> PAC bond offer a fixed redemption schedule as long as prepayments on thecollateral stays within a specific range = PAC collar

IO/PO-structure: strips the MBS into two components- interest-only (IO): receives only the interest payment on the underlying MBS -> sum

of all interest payments depends on the timing of principal payments interest rates fall = faster prepayment = less interest payments over the life of the

MBS = IO depreciates in value interest rates rise = slower prepayments = more interest payments over the life ofthe MBS = IO appreciates in value-> are bullish securities with negative duration

- principal-only (PO): receives only the principal payment on the underlying MBS ->sum of all payments are constant but the timing is uncertain interest rates fall = contraction risk interest rates rise = extension risk = PO loses value (principal is paid later, discountrate increases)

8 Fixed-Income Derivatives

fixed-income derivatives = instruments whose value derives from a bond price, interest rateor other bond market variable

Forward Contracts

Forward rate agreements (FRAs) = over-the-counter financial contracts that allowcounterparties to lock in an interest rate starting at a future time (buyer = borrowing rate ->

benefits from an increase in rates, seller = lending rate -> benefits from a fall in rates)

Notation: 1 x 4 (settlich in one month on three-month LIBOR)-> 1 first settlement date, 4 time to final maturity on the settlement date (in one month) the payment to the long involves the net value of thedifference between the spot rate and of the locked-in forward rate -> settled in cash, noupfront investment

VT = (STF) * * Notional * PV($1)

a short position in an FRA = equivalent to borrowing short-term to finance a long-term

investment -> similar to a long position in a bondduration = difference between the durations of the two legs = (for a short FRA)

STspot rateF locked-in forward rate the period to which the Index appliesPV($1) = $1/ (1 + ST)


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Dollar Duration (DD) = * Notional * PV($1)

Futures

traded on organized exchanges

Eurodollar Futures tied to a forward LIBOR rate (e.g. Euribor futures denominated in Euro, Euroyen futures

denominated in Japanese yen) are akin to FRAs involving three-months forward rates starting on a wide range of dates,up to 10 years into the future -> settled in cash, constant duration = 3 month

Value of one contract Pt = 10,000 * [100 * 0.25(100-FQt)]= 10,000 * [100 * 0.25 Ft]

Settlement value Pt = 10,000 * [100 * 0.25 S t]

increase in interest leads to a decrease in price of the contract (negative correlation, moves

like a bond) -> funds have to be provided precisely when the borrowing cost or reinvestmentrate is higher convexity adjustment: adjustment of the futures rate/forward rate due to the increasedvolatility of marked-to-market transactions

Futures rate = Forward rate + (1/2)2 t1 t2

T-Bond-Futures tied to a pool of Treasury bonds that consists all bonds with a remaining maturity greater

than 15 years (noncallable) -> similar contracts on shorter rates (2-, 5-, 10-years Treasurynotes) settled by physical delivery -> conversion factor (CF) to ensure interchangeability betweenthe deliverable bonds multiplies the futures price for payments to the short, attempts toequalize the net cost of delivering the various eligible bonds

minimizes the possibility of market squeezes = when holders of the short positioncannot acquire or borrow the securities required under the terms of contract high coupon bonds have higher CFs is not perfect when the term structure is not flat or when the bonds do not trade attheir theoretical prices

short position delivers bond, receives futures price times conversion factor specific to thedelivered bond (set by the exchange at initiation of the contract for each bond) + accrued

interest

Cost = PriceFutures Quote * CF

cheapest-to-deliver (CTD) = bond with the lowest net cost the longer the maturity, the greater the difference (lower CF) -> net cost is greater forlong-term bonds -> favours short-term bonds for deliveryequilibrium futures price:

Fte-rt

= StPV(D)

FQt quoted Eurodollar futures priceFt interest rate0.25 three month maturitySt spot rate

volatility of the change in the short-term ratet1 time to maturity of the futures contractt2maturity of the rate underlying the futures contract

St gross price of the CTD

PV(D) present value of the coupon payments


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duration of the futures contract is given by that of the CTD -> approximation as the shortonly has the option to deliver the CTD -> option should depress the futures price (long isshort the option and therefore requires a lower acquisition cost)

Swaps

= agreements by two parties to exchange cash flows in the future according to a prearrangedformula interest rate swaps: payments tied to an interest rate

- fixed-for-floating swap: one party commits to pay a fixed percentage of notionalagainst a receipt that indexed to a floating rate (typically LIBOR) -> risk = change inthe level of rates

- basis swaps: both payments are indexed by a floating rate -> risk = change in thespread between reference rates

Instruments in terms of actual cash flows, swap payments are typically netted against each other

Quotation quoted in terms of spreads relative to the yield of similar-maturity Treasury notese.g. 10 year swap spread as 31/34 bp against LIBOR -> 3 bp spread- dealer is willing to pay: current yield + 0.31 against receiving LIBOR- dealer is willing to receive: current yield + 0.34 against paying LIBOR swap should trade at a positive credit spread to Treasuries

interest rate swap rate market is by far the largest derivative market in terms of notional ->very liquid market, market quotations for the fixed leg have become benchmark interest rates basis for the swap curve/ par curve (is equivalent to yields on bonds selling at par) ->floating leg indexed to LIBOR carries credit risk -> swap curve is normally higher than the

par curve for government bonds in the same currency

Pricing either by taking the difference between two bond prices or valuing a sequence of forwardcontracts

swap is equivalent to a long position in a fixed-rate bond with similar couponcharacteristics and maturity offset by a short position in a floating rate note (FRN)

value of the swap (V) = BFBf

just before reset: the value of the FRN behaves exactly like a cash investment, as thecoupon for the next period will be set to the prevailing interest rate -> market value is close to

BF value of the fixed rate bondBfvalue of the floating rate note -> should be close to par


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face value -> just after reset: the FRN behaves like a bond with a six-month maturity fluctuations in the market value of the FRN should be small

as difference between two bond prices:- at initiation the swap coupon is set to the prevailing par yield -> B f = face value (= 100)

swap value = 0 -> is like a forward contract at initiation- after the swap is consummated, the value will be affected by interest rates (interest rates fall:swap will move in-the-money -> receives higher coupons than prevailing market yields) -> BFwill increase, Bf will barely change- duration of a receive-fixed swap is similar to that of a fixed-rate bind, including the fixedcoupons and principal (included in duration calculation even though it is not exchanged!) atmaturity (duration of the floating leg is close to 0)

as sequence of forward contracts:- value of long forward contract (benefits of rates going up)

- swap: receives a fixed-rate -> looses money if rates go up -> sign of the equation is reversed interest rate swaps can be priced and hedged using a sequence of forward rates -> in

practice the practice of daily marking-to-market futures induces a slight convexity bias infuture rates, which have to be adjusted downward to get forward rates

Options

large variety of fixed-income options, often embedded in many securities underlying can be a yield or a price

Caps and Floors

Cap = call option on interest rates with value:

are purchased jointly with the issuance of floating-rate notes that pay LIBOR plus a spreadon a periodic basis for the term of the note -> issuer ensures that the cost of capital will notexceed the capped rate such caps are really a combination of individual options = caplets payment on each caplet is determined by the CT, the notional, and an accrual factor ->

payments are made in arrears (= at the end of the period) typically price using a variant of the Black model, assuming that interest rate changes arelognormal -> value of the cap is set equal to a portfolio of N caplets (= European-styleindividual options on different interest rates with regularly spaced maturities)

ninotional amount for maturity iFicurrent forward rateK prespecified rateRispot rate (discrete compounding) time

K = iC cap rateiTrate prevailing at maturity

ci price of a capletF current forward rate for period t i to ti+1K current cap ratePV($1) discount factor to time ti+1


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volatility is that of the forward rate between now and the expiration of the option contract(ti) -> quoted as one number for all caplets within a cap = flat volatilities (see spot volatilities= quotes separately for each forward rate in the caplet) cap is at-the-money, if the cap rate is the same as the prevailing swap rate (near caplets areout-of-the money, distant caplets are in-the-money)

Floor = put option on interest rate with value:

Collar = a combination of buying a cap and selling a floor decreases the net cost of purchasing the cap protection when the cap and floor rates converge to the same value (K = iC = iF), the overall debt cost

becomes fixed instead of floating -> collar is the same as a pay-fixed swap, which is theequivalent of put-call-parity

Long Cap (iC = K) + Short Floor (iF = K) = Long Pay-Foxed Swap

Swaptions= OTC options that give the buyer the right to enter a swap at a fixed point in time at specifiedterms, including a fixed coupon rate

European swaption = exercisable on a single date at some point in the future -> owner hasthe right to enter a swap with a specific rate and term (e.g. 1Yx5Y swaption = in one year along or short position in a 5 year swap)

European 6/1 put swaption or one into five-year payer option: company wants to swap thefloating payments (= from issuing a five-year floating rate debt in one year) into fixed

payments -> purchase a swaption that will give it the right to create a five-year pay-fixedswap -> if the prevailing swap rate is higher than the agreed rate = exercise the swaption equivalent to a one-year put option on a six-year bond (creates a profit if rates rise andtherefore bond value falls)

Value of the option at expiration:

cf. swaption that gives the right to receive fixed is akin to a call option on a bond used for various purposes are typically priced using a variant of the Black model, assuming that interest rates arelognormal -> value of the swaption is equal to a portfolio of options on different interest rateall with the same maturity are traded in terms of volatilities instead of option premiums (in practice) -> applicableforward rate starts at the same time as the option, with a term equal to that of the option

Exchange-Trade Options options on Eurodollar futures and on T-bond futures

Options on Eurodollar futures give the owner the right to enter a long or short position in Eurodollar futures at afixed price

K = if floor rateiTrate prevailing at maturity

V(i) value of a swap to pay a fixed rate i iTprevailing swap rate for the swap maturityK locked-in swap rate


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payoff put option:

-> in addition to the cash payoff the option holder enters a position in the underlyingfutures = put, which creates a short position after exercise with the counterparty takingthe opposite position -> futures are settled daily, value of contract = 0-> payoff is equivalent to that of a cap on rates: Eurodollar future equivalent a capleton LIBOR with the difference that options on Eurodollar futures are American styleand that payments are paid at the expiration date of the Eurodollar futures optionsinstead of in arrears

Options on T-bond futures give the owner a right to enter a long or short position in futures at a fixed price payoff on a call option

investors who thinks that rates will fall/ bond market will rally = buy a call on T-bond futures -> participate in the upside, without the downside risk

Product Long (buy) Short (sell)

Fixed/Floating Swap Pay fixedReceive floating

Pay floatingReceive fixed

Cap Pay premiumReceive Max(iic,0)

Pay Max(iic,0)Receive premium

Floor Pay premiumReceive Max(iFi,0) Pay Max(iFi,0)Receive premium

Put Swaption

(payer option)

Pay premiumOption to pay fixed and receivefloating

Receive premiumIf exercised: pay floating andreceive fixed

Cal Swaption

(receiver option)

Pay premiumOption to pay floating andreceive fixed

Receive premiumIf exercised: pay fixed andreceive floating

9 Equity, Currency, and Commodity Markets

Common stocks/Equities= securities that represent ownership in a corporationbonds are senior to equities (= have a prior claim to the firms assets in case of

bankruptcy) equities represent residual claims to what is left of the value of the firm after bonds, loansand other contractual obligations have been paid off limited liability: the most shareholders can lose is their original investment (cf. owners ofunincorporated businesses: creditors have claim on personal assets)

preferred stocks: differ from common stocks because they promise to pay a specific streamof dividends

K strike priceFQTprevailing futures price quoteat maturity


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-> behave like a perpetual bond/consol -> but failure to pay the dividends does not result indefault -> corporations must withhold dividends to common stock holders until the preferreddividends (cumulative preferred dividends = all current and previous postponed dividends)have been paid out (preferred stocks are junior to bonds but senior to common stocks)-> usually have no voting rights

-> are not tax-deductible expenses but have an offsetting tax advantage (corporationsreceiving preferred dividends only pay taxes on 30% of the amount received -> lowersincome tax burden)-> market capitalization and trading volumes is much lower than that of common stocks

Valuation extremely difficult value derives from future benefits (stream of future cash flows orfuture stock price)

Gordon-growth model

firms value can be assessed using the net present value formula like a bond, as long as thediscount rate > growth rate -> growth rate of dividends is uncertain and the required discountrate is not clear, dividends are not always paid out large variations in equity prices can arise from small changes in the discount rate or in thegrowth rate of dividends -> large volatility of equities risk and expected return of the equity depends on the underlying business fundamentals aswell as on the amount of leverage/debt in the capital structure for financial intermediaries the value of underlying assets can be measured precisely ->

value the equity from the underlying assets and the cost of borrowing -> more akin to thepricing of a derivative from the price of the underlying

Convertible Bonds and Warrants

have option like features exercise of convertible bonds/warrants entails the creation of new shares as the option issold by the corporation itself -> existing shares are said to be diluted by the creation of newshares

Warrants = long-term call options issued by a corporation on its own stock-> typically created at the time of a bond issue -> trade separately from the bond to which

they were originally attached-> exercised = cash inflow to the firm which issues more shares

Convertible Bonds = bonds issued by a corporation that can be converted into equity atcertain times using a predetermined exchange ratio -> are equivalent to a regular bond plus awarrant

allows the company to issue debt with a lower coupon than otherwise conversion rate = rate with which the conversion can take place (e.g. conversion rare 10 =

one bond converted into 10 shares)

conversion price = strike price of the option = bond price / conversion rate

conversion value = current stock price time the conversion ratio -> convertible bond valuemust be higher than the price of an otherwise identical straight bond and the conversionvalue

D dividend

r constant discount rateg growth rate


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-> high value of stock price = default of firm unlikely = straight bond price constant(reflecting the discounting of cash flows at the risk free rate) -> option is certain to beexercised: convertible value = close to conversion value-> low value of stock price = default of firm likely = drop of straight bond price (reflecting theloss upon default) -> option will most certainly not be exercised: convertible value = close to

straight bond price corporation will typically issue the convertible deep out-of-the-money convertible is also sensitive to interest rate riskValuation:

Warrants can be valued by adapting standard option price models to th dilution effect of newsharesValue at origin:

After dilution:

= equivalent to

Warrant can be valued by standard option model with asset price equal to the stock priceplus the warrant proceeds, multiplied by n with the usual parameters

then unit asset value = V0/N = S0 +M/N*W0 -> this must be solved iteratively => W0 onboth sides , if M =is small to current float , or number of outstanding shares N

On complication is that most convertibles are also callable at the discretion of the firm convertible securities called for several reasons ->

First: Issue can be called to force conversion into common stocks when stock price is highenough -> bondholders have typically a month during which they can still convert -> its aforced conversion -> this call feature gives corporation more control over conversion ->allows company to raise equity capital by forcing bondholders to pay exercise price

Second: The call may be exercised when option value is worthless and firm can refinance its

debt at lower coupon -> similar to a call of nonconvertible bond -> except convertible must bebusted-> occurs if stock price much lower than conversion price

V0 = NS0 + MW0

N Outstanding SharesM Outstanding WarrantsS0 Initial Stock PriceW0 up-front value of Warrant

WT = MAX(ST-K,0) = (ST-K) = ((VT +MK)/(N+M)K) -> > 0

WT = ((VTNK ) / (N+M)) = /(N+M) *((VTNK ) = N/(N+M)*( VT/N-K)

n = N/(N+M)

W0 = n* c (S0 + M/N W0, K, , , r, d)

W0 = c (S0 + M/N W0, K, , , r, d)


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In term of risk factors, a long position in a convertible bond is exposed to increasing interestrates and credit spreads -> e.g. regular corporate bonds but also to factors that decrease thevalue of the embedded call -> e.g. decreasing stock price and decreasing implied volatility.

Equity Derivatives

Stock Index FuturesAre actively traded all over the world -> turnover corresponding to the notional amount isoften greater as amount of trading in physical stocks in same market-> success is explainable

by their versatility for risk management -> they allow investors to manage efficiently theirexposure to broad stock market movements -> speculators can take directional bets withfutures on the upside or downside-> hedgers find that futures provide cost-efficient method to

protect against price risk most active contract = S&P 500 future contract on ChicagoMercantile Exchange (CME) -> notional is defined as $250 times index level contracts arecash settled -> do not involve delivery of underlying stocks at expiration -> future contract is

priced according to the usual cash-and-carry relationship

Single Stock FuturesIn late 2000 USA passed legislation authorizing trading in single stock futures -> are futurecontracts on individual stocks-> Europe and elsewhere already trading -> USA startedelectronic trading in November 2002 -> each contract gives obligation to buy or sell 100shares of the underlying -> settlement usually involves physical delivery -> have manyadvantages-> positions can be established more efficiently due to their low marginrequirements, generally 20% of cash value -> also short selling eliminates the costs ofinefficiencies associated with stock loan process-> other than physical settlement these

contracts trade like stock index futures

Equity OptionsOptions can be traded on individual stocks, on stock indices, or on stock index futures.In USA stock options trade on Chicago Board Options Exchange (CBOE) -> Each optiongives right to buy or sell a round lot of 100 shares -> settlement involves physical delivery->traded options are typically American-style their valuation should include possibility ofearly exercise -> in practise value do not differ from European-style option -> when stock

pays no dividend values are same for more precision use of numerical models such asbinominal tree -> in USA most active index option is S&P 100 and S&P 500 index traded on

CBOE

S&P 100 is American-style , S&P 500 is European-style -> options are cash settled-> each contract is for $100 times the value of index-> European options on stocks can bepriced using Black-Scholes formula, using y as the dividend yield on the index -> options onS&P 500 stock index also very popular-> give right to enter long or short futures position at afixed price -> exercise is cash settled

Equity SwapsIs an agreement to exchange cash flows tied to return on a stock market index exchange for afixed or floating rate of interest-> e.g. swap that provides the return of S&P 500 index everysix months in exchange for payment of LIBOR plus spread ->swap will be typically priced ->so as to have zero value at initiation-> equity swaps can be valued as portfolios of forward

contracts, as in case of interest swaps -> pricing method of currency swaps can also be used -> are used by investment managers to acquire exposure to, e,g, an emerging stock marketwithout to invest in the market itself -> can be used to skirt restrictions on foreign investments

Fte-r = Ste

-y y dividend vield defined by unit of time


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Variance SwapsIs forward contract on variance -> payoff:

Can be written on underlying asset -> but most common for equities or equity indices-> allowtrades based on direct views on variance -> long positions are bets on high volatility -> initialvalue of contract is zero -> the widely quoted VIX index is fair strike price for a varianceswap on S&P 500 index, quoted as volatility -> market value for outstanding variance swapwith = T t days remaining to maturity is

contracrs allow also correlation trading -> e.g. index of two stocks -> variance swap isavailable for each constituent stock as well as for index -> realized variance depends ontwo variances and the correlation coefficient -> all else equal -> higher correlation =higher portfolivariance -> long correlation trade would buy a variance swap on the indexand short variance swaps on components -> if correlation increases long position shouldgain more than short positions

Currency Markets=forex -> enormous trading activity -> spot transactions are exchanges of two currencies forsettlement as soon as it is practical -> typically two business days -> ca 35% of tradingvolumeOutright forward contracts = agreements to exchange two currencies at future date -> ca 12%Forex swaps = involve two transactions, an exchange of currencies on given date and areversal at a later date -> 35% -> typically short-term nature -> do not confuse with long-termcurrency swaps = involve stream of payments over longer horizonsin addition market also includes OTC forex options and exchanged-traded derivates ->Most active currency futures traded on CME -> settled by physical delivery -> CME alsotrades options on currency futures -> currency forwards, futures, and options can be pricedaccording to standard valuation models-> specify income payment as continuous flowCurrencies are generally quoted in European terms = in units of foreign currency per dollar ->Two exceptions: GBP, EUR -> quoted in American terms = dollars per unit of foreigncurrency

Currency Products

currency markets offer full range of financial instruments -> one type of instrument isspecific -> quanto = quantity-adjusted derivate -> payoff defined by variables associatedwith one currency but is paid in another currency -> e.g. U.S investor consider buyingJapanese stocks represented by Nikkei 225 index -> normally stock buying involves a

position in stocks and in Japanese yen currency -> could have situation where Nikkeiindex increases in value but gain is wiped out by fall in value of yen -> to avoid currencyrisk investor could buy quanto forward contract on index with expiry in one year ->contract has notional in dollars and Forward price in yen -> payout at expiration dependson the yen value of index

forward price of quanto depends on usual forward price in foreign currency but also thecovariance between movements in the yen stock price and in the dollar price of the yen

VT= (t0,TKV) NN national Amount realized variance over lifeof contractKV strike price or forward

price= 252/[ln(Si/Si-1)]

Vt= Ne-r[w(t0,tKV) + (1w) ( Kt-KV)]

t0,t elapsed variance beween initial time t0 andcurren time rw is fraction of days elapsed since t0Kt is current forward price

VT

=N (ST-F )


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Currency SwapsAre agreements by two parties to exchange a stream of cash flows in different currenciesaccording to prearranged formula

Instruments

Consider two counterparties, A and B that can raise funds either in dollars or in yen -> A wantto raise dollars and B wants to raise yen -> company A has absolute advantage in the twomarkets -> B has comparative advantage in raising dollars -> provides basis for swap -> if

both raise funds in desired markets cost can be higher as than both companies raise funds inthe market their they have comparative market -> both parties benefit from swap -> paymentsare typically netted for interest rate swap -> because currency is same -> not the case forcurrency swaps -> full interest payments are made in different currencies -> in addition atinitiation and termination there is exchange of principal in different currencies

PricingAs with interest rate swaps -> swaps can be priced using either of two approaches -> taking

the difference between two bonds or valuing a sequence of forward contracts -> swap isequivalent to a long position in a fixed-rate and a short-position -> value of swap is that oflong yen bond minus dollar bond

Initial value of swap is zero -> assuming flat term structure for both countries and no creditriskThree conditions under which swap will be in the money

If value of yen appreciates If yen interest rate falls If dollar interest rate goes upSwap exposed to three risk-factors: spot-rate, and two interest rates -> the latter exposuresare given by the duration of equivalent bond

A position in a receive-foreign currency swap is equivalent to a long position in foreigncurrency bond offset by a short position in a dollar bond

Swap can be alternatively valued as a sequence of forward contracts

Extending to multiple maturities

Commodities

ProductsCommodities are typically traded on exchanges -> contracts include spot, futures and optionson futures-> also OTC market for long-term commodity swaps -> payments are tied to priceof a commodity against a fixed or floating rateCommodity contracts can be classified into

Agricultural products ( including grains and oilseeds (corn, wheat, soybean), food andfiber (cocoa, coffee, sugar, orange juice)

Livestock and meat (cattle, hogs)

FQ=F exp(pSTFX)

V = S($/Y)P (Y)-P($)

Vi= (Fi-K)exp(-rii) ri dollar interest rateFi prevailing forward rate in $/yenK locked in rate of exchange

V = ni(Fi-K)exp(-rii)


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Base metals (aluminium, copper, nickel, and zinc) Precious metals (gold, silver, platinum) Energy products (natural gas, heating oil, unleaded gasoline, crude oil)Goldmann Sachs Commodity Index (GSCI) is broad production-weighted index ofcommodity price performance -> composed of 24 liquid exchange-traded futures contracts ->

contains 72% energy products, 7% industrial metals, 2% precious metals, 14% agriculturalproducts, and 5% livestock products -> CME trades futures and option contracts on GSCI ->few last years markets developed for electricity products -> more recently weather derivates

Pricing of FuturesCommodities differ from financial assets in two notable dimensions

May be expensive or impossible to store May generate a flow of benefits that are not directly measureableFirst dimension involves cost of carrying physical inventory of commodities -> for mostfinancial instruments this cost is negligible -> for bulky commodities this cost may be high ->other commodity cannot be stored easilySecond involves benefit from holding the physical commodity -> e.g. company thatmanufactures copper pipes benefits from an inventory of copper -> this flow is also calledconvenience Yield for holder -> for financial asset this flow would be a monetary income

payment for investor -> when asset can be lent out for profit the yield represents lease rate ->Consider storage cost only -> cash and carry relationship should be modified -> compare two

positions -> in the first we buy the commodity spot plus a pay up-front the present value ofstorage costs -> in the second -> enter forward contract and invest the present value of theforward price -> since two positions are indentical at expiration -> must have same initialvalue

Alternatively -> storage costs are incurred per unit of time and defined as c

Due to these costs forward rate should be much greater than spot rate

Convenience yield can be expressed as y per unit of time -> y represents the net benefit fromholding the commodity, after storage costs

Actualization factor may have economically identifiable meaning -> reflecting demand andsupply conditions in the cash and futures markets -> can be viewed as plug-inFuture prices are generally increasing with maturity -> reflecting time value of money,storage costs and low convenience yields -> some irregularities reflecting anticipatedimbalances between demand and supply

Futures and Expected Spot PricesInteresting issue is whether todays futures price gives best forecast of future spot price -> ifso it satisfies the expectations hypothesis

Ft e-r = S t+ PV[C] e

-r present value factor

Ft e-r = S te

c

Ft e-r

= S te-y

e-y

actualization factor

Ft = Et[ST]


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The reason this relationship may hold is as follows:

One-year oil future price F = $20.47 -> if market forecasts that oil prices in one year willbe at $25, one could make profit by going long a futures contract at the cheap future price,waiting a year, then buying oil for cheap $20.47 and reselling at the higher price of $ 25 -> deviations from this relationship imply speculative profits

To be sure, these profits are not-risk free -> they may represent some compensation forrisk -> e.g. market is dominated by producers who want to hedge by selling oil futures ->F will be abnormally low compared with expectations -> relationship between futures

prices and expected spot prices can be complex

For financial assets for which arbitrage between cash and futures is easy -> the future orforward rate is solely determined by the cash-and-carry relationship

For commodities the arbitrage may not be so easy -> future prices may deviate from thecash-and carry relationship through this convenience yield factor -> such prices reflectexpectations of futures spot prices, as well as speculative and hedging pressures

Market is said to be in contango when futures price trades at a premium relative to spot priceMarket is said to be in backwardation when forward prices trade at discount relative to spot

priceWith backwardation the future price trends to increase as the contracts near maturity -> insuch a situation a roll-over strategy could be profitable-> roll-over strategy involves buying a long maturity contract, waiting, and then selling it at ahigher price in exchange for buying a cheaper longer-term contractStrategy is comparable to riding the yield curve when upward-sloping = involves buying longmaturities and waiting to have yields fall due to the passage of time

Markets are in contango if spot prices are lower than forward prices. Markets are inbackwardation if spot prices are higher than forward prices. Backwardation occurs when there

is a high current demand for the commodity, which implies high convenience yields

10 Introduction to Market Risk

market risk is primarily measured with position-based risk measures (e.g. value-at-risk) Value-at-risk (VAR) = a statistical measure of total portfolio risk, based on the mostcurrent positions, which takes into account portfolio diversification and leverage risk managers should consider the entire distribution of profits and losses over thespecified horizon -> in practice, the distribution is summarized by the worst loss at a specifiedconfidence level stress-testing = identifies potential losses under extreme market conditions

Types of Financial Risksfinancial risk= market risk, credit risk, operational risk

market risk= the risk of losses due to movements in financial market prices or volatilities-> usually includes liquidity risk = risk of losses due to the need to liquidate positions tomeet funding requirements -> not amendable to formal quantification credit risk = the risk of losses due to the fact that counterparties may be unwilling orunable to fulfil their contractual obligations-> settlement risk (=theriskthat acounterpartydoes not deliver asecurityor itsvaluein cashas per agreement when the security was traded after the othercounterpartyor counterpartieshave already delivered security or cash value as per the trade agreement) / Herstatt risk (=

cross-currency settlement risk that arises where the working hours of inter-bank fund transfersystems do not overlap due to time zone differences)
http://en.wikipedia.org/wiki/Riskhttp://en.wikipedia.org/wiki/Riskhttp://en.wikipedia.org/wiki/Riskhttp://en.wikipedia.org/wiki/Counterpartyhttp://en.wikipedia.org/wiki/Counterpartyhttp://en.wikipedia.org/wiki/Counterpartyhttp://en.wikipedia.org/wiki/Security_(finance)http://en.wikipedia.org/wiki/Security_(finance)http://en.wikipedia.org/wiki/Security_(finance)http://en.wikipedia.org/wiki/Value_(economics)http://en.wikipedia.org/wiki/Value_(economics)http://en.wikipedia.org/wiki/Value_(economics)http://en.wikipedia.org/wiki/Counterpartyhttp://en.wikipedia.org/wiki/Counterpartyhttp://en.wikipedia.org/wiki/Counterpartyhttp://en.wikipedia.org/wiki/Tradehttp://en.wikipedia.org/wiki/Tradehttp://en.wikipedia.org/wiki/Tradehttp://en.wikipedia.org/wiki/Counterpartyhttp://en.wikipedia.org/wiki/Value_(economics)http://en.wikipedia.org/wiki/Security_(finance)http://en.wikipedia.org/wiki/Counterpartyhttp://en.wikipedia.org/wiki/Risk


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operational risk= the risk of loss resulting from failed or inadequate internal processes,systems and people or from external events-

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Jorion ZusaFa