Risk Factor Mapping and the Market Risk Loss Operator

Before you can compute the VaR of a trading book, you need to answer a more basic question: what causes the book to gain or lose value? The answer is the risk factor map — an explicit function $V_t = g(\tau_t, \mathbf{Z}_t)$ linking portfolio value to time and to a small vector of market observables (equity prices, yield curve points, FX rates, implied volatilities). From this map comes the loss operator $l^{[t]}(\mathbf{x})$: given a risk-factor change scenario $\mathbf{x}$, it computes the loss. The loss distribution is then the distribution of $l^{[t]}(\mathbf{X}_{t+1})$ over the distribution of tomorrow's risk-factor changes.

For non-linear books (options, callable bonds), computing $l^{[t]}$ exactly requires full revaluation under every scenario — computationally expensive. The delta and delta–gamma (first- and second-order Taylor) approximations reduce this to matrix algebra. For a bond portfolio, the delta approximation recovers duration and the delta–gamma approximation recovers convexity — connecting the general risk-factor framework to classical fixed-income analytics.

Learning objectives. By the end of this note you should be able to: