Small banks face rate options valuation model change

Negative rates seen in a number of jurisdictions are causing the Black model to break down when valuing embedded interest rate options. Smaller banks had hoped the storm would blow over, but are now facing an overhaul of their valuation models

Negative interest rates have forced many traders to think differently. Normally, a party lending to another is compensated for its opportunity cost and the credit risk it is taking. However, with quantitative easing pushing interest rates close to zero or even into the negative in the eurozone, banks have had to consider some of the stranger effects it can cause on their valuation models.

It is a particular challenge for structured products with embedded interest rate options, as the most widely used model for valuing the derivatives assumes rates cannot go below zero. While large banks tackled this problem when it first emerged in 2012, smaller banks have been hiding out in the hope that the issue would blow over. But with the original European Central Bank (ECB) negative rate move in June 2014 having been extended by 10 basis points to 0.3% on December 3, 2015 - despite an assurance from ECB president Mario Draghi that it had already reached its "lower bound" on October 22 - and with other European jurisdictions in the same boat, smaller banks across the continent are now having to take the plunge and consider an overhaul of their valuation models.

"Many market participants among the smaller houses tried to ignore the problem for as long as possible hoping it will go away," says Mark Beinker, head of unit for applied financial engineering at quantitative consultancy d-fine in Frankfurt. "But in fact it is getting worse. So they have started to do something."

As quotes may only be available for certain interest rate option strikes and maturities, when valuing at-the-money derivatives, many banks calculate the implied volatility, build a volatility smile, interpolate for the other strikes and convert the data into a price via the Black model, created by economist Fischer Black in 1976. For out-of-the-money pricing, the stochastic alpha, beta, rho (SABR) model is used as an extension of the Black model.

A variant of the original Black-Scholes model from 1973, the Black model uses a logarithm of returns that are normally distributed, known as a lognormal distribution, which assumes that price cannot go below zero. This is useful for modelling the value of assets such as equities where there cannot be a negative value.

However, in a negative rate environment, banks pricing instruments in Danish krone, euro, Swedish krona and Swiss franc will find their model has broken. Bank models will be telling them that put options bought below a 0% strike had no value, for instance, when it had already moved past the point of zero.

If a model does crash, the impact for the valuation team will depend on the quality of the code in the system. It might simply refuse to accept negative forward rates, or if there is a process for identifying and alerting exceptions - that is, items that are not processed successfully - it might tell the middle office to check that its inputs and curves are correct. But given the Black model cannot handle negative rates, an overhaul is needed to allow for accurate valuations.

The extent to which sell-side firms have been affected depends on their experience in other markets, their size and the resources they can allocate to their valuation teams. Thomas Decouvelaere (pictured), co-head of pricing for Europe at Societe Generale Corporate & Investment Banking (SG CIB), says his firm's experience in other markets where short-term rates have turned negative in the past has allowed it to adapt its models to suit the current environment.

But smaller banks have struggled, says Daniele Marangelli, head of quantitative valuation and risk services at technology provider FIS in Paris: "It was quite a challenge, especially for an organisation that didn't have a lot of experience.Big banks are very quick to react and find a solution. But the ones that are not so expert ended up with halfway solutions."

If a bank is to change the way it prices implied volatility, it has three real alternatives. The first is to switch to using a normal or Gaussian model, which allows rates to move from the positive to negative side with the same probability.

This would be more of a culture shift for US banks than for Europeans: "US banks have had a tendency to value derivatives or options using the Black model - that is, assuming the distribution is lognormal - while European banks have historically used the Gaussian model, which assumes a normal distribution around the fall out of rates," says SG CIB's Decouvelaere.

The normal distribution can be helpful in low-rate environments. Decouvelaere notes that using lognormal distribution assumes that when rates are low, volatility will be lower because it's a percentage of the level of rates. However, this is not how rates move in practice.

"When rates are between 2% and 8% they have a tendency to move in a Gaussian manner - that is, the volatility of rates does not depend linearly on the level of rates," he says.

It is not only the model that is different, there is considerable operational risk – Dmitry Pugachevsky, Quantifi

However, in other instances, the lognormal model can represent the instrument's characteristics more effectively.

"When rates are very close to zero but not negative then the volatility decreases, which is something we have seen in Japan," says Decouvelaere. "Although rates could be slightly negative, the volatility smile tends to be slightly Black. So the choice between normal and lognormal is not a no-brainer. It is a choice you have to make," he adds.

Luke Kotchie, head of interest rate options trading at JP Morgan in London, says that despite the challenges posed by negative rates, there is still some interest in retaining the lognormal Black model.

"The level of euro rates and volatilities, combined with market inventory and the current supply and demand idiosyncrasies, makes modelling interest rates challenging from both a static and dynamic perspective. I would not advocate the use of any hard boundary model, but there is certainly something special, perhaps sticky about the conventional lower boundary in rates," says Kotchie.

There is a way to retain the Black model - simply shift the strike, by taking the negative interest rate and adding a buffer that takes it into positive territory. It is not a perfect solution and is one that is likely to annoy quant purists, but it does remove the immediate operational problem, and has proved to be a popular solution.

As the size of the shift is predetermined, the interest rate with a shifted lognormal model can be negative but within a limit.

"This is the most interesting and appealing advantage of the shifted lognormal model," says FIS's Marangelli. "It puts a limit - based on the size of the shift - on the negative value of the forward interest rate, which is more or less in line with the history of interest rates. Another advantage is that most of the systems in the market used by practitioners have used the Black lognormal model, and moving to a shifted lognormal version of it is the most straightforward way to solve the issue."

However, as it is essentially a workaround, it has its drawbacks. For instance, a shifted model introduces a completely new parameter that needs to be fed into the calculation.

"The shift is just subtracted from the distribution and then you get your regular lognormal option formula with adjusted forward and strike," says Dmitry Pugachevsky, director of research at analytics vendor Quantifi in New York. "When you trade with someone you have to quote both volatility and this shift."

There is no standard way within the market to dynamically calculate the shift and so it has to be arbitrarily chosen, plus it might have to be changed again if rates drop further. The size of the shift can also differ by currency or instrument, as these need to be aligned with the bank's peers.

"That is the main con," says FIS's Marangelli. "[But] at the moment, market operators are happy to do this given the positive elements of using shifted lognormal."

A third option is to use options prices from the interdealer broker market, rather than dealing with the formula itself. Brokers do not quote all maturities, however, so the bank would have to interpolate prices for the gaps. However, Marangelli warns this can be a risky process.

"You need to interpolate and interpolation in prices is very dangerous because these are highly non-linear prices," says Marangelli. "You will end up with a gross approximation and inaccuracy."

Granular shift

Once a firm has decided which model to use it then has to actually make the change, which can be an enormous operational burden in itself. For example, if a firm is quoting swaptions using normal instead of lognormal volatility, its systems need to understand that the quote will be in different terms.

"Normal volatility is around 100 times smaller than lognormal, as it is quoted in basis points not percentage like lognormal. You have to understand that the ‘100' you see is 100bp and that is 1% lognormal volatility, so there is a need to rework the middle office. The whole workflow should then change. It is not only the model that is different, there is considerable operational risk," says Quantifi's Pugachevsky.

It is not only the valuation model that is affected by negative rates: SABR also struggles to cope with very low, zero or negative strikes.

Changing models could also negatively impact the valuation of existing products with embedded interest rate options, particularly those with 0% floors.

"I have heard of losses on re-marking of zero floors," says SG CIB's Decouvelaere. "If you had booked a constant maturity swap (CMS) trade five years ago when rates were at 5% or 6%, the probability of rates going below zero was nil. You might have paid to a client CMS floored at zero capped at five, and valued this in the Black model. A few years down the road rates are much closer to zero and you need to re-mark your books. That potentially implies a loss on the zero floor coming from this increased probability of rates being negative," he adds.