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Interest Rates and Discounting

This section briefly reviews various terms related to the concept of time value of money. Terms covered are 'Discounting', 'Stochastic discount factor', 'Discount bond',`'Bank account' and 'Numeraire'.

Discounting is to obtain the present value of future cash flows. It is the act of representing future cash flows as an equivalent immediate cash amount. Depending on the type of future cash flows, the discount rate can be a risk-free interest rate ($ r(t) $) for risk-free investments, or a rate ($ \mu (t) $) with some risk premium over and above the risk-free rate for risky investments.

A stochastic discount factor is a stochastic process that depends on the stochastic evolution of future interest rates:

$ D(t,T)=e^{-\int_t^T r(s)ds}, \qquad (1) $

where $ D(t,T) $ is a stochastic discount factor at time $ t $ with maturity $ T $, and $ r(s) $ is future interest rate at time $ s $.

A discount bond $ P(t,T) $ (also known as zero coupon bond) is a theoretical construct used to discount future cash flows. It has a price known at time $ t $. The relation between discount bond and stochastic discount factor is:

$ P(t,T)=\mathrm{E} \left [ D(t,T) \mid \mathcal{F}_t \right ], \qquad (2) $

i.e. mathematically, discount bond is the equivalent of the risk-neutral expectation of the stochastic discount factor.

A bank account is a process that starts with a unit cash amount and grows at the risk-free rate. Mathematically, it is

$ B(t,T) = B(0)e^{-\int_t^T r(s)ds}, \qquad (3) $

where $ B(t,T) $ is a bank account process. It is the solution to the following deterministic process

$ dB(t) = r(t)\,B(t)dt, \quad B(0) = 1. \qquad (4) $

A numeraire is any positive non-dividend-paying asset. We can choose a numeraire to normalize other assets for more convenient pricing. The choice of a numeraire induces a particular measure associated with this numeraire, and normalized assets are Martingales under this numeraire measure.

A typical numeraire example in option pricing is to use the bank account as numeraire. When asset prices are normalized by the bank account, they are Martingales under the bank account numeraire measure (also known as the risk-neutral measure). Under this bank account measure, the original assets grow at the risk-free rate. This is also why we have the risk-free rate as the new drift term in the Black-Scholes PDE. An example of stock processes using the bank account as numeraire can be found at Geometric Brownian Motion.

Discount numeraire in ThetaML

In the following, we will illustrate by example the concept of discounting in ThetaML via our example discount numeraire EUR.

The following ThetaML model computes a discount factor EUR that is initially fixed at 1 EUR. The value of 1 EUR is defined in the currency unit Euro, i.e. 1 EUR = 1 Euro. The future values of the discount factor EUR decays exponentially at a constant interest rate r.

By dividing security prices by the value of one EUR ,we normalize the security prices in units of Euro coins.

model DiscountFactor
  %This model computes the discount factors under the assumption
  %of constant interest rates; the interest rate 'r' is for debt
  %securities denominated in the currency Euro;
  %the discount factors 'EUR' is a process variable, 'EUR' implicitly incorporate 
  %scenario and time indexes 
  import r    "Constant interest rate"
  export EUR  "Discount factor process"
  %initialize the discount factor at 1 Euro
  EUR = 1
  %'loop ... inf' is an infinite loop; this infinite loop computes an interest
  %rate process of an arbitrary length; the lifetime of the infinite loop is
  %automatically extended to the desired length depending on a specific pricing
  loop inf
    %the ThetaML command 'theta' passes time by '@dt' units
    %the ThetaML parameter '@dt' denotes an arbitrary time unit; its specific
    %value depends on a specific pricing application
    theta @dt
    %the discount factor decays at a rate of 'r' for the time interval '@dt'
    EUR = EUR * exp(-r * @dt)