# Pragmatic Note on Volatility Model When Rates Turn Negative

In the interest rate volatility market, market participants often employ two sets of models – one for quoting the market (ie map the traded option prices in volatility terms using Black model initially and migrated to shifted Black or Bachelier in recent years), another for the modelling of the underlying volatility dynamics e.g. used to model of volatility smile (using SABR and maybe Hull-White). Any model with implicit lognormal assumption will fail under zero or negative rates. As the interest rates for much of the developed world has fallen to close or below zero post 2008 global financial crisis and 2012 European Sovereign debt crisis, early models and conventions need to be adjusted. Here found some of my notes in this topic.

Changes to the Option Quotation Model

The Black model assumes log normal distribution of the forward rate with ${dF = \sigma_B F dW}$. It was market convention when trading rate volatility products in the 90s and 00s. The price of a European call ${C}$ and put option ${P}$ are $\displaystyle C=D[F\Phi(d_1) - K\Phi(d_2)]$ $\displaystyle P=D[K\Phi(-d_2) - F\Phi(d_1)]$

where $\displaystyle d_1 = \frac{ln(F/K)+\sigma_B^2 T/ 2}{\sigma_B \sqrt{T}}$ $\displaystyle d_2=d_1- \sigma_B \sqrt{T}$ ${\Phi()}$ is the cumulative normal distribution function. ${\sigma_B}$, ${F}$, ${K}$, ${T}$ are the Black volatility, forward rate, strike and the time-to-expiry of the option respectively. ${D}$ is a discount factor proportional to the amount of derivatives being held. For example, in the case of a caplet with a notional of ${N}$, time interval of ${\delta}$ between the rate reset and payment date and discount factor ${df}$ at the payment date, ${D=N*\delta*df}$. In the case of swaption, ${D = N*A}$ if we define the annuity ${A}$ as sum of discount factor and time interval for each payment period.

When the interest rate being considered (either the forward or the strike) is very low, the solution of the Black model becomes very unstable. When either becomes zero or negative, the solution of the Black model is undefined. We can resolve this either by adopting a shifted Black model or a Bachelier (normal) model.

Solution 1: Shifted Black Model

The idea of shifted Black model is very simple. We pick a “low enough” interest rate level that we do not expect the interest rate would ever fall below (say -3%). We then add this amount to all the inputs. Though there are two main drawbacks: 1. The shifted Black volatility is depend on the shift amount. If two dealers quoting volatilities with different shift amount, conversion has to be done before levels are being compared. 2. The process to select the shift amount can be a bit arbitrary.

Solution 2: Bachelier (Normal) Model

The alternative is to use Bachelier’s century old model which assumes that the forward rate follows normal distribution ${dF = \sigma_N dW}$. It is also known as Normal Model. The European call and put option prices become $\displaystyle C=D[(F-K) \Phi(d) + \sigma_N \sqrt{T} \phi(d)]$ $\displaystyle P=D[(F-K) \Phi(d) + \sigma_N \sqrt{T} \phi(d)]$ $\displaystyle d=(F-K)/(\sigma_N \sqrt{T})$

with ${\phi()}$ being the probability density function of normal distribution.

Some quick notes on normal volatility: first, normal volatility is shift-invariant as ${dF}$ does not depend of the forward rate. This will become relevant later on. Second, the market convention for normal volatility is to quote in bps (rather than as percent in the case of Black volatility).

Brief Review of the SABR Vol Model

SABR is a popular volatility model for interest rate products. Take swaption as an example, dealer quotes are available often for just the at-the-money and a few popular out-of-the-money strikes (say, ATM, ATM ${\pm}$100bp, ATM ${\pm}$200bp etc) for a given swaption expiry date and tenor. If we need to find the volatility at a strike different from the known values, we can use SABR as a volatility smile model to handle the interpolation.

SABR is a four-parameter model with parameters ${\alpha, \beta, \nu, \rho}$ with ${\alpha \geq 0}$, ${0 \leq \beta \leq 1}$, ${\nu \geq 0}$ and ${-1 < \rho <1}$. $\displaystyle \begin{array}{rcl} dF &=& \hat{\alpha} F^\beta \, dW_1 \\ d\hat{\alpha} &=& \nu \hat{\alpha} \, dW_2, \quad \hat{\alpha}(0)=\alpha \\ dW_1 dW_2 &=& \rho dt \end{array}$ ${\beta}$ is often chosen by the user instead of backing out from the data. Setting ${\beta}$ to zero is a special case. The instantaneous change of the forward rate does not depend on its current value in this situation and this is known as a stochastic normal SABR model. ${\alpha}$ is a volatility-like parameter and ${\nu}$ is a vol-of-vol like parameter. ${\rho}$ is the correlation coefficient between two sources of random noises. The effect of change different SABR parameter is shown in the series of charts below. The effect of changing different SABR parameters to quoted Bachelier volatility. The base case assume ${\alpha=0.001, \beta=0.5, \nu=0.2, \rho=0.3, fwd=0.03, t=5}$ and one of the parameter is being changed each time

Hagen et.al. derived closed-form approximations which map the SABR parameters to either Black or Bachelier volatilities. Define ${T}$ as the option expiry, ${z}$ and ${\chi(z)}$ as $\displaystyle z = \frac{\nu}{\alpha} (fK)^{\frac{1-\beta}{2}} \log{\frac{F}{K}}$ $\displaystyle \chi(z) = \log{\frac{{\sqrt{1-2\rho z+z^2}+z+\rho}}{1-\rho}}$

SABR to Black Volatility: $\displaystyle \sigma_B(F,K) = \frac{\alpha \left[ 1+\frac{(1-\beta)^2 \alpha^2}{24 (FK)^{1-\beta}}+\frac{\alpha \beta \nu \rho}{4 (FK)^{(1-\beta)/2}}+ \frac{2-3 \rho^2}{24} \nu^2 \right] T}{ (F\beta)^{(1-\beta)/2} \{1+\frac{(1-\beta)^2}{24}\log^2{\frac{F}{K}}+\frac{(1-\beta)^4}{1920}\log^4{\frac{F}{K}}\} } \frac{z}{\chi(z)} \ \ \ \ \ (1)$

SABR to Bachelier Volatility: $\displaystyle \begin{array}{rcl} c_1 &=& \dfrac{1+\frac{1}{24}\log^2\frac{F}{K}+\frac{1}{1920}\log^4\frac{F}{K}}{1+\frac{(1-\beta)^2}{24}\log^2\frac{F}{K}+\frac{(1-\beta)^4}{1920}\log^4\frac{F}{K}} \\ c_2 &=& \left[1+\frac{-\alpha^2 \beta (2-\beta)}{24 (FK)^{1-\beta}}+\frac{\alpha \beta \nu \rho}{4 (FK)^{(1-\beta)/2}}+\frac{2-3\rho^2}{24}\nu^2 \right] T \end{array}$ $\displaystyle \sigma_N(F,K) = \alpha (FK)^{\beta/2}\, c_1 \, c_2 \, \frac{z}{\chi(z)} \ \ \ \ \ (2)$

Continue with our swaption example. We first calculate the implied volatilities for the list of quoted swaptions with our chosen option model. We then fit the paired strike and implied volatility list to the approximation to Eq 1 or 2 by constrained optimisation to obtain the SABR parameters.

Hague applied singular perturbation technique to derive the above approximations. For points far away from ATM strike, the solutions can deviate for not insignificant amount when compared with solutions obtained by solving the SDEs numerically (e.g. by Monte Carlo).

Changes to Vol Model

Coming back to the topic of negative rates. For those implementations with Hull-White, no adjustment is required as Hull-White assumes the forward rate to follow a normal distribution ${dF = (\theta-\alpha F)\,dt+\sigma\,dW}$. Negative rate would not be an issue Rates can be negative and no adjustment is required.

SABR volatility model is however affected when the rates are approaching or below zero. As ${dF \propto F^\beta}$, SABR model is undefined for ${0<\beta \leq 1}$ if the rates becomes zero or negative. There are three common solutions, corresponding to setting ${dF}$ to ${F+x}$, ${1}$ and ${|F|^\beta}$ respectively.

Solution 1: Shifted Model

Again we can pick a “low enough” interest rate level we do not expect the interest rate would ever reach. Add that amount to all the inputs (forward rate, strikes). As before, determining such adjustment can be a bit arbitrary.

Solution 2: Stochastic Normal SABR

Turning the SABR model into the stochastic normal SABR model special case by setting ${\beta}$ to 0. In such case, term ${c_1}$ of Eq 2 becomes 1. The log forward or log strike calculations become unnecessary. Thus it can handle negative interest rate without any further modification. However, the increment of forward rate ${dF}$ is now independent to its current value. For some, it is arguable whether this is conceptually appropriate .

Solution 3: Modified Models e.g. Free SABR

Antonov et.al.  modifies the SDE by taking the absolute value of the forward rate: ${dF = \hat{\alpha} |F|^\beta dW_1}$. It introduces a fairly sharp peak in for the probability density around zero. It can be explained as the market’s natural tendency to price high chance that the interest rate would hover not below zero. By now, most would agree the crucial role of the central banks in driving the short term rate to negative. A case in point, the short end of the Euro curve has been below zero (around -25 bps or lower) for a number of years since 2016. The market expectation is likely to peak around this negative ECB policy rate. There is no flexibility for this model to show the pdf peak down below. FreePDF (blue solid). Graph is from 

Note 1 “Managing Smile Risk”, Hagan, Kumar, Lesniewski & Woodward, Wilmott Magazine, 2002.

Note 2 “Stability of the SABR model”, Deloitte, 2016.

Note 3 “The Free Boundary SABR: Natural Extension to Negative Rates”, Antonov, Konikov & Spector, published in SSRN, 2015