SENSITIVITY AND RISK FACTORS
Sensitivities are ratios of the variation of a target variable, such as interest margin or change of the mark-to-market values of instruments, to a shock of the underlying random parameter driving this change. This property makes them very convenient for measuring risks, because they link any target variable of interest to the underlying sources of uncertainty that influence these variables. Those sources of uncertainty are underlying assets for derivatives or market parameters. They are called "risk factors."
The basic formula for understanding sensitivities and their limitations is the Taylor expansion formula. The formula expresses the value of any function fix) as equal to its value when the argument has value xQ plus various terms that depend on the difference between x and x . The formula is primarily used for small variations around x , otherwise it becomes a proxy. The function is commonly the value of an asset, such as an option.
The Taylor expansion formula was introduced in Chapter 12, when introducing Ito lemma for a function of two variables. With a single variable, it simplifies into:
Notations are abbreviated by writing the small variation of x as:
The formula says that, with deterministic variables, a small change of the variable results in a change than can be approximated by the first term which is a function of Ax. A better approximation is obtained by using the second-order term.
The first derivative applied to Ax is the sensitivity, or the first-order term. The second-order term is the change of the sensitivity when Ax is not negligible. The second-order derivative is the first derivative of the sensitivity. The Taylor series makes explicit the change of value of instruments as a function of the first-, second-, third-order derivatives and so on. For large changes, it is preferable to consider additional terms, beyond first-order terms, of Taylor expansion of the equation relating the value to these parameters. In all cases, using terms beyond the first derivative provides better proxies of any non-linear relation between value and risk factors.
When the second-order terms can be neglected, the instrument is called "linear" because its variation of value relates linearly to the risk factors. Options are non-linear instruments because of the "kink" of the payoff when the underlying asset has a value equal to the strike price. The option value shows smoother variations than payoff because of the time value of options, but the delta changes significantly with the underlying value.
The above formula can be extended to any number of variables. For example, if we have two variables, the formula extends and seems more complex, but first-order changes are always the first derivatives. The generic formula with two variables is well known because it serves as a starting point for the Ito lemma. In the two-variable case, the function is fix, y), and when moving x and y slightly away from the initial point (x , y), the variation of the function is approximated by the Taylor expansion. Derivatives are now partial derivatives with respect to either one of the variables, and so on, as follows. The function and partial derivatives are calculated at the starting point (x , y0) and Ax — x - xQ and Ay — y - yQ.
Typical Greek letters are notably used for options, but extend to other assets as well. In general, value depends on several factors, for which first-order and second-order derivatives are calculated with respect to each of the risk factors. Considering the example of an equity option:
• delta or 8 is the first-order change
• gamma, or y, is the second-order derivative: it measures by how much the sensitivity changes when the variations of Ax are not small enough to ignore the second order
• vega, v, is the variation of value of an option due to a change of volatility: volatility is a measure of the instability of a risk factor, which is detailed in the next chapter.
Market risk models widely use sensitivities, these "Greek letters," relating market instrument values to the underlying risk factors that influence them. Delta-VaR for market risk and linear instruments relies on sensitivities or first-order proxies of change of the value of positions exposed to market risk.
Sensitivities apply as well to non-market instruments. For example, the variable rate gap model is the sensitivity of the net interest income, Nil, to a shock of an interest rate in the banking portfolio6. The gamma also exists for Nil because there are options embedded in banking products.
First-order sensitivities have two major drawbacks:
• They always refer to a given change of risk factors (such as a 1% shift of interest rates), without considering that some parameters are quite unstable (or "volatile") while others are not.
• They depend on the prevailing conditions, and are "local" measures. If the market conditions change or if time drifts, the sensitivities are not constant. This is notably true for options where a small change of the underlying asset, such as a stock, might result in significant changes of sensitivities.
The chapter on market VaR (Chapter 35, Delta-normal VaR) uses the sensitivities of a forward foreign exchange contract for calculating VaR, and starts with the decomposition into "elementary" positions depending on each one of the risk factors, the spot exchange rate and the two interest rates of each currency, using sensitivities with respect to these risk factors.
SENSITIVITIES AND RISK CONTROLLING
Risk controlling implies controlling sensitivities to risk by taking offsetting positions to the same risk factors. The size of the hedging position depends on the relative sensitivities of the hedged instruments and the hedging instruments, as it was the case for hedging a short position on an option, for example. Delta hedging refers to offsetting the option value by a position on the underlying instrument, as a direct implication of forming risk-free portfolios for option valuation purpose. Since an option has several sensitivities, other sensitivities than delta remain un-hedged, such as the vega. Hedging positions should consider all sensitivities, notably delta, gamma and vega for options.
When the hedging instruments do not depend exactly on the same factors as the hedged position, there is basis risk, implying that the two positions do not offset exactly (for example when using traded futures on assets different than those being hedged but which are correlated). In hedge accounting, a hedge is recognized based on its effectiveness. Effectiveness is measured by the adequacy of the hedging instruments in offsetting variations of value of the hedged instrument, which depends on how much is offset, for how long and on the correlation between the risk factors of the two instruments.
Portfolios of instruments are sensitive to potentially hundreds of risk factors. All interest rates, exchanges rates, stocks, etc. Risk management implies considering sensitivities to all factors, while individual instruments are sensitive only to a fraction of all risk factors.