Bond Immunisation (Coursework Sample)

Bond Immunisation
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OUTLINE OF THE FINANCIAL MODELS:
Black-Derman-Toy (BDT) Model
BTD is a short rate model used in pricing of bond options, swaptions and other interest rate derivatives. It uses a single factor-the short rate-which determines the future evolution of all interest rates.
Under this model, using a binomial lattice, you need to calibrate the variables to fit the current term structure (yield curve) and the volatility structure for interest rate caps (Sylvester & Qing, 2007).
The general formula for the model is:
Dln(r) = {ϴt + (σ’t/ σt) lnr} DT + σtdWt where:
r= the instantaneous short rate at time t,
Ï´t= value of the underlying asset at option expiry,
σt= instant short rate volatility, and
Wt=a standard Brownian motion under a Risk-neutral probability measure.
For constant (time independent) short rate volatility, , the model is:
Dln(r) =ϴtdt + σdWt
According to Sylvester & Qing (2007), the standard BDT model is constructed algorithmically to be consistent with both the existing term structure of zero-coupon yields, and (optionally) the term structure of yield volatilities. That is, the main aim of the tree building procedure is to derive a binomial representation for the level of the short-term interest rate such that zero-coupon bond prices computed from the tree are exactly equal to the set of zero coupon prices that are directly observable in the market. If desired, the model can also be constructed so that the implied interest rate distribution at each time step matches an observed interest rate volatility curve. Sylvester & Qing (2007) define a zero-coupon bond as a single certain cash flow occurring at known time, t, in the future.
The model built in this assignment is based on one factor, the yield rates, which have the numerous advantages including the ease of implementation, less time consuming and ease of calibration.
For this model to hold, the following assumptions are made:
The changes in all bond yields are perfectly correlated.
Expected returns on all securities over one period are equal
The short rates are log-normally distributed. This prevents negative short rates and has the advantage of volatility input being in percentage form.
There exist no taxes or transaction costs.
Fisher-Weil Duration Model
This model describes the duration of a financial asset that consists of fixed cash flows such as bonds. The duration of such an asset is the weighted average of the times until the fixed cash flows are received (Sylvester & Qing, 2007). When an asset is considered as a function of yield, duration also measures the price sensitivity to yield, the rate of change of price with respect to yield or the percentage. Macaulay duration refers to the weighted average time until cash flows are received, and is measured in years. It is the price sensitivity and is the percentage change in price for a unit change in yield. When yields are continuously compounded Macaulay duration and modified duration will be numerically equal. When yields are periodically compounded Macaulay and modified duration will differ slightly, and there is a simple relation between the two. Modified duration is used more than Macaulay duration.
The duration has two basic assumptions when it is applied in the interest rate management:
Yield curve is flat which means the interest rates are the same for different bonds.
There is no taxation or transaction cost.
For bonds with fixed cash flows a price change can come from two sources:
The passage of time (convergence towards par). This is of course totally predictable, and hence not a risk.
A change in the yield. This can be due to a change in the benchmark yield, and/or change in the yield spread.
The yield-price relationship is inverse, and we would like to have a measure of how sensitive the bond price is to yield changes. The modified duration is a measure of the price sensitivity to yields and provides a linear approximation.
Macaulay duration and modified duration are both termed "duration" and have the same (or close to the same) numerical value. Macaulay duration is a time measure with units in years, and really makes sense only for an instrument with fixed cash flows. Fo...