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LO 11.6: Define option-adjusted spread (OAS) and apply it to security pricing.
The option-adjusted spread (OAS) is the spread that makes the model value (calculated by the present value of projected cash flows) equal to the current market price. In the previous CM T example, the model price was equal to $1,466.63. Now assume that the market price of the CM T swap was instead $1,464.40, which is $2.23 less than the model price. In this case, the OAS to be added to each discounted risk-neutral rate in the CM T swap binomial tree turns out to be 20 basis points. In six months, the rates to be adjusted are 7.25% in the up node and 6.75% in the down node. Incorporating the OAS into the six-month rates generates the following new swap values:
($2,500 x 0.6) + ($0 x0.4)
+ 0.0745 1
+ $1,250
$2,696.13
($ 0 x 0 .6 )+ (-$ 2 ,5 0 0 x 0 .4 )
t | 0.0695
2
$1,250
-$ 2 ,2 1 6 .4 2
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Notice that the only rates adjusted by the OAS spread are the rates used for discounting values. The OAS does not impact the rates used for estimating cash flows. The final step in this CM T swap valuation is to adjust the interest rate used to discount the price back to today. In this example, the discounted rate of 7% is adjusted by 20 basis points to 7.2%. The updated initial CM T swap value is:
($2,696.13 x 0.76) + (-$2,216.42 x 0.24)
+ 0.072 1
$1,464.40
Now we can see that adding the OAS to the discounted risk-neutral rates in the binomial tree generates a model price ($1,464.40) that is equal to the market price ($1,464.40). In this example, the market price was initially less than the model price. This means that the security was trading cheap. If the market price were instead higher than the model price we would say that the security was trading rich.
T i m e S t e p s

LO 11.8: Calculate the value o f a constant maturity Treasury swap, given an interest rate tree and the risk-neutral probabilities.
In addition to valuing options with binomial interest rate trees, we can also value other derivatives such as swaps. The following example calculates the price of a constant maturity Treasury (CMT) swap. A CM T swap is an agreement to swap a floating rate for a Treasury rate such as the 10-year rate.
Example: CM T swap
Assume that you want to value a constant maturity Treasury (CMT) swap. The swap pays the following every six months until maturity:
/
$ 1,000,000
2
X (y
V C M T
Tc m t ls a semiannually compounded yield, of a predetermined maturity, at the time of payment (y( is equivalent to 6-month spot rates). Assume there is a 76% risk-neutral probability of an increase in the 6-month spot rate and a 60% risk-neutral probability of an increase in the 1-year spot rate.
Fill in the missing data in the binomial tree, and calculate the value of the swap.
Figure 5: Incomplete Binomial Tree for CM T Swap
Int. rate = 7.25% Swap price = ?
0.76
Int. rate = 6.75% Swap price = ?
Int. rate = 7.50% Swap price = ?
0.60
Int. rate = 7.00% Swap price = ?
Int. rate = 6.50% Swap price = ?
0.40
Today
Six Months
One Year
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Answer:
In six months, the top node and bottom node payoffs are, respectively:
payoff^ = ———— x (7.25% 7.00%) = $1,250
$ 1,000,000
payoff^ = ————-x (6.75% 7.00%) = $1,250
$ 1,000,000
rr
rr
2
2
^
^
Similarly in one year, the top, middle, and bottom payoffs are, respectively:
r r
$ 1, 000,000
/ r r i r A f t /
payoff2ju = ————-x (7.50% 7.00%) = $2,500
^
^ payoff2jM = ————-x (7.00% 7.00%) = $0
/ r r / w w w
r
r
$ 1,000,000
2
rr
$ 1,000,000
payoff2 L = ————-x (6.50% 7.00%) = $2,500
2
The possible prices in six months are given by the expected discounted value of the 1-year payoffs under the risk-neutral probabilities, plus the 6-month payoffs ($1,250 and
$1,250). Hence, the 6-month values for the top and bottom node are, respectively:
l,u Vi
V
1>L
($2,500×0.6)+ ($0x0.4)
1+ 0.0725
+ $1,250 = $2,697.53
($ 0 x 0 .6 )+ (-$2,500×0.4)
+ 0.0675 1
-$1,250 = -$2,217.35
Today the price is $1,466.63, calculated as follows:
w ^ V Q — —————————ri'”r\~v———————- $1,466.63
($2,697.53 x 0.76) + ($2,217.35 x 0.24)
.
1+ 0.07
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Topic 11 Cross Reference to GARP Assigned Reading – Tuckman, Chapter 7
Figure 6 shows the binomial tree with all values included.
Figure 6: Completed Binomial Tree for CM T Swap
0.60
Int. rate = 7.50%
Swap price = $2,500.00
Int. rate = 7.00%
Swap price = $1,466.63
0.76
0.24
Int. rate = 7.25%
Swap price = $2,697.53
Int. rate = 6.75%
Swap price = -$2,217.35
Int. rate = 7.00%
Swap price =
$0
Int. rate = 6.50%
Swap price = -$2,500.00
0.40
Today
Six Months
One Year
O p t i o n -Ad j u s t e d S p r e a d

LO 11.7: Describe the rationale behind the use o f recombining trees in option pricing.
In the previous example, the interest rate in the middle node of period two was the same (i.e., 6.34%) regardless of the path being up then down or down then up. This is known as a recombining tree. It may be the case, in a practical setting, that the up then down scenario produces a different rate than the down then up scenario. An example of this type of tree may result when any interest rate above a certain level (e.g., 3%) causes rates to move a fixed number of basis points, but any interest rate below that level causes rates to move at a pace that is below the up states fixed amount. When rates move in this fashion, the movement process is known as state-dependent volatility, and it results in nonrecombining trees. From an economic standpoint, nonrecombining trees are appropriate; however, prices can be very difficult to calculate when the binomial tree is extended to multiple periods.
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C o n s t a n t M a t u r i t y T r e a s u r y S w a p

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.