Sensitivities: DV01, Vega and Key Rate Sensitivities

In market risk management, you track how the market value of each position changes and monitor it against limits. Sensitivities are the quantities used for this, and they are also added up across desks and the whole firm. The same quantities also feed the standardized approach, the method that calculates regulatory capital from sensitivities.

Sensitivities and risk factors

Interest rates, exchange rates, stock prices and other market variables are called risk factors when they drive the value of a position. When interest rates move, the rate used to discount future cash flows changes. When exchange rates or stock prices move, the amounts of the cash flows themselves change. Either way, the present value moves¹. Which risk factors a position responds to depends on the type of position.

The options row follows the US Federal Reserve's examination manual, and the other rows follow the categories in a Bank of Japan training document².

A sensitivity is the change in market value when you move just one risk factor by a small amount, hold everything else fixed and revalue the position³. Moving interest rates gives DV01, moving the price of an option's underlying asset gives delta, and moving volatility gives vega. The sensitivities of options are called the Greeks, and their definitions and signs are covered on the page The Greeks.

Treating interest rates as a single risk factor is too coarse. The 1-year rate and the 10-year rate do not necessarily move by the same amount, and the shape of the yield curve (or simply the curve) also changes. So the present value is viewed as a function of the rate for each maturity, $r_1, r_2, \dots, r_n$, and each rate is treated as a separate risk factor⁴.

How to calculate DV01 and key rate sensitivities

DV01 is the change in market value when the rates for all maturities rise by 1bp (0.01%) at the same time. With $V$ as the market value of the position and $r_t$ as the $t$-year rate, it is written as follows⁴.

$$\text{DV01} = V(r_1 + 0.01\%, \dots, r_n + 0.01\%) - V(r_1, \dots, r_n)$$

The change in market value when only the rate for one maturity $t$ rises by 1bp is called the key rate sensitivity⁵.

$$s_t = V(r_1, \dots, r_t + 0.01\%, \dots, r_n) - V(r_1, \dots, r_n)$$

The Bank of Japan training document calls the same quantity as DV01 the BPV (basis point value) and the key rate sensitivity the GPS (grid point sensitivity). Using a worked example of a 5-year bond, the document shows that the sum of the key rate sensitivities for maturities from 1 year to 5 years matches the BPV⁶.

$$\text{DV01} \approx \sum_{t} s_t$$

In general, the sum of the key rate sensitivities is only approximately equal to the DV01. When all maturities move at once, a move in the rate for one maturity can change the sensitivity to another maturity (a cross effect). Each $s_t$, computed separately one maturity at a time, does not include this effect. For a 1bp move, the effect is very small. For a bond like the one in the Bank of Japan example, where each cash flow is discounted using only the rate for its own maturity, there is no such effect, so the sum matches exactly. On this page, a sensitivity is positive when the market value increases as rates rise.

Key rate sensitivities and changes in the shape of the curve

Consider a hypothetical example in which V Bank's interest rate positions are expressed as key rate sensitivities. At the short maturities, the market value increases when rates rise; at the long maturities, it decreases when rates rise.

Adding the five maturities gives 0.2 + 3.8 − 0.5 − 3 − 0.5 = 0, so the DV01 is close to zero. If all rates move by the same amount, V Bank's market value barely changes.

Figure description

The figure joins V Bank's key rate sensitivities at the 1-, 2-, 5-, 10-, and 20-year points with a line. The vertical axis is the change in market value (million yen) when the rate for that maturity rises by 1bp. The vertical lines at 2 and 10 years mark the maturities with the largest imbalances: at 2 years the market value increases when rates rise, and at 10 years it decreases. The line between points is only a visual aid and does not represent values at intermediate maturities. Imbalances such as +3.8 and −3 offset each other, so the total is 0 million yen. The figure shows the imbalance at each maturity, which the total DV01 alone hides.

When rates move by different amounts at different maturities, you can approximate the change in market value by multiplying each key rate sensitivity by the move in that maturity's rate (in bp) and adding up the results⁸. $\Delta r_t$ is the move in the $t$-year rate in bp.

$$\Delta V \approx \sum_{t} s_t \times \Delta r_t$$

Here is the calculation for a move in which short-maturity rates stay unchanged and only long-maturity rates rise (a steepening), and for the opposite move in which only long-maturity rates fall.

If long-maturity rates rise, V Bank loses 37.5 million yen; if they fall, it gains 37.5 million yen. Even when the DV01 is close to zero, positions at different maturities offset each other against a move in the level of rates, but not against a change in the slope of the curve⁹. If you look only at DV01, this position appears to carry no interest rate risk. The risk from changes in the shape of the curve becomes visible only when you lay out the key rate sensitivities side by side.

Vega on a grid of maturities and strikes

Vega is the change in an option's market value when implied volatility rises by 1 percentage point³. Implied volatility takes a value for each strike price and maturity of an option, and together these values form the volatility surface. The surface is not flat, and its shape also changes over time¹⁰. For this reason, risk management often moves the surface and revalues positions¹¹. Vega, too, shows where a position is exposed to volatility moves when it is aggregated separately for each combination of maturity and strike, rather than as a single total.

Consider a hypothetical example in which the vega of V Bank's interest rate options is aggregated on a grid of maturities and strikes. Each value is the change in market value when only the implied volatility in that cell rises by 1 percentage point.

The change when volatility in every cell rises by 1 percentage point is the total of −0.5 million yen, so vega looks small. But if only 1-year volatility rises by 2 percentage points, V Bank gains +1 × 2 = +2 million yen, and if only 5-year volatility rises by 2 percentage points, the change is −1.5 × 2 = −3 million yen, a loss. V Bank is effectively long short-maturity options and short long-maturity options. The strike columns also differ. If only volatility at the low strike rises by 2 percentage points, the change is +0.1 × 2 = +0.2 million yen; if only volatility at the high strike rises by 2 percentage points, it is −0.3 × 2 = −0.6 million yen. A change in the shape of the smile, the pattern of volatility across strikes, also produces gains or losses.

Adding up sensitivities and setting limits

Sensitivities to the same risk factor can be added up across positions and desks. If V Bank's 10-year figure of −3 million yen is the sum of −5 million yen at the bond desk and +2 million yen at the swap desk, then at the firm level the positions offset each other by 2 million yen against moves in the 10-year rate. The standardized approach in the Notice also starts its calculation by offsetting (netting) sensitivities to the same risk factor across positions¹². The size of a position (its notional amount) does not reflect how much the price of a derivative tends to move, which makes it hard to compare across products and desks. Sensitivities, by contrast, can be added up in the same units for each risk factor³.

Adding up sensitivities to different risk factors ignores the correlation between the movements of those risk factors. The last section explains how correlation is handled.

Sensitivities are also used as the units for setting limits. The US Federal Reserve's examination manual (1998) lists the following limits that banks commonly use, and how they are operated. For option positions, there are limits on delta, gamma, vega, theta and rho, and there are limits by maturity to guard against non-parallel moves in the curve. Limits are allocated to desks and traders, and the bank puts in place arrangements to detect and deal with breaches¹³. A 2012 Bank of Japan training document gives examples of managing the positions of the markets division using a 10bp change in rates for all maturities and a 1bp change for each maturity¹⁴.

Limits of sensitivities and how to fill them

A sensitivity is a first-order approximation for small moves in a risk factor. When the price of the underlying asset moves by $\Delta S$ and volatility moves by $\Delta \sigma$, the change in an option's market value can be approximated with delta $\delta$, gamma $\Gamma$ and vega as follows¹⁵. Vega is per 1%, so express $\Delta \sigma$ in percentage points.

$$\Delta V \approx \delta \Delta S + \frac{1}{2} \Gamma (\Delta S)^2 + \text{vega} \times \Delta \sigma$$

The gamma term works through the square of the move, so the larger the move, the further the result departs from an approximation using delta alone. For example, take a position with a delta of 0 and a gamma of −100,000, with market value and price both in yen. If the price of the underlying asset moves 1 yen in either direction, the loss is 0.5 × 100,000 × 1 = 50,000 yen, but if it moves 10 yen, the loss is 0.5 × 100,000 × 100 = 5 million yen. With a gamma of +100,000, the same moves produce gains of 50,000 yen and 5 million yen. Sensitivities such as delta and DV01 on their own cannot capture this curvature in the price (convexity)³.

Sensitivities have two other limitations. First, they do not include how much prices tend to move (volatility)³, nor how risk factors move together (correlation)⁹. Second, even if you assume a scenario such as "rates rise by 10bp," a sensitivity does not tell you how likely that scenario is to happen¹⁶.

These limitations are covered by other measures. The variance-covariance method for VaR (value at risk) assumes that sensitivities are constant and that risk factors follow a normal distribution. On that basis, it combines the sensitivities with the size of the risk factor moves and their correlations to find the loss that can occur with a given probability. For positions whose sensitivities are not constant, this method is an approximation¹⁷. The gains and losses from large moves whose probability is unknown (a rise in rates of 100–200bp, a 50% fall in stock prices and so on) are found by revaluing positions in stress testing¹⁶. The calculation method is covered on the page Value at Risk (VaR).

Sensitivities are also used to calculate regulatory capital. The standardized approach that the Notice uses to calculate capital for market risk combines key rate sensitivities to interest rates and vega using risk weights and correlations, and adds a charge for what delta cannot explain (curvature risk)¹⁸. Interest rate sensitivities are calculated for 10 maturities from 0.25 years to 30 years, and interest rate vega is calculated by option maturity (five maturities from 0.5 years to 10 years) and other dimensions¹⁸.

Notes

1. Bank of Japan, Financial System and Bank Examination Department, Center for Advanced Financial Technology, "Ⅱ．市場リスクの把握と管理" (II. Understanding and managing market risk) (June 2012; hereafter the BOJ training material), "（参考）リスクファクター" ((Reference) Risk factors), page 11 (in Japanese). ↩

2. BOJ training material, "（参考）金融商品とリスクファクター" ((Reference) Financial products and risk factors), page 12 (in Japanese). The options row is from Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual," Section 2010.1 "Market Risk" (February 1998; hereafter the examination manual), "Basic Measures of Optionality," page 4. ↩

3. Examination manual, "Nominal Measures," "Factor-Sensitivity Measures," and "Basic Measures of Optionality," pages 3–4. Quoting vega per 1% matches the unit used on this site's Greeks page. ↩ ↩² ↩³ ↩⁴ ↩⁵

4. BOJ training material, "（１）ＢＰＶ（ベーシス・ポイント・バリュー）" ((1) BPV (basis point value)), page 13 (in Japanese). The document writes present value as a function of the rates for each maturity, $r_1, \dots, r_n$. ↩ ↩²

5. BOJ training material, "（２）ＧＰＳ（グリッド・ポイント・センシティビティ）" ((2) GPS (grid point sensitivity)), page 14 (in Japanese). ↩

6. BOJ training material, "ＢＰＶ、ＧＰＳの計算例" (Worked example of BPV and GPS), page 15 (in Japanese). The example is a 5-year bond with a principal of 10 billion yen and a 1.5% coupon, with the spot rate for each maturity raised by 1bp. ↩ ↩²

7. Financial Services Agency, "銀行法第十四条の二の規定に基づき、銀行がその保有する資産等に照らし自己資本の充実の状況が適当であるかどうかを判断するための基準" (Standards for determining whether a bank's capital adequacy is appropriate in light of the assets it holds, under Article 14-2 of the Banking Act, our translation; FSA Notice No. 19 of 2006, consolidated version updated on August 24, 2026; hereafter the Notice), Article 283-2 (page 490) and Article 283-3, item 1 (page 490) (in Japanese). ↩

8. Examination manual, "Scenario Simulations," page 5. This is a method of finding the change in value under a scenario by applying changes in risk factors to sensitivities such as DV01. ↩

9. Examination manual, pages 1–2. It describes "yield curve" risk, where long and short positions at different maturities are hedged against a change in the level of rates but not against a change in the shape of the curve. The dependence of interest rate risk on the correlation of sensitivities is on page 1. ↩ ↩²

10. Martin Haugh, "Black-Scholes and the Volatility Surface" (Columbia University), page 4. The source treats implied volatility as a function $\sigma(K, T)$ of the strike $K$ and maturity $T$, and shows that the surface is not flat and changes substantially over time. ↩

11. The same source, page 5. It says that "bumping" or "stressing" the volatility surface is commonly performed for risk management purposes. ↩

12. The Notice, Article 282-2, paragraph 2 (page 480) (in Japanese). ↩

13. Examination manual, "Market-Risk Limits," pages 6–7. ↩ ↩²

14. BOJ training material, "（３）シナリオに基づくリスク量の把握（続き）" ((3) Measuring risk based on scenarios (continued)), page 18 (in Japanese). ↩

15. Martin Haugh, "Black-Scholes and the Volatility Surface" (Columbia University), "Delta-Gamma-Vega Approximations to Option Prices," pages 11–12. The source writes delta as $\delta$, gamma as $\Gamma$ and vega as vega. ↩

16. BOJ training material, "（３）シナリオに基づくリスク量の把握" ((3) Measuring risk based on scenarios) and "（続き）" (continued), pages 17–18 (in Japanese). The document states that the probability of an assumed scenario occurring is unknown, and gives stress testing examples of a 100–200bp rise in rates and a 50% fall in stock prices. ↩ ↩²

17. BOJ training material, "Ａ．分散共分散法" (A. The variance-covariance method), pages 23–24, and "留意事項②" (Points to note 2), page 28 (in Japanese). ↩

18. The Notice, Article 281 (page 479), Article 282-2, paragraphs 3–5 (pages 480–481), Article 282-3, paragraph 2 (pages 481–482), and Article 283, item 1(a)(2) and item 2(b) (pages 484–486) (in Japanese). ↩ ↩²