This Chapter Covers

The definition of market risk.

General approaches to the measurement of market risk.

Regulatory requirements concerning market risk.

The definition of valueatrisk (VaR).

Historical and Monte Carlo simulation of VaR.

The scaling of VaR between different time horizons.

A comparison of historical simulation and Monte Carlo methods.
Introduction
Every financial institution with a portfolio exposed to market risk should have a model in place which is designed to measure that risk. Such a model allows one to control and to limit the market risk taken by each desk or by the traders and to charge each portfolio position a cost of capital required to cover its market risk. Success in meeting these objectives serves the interests of the stakeholders in the firm.
The measures of market risk employed by financial institutions are usually based on the mathematical concept of valueatrisk (VaR). Roughly speaking, VaR is defined as the maximum loss of a portfolio over some target period that will be not exceeded with a specified probability, or confidence level. Various techniques have been developed to estimate the VaR of trading portfolios. Some, like the variance–covariance approach, are outdated and rarely used in practice as they are incompatible with regulatory requirements to capture the nonlinear behavior of derivative instruments and they ignore event risk. The two main approaches currently used in financial institutions are historical simulation and Monte Carlo methods.
In the present overview we briefly describe the main ideas of market risk modeling and present the two main approaches to such modeling in more detail. As the simple forms of both historical simulation and the Monte Carlo have some undesirable properties, we show how both approaches can be improved. Finally, we compare the benefits and the drawbacks of the different approaches.
Market Risk
According to the Basel II framework, the banks must marktomarket their trading books at least daily, which means that they must revalue all trading book positions at readily available closeout prices in orderly transactions that are sourced independently (Basel Committee on Banking Supervision (BCBS), 2006). Market risk is the risk that the market value of positions may change in the future. It is of little consequence to investors who purchase financial instruments with the intention of holding them until maturity or for long time periods.
The market value of trading book positions depends on:

risk factors that are observable on the market, such as share and commodity prices, interest rates, credit spreads and foreign exchange rates;

additional pricing parameters like volatilities and correlations that are not directly observable and which must be implied from the market prices of financial instruments.
All pricing parameters contribute to the market risk of positions, but the set of pricing parameters that must be captured in a particular market risk model depends on the regulatory requirements.
The same framework sets capital requirements to cover potential losses resulting from market risk in the trading book. These capital requirements are formulated in terms of several risk measures which capture different types of market risk.
How to Measure Market Risk
Due to the stochastic nature of financial markets, it is obviously not possible to predict exactly the future value of a portfolio. However, knowing the past behavior of the risk factors that drive the market value of the portfolio, it is possible to generate a stochastic set of possible scenarios for the future value of the portfolio. The idea is as follows:

we describe the past behavior of underlying risk factors by means of probability distributions;

we generate stochastic forecasts of the future behavior of the risk factors using their probability distributions;

we revalue the portfolio for each forecast of the risk factors.
In this way we obtain a probability distribution of the possible future values of the portfolio. The risk measures can now be defined as special characteristics of this distribution.
The risk measure used in the Basel II capital requirements is the valueatrisk, or VaR, which is defined in the following way. Let us denote by the value of the portfolio at time t and by the value at the end of the forecast period [t, t + h]. Then the loss distribution at time t for the forecast horizon h is defined by
Here we take the time value of money into account, i.e. is a discounting factor such that is the value of at time t.
VaR is defined as a threshold value such that the probability that loss on the portfolio over the given time horizon will exceed this value is the given probability level, 1 – α, i.e. valueatrisk is the negative of the α quantile of the loss distribution
Note that in this notation losses correspond to negative values and profits correspond to positive values of . Furthermore VaR is positive.
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