Interest Rates – Sisi Tang

The attachment contains a toy RFR (e.g. SOFR, ESTR) )curve calibrator and a Libor-type pricer.

It is called a toy because the day count convention and effective/settlement date are not accurately considered. Also, the fine structure of the front end of the RFR curve (short tenors) is not handled. To handle front-end fine structure, short maturity instruments should be used as calibration instruments for example FRA (forward rate agreements). In this toy example, only par swap rates are used.

A key part of an interest rate calibrator is an optimizer or a solver (bootstrapping case only). In this toy example, Excel Solver is used. This is a widely available tool as long as you have Excel. See the Microsoft help page for how to turn on Solver: Load the Solver Add-in in Excel.

Note 1: The calibrator part is set up for a swap whose floating leg pay frequency is annual only, e.g., ESTR and SOFR, because the cashflows are evaluated out of 1-year forward rate. If you were to strip a Libor curve, you have to make the tenors granular enough to cover every payment (every quarter in the Libor case) and add more interpolation pillars.

Note 2: The RFR rate is compounded daily. The actual interest rate paid is a backward-looking backward rate. But it is still reasonable to use a forward rate to evaluate cashflows because any compounding format describes borrowing money for a certain period. The key variable that drives the whole curve is the discount factor, which could be equivalently expressed as forward rate, zero rate, etc.

February 14, 2023February 15, 2023

Black’s Model for Swaption

The quoting convention of the swaption volatility is annualized normal vol under annuity measure. Here, normal is contrary to log-normal. It is also called Black’s (vs Balck-Scholes) or Bachelier’s model. We will use formulas to explain what it means.

Note 1: Black’s model is used for quoting only in modern financial modeling. It doesn’t mean the dynamics of interest rates are modeled this way. A commonly used model for swaption is SABR (stochastic alpha beta rho).

Note 2: The formula below ignores nuances of the effective date, settlement date, and day count convention.

Notations:

$\hat{T}$ : maturity of swaption

$\Delta$ : tenor (maturity) of the underlying swap

$\mathbb{Q}$ : risk neutral measure

$A$ : annuity or annuity measure where annuity matches the floating leg pay frequency of the underlying swap

$K$ : strike

$P(0,t)$ : discount factor from $t (t \ge 0)$ to $0$

$S_{t}(\hat{T}, \hat{T}+\Delta)$ : par swap rate for a swap contract from $\hat{T}$ to $\hat{T}+\Delta$ seen at $t$ . We define a short-hand $S_t := S_t(\hat{T} - \hat{T}+\Delta)$ when there is no confusion.

$\mathbb{E}_t^{\mathbb{P}}$ : conditional expectation on $\mathcal{F}_t$ for filtration $(\Omega,\mathcal{F}, (\mathcal{F}_t)_{t\ge 0},\mathbb{P})$

$PV_{\mbox{something}}$ : present value of something.

$V_{\mbox{something}}(t)$ : value of something at time $t$ .

Below, we set up the formula for PV of a payer swaption assuming the notional is $1. For a more general notional, just scale the formula up by the notional. In a payer swaption, the purchaser has the right but not the obligation to enter into a swap contract where they become the fixed-rate payer and the floating-rate receiver.

$\small \begin{aligned} PV_{\mbox{payer}} =& \mathbb{E}_0^{\mathbb{Q}}\left[\left(V_{\mbox{float}}(\hat{T})-V_{\mbox{fixed}}(\hat{T})\right)^+P(0,\hat{T})\right] \\ =& \mathbb{E}_0^{\mathbb{Q}}\left[ \left(S_{\hat{T}}(\hat{T},\hat{T}+\Delta)A(\hat{T})-KA(\hat{T})\right)^+P(0,\hat{T})\right] \\ = & A(0) \mathbb{E}^{A}_{0} \left[\left(S_{\hat{T}}(\hat{T},\hat{T}+\Delta) - K\right)^+\right] \end{aligned}$

The last line of the above formula is a change of numeraire from risk-neutral measure to annuity measure.

Since swaption vol is quoted as Bachelier’s vol in annuity measure, the dynamic of interest rate under this measure is the following:

$\small \textrm{d} S_t = \sigma \textrm{d} W_t,$

where $W_t$ is a standard Brownian motion under annuity measure and $\sigma$ is constant.

Hence (by integrating both sides from $0$ to $\hat{T}$ ), denoting by $\mathcal{N}(\mbox{mean}, s.d.)$ a normal distribution, we have

$\small S_{\hat{T}} \sim \mathcal{N}(S_0, \sigma\sqrt{\hat{T}}).$

After plugging in the Gaussian density, the annuity measure expectation becomes

$\small \begin{aligned} & \mathbb{E}^{A}_{0} \left[\left(S_{\hat{T}} - K\right)^+\right] \\ = & \int_{K}^{+\infty} (x-K) \frac{1}{\sigma \sqrt{2\pi\hat{T}}} e^{-\frac{(x-S_0)^2}{2\sigma^2\hat{T}}} \textrm{d} x \\ = & \sigma \sqrt{\hat{T}} \cdot \varphi (d_1) + (S_0 - K) \cdot \Phi(d_1) \end{aligned} ,$

where

$\varphi$ is the probability density function of the standard Gaussian distribution

$\Phi$ is the cumulative density function of the standard Gaussian distribution

$d_1 := \frac{S_0 - K}{\sigma \sqrt{\hat{T}}}.$

Finally,

$\small PV_{\mbox{payer}} = A(0) \left[\sigma \sqrt{\hat{T}} \cdot \varphi (d_1) + (S_0 - K) \cdot \Phi(d_1) \right]$

The Greeks are as follows:

Delta = $A(0) \Phi(d_1)$

Vega = $A(0) \varphi(d_1) \sqrt{\hat{T}}$

Gamma = $A(0)\frac{\varphi(d_1)}{\sigma \sqrt{\hat{T}}}$

The vega-gamma relationship is:

$\small \mbox{Vega} =\hat{T}} \sigma \mbox{Gamma}$

The value of annuity $A(0)$ depends on discounting curve, but one can approximate it by tenor ( $\Delta$ ) in years, i.e., no discounting. The swaption Bachelier’s/normal/Black’s vol is quoted in the unit of bps. For example, 107 means 107bps, i.e., 0.0107. Below we provide two examples of PV: one ATM and one ITM.

For a receiver swaption, PV and greeks can be calculated in the same manner.

July 4, 2020March 8, 2021

Interest Rates Basic

Suppose $b>a>0$ . Let $P(a,b)$ be the discount factor discounting 1 dollar paid at $b$ to time $a$ . Let $T_1, T_2, \ldots, T_{N+1}$ be the floating rate reset date of the floating leg in a swap. For the case of LIBOR 3M, $T_i = \frac{i}{4}$ if we assume the day count convention is that 3 months is $\frac{1}{4}$ of a year. Define $\alpha_n = T_{n+1} - T_{n}$ .

Forward LIBOR Rate

Let L_n(t) be the forward LIBOR rate for period $[T_n, T_{n+1}]$ seen at time $t$ for $t < T$ . Then,

$L_n(t,T) := \frac{1}{\alpha_n}\left(\frac{P(t,T_n)}{P(t,T_{n+1})}-1\right).$

This definition makes sense if we seen it as:

$1+\alpha_nL_n(t) = \frac{1}{P(T_n, T_{n+1})} = \frac{P(t,T_n)}{P(t,T_{n+1})}.$

Instantaneous Forward Rate

Instantaneous forward rate is the forward rate with time interval goes to infinity.

$\begin{aligned}f(t,T)&=\lim_{\delta\rightarrow 0}\frac{1}{\delta}\left(\frac{P(t,T)}{P(t,T+\delta)}-1\right) \\ & = -\frac{\partial \log P(t,T)}{\partial T}\end{aligned}.$

Short Rate

Short rate $r(t)$ is an instantaneous interest rate such that if $r$ is deterministic

$P(t,T) = e^{-\int_t^Tr(s)\textnormal{d}s}.$

Or if $r$ is stochastic

$P(t,T) = \mathbf{E}\left[e^{-\int_t^Tr(s)\textnormal{d}s}|\mathcal{F}_t\right].$

Forward Par Swap Rate

In finance, ‘par’ usually means ‘equal’. Par swap rate is the fixed rate such that the floating leg and the fixed leg of the swap make the contract has value 0. We denote $S_n(t)$ by the forward par swap rate on tenor $[T_n, T_{N+1}]$ . Suppose the pay frequency of the fixed leg is the same as the floating leg (otherwise redefine the below annuity as one that matches the fixed leg pay frequency). Let $A_n(t)$ represents the present value of annuity, i.e.,

$A_n(t) = \sum_{k=n+1}^{N+1} \alpha_{k-1}P(t,T_k).$

Then the present value of the fixed leg is

$PV_{\mbox{fix}} = S_n(t)A_n(t)$

The present value of the floating leg is

$\begin{aligned}& PV_{\mbox{float}} \\ = & \sum_{k=n}^N L_k(t)\alpha_k P(t,T_{k+1}) \\ =& \sum_{k=n}^N \frac{1}{\alpha_k}\left(\frac{P(t,T_k)}{P(t,T_{k+1})}-1\right) \alpha_k P(t,T_{k+1}) \\ =& \sum_{k=n}^N \left(P(t,T_k) - P(t,T_{k+1})\right) \\ =& P(t,T_n) - P(t,T_{N+1}) \end{aligned}$

By letting $PV_{\mbox{fix}} = PV_{\mbox{float}}$ , we have that

$S_n(t) = \frac{P(t,T_n) - P(t,T_{N+1})}{A_n(t)}.$

Category: Interest Rates

Toy interest rate curve calibrator

Black’s Model for Swaption

Interest Rates Basic