“We have to tell customers when the mark-to-market value goes down, so you can imagine their reaction if you explain to them that it’s due to a model change,” Ligato said in an email.
Kuniyuki Hirai, head of trading for the Americas at MUFG Bank, recalls similar headaches when the Bank of Japan went negative in 2016.
“The tension was pretty high,” he said. “We had a lot of swap agreements with our clients in Tokyo. Nobody knew the BOJ would go negative in advance.”
Hirai says MUFG was lucky its models were built in-house, so they were able to make changes on the fly. Yet for some formulas, the firm just had to wing it.
“We went into the formulas, and instead of having one fair value curve, we installed lower bands that can handle negative spreads,” Hirai said of the firm’s model for options on swaps, or swaptions. “We tried to analyze the possibilities, but to be honest we didn’t have a very concrete number.”
Derivatives traders in equities, on the other hand, have it relatively easy by comparison. While their version of Black-Scholes is capable of accepting negative inputs, many have built-in limits on them. So for those, it’s more a matter of software reprogramming. For example, if you try to enter a negative risk-free rate into this online version, here’s what you get:
Tweaking the code, however, doesn’t solve all the issues that go along with negative rates. Pier Giuseppe Giribone, a financial engineer at Banca Carige SpA, points out that sub-zero risk-free rates can cause “approximation errors” for certain kinds of American-style options, which may lead them to become mispriced compared with those used in Europe.
Take an American call option on a stock without dividends, for example. Its value will “only equal the European option when the risk-free rate is positive,” said Giribone, who co-wrote a paper on the subject. “Many automatic pricing systems fail to calculate the right fair value in that scenario.”
For exotic contracts, traders need to turn to more complicated models. One is the Hull-White model, which is more accurate and better able to implement market information, but is also difficult to calibrate.
Oh, and you’ll also want to have your favorite quant on speed dial. That’s because Hull-White needs to be run in a programming language like MATLAB or R, rather than in Excel like Bachelier. And cross fingers that your counterparty has the same inputs, or else there could be a lot of back and forth.