Fixed Income Part 1 Flashcards
How does GSW bootsrap ZC yield
use the nelson siegel Swenson approach and assume zero coupon yield can be written as a formula with 6 paramters: beta 0 - beta 3, T1 and T2
Thus, for a given set parameters the yield curve is specified for
all maturities
Each day, they estimate parameters by minimizing (duration-weighted) squared deviations between actual and predicted bond prices
Advantages of future options over spot prices
Advantages of futures option over spot options
Unambiguous, easily available futures prices.
(Especially relevant if underlying traded OTC.)
Futures markets often more liquid than the underlying markets.
See, e.g., Treasuries
Summary of Interest Rates:
Treasury rates: Earned on Treasury bills and bonds, instruments issued by governments. The government is borrowing from you. Usually there is negligible risk of not paying back.
LIBOR: Short-term unsecured borrowing rate between AA and higher rated banks, between 1 day and 1 year; used as a benchmark for many assets, e.g., swaps.
Fed Funds Rate (FFR): Overnight unsecured rate based on borrowing/lending of money re-serves US institutions must hold at the Fed. EONIA is the same for the Eurozone.
Overnight indexed swap (OIS): The fixed rate you can exchange with FFR/EONIA
Repo rates: Secured borrowing rate with a collateral.
Why is the pricing of interest rates derivatives trickier?
Pricing interest rate derivatives is trickier than equity or currency derivatives because (a) the behaviour of an interest rate is more complicated than that of, e.g., a stock (there is mean reversion), (b) many products require a model of the behaviour of the entire yield curve, (c) volatilities for different points on the yield curve are different, and (d) the interest rate is used for discounting and defining the payoff
Do we get for maturities up to one year zero coupon yields directly?
Yes! We typically get ZC yields almost directly from treasury bills or LIBOR rates [OBS: Treasury = U.S. Government, Bill = Bonds with initial maturity <= 1 year)
For longer maturities we need to bootstrap ZC yields from coupon paying bonds.
Who is GSW
Gurkaynak, Sack, & Wright (2006) (GSW) are the first to provide a long history of good daily estimates of the US yield curve.
Updated on daily basis on Fed’s webpage. In their data set the XX-year continiously compounded ZC yield is denoted SVENYXX.
What is GSW yield curve / How do they bootstrap?
To bootstrap zero-coupon yields, GSW use the Nelson-Siegel-Svensson approach.
For a given set of 6 parameters the yield curve is specified for all maturities. Each day they estimate parameters by minimizing (duration-weighted) squared deviations between actual and predicted bond prices.
What are advantages of futures option over spot options?
Umabigious, easily available futures prices (epsecially relevant if underlying traded OTC)
Futures markets often more liquid than the underlying markets (e.g. treasuries)
Future Option Price with BS
The price at time t of a European futures call option with payoff [Ft,t* - K] at time T is
e^-r(T-t) (Ft, T*N(d1) - KN(d2))
The dividend yield magically disappeard in the pricing formula because the impact of the dividend yield is already there through the futures price. The volatility sigma is the volatility of the futures price. We have to discount to today since we pay for the future at time T.
If T is the same for futures call option and the underlying future payoff is [Ft,t - K] = [St - K]. In formula we now can put instead of spot price the future price as an input.
Bond Option
A European call on a bond Bt with strike K pays [Bt - K]
We assume that interest rates are potentially stochastic.
First we calculate the forward price.
This forward price, the interest rate, volatility of forward and strike Price K (which is FV + coupon payments) ar plugged into BS.
BS assumes that the bond price is lognormally distrbuted at the options maturity.
What is the price and the use of a cap?
A cap is an insurance against high interest rates. Purchasing the contract costs something. If interest rates end up being low, payoff is zero, while if interest rates end up being high, payoff is positive.
A cap pays Lδ_k [R_k-R_K ]
L = principal/notional L
δ_k=t_(k+1)-t_k = the accrual period length
R_k = the floating rate, set at t_k for the period (often Libor)
R_K = cap rate (with same compounding frequency as the frequency of resets)
If libor > cap rate: positive difference, cap owner gets payment
Caplet and Floorlet Pricing
A caplet can be seen as a European call option at time t_k+1
A caplet pays: Lδ_k [R_k-R_K]
A cap is a collection of caplets.
A floorlet can be seen as a put option. We can price it with BS by put-call parity.
A floorlet pays: Lδ_k [R_K-R_k].
A floorlet is part of floor.
BS assumes that the interest rate underlying each caplet are lognormal distributed.
What is another approach for caplet pricing instead BS?
A cap is a PF of zero coupon bond options.
We can price caps as soon as we can price zero coupon options.
Volatility and interest rate caps
Typical broker implied flat volatility quotes for US dollar caps and floors.
Flat volatility means that the same volatility is sued for floors and caplets.
What is a Swaption?
A swaption is the option to enter at some future maturity a swap whose fixed rate is chosen today.
The value of the option to enter a payer swap with fixed sK is the value of receiving
Lδ_i [sT0-sK].
m times a year, with accrual period δ_i= 1/m for n years starting at T0, until Tnm = T0 + n
L = notional swap
Swaption pays L 1/m [sT0-sK].
