2020/07/16

Option Model Pricing When Rates Are Negative

by Lucido Group

Interest rates around the world are now at historic lows, and are set to remain there for an extended period of time. Yen portfolio managers have had to contend with low and negative rates for many years, however interest rates in USD and AUD are now approaching zero. And it is not just interest rate markets that are facing negative prices: the front futures contract for crude oil turned negative in April. This pricing environment causes serious problems for many option pricing models.

Interest Rate Option Pricing

The most common family of option pricing models relies on underlying prices (interest rates for interest rate options) following a log-normal distribution. These models cannot operate when underlying prices are negative, and their accuracy suffers as prices approach zero.

Findur has support for models that will operate in this pricing environment. For instruments in the interest rates option family, such as caps, floors and collars, Findur has a pricing model called Normal. This is known more widely as the Bachelier model, named for the man who first described it more than 100 years ago. Underlying prices in the Bachelier model follow a normal distribution, so prices can be negative. The model requires normal volatilities, as opposed to log-normal vols required by the Black Scholes model. These volatilities are expressed in basis points, so there is a change required to the configuration of the volatility surface. Also, users need to pay attention to the vega shift on the surface to ensure vega is reported for a 1 basis point shift, and not a 1 percentage point shift.

Swaption Pricing

Swaption pricing is also affected by the low and negative interest rate environment. Most users of Findur with a swaption portfolio use a Black Swaption model for pricing, supported by a log-normal volatility cube. There is support for pricing Bermudan swaptions using a variant of the model called Black Swaption Bermudan.

Just as Findur has a Bachelier normal model for interest rate options, there is an equivalent Bachelier normal model for pricing swaptions. The model is called Normal Swaption in Findur and expects normal volatilities. This model can support American and European swaptions. It cannot be used to price Bermudan swaptions. In particular, it produces wayward results for Bermudan swaptions that deliver into a swap with a start date in the past. This is a frequent pricing requirement in the market, as it is needed to handle the optionality embedded in a decayed cancellable swap.

A popular choice of pricing model, even before rates were driven so low, has been Hull-White’s ‘Tree model’. This is a well-known, mean-reverting trinomial tree model. In lieu of a volatility cube, it has two volatility parameters, alpha and sigma.

Alpha and sigma are identified using a calibration routine. The typical routine involves using actively traded European swaptions to minimize the sum of squared pricing errors to a baseline Black swaption pricing model. Naturally, if the Black model pricing is no longer precise, the calibration routine must use an alternative model. The Bachelier variant is one model we have used with satisfactory results.

Implementation Considerations

It is easy to underestimate the effort and the complexity of this seemingly minor shift in the market. If you would like more information, reach out!

Any evaluation of alternative option pricing models should include the following tasks –

  • Identify market data vendor for normal volatilities
  • Configure normal and Hull-White volatility surfaces
  • Enable new pricing models
  • Configure Hull-White calibration
  • Create new data points in MDT Admin to import volatilities
  • Configure MDT Mapper
  • Configure EOD to import volatilities and perform calibration daily
  • Perform pricing model validation testing
  • Update trade templates
  • Update existing trades
    • Either amend existing instruments to use new pricing models, or
    • Map pricing models in simulation definition(s)

The following tasks may also be subject to review –

  • Review pricing of credit risk, market risk and scenario analysis
  • Review compliance rules
  • Review Greek sensitivities and P&L detail
  • Review impact on hedge accounting, including hedge effectiveness tests
  • Review inbound trade interfaces
  • Review outbound post-trade interfaces