The Science of Term Structure Models

Pricing of interest contingent claims is covered in this chapter relative to a set of underlying securities. Here, the most popular measure of deviation of market prices is the Option-Adjusted Spread (\(OAS\)).

Rate and Price Tree

Let the spot rates for the six-month and one-year be given as 5% and 5.15% respectively. Then six months later, the six-month spot rate will be 4.5% or 5.5% as depicted in the following binomial tree:

$$ \begin{array} \hline {} & {\small \frac { 1 }{ 2 } } & 5.5\% \\ 5\% & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\ {} & {\small \frac { 1 }{ 2 }} & 4.5\% \\ \end{array} $$

The columns represent dates. In our example, the six-month rate is 5% today often called date 0. Six months later shall be date 1 and so on. The 5.5% state is the up-state while the 4.5% is the down-state.

Generally, over a period, discounting the expected value and taking the expectation of the discounted values are the same. However, over many periods, the two – expected value and expectation of the discounted values – are different.

The approach taken by the short rate models (taking the expectation of discounted values) is the correct one, thus the choice of the term expected discounted value. In this approach, securities with assumed prices are referred to as underlying securities to differentiate them from the contingent claims priced by the arbitrage argument.

In a rate and price tree, the probability of moving either up or down the tree is used to compute the average or expected values.

Arbitrage Pricing of Derivatives

To price a security by arbitrage, its replicating portfolio is determined and then priced. In this context, cash flows do not depend on the level of rates; therefore, the construction of the replicating portfolio is relatively simple.

Assuming we are interested in pricing a portfolio whose maturity date is six months from now, to purchase $1000 face value of a six month zero at $975.

To price the option by arbitrage, you need to construct a portfolio on date 0 of underlying securities that is six month and one-year zero-coupon bonds; it should be worth $0 in the up-state on date 1 and $3 in the down-state. The values must satisfy the following equations:

$$ \begin{array} \hline {} & {\small \frac { 1 }{ 2 } } & 0 \\ ??? & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\ {} & {\small \frac { 1 }{ 2 }} & 3 \\ \end{array} $$

Where

$$ { F }^{ .5 }+0.97324{ F }^{ 1 }=$0\quad \quad \quad \quad \quad \left( 1 \right) $$

$$ { F }^{ .5 }+0.97800{ F }^{ 1 }=$3\quad \quad \quad \quad \quad \left( 2 \right) $$

From the equation, in the up-state, the value of the replicating portfolio’s maturing six-month zero is its face value. The value of the once one-year zeros, now six months zeros, is 0.97324 per dollar face value. The left-hand side of the (\(1\)) equation denotes the value of the replicating portfolio in the up-state, the value must be equal to $0. The (\(2\)) equation denotes the value of the replicating portfolio in the downstate.

Thus:

$$ { F }^{ .5 }=-$613.3866\quad and\quad { F }^{ 1 }=$630.25 $$

It is a requirement by the law of one price the option’s price be equal to the price of the replicating portfolio.

The probabilities of \(\frac { 1 }{ 2 } \) for the up and down states are the assumed true or real-world probabilities. There are,however, other probabilities called risk-neutral probabilities that cause the expected discounted values to equal the market price.

Risk-Neutral Pricing

This method modifies an assumed interest rate process such that any contingent claim can be priced without having to construct and price its replicating portfolio. It is an effective way of pricing many claims under the same assumed rate process.

Since the argument is complex, it is broken down into the following 4 steps:

Step 1: A security is priced by arbitrage if one can contrast a portfolio that replicates its cash flow.

Step 2: Imagine an economy identical to the true economy with respect to current bond prices and the possible value of the six-month rate over time. The investors in the imaginary economy need to be risk neutral.

Step 3: The price of the option in the imaginary economy is computed by expected discounted value.

Step 4: The price of an option should not depend on investors risk preferences.

Arbitrage Pricing In a Multi-Period Setting

While maintaining a binomial assumption, when an up move followed by a down move does not give the same rate as a down move followed by an up move, the tree is said to be non-recombining. This tree can be difficult or even impossible to implement.

By definition, expected discounted value under risk-neutral probabilities must produce market prices. This means that any derivative security that depends on the six-month rate in six months and in one year may be priced by computing its discounted expected value along the rate and price tree, with the difference between the true and risk-neutral probabilities being described in terms of drift.

Just as a replicating portfolio can reproduce the cash flow of a security, the composition of this replicating portfolio depends on the date and state. Therefore, the replicating portfolio must be adjusted as time passes and as interest rate changes. This process is what is known as dynamic replication;the contrast of this is static replicating strategies where the replicating coupon bond does not change over time or with the value of interest rate.

Pricing a Constant-Maturity Treasury Swamp

To price a particular derivatives security, say $1,000,000 face value of a stylized constant-maturity treasury (\(CMT\)) swamp struck at 5%, the swap pays:

$$ $1,000,000\frac { { y }_{ CMT }-5\% }{ 2 } $$

Until maturity, the payment is made every six months. \({ y }_{ CMT }\) is a semiannually compounded yield of predetermined maturity on the payment date.

Note: the expected cash flow of the \(CMT\) swap under the real probabilities is zero. Therefore, the discounted value of these expected cash flows is zero.

Option-Adjusted Spread

This is a spread such that on computing discounted values relative to risk-neutral rates plus the spread, a security’s market value equals to its model price. \(OAS\) is a measure of a relative value of a security. That is, its market price relative to its model value. When calculating the value with an \(OAS\) spread, rates are only shifted for the purpose of discounting, and not for the purpose of computing cash flows.

For a positive security’s \(OAS\), its market price is less than the price of its model hence cheap security trades. On the other hand, a negative \(OAS\) implies rich security trades.

Under the risk-neutral process, a security with an \(OAS\) spread has its expected return as the sum of its \(OAS\) per period and the short-term rate. This is another \(OAS\) spread’s relative value implication. Should the expected value of a security be discounted by a particular rate per period, then this is equivalent to the rate per period of the security’s earning.

Profit and Loss Attribution with an \(OAS\)

To introduce profit and loss attribution, in a one-factor model where the market price of a security at time t and a factor value of \(x\) can be written as \({ P }_{ t }\left( x,\quad OAS \right)\). Using a first-order Taylor approximation, the change in price of the security will be given as:

$$ dP=\frac { \partial P }{ \partial x } dx+\frac { \partial P }{ \partial t } dt+\frac { \partial P }{ \partial OAS } \partial OAS $$

We then divide the above equation by price and then take expectations:

$$ E\left[ \frac { dP }{ P } \right] =\frac { 1 }{ P } \frac { \partial P }{ \partial x } E\left[ dx \right] +\frac { 1 }{ P } \frac { \partial P }{ \partial t } dt\quad \quad \quad \quad \quad \left( a \right) $$

The expected change of in the \(OAS\) is assumed as zero \(OAS\) is constant over the security’s life. Taking expectations based on risk-neutral process, security priced according to the model:

$$ E\left[ \frac { dP }{ P } \right] =rdt\quad \quad \quad \quad \quad \left( b \right) $$

Therefore:

$$ \frac { dP }{ P } =\left( r+OAS \right) dt+\frac { 1 }{ P } \frac { \partial P }{ \partial x } \left( dx-E\left[ dx \right] \right) +\frac { 1 }{ P } \frac { \partial P }{ \partial OAS } dOAS $$

Multiplying the entire equation by \(P\):

$$ dP=\left( r+OAS \right) dt+\frac { \partial P }{ \partial x } \left( dx-E\left[ dx \right] \right) +\frac { \partial P }{ \partial OAS } dOAS $$

The implication here is that, a security’s P&L may be divided into a component due to: time passage, factor changes, and the \(OAS\) changes.

Reducing the Time Step

Even though this method assumes that the time that elapses between dates of the tree is 6 months, the methodology can easily be adapted to any time step of \(\Delta t\) years. The discounting must be done over the appropriate time interval, that is, if the rate of term \(\Delta t\) is \(r\), discounting mean dividing by \(1+r\quad \Delta t\)

The main reasons for choosing time steps smaller than 6 months are:

  • Reducing the time step ensures that all cash flows are close in time to some date in the tree
  • Reducing the time step can fill the tree with enough rates to price contingent claims with sufficient accuracy

Note: while smaller time steps generate more realistic interest rate distributions, smaller steps aren’t always desirable. This mainly because, the greater the number of computation in pricing a security, the more attention needs to be paid to numerical issues like round-off error.

Fixed Income versus Equity Derivatives

Generally, models created for stock markets cannot be adopted for use in fixed income markets without modifications. Under the assumption that stock price evolves according to a random process and that short-term interests rate is constant, it is possible to form a portfolio of stocks and short-term bonds that replicate the payoffs of an option. By arbitrage argument, the price of the option must equal the known price of the replicating portfolio.

While the short-term interest rate is constant, price follows some constant random process. From this:

  • bond prices would naturally approach par;
  • price volatilities would not be assumed to be constant; and
  • the interest rate is assumed to evolve in a particular way.

Making the assumption of rate over time is not helpful for two reasons. First, the coupon bond prices depend on the shorter-term rate and secondly, pricing a bond on a given year requires assumptions about the bond’s future possible prices. From this, one has to make assumptions about the evolution of the entire term structure of interest rate to price bond options and other derivatives.

Modeling the evolution of the short-term rate is sufficient, combined with arbitrage arguments to build a model of the entire term structure. Despite the importance of Black-Scholes-Merton analysis, fixed income demand special attention.

Practice Questions

1) A $10 million face value of a stylized constant-maturity treasury (\(CMT\)) swap is struck at 10%. It is a one-year \(CMT\) swap on the six-month yield in 0.5% increments. Calculate the possible values of the \(CMT\) swap after one year.

  1. $50,000, $0 and -$50,000
  2. $25,000, $0 and -$25,000
  3. $100,000, $0 and -$100,000
  4. None of the above

The correct answer is A.

The \(CMT\) swap pays:

$$ $10,000,000\frac { { y }_{ CMT }-10\% }{ 2 } $$

In 6 months, the state 1 and 0 payoffs are, respectively:

$$ $10,000,000\frac { 10.5\%-10\% }{ 2 } =$25,000 $$

$$ $10,000,000\frac { 9.5\%-10\% }{ 2 } =-$25,000 $$

In 1 year, the state 2, 1 and 0 payoffs are:

$$ $10,000,000\frac { 11\%-10\% }{ 2 } =$50,000 $$

$$ $10,000,000\frac { 10\%-10\% }{ 2 } =0 $$

$$ $10,000,000\frac { 9\%-10\% }{ 2 } =-$50,000 $$


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