13.1 Overview

Monte Carlo is a valuation technique that provides a direct and robust simulation of interest rate paths and provides for market value and Value-at-Risk (VaR) calculations. Monte Carlo becomes a necessary tool in financial markets for solving problems when other methods are unavailable for problems in high dimensions, simulation, and optimization. For Oracle Asset Liability Management (ALM) and Funds Transfer Pricing (FTP) processes, Monte Carlo is a particularly useful tool for valuing instruments with uncertain cash flows. Examples of such instruments include:

• Callable Debt
• Capped Loans
• Prepayable Mortgages

The Monte Carlo Rate Generator is a calculation engine that forecasts future rate changes within a Stochastic Process. Central to the understanding of the rate generator is the acknowledgment that a rate forecast will always be imperfect. This means that future rates will not fully match the prognosis given by the model. However, it is possible to quantify the uncertainty of future interest rates or, in other terms, to forecast a probability distribution of interest rates.

Economic theory tells us that there are two types of forecasts:

• Forecast of the real interest rates, based on a “subjective” assessment of the economy.
• Forecast of the risk-neutral interest rates, based on the original yield curve and the no-arbitrage condition.

The two types of prognosis will not necessarily match. A bank would typically use the first type of rate to model future income because it wants its income forecast to be as close as possible to the actual future income. A bank would typically use the second type of rate to calculate present and future market value because market value depends not only on the rates but also on the degree of risk-aversion of each agent in the economy. “Risk-neutral rates” are a theoretical construction that enables us to calculate rates as if nobody were risk-averse.

Many types of analysis in interest rate management require computing the expected value of a function of the interest rate. One example is to calculate the probability that portfolio loss is within a certain range. The probability of such an event is nothing but the expected value of the indicator function of this occurrence, which is worth one if the event is true and zero otherwise. Another example is to compute the market value of a derivative instrument.

In mathematical terms, the market value of a security that pays a cash flow at time T is equal to the expected value of the product of the stochastic discount factor at time T and the cash flow, that is:

Market Value = E [Discount Factor * Cash Flow]

where the Stochastic Discount factor is equal to the present value (along with a rate scenario) of one dollar received at time T. It is therefore a function of the rate.

The goal of term structure models is to forecast probability distributions of interest rates under which the expected value is defined.

Most term structure models used in practice, and all term structure models available in the system, are single-factor models of the short term interest rate. Short Rate modeling prevails because the problem of correlating multiple factors is greatly simplified. With single-factor models, the value in the future depends only on the value at the current time, and not on any previous data. This property is referred to as a Markov process.

Monte Carlo is the most popular numerical technique to compute an expected value, in our case market value and Value-at-Risk. The methodology consists of generating rate scenarios using random numbers, computing a function of the rates for each scenario, and then averaging it. Market value is the average across all scenarios of the sum of all cash flows discounted by the (scenario-specific) rate. Value-at-Risk is the maximum loss in value over a specific horizon and confidence level.

Monte Carlo simulation works forward from the beginning to the end of the life of an instrument and can accommodate complex payoffs, for instance, path-dependent cash flows. The other numerical methods, such as lattice and finite difference, cannot handle the valuation of these path-dependent securities. The drawback of Monte Carlo is its slow convergence compared to other methods. We address this problem by implementing better random sequences of random numbers, namely Low-Discrepancy Sequences. Monte Carlo has better performance than other techniques, however, when the dimension of a problem is large.