Negative Rates in a Multi Curve Framework - Cap Pricing and Volatility Transformation

Abstract

The SABR model has for a long time been an invaluable tool for capturing the volatility
smile and to price nancial derivatives not quoted in the market. However, the current
negative rate environment in the EUR market has led to numerous challenges for nancial
institutions. One of the most problematic issues is that the SABR model used for
volatility interpolation and extrapolation fails when rates are negative. A related issue
is that SABR based techniques for transformation of cap volatilities between dierent
tenors, do not work anymore.
This thesis describes how to use an extension of the SABR model, known as the shifted
SABR, to solve these issues. Using three dierent methods, the shifted SABR model is
calibrated to EUR cap volatilities based on 6 month EURIBOR. However, market standard
is to quote a mixture of 3 month and 6 month cap volatilities, thus the volatilities
have to be transformed to a common tenor before calibration. To this end we have developed
a technique for volatility transformation in a multi curve framework when rates
are negative. The concept is derived by applying It^o's formula on an arbitrage relation
between shifted forward rates.
The results show that two of the methods for calibrating the shifted SABR model have
a good t to liquid contracts. However, the methods vary in performance capturing far
OTM caps. Our developed volatility transformation technique also works well, and the
sensitivity to the potentially unknown correlation between forward rates is low.
The implications are that the extension of the SABR model to the shifted SABR model
works ne, both in terms of pricing of caps and volatility transformation in a multi curve
framework. Although, it comes to the cost of some additional complexity of extending
formulas and arbitrage relations, that used to hold in a positive rate environment.

Popular Abstract

Popular Science Summary
Negative Rates when dealing with Caps
Negative rates have become a more common view on the leading markets.
This creates problems for nancial institutions, as some of their existing
models break down. We have analyzed three methods, used for calibration
to caps, and developed a new technique that deals with some of the problems
related to negative rates.
In June 2014 the European Central Bank (ECB) did something extremely unusual seen
from a historical perspective: they applied negative deposit rates. For nancial institutions
and investors this was an entirely new situation. Historically, negative rates have
been regarded as impossible. The economic intuition behind this is that negative rates
actual means you have to pay for lending money.
The motivation for negative interest rates diers depending on the national bank. The
main reason in the Eurozone and Sweden is to ght the growing threat of de
ation. Policymakers
want to make it less attractive to save money. As such lowering the interest
rate on savings should potentially make it more attractive to invest the money instead.
Financial institutions use statistical models in dierent areas such as pricing and risk
validation. An example is the SABR model which for a long time has been an invaluable
tool for capturing the volatility smile and to price cap contracts not quoted in the
market. However, the current negative rate environment in the European market has
led to numerous challenges for nancial institutions. One of the most problematic issues
is that the SABR model used for volatility interpolation and extrapolation fails when
rates are negative. A related issue is that SABR based techniques for transformation of
cap volatilities, do not work anymore.
Our work1 focuses on the pricing of cap contracts available on the European market.
Caps are contracts based on a set of simpler contracts known as caplets. Each caplet
provides the holder an insurance guaranteeing that the rate connected to the caplet won't exceed a predetermined level. Caps are quoted in terms of volatilities, which can
be seen as a measure for the variability of the underlying rates. Moreover, EUR cap
contracts are oered on dierent underlying rate index. Depending on the maturity of
the cap, it may be based on 3 or 6 months rates. In addition, the discount curve used
when pricing the caps, is not constructed from the underlying indexes. This is known as
the multi curve setting and brings additional complexity to the calibration when trying
to unify caps under the same index rate, i.e. volatility transformation to the same tenor.
A suggested solution to the problem is to use an extension of the SABR model, known
as the shifted SABR. The extended model adds a positive shift to the negative rate, so
that the shifted process is modeled by the ordinary SABR. In our study we use three
dierent methods to show how to calibrate the shifted SABR model to cap volatilities
based on 6 months EURIBOR. Nonetheless, market standard is to quote a mixture of
cap volatilities, thus the volatilities have to be transformed to a common tenor before
calibration. To this end we have developed a technique for volatility transformation in
a multi curve framework when rates are negative. The concept is derived by applying
It^o's formula on a arbitrage relation between shifted forward rates2.
The results show that two of the methods for calibrating the shifted SABR model have a
good t to liquid contracts. Though, the methods vary in performance capturing cheap
caps which are out of the money. Our developed volatility transformation technique also
works well, and the sensitivity to the potentially unknown correlation between forward
rates is low.
The implications of the study are that the extension of the SABR model to the shifted
SABR model works ne, both in terms of pricing of caps and volatility transformation
in a multi curve framework. Although, it comes to the cost of some additionally complexity
of extending formulas and arbitrage relations, that used to hold in a positive rate
environment.

• author: Jönsson, Mattias and Såmark, Ulrica
• supervisor: Magnus Wiktorsson
• organization: Mathematical Statistics
• course: FMS820 20162
• year: 2016
• type: H2 - Master's Degree (Two Years)
• subject: Mathematics and Statistics
• keywords: Multi Curve, Volatility Transformation, Negative Rates, shifted SABR, non-standard Tenor, Caps, EUR Market
• language: English
• id: 8872943
• date added to LUP: 2016-05-18 10:52:27
• date last changed: 2016-05-18 10:52:27