Overview

Market risk is the risk of loss from changes in traded asset prices, interest rates, exchange rates, and implied volatilities — the "mark-to-market" risk that regulators require banks to quantify and capitalise daily. This topic develops the statistical machinery for doing that calculation rigorously.

The central problem: a trading book's value

V_t

is a complex, potentially non-linear function of hundreds of risk factors

Z_t

(equity prices, yield curve points, FX rates, vols). When risk factors move, the loss is

L_{t+1} = -(V_{t+1} - V_t)

. Computing the distribution of

L_{t+1}

— specifically its 99th percentile (VaR) or expected tail loss (ES) — is simultaneously a financial modelling problem (how does

V

depend on

Z

?) and a statistical problem (what is the distribution of

\Delta Z

?).

This topic answers both questions in sequence: first, how to map a portfolio to risk factors and approximate the loss operator; second, how to estimate the loss distribution using variance–covariance, historical simulation, or Monte Carlo methods; third, how to validate the model using backtests. The regulatory side — Basel VaR capital charges and FRTB's switch to ES — appears throughout as the institutional context that makes all of this load-bearing.

By the end of this topic you should be able to:

define the loss operator $l^{[t]}(\mathbf{x})$ and derive its delta and delta–gamma approximations;
explain the variance–covariance (delta-normal) method and its assumptions;
describe historical simulation, filtered historical simulation (GARCH-rescaled), and Monte Carlo approaches;
state the square-root-of-time scaling rule and identify when it fails;
apply the Kupiec POF test and Christoffersen conditional coverage test;
interpret the Basel traffic-light system and explain why FRTB moved from VaR to ES.

In this lesson

Contents
The big picture

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