formula for physical DV01 of interest rate swap

Question

Most answers to the question "what is the dv01 of an interest rate swap" are along the lines of: "compute the difference between the price of the swap and its price using a curve perturbed by 1 basis point". While i agree with this answer, I wanted to link this to a formula that I believe expresses the dv01 as a function of the relevant discount factors throughout the life of the swap and the day count fractions for each period. Can someone help point towards this version?

Answer 1

There are two items that must be clarified with respect to your question:

1. Are you assuming an interest rate swap (IRS) at mid-market, i.e. at-the-money (ATM) or an off-market IRS with some unknown net present value (PV)?
2. Are you interested in a risk approximation or a more accurate formula that reflects the truest risk sense of a market curve shifted up or down?

If you are assuming a mid-market IRS and want an approximation then MarinD's answer should suffice. This can also be seen for example by differentiating the fixed leg PV formula with respect to the fixed rate: $$\text{Analytic delta of mid-market IRS} = \frac{\partial P_{\text{fixed}}}{\partial R} = N \sum_{i=1}^{n_1} d_i v_i$$ (using wikipedia's IRS formula notation)

If you have an off-market IRS then you must add in the delta associated with the PV. In my experience a good approximation for a par tenor IRS is to divide the PV by 10,000 (bps) and multiply by the tenor divided by two (which approximates the annuity PV's cashflows being exchanged over the tenor of the swap). E.g. a 10Y IRS with a PV of \$1mm will correspond to \$500 total delta per bp. Since PV has already factored in discount factors when arriving at the PV there is no need to reassert them again. $$\text{Analytic delta of off-market IRS} = \frac{\partial P_{\text{fixed}}}{\partial R} + Delta(PV)= N \sum_{i=1}^{n_1} d_i v_i + \frac{PV * tenor}{10,000 * 2}$$

Neither of these are risks in the true sense of market moves because they both assume the market remains constant whilst the fixed rate on the contract moves. It is possible to derive the formula you seek in analytic terms but is rather complicated. And also to apply. The book Pricing and Trading Interest Rate Derivatives covers what you are after in the chapter "analytic cross-gamma" since these derivations are required to calculate the real gamma on IRSs.

Answer 2

Short answer

\begin{equation} DV01(t) = \sum_{j=1}^N \alpha_j Z_t(t_j) \end{equation} with

• $DV01(t)$ the DV01 of the swap at time t
• $j$ the period number
• $\alpha_j$ the fraction of a year of the period $j$
• $Z_t(t_j)$ the discount factor from $t$ to $t_j-t$

Simple proof

First let's define $\alpha_j$ as the fraction of a year of the period $j$ (time between two swap payments)

Let $Z_t(T)$ be the value of a zero-coupon bond of maturity $T$ at time $t$ (ie discount factor from $t$ to $T-t$). Then $Z_0(t)$ is today discount factor for maturity $t$.

The value of a swap $V_{swap}$ is the difference of its floating leg value $V_{float}$ and its fixed leg value $V_{fix}$.

At any time $t$, for an $N$-year swap between its start date $t_0$ and its end date $t_N$: \begin{equation} V_{float}(t) = Z_t(t_0) - Z_t(t_N) \end{equation} and \begin{equation} V_{fix}(t) = R_{fix} \times \sum_{j=1}^N \alpha_j Z_t(t_j) \end{equation} with $R_{fix}$ the fixed rate of the swap.

The value of the swap is \begin{align} \begin{split} V_{swap}(t) &= V_{float}(t) - V_{fix}(t) \\ &= Z_t(t_0) - Z_t(t_N) - R_{fix} \times \sum_{j=1}^N \alpha_j Z_t(t_j) \end{split} \end{align}

Let us denote the DV01 of the swap by $DV01(t)$ at time $t$. It is defined as the partial derivative of the swap value with respect to the fixed rate of the swap $R_{fix}$ \begin{equation} DV01(t) = \frac{\partial V_{swap}(t)}{\partial R_{fix}} = \sum_{j=1}^N \alpha_j Z_t(t_j) \end{equation}

(Note that the minus sign can be added or removed depending if you are paying or receiving the swap)