Temp_store

LO 11.5: Explain how the principles o f arbitrage pricing o f derivatives on fixed income securities can be extended over multiple periods.
There are three basic steps to valuing an option on a fixed-income instrument using a binomial tree:
Step 1: Price the bond value at each node using the projected interest rates. Step 2: Calculate the intrinsic value of the derivative at each node at maturity. Step 3: Calculate the expected discounted value of the derivative at each node using the
risk-neutral probabilities and working backward through the tree.
Note that the option cannot be properly priced using expected discounted values because the call option value depends on the path of interest rates over the life of the option. Incorporating the various interest rate paths will prohibit arbitrage from occurring.
Example: Call option
Assume that you want to value a European call option with two years to expiration and a strike price of $100.00. The underlying is a 7%, annual-coupon bond with three years to maturity. Figure 3 represents the first two years of the binomial tree for valuing the underlying bond. Assume that the risk-neutral probability of an up move is 0.76 in year 1 and 0.60 in year 2.
Fill in the missing data in the binomial tree, and calculate the value of the European call option.
Professor’s Note: Since the option is European, it can only he exercised at maturity.
Figure 3: Incomplete Binomial Tree for European Call Option on 3-Year, 7% Bond
Int. rate = 3.00% Bond price = ? Option value = ?
0.76
Int. rate = 5.99% Bond price = ? Option value = ?
0.24
Int. rate = 4.44% Bond price = ? Option value = ?
Int. rate = 8.56% Bond price = $98.56 Option value = ?
0.60
Int. rate = 6.34% Bond price = ? Option value = ?
0.40 0.60
Int. rate = 4.70% Bond price = ? Option value = ?
0.40
Today
End of 1 year
End of 2 years
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Answer:
Step 1: Calculate the bond prices at each node using the backward induction methodology.
At the middle node in year 2, the price is $100.62. You can calculate this by noting that at the end of year 2 the bond has one year left to maturity:
N = 1; I / Y = 6.34; PMT = 7; FV = 100; CPT – PV = 100.62
At the bottom node in year 2, the price is $102.20:
N = 1; I / Y = 4.70; PMT = 7; FV = 100; CPT > PV = 102.20
At the top node in year 1, the price is $100.37:
($103.36 x 0.6) + ($107.62 x 0.4)
1.0599
= $100.37
At the bottom node in year 1, the price is $103.65:
($107.62 x 0.6) + ($109.20 x 0.4)
1.0444
= $103.65
Today, the price is $105.01:
($107.37 x 0.76) + ($110.65 x 0.24)
1.03
$105.01
As shown here, the price at a given node is the expected discounted value of the cash flows associated with the two nodes that feed into that node. The discount rate that is applied is the prevailing interest rate at the given node. Note that since this is a European option, you really only need the bond prices at the maturity date of the option (end of year 2) if you are given the arbitrage-free interest rate tree. However, its good practice to compute all the bond prices.
Step 2: Determine the intrinsic value of the option at maturity in each node. For example, the intrinsic value of the option at the bottom node at the end of year 2 is $2.20 = $102.20 – $100.00. At the top node in year 2, the intrinsic value of the option is zero since the bond price is less than the call price.
Step 3: Using the backward induction methodology, calculate the option value at each
node prior to expiration. For example, at the top node for year 1, the option price is $0.23:
($0.00×0.6)+ ($0.62×0.4) ——————————– = $0.23
1.0599
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Figure 4 shows the binomial tree with all values included.
Figure 4: Completed Binomial Tree for European Call Option on 3-Year, 7% Bond
Int. rate = 3.00% Bond price = $105.01 Option value = $0.45
0.76
Int. rate = 5.99% Bond price = $100.37 Option value = $0.23
0.24
Int. rate = 4.44% Bond price = $103.65 Option value = $1.20
Int. rate = 8.56% Bond price = $98.56 Option value = $0.00
0.60
Int. rate = 6.34% Bond price = $100.62 Option value = $0.62
Int. rate = 4.70% Bond price = $102.20 Option value = $2.20
0.40
Today
End of 1 year
End of 2 years
The option value today is computed as:
($0.23 x 0.76) + ($ 1.20 x 0.24)
1.03
= $0.45
Recombining and Nonrecombining

LO 11.4: Distinguish between true and risk-neutral probabilities, and apply this difference to interest rate drift.
Using the 0.5 probabilities for up and down states as shown in the previous example may not produce an expected discounted value that exactly matches the market price of the bond. This is because the 0.5 probabilities are the assumed true probabilities of price movements. In order to equate the discounted value using a binomial tree and the market price, we need to use what is known as risk-neutral probabilities. Any difference between the risk-neutral and true probabilities is referred to as the interest rate drift.
U s i n g t h e R i s k -N e u t r a l In t e r e s t R a t e T r e e
There are actually two ways to compute bond and bond derivative values using a binomial model. These techniques are referred to as risk-neutral pricing.
The first method is to start with spot and forward rates derived from the current yield curve and then adjust the interest rates on the paths of the tree so that the value derived from the model is equal to the current market price of an on-the-run bond (i.e., the tree is created to be arbitrage free). This is the method we used in the previous example. Once the interest rate tree is derived for an on-the-run bond, we can use it to price derivative securities on the bond by calculating the expected discounted value at each node using the real-world probabilities.
The second method is to take the rates on the tree as given and then adjust the
probabilities so that the value of the bond derived from the model is equal to its current market price. Once we derive these risk-neutral probabilities, we can use them to price derivative securities on the bond by once again calculating the expected discounted value at each node using the risk-neutral probabilities and working backward through the tree.
The value of the derivative is the same under either method.
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LO 11.3: Define risk-neutral pricing and apply it to option pricing.

LO 11.2: Construct and apply an arbitrage argument to price a call option on a zero-coupon security using replicating portfolios.
The binomial interest rate model is used throughout this topic to illustrate the issues that must be considered when valuing bonds with embedded options. A binomial model is a model that assumes that interest rates can take only one of two possible values in the next period.
This interest rate model makes assumptions about interest rate volatility, along with a set of paths that interest rates may follow over time. This set of possible interest rate paths is referred to as an interest rate tree.
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Binomial Interest Rate Tree
The diagram in Figure 1 depicts a binomial interest rate tree.
Figure 1: 2-Period Binomial
Today
Period 1
Period 2
To understand this 2-period binomial tree, consider the nodes indicated with the boxes in Figure 1. A node is a point in time when interest rates can take one of two possible paths an upper path, U, or a lower path, L. Now consider the node on the right side of the diagram where the interest rate z2 rate, z’0, follows the lower path from node 0 to node 1 to become q L, then follows the upper of the two possible paths to node 2, where it takes on the value z’2 LU. At the risk of stating the obvious, the upper path from a given node leads to a higher rate than the lower path. Notice also that an upward move followed by a downward move gets us to the same place on the tree as a down-then-up move, so i2
appears. This is the rate that will occur if the initial
= i2 jjl *
The interest rates at each node in this interest rate tree are 1 -period forward rates corresponding to the nodal period. Beyond the root of the tree, there is more than one 1-period forward rate for each nodal period (i.e., at year 1, we have two 1-year forward rates, q ^ and q L). The relationship among the rates associated with each individual nodal period is a function of the interest rate volatility assumption of the model being employed to generate the tree.
Constructing the Binomial Interest Rate Tree
The construction of an interest rate tree, binomial or otherwise, is a tedious process. In practice, the interest rate tree is usually generated using specialized computer software. There is one underlying rule governing the construction of an interest rate tree: The values for on-the-run issues generated using an interest rate tree should prohibit arbitrage opportunities. This means that the value of an on-the-run issue produced by the interest rate tree must equal its market price. It should be noted that in accomplishing this, the interest rate tree must maintain the interest rate volatility assumption of the underlying model.
Valuing an Option-Free Bond W ith the Tree, Using Backward Induction
Backward induction refers to the process of valuing a bond using a binomial interest rate tree. The term backward is used because in order to determine the value of a bond at node 0, you need to know the values that the bond can take on at node 1. But to determine the values of the bond at node 1, you need to know the possible values of the bond at node 2,
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compounding periods, the current value of the bond and so on. Thus, for a bond that has is determined by computing the bonds possible values at period A^and working backward to node 0.
Consider the binomial tree shown in Figure 2 for a $ 100 face value, zero-coupon bond, with two years remaining until maturity, and a market price of $90,006. Starting on the top line, the blocks at each node include the value of the bond and the 1 -year forward rate at that node. For example, at the upper path of node 1, the price is $93,299, and the 1-year forward rate is 7.1826%.
Figure 2: Valuing a 2-Year, Zero-Coupon, Option-Free Bond
$90,006 4.5749%
$93,299 7.1826%
$94,948 5.3210%
$ 100.00
$ 100.00
$ 100.00
$ 100.00
Today
End of 1 year
End of 2 years
Know that the value of a bond at a given node in a binomial tree is the average of the present values of the two possible values from the next period. The appropriate discount rate is the forward rate associated with the node under analysis.
Example: Valuing an option-free bond
Assuming the bonds market price is $90,006, demonstrate that the tree in Figure 2 is arbitrage free using backward induction.
Answer:
Consider the value of the bond at the upper node for period 1,
Vi,u
($100×0.5)+ ($100×0.5)
1.071826
$93,299
Similarly, the value of the bond at the lower node for period 1,
L is:
V
1+
($100×0.5) + ($100×0.5)
1.053210
$94,948
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Now calculate Vq, the current value of the bond at node 0:
= ($93.299×0.5) + ($94.948×0.5) =
1.045749
Since the computed value of the bond equals the market price, the binomial tree is arbitrage free.
Professors Note: When valuing bonds with coupon payments, you need to add the coupons to the bond prices at each node. For example, with a $100 face value, 7% annual coupon bond, you would add the $7 coupon to each price before computing present values. Valuing coupon-paying bonds with a binomial tree will be illustrated in LO 11.5.

LO 11.1: Calculate the expected discounted value o f a zero-coupon security using a binomial tree.

Regression analysis focuses on yield changes among a small number of bonds. Empirical approaches, such as principal components analysis (PCA), take a different approach by providing a single empirical description of term structure behavior, which can be applied across all bonds.

PCA attempts to explain all factor exposures using a small number of uncorrelated exposures which do an adequate job of capturing risk.

For example, if we consider the set of swap rates from 1 to 30 years, at annual maturities, the PCA approach creates 30 interest rate factors or components, and each factor describes a change in each of the 30 rates. This is in contrast to regression analysis, which looks at variances of rates and their pairwise correlations.

PCA sets up the 30 factors with the following properties:
1. The sum of the variances of the 30 principal components (PCs) equals the sum of the variances of the individual rates. The PCs thus capture the volatility of the set of rates.
2. The PCs are not correlated with each other.
3. Each PC is chosen to contain the highest possible variance, given the earlier PCs.

The advantage of this approach is that we only really need to describe the volatility and structure of the first three PCs since the sum of the variances of the first three PCs is a good approximation of the sum of the variances of all rates. Thus, the PCA approach creates three factors that capture similar data as a comprehensive matrix containing variances and covariances of all interest rate factors. Changes in 30 rates can now be expressed with changes in three factors, which is a much simpler approach.

When setting up and establishing regression-based hedges, there are two schools of thought. Some regress changes in yields on changes in yields, as demonstrated previously, but an alternative approach is to regress yields on yields.

Using a single-variable approach, the formula for a change-on-change regression with dependent variable y and independent variable x is as follows:

Ayt = ol + (3Axt + Aet
where: Ayt = yt – yt-i Axt = xt – xt l

Alternatively, the formula for a level-on-level regression is as follows:
yt = a + (3xt + t

With both approaches, the estimated regression coefficients are unbiased and consistent; however, the error terms are unlikely to be independent of each other. Thus, since the error terms are correlated over time (i.e., serially correlated), the estimated regression coefficients are not efficient. As a result, there is a third way to model the relationship between two bond yields (for some constant correlation < 1):

t = Pt-l + vt

This formula assumes that todays error term consists of some part of yesterdays error term, plus a new random fluctuation.

LO 10.5: Calculate the face value o f multiple offsetting swap positions needed to carry out a two-variable regression hedge.
Regression hedging can also be conducted with two independent variables. For example, assume a trader in euro interest rate swaps buys/receives the fixed rate in a relatively illiquid 20-year swap and wishes to hedge this interest rate exposure. In this case, a regression hedge with swaps of different maturities would be appropriate. Since it may be impractical to hedge this position by immediately selling 20-year swaps, the trader may choose to sell a combination of 10- and 30-year swaps.
The trader is thus relying on a two-variable regression model to approximate the relationship between changes in 20-year swap rates and changes in 10- and 30-year swap rates. The following regression equation describes this relationship:
Ayt20 = ol + (310Ayt10 + (330Ayt30 + et
Similar to the single-variable regression hedge, this hedge of the 20-year euro swap can be expressed in terms of risk weights, which are the beta coefficients in the above equation:
(F10 xD V O l10) (F20 x DVO l20)
(- F 30x DVQ130) (F20x DV0120)
change in 10-year swap rate, 3
change in 30-year swap rate, 3
The trader next does an initial regression analysis using data on changes in the 10-, 20-, and 30-year euro swap rates for a five-year time period. Assume the regression output is as follows:
Number of observations R-squared Standard error
1281 99.8% 0.14
Regression Coefficients Alpha Change in 10-year swap rate Change in 30-year swap rate
Value -0.0014 0.2221 0.7765
Standard Error
0.0040 0.0034 0.0037
Given these regression results and an illiquid 20-year swap, the trader would hedge 22.21% of the 20-year swap DV01 with a 10-year swap and 77.65% of the 20-year swap DV01 with a 30-year swap. Because these weights sum to approximately one, the regression hedge DV01 will be very close to the 20-year swap DV01.
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The two-variable approach will provide a better hedge (in terms of R-squared) compared to a single-variable approach. However, regression hedging is not an exact science. There are several cases in which simply doing a one-security DV01 hedge, or a two-variable hedge with arbitrary risk weights, is not appropriate (e.g., hedging during a financial crisis).
Level and Change Regressions

LO 10.4: Calculate the face value o f an offsetting position needed to carry out a regression hedge.
Defining FR and FN as the face amounts of the real and nominal bonds, respectively, and their corresponding DVOls as DV01^ and DV01^, a DV01 hedge is adjusted by the hedge adjustment factor, or beta, as follows:
FR = FN x
PV01N DV01r
X(3
Now that we have determined the variability between the nominal and real yields, the hedge can be adjusted by the hedge adjustment factor of 1.0198:
xl.0198
$82.55 million
\
/
.068 0.084
V0
/
lOOMx
This regression hedge approach suggests that for every $ 100 million sold in T-bonds, we should buy $82.55 million in TIPS. This will account for hedging not only the size of the underlying instrument, but also differences between nominal and real yields over time.
Note that in our example, the beta was close to one, so the resulting regression hedge did not change much from the DV01-neutral hedge. The regression hedge approach assumes that the hedge coefficient, (3, is constant over time. This of course is not always the case, so it is best to estimate the coefficient over different time periods and make comparisons.
Two other factors should be also considered in our analysis: (1) the R-squared (i.e., the coefficient of determination), and (2) the standard error of the regression (SER). The
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R-squared gives the percentage of variation in nominal yields that is explained by real yields. The standard error of the regression is the standard deviation of the realized error terms in the regression.
Two-Variable Regression Hedge

LO 10.3: Calculate the regression hedge adjustment factor, beta.
In order to profit from a hedge, we must assume variability in the spread between the real and nominal yields over time. As mentioned, least squares regression is conducted to analyze these changes. The alpha and beta coefficients of a least squares regression line will be determined by the line of best fit through historical yield data points.
where: AytN = changes in the nominal yield AytR = changes in the real yield
Recall that alpha represents the intercept term and beta represents the slope of the data plot. If least squares estimation determines the yield beta to be 1.0198, then this means that over the sample period, the nominal yield increases by 1.0198 basis points for every basis point increase in real yields.