Skip Headers

Oracle Financial Services Reference Guide
Release 12.1
Part Number E13528-03
Go to Table of Contents
Contents
Go to previous page
Previous
Go to next page
Next

Transfer Pricing Option Costs

This chapter describes how Oracle Transfer Pricing goes about calculating option costs. The chapter begins with an introduction to option costs and subsequently describes option cost theory and the calculation architecture.

This chapter covers the following topics:

Overview of Transfer Pricing Option Costs

The purpose of option cost calculations is to quantify the cost of optionality, in terms of a spread over the transfer rate, for a single instrument. The cash flows of an instrument with an optionality feature change under different interest rate environments and thus should be priced accordingly.

Consider a mortgage that may be prepaid by the borrower at any time without penalty. Here the lender has, in effect, granted the borrower an option to buy back the mortgage at par, even if interest rates have fallen in value. Thus, this option has a cost to the lender and should be priced accordingly.

Another example of an instrument with an optionality feature is an adjustable rate loan issued with rate caps (floors) which limit its maximum (minimum) periodic cash flows. These caps and floors constitute options.

When banks give such options to their borrowers, they raise the bank's cost of funding the loan and affect the underlying profit. Consequently, banks need to use the calculated cost of options given to their borrowers in conjunction with the transfer rate to analyze profitability.

Oracle Transfer Pricing uses the Monte Carlo technique to calculate the option cost. The application calculates and outputs two spreads, and the option cost is calculated indirectly as a difference between these two spreads.

The option cost is derived as follows:

option cost = static spread - OAS

The static spread is equal to the margin, and the OAS to the risk-adjusted margin of an instrument. Therefore, the option cost quantifies the loss or gain due to risk.

You can calculate option costs using the Transfer Pricing Process rule. See: Transfer Pricing Process Rules, Oracle Transfer Pricing User Guide.

Understanding the Option Cost Calculation Architecture

This description of the option cost calculation architecture makes use of an example and assumes:

Related Topics

Overview of Transfer Pricing Option Cost

Defining the Static and Option-Adjusted Spreads

You can define neither the static nor the option-adjusted spread directly, as they are solutions of two different equations. Therefore, the system solves a simplified version of the equations. The static spread is the value ss that solves the following equation:

the picture is described in the document text

Here:

In the Monte Carlo methodology, the option-adjusted spread is the value OAS that solves the following equation:

the picture is described in the document text

Here:

Related Topics

Understanding Option Cost Calculation Architecture

Option Cost Calculations Example

In this example, the transfer pricing yield curve is the Treasury curve. It is flat at 5%, which means that the forward rate is equal to 1%. This example uses only two Monte Carlo scenarios:

The average of these two stochastic rates is equal to 5%.

The instrument record is two year adjustable, paying yearly, with simple amortization. Its rate is Treasury rate plus 2%, with a cap at 7.5%. Par value and market value are equal to $1.

For simplicity, this example assumes that the compounding period used for discounting is equal to a year, for example:

Delta t = 1

The static spread is the solution of the following equation:

1 = [0.07 / (1 + 0.05 + SS) ] + [(1 + 0.07) / (1 + 0.05 + SS)2]

The static spread is supposed to be equal to the margin. In this example:

static spread = coupon rate - forward rate =7%-5%=2%

Substituting this value (2% or .02) in the right side of the above equation yields:

the picture is described in the document text

This is equal to par, which proves that the static spread is equal to the margin.

The OAS is the solution of the following equation:

the picture is described in the document text

By trial and error you get a value of 1.88%.

To summarize:

option cost = static spread - OAS = 2%-1.88% = 12 basis points

Related Topics

Understanding Option Cost Calculation Architecture

Option Cost Calculations Process Flow

The following graphic represents the option cost calculations process flow.

the picture is described in the document text

This exposition focuses only on the following steps:

Related Topics

Understanding Option Cost Calculation Architecture

Calculating Forward Rates

The cubic spline interpolation routine first calculates smoothed, continuously compounded zero-coupon yields Y(j) with maturity equal to the end of month j. The formula for the one-month annually compounded forward rate spanning month j + 1 is:

fj = exp [ (Yj) (j + 1) - Y(j) j ] -1

Related Topics

Option Cost Calculations Process Flow

Calculating Static Spread

You can calculate the static spread using the Newton-Raphson algorithm. If Newton-Raphson algorithm does not converge, which can happen if cash flows alternate in sign, you can revert to a brute search algorithm. However, this algorithm is much slower.

You can control the convergence speed of the algorithm by adjusting the value of the variable OptionCostSpeedFactor. This variable is defined through a profile option, Option Cost Speed Factor.

The default value is equal to one. A lower speed factor provides more accurate results. In all experiments, a speed factor equal to one results in a maximum error (on the static spread and OAS), which is lower than half a basis point.

To recap the Newton-Raphson algorithm, let x be the static spread. At each iteration m, the function F(m) is defined by the following equation:

Equation A

the picture is described in the document text

The algorithm is:

the picture is described in the document text

For performance reasons, the code utilizes a more complicated algorithm, albeit similar in spirit. This is the reason why the specific values for tol and MaxIterations, or details on the brute search are mentioned above.

Related Topics

Option Cost Calculations Process Flow

Calculating OAS

For fixed rate instruments, such as instruments having the same deterministic cash flows as the stochastic cash flows, the OAS is by definition equal to the static spread. This statement is true in the case of continuous compounding. For discrete compounding this approximation has a negligible impact on the accuracy of the results.

The OAS is also calculated with an optimized version of Newton-Raphson algorithm. See: Calculating Static Spread.

Note: While calculating OAS, the following substitution is made in the Newton-Raphson method: OAS = x(m)

the picture is described in the document text

Related Topics

Option Cost Calculations Process Flow

Option Cost Theory

According to the option cost theory, when you select the market value of an instrument to equate the discounted stream of cash flows, the static spread is equal to margin and the OAS to the risk-adjusted margin of the instrument.

This exposition of option cost theory assumes that you have good knowledge of no arbitrage theory, and requires you to note these assumptions and definitions:

More precisely,

the picture is described in the document text

Equation B

the picture is described in the document text

Related Topics

Overview of Transfer Pricing Option Cost

Equivalence of the Option Adjusted Spread and Risk-Adjusted Margin

In a no-arbitrage economy with complete markets, the market value at time t of an instrument with cash flow rate C(t)is given by:

the picture is described in the document text

If expectation is taken with respect to the risk-neutral measure, the expected change in value is given by:

the picture is described in the document text

The variation in value is, therefore, equal to the expected value of the change dV plus the change in value of a martingale M in the risk-neutral measure:

dV(t) = Et[dV] + dM = rVdt - Cdt + dM

If I is the market value of the money market account in which cash flows are reinvested then:

the picture is described in the document text

Note that unlike V, this is a process of finite variation. By Ito's lemma:

dl = rIdt + Cdt

Let Sbe the market value of a portfolio composed of the instrument plus the money market account. We have:

dS = dV + dI = rSdt + dM

S(0) = V(0)

In other words, the portfolio, and not the instrument, earns the risk-free rate of return. An alternate representation of this process is:

dS/S = rdt + dN

Here N is another martingale in the risk-neutral measure. The expected value of the portfolio is then:

the picture is described in the document text

Here, <N, N> is the quadratic variation of N. This is equivalent to:

the picture is described in the document text

To define the martingale:

Z = eN-(1/2)<N,N>

This represents the relative risk of the portfolio with respect to the standard money market account, that is, the account where only an initial investment of V(0) is made. Then

the picture is described in the document text

In other words, the expected future value of the portfolio is equal to the expected future value of the money market account adjusted by the correlation between the standard money market account and the relative risk. Assuming complete and efficient markets, banks can fully hedge their balance sheet against this relative risk, which should be neglected to calculate the contribution of a particular portfolio to the profitability of the balance sheet. Therefore:

the picture is described in the document text

In this example, the risk-adjusted rate of return of the bank on its portfolio is equal to the risk-free rate of return.

Now suppose that another instrument offers cash flows C' > C.

Assuming complete and efficient markets, the market value of this instrument is:

the picture is described in the document text

the picture is described in the document text

The value of the corresponding portfolio is denoted by S' > S.

By analogy with the previous development, we have:

the picture is described in the document text

Again, the risk-adjusted rate of return of the bank on its portfolio is equal to the risk-free rate of return. Suppose now that markets are incomplete and inefficient. The bank pays the value V(0) and receives cash flows equal to C'. We have:

the picture is described in the document text

By definition of the total risk-adjusted return for Equation B, we have:

Equation C

the picture is described in the document text

Therefore, by analogy with the previous development,

dS' = (r + m)S' + dM

S(0) = V(0)

This can be decomposed into

Equation D

dV' = (r + m) V' dt - C' dt + dM'

Equation E

dS' = dV' + dI'

dI' = rI' dt + C' dt

Equation F

V' (0) - V(0)

The solution of Equation D and Equation F is:

the picture is described in the document text

By the law of large numbers, Equation D and Equation F result in:

OAS = m

In other words, the OAS is equal to the risk-adjusted margin.

Related Topics

Option Cost Theory

Equivalence of the Static Spread and Margin

Static spread calculations are deterministic. Therefore, they are a special case of the equations in the previous section where all processes generally are equal to their expected value, and the margin p is substituted for the risk-adjusted margin m. The equivalent of Equation C is then:

the picture is described in the document text

Here f is the instantaneous forward rate.

The equivalent of Equation D and Equation F is then:

Equation G

dV' = (r + m)V' dt - C' dt

Equation H

dI' = rI' dt + C' dt

dS' = dV' + dI'

Equation I

V'(0) = V(0)

The solution of Equation G and Equation I is:

Equation J

the picture is described in the document text

Comparing Equation J and Equation A:

ss = p

In other words, the static spread is equal to the margin.

Related Topics

Option Cost Theory

Option Cost Model Usage Hints

The option cost calculation model is flexible and you can calibrate the calculations to your needs. See:

Related Topics

Overview of Transfer Pricing Option Cost

Nonunicity of the Static Spread

Nonunicity of the static spread means that sometimes more than one value can solve the static-spread equation. However, such cases extremely rare.

Take an instrument with a market value of $0.445495 for example. Suppose the instrument has two cash flows. The following table shows the value of the cash flows and the corresponding discount factors (assuming a static spread of zero).

Value of the Cash Flows and the Corresponding Discount Factors
Events Time Cash Flow Value Discount Factor (static spread = 0)
First Event 1 1 0.9
Second Event 2 -0.505025 0.8

The continuously compounded static spread solves the following equation:

0.9 Exp (-ss) -0.8 * 0.505025 Exp (-2ss) -0.445494 = 0

There are two possible solutions for the static spread:

Related Topics

Option Cost Model Usage Hints

Calibrating the Accuracy of Option Cost Calculations

If you desire a better numerical precision than the default precision, you can take two actions:

Both actions increase the calculation time.

Related Topics

Option Cost Model Usage Hints