# Newsletter of the Bachelier Finance Society

## Volume 8, Number 4, September 2016

**JOB POSTINGS**

The aim of these postings is to create a forum for the dissemination of information on academic and industrial positions related to mathematical finance, across different disciplines and different geographical regions. Please submit any job advertisements you are aware of to: jobads@bachelierfinance.org, preferably in plain text and sending the link to the website containing all the information. Updates and new items appear continuously at: http://www.bachelierfinance.org/forum/jobs/.

**Lecturer in Mathematical Finance**

Oxford University

Deadline: September 19, 2016

**Professorship (W3)**

Technische Universität Dortmund

Deadline: September 22, 2016

**Juniorprofessur (W1)**

Universität Mannheim

Deadline: September 30, 2016

**Two Fellowships**

Radboud University

Deadline: September 30, 2016

**Chair of Quantitative Financial Risk**

Heriot-Watt University

Deadline: October 2, 2016

**PostDocs**

Heidelberger Institut für Theoretische Studien

Deadline: October 23, 2016

**Associate Editor**

Springer-Verlag GmbH

**LEADING ARTICLE**

### Negative Rates: The Challenges from a Quant Perspective

#### Fabio Mercurio*

##### Introduction

There are many instances in the past and recent history where Treasury bills and even sovereign bonds happened to trade at a negative yield. For instance: i) in November 1998, the yield of Japan's six-month Treasury bills fell to minus 0.004 percent; ii) in November 2009, 3-month US T-Bills were trading at minus 0.03 percent after market supply shrunk. Traders were so eager to carry healthy assets in their books that they were willing to pay an extra premium for that; iii) in February 2016, the Japanese government sold 2.2 trillion yen of bonds at an average yield of minus 0.024 percent, followed in July by the German government that issued ten-year bonds whose yield recorded a historic low of minus 0.05 percent. Also in July, the yields of all Swiss government bonds up to fifty years turned negative, with the one-year bond yield falling as low as about minus 1 percent.

The interest rates in the interbank money and LIBOR markets for major currencies also ventured into negative territory. Central banks pushed rates below zero and started charging a fee on the deposits banks held with them, in an effort to incentivize banks to lend money and stimulate economic growth. Not only the overnight rates but even the LIBORs went negative. In Europe, the first currency to experience negative rates was CHF in August 2011, followed by DKK in July 2012, EUR in August/September 2014 and SEK in February 2015. In Asia, the SGD six-month LIBOR plummeted to near minus 1 percent in August 2011.

##### Practical issues in interest-rate modeling

The appearance of negative rates in the market did not urge the creation of new dynamical models. Paradoxically, the main interest-rate models used in the industry were already based on either the Gaussian or the Shifted-Lognormal (SL) distribution, both having negative real intervals in their support. Think for instance of the Hull-White (1990) one-factor risk-neutral dynamics of the short rate, or the SL LIBOR market model. What used to be a drawback of these models, suddenly turned into an advantage, and in fact became a necessary requirement to accommodate market data [1].

That being said, the transition to a negative-rate regime has not been harmless, and several methodologies have been impacted by negative rates. I analyze them in the following along with the challenges quants had to face to adapt their models to the new environment.

**Yield-curve construction: **A typical no-arbitrage condition that was imposed in yield-curve constructions was that forward rates had to be positive, or equivalently that discount factors had to be smaller than 1 and decreasing for increasing maturities. Under negative rates, these constraints are no longer needed and have to be removed from pricing routines. Stripped forwards are allowed to be negative and discount factors can be larger than 1 or increasing in maturity for a period of time.

Removing the positivity constraint can also have implications in the choice of an optimal curve-interpolation method. For instance, a monotonic cubic spline that enforces, by construction, positivity of forward rates can no longer be the preferred solution. At the same time, other types of splines that allow for negative values, still need careful implementation and fine tuning to make sure negative forwards do not appear where they should not.

**Volatility quotes: **The wide-spread convention for volatility quotes was to use Black volatilities. The Black volatility for a given swaption is defined as the unique value of the volatility parameter to plug into Black's swaption formula to match the corresponding market price. The problem with Black's formula is that it is undefined (if we want prices to be real numbers, and we do!) when either the swap rate or the strike are negative or zero. So, when the swap rate is positive (resp. negative), Black's volatility can not be calculated for a negative (resp. positive) strike.

The market circumvented this limitation by using either normal or SL volatilities. A normal (resp. SL) volatility is the unique volatility parameter to plug into Bachelier's (resp. shifted Black's) formula to match the corresponding market price. Brokers are currently quoting both normal and SL volatilities with their associated shifts for currencies such as EUR, CHF, SEK, JPY and DKK, along with their (forward) premiums. While normal volatilities are unambiguously defined, SL volatilities must be associated with a shift parameter. A typical shift parameter is 2 percent, but different shifts may be used for different currencies or pairs of maturities and tenors.

Traders have been using normal volatilities for a long time, even before the financial crisis. In fact, besides being defined for negative rates as well, normal volatilities and Bachelier prices present the following advantages: i) normal volatilities tend to be much more stable than Black volatilities, which are prone to large intraday fluctuations because of their sensitivity to variations of the underlying swap rates; ii) Bachelier prices are symmetric with respect to the ATM strike in the following sense. Consider a receiver and a payer swaption with the same maturity and tenor, the first with strike equal to ATM minus *x* bp, the second with strike equal to ATM plus *x* bp. If these two swaptions have the same normal volatility then they also have the same (Bachelier) price. This is not true for Black volatilities, which is a problem because swaption smiles are quoted in terms of absolute moneyness, that is, the difference between strike and ATM level.

Normal or SL volatilities, supplemented with standard interpolation/extrapolation techniques, can then be used to complete a volatility cube, filling in, in particular, missing quotes at negative strikes.

**Smile construction: **A standard approach used by the market for building swaption volatility cubes was based on the SABR functional form, which relies on the assumption of a positive distribution for the underlying swap rate. To deal with negative rates, practitioners then decided to move to a shifted SABR model, which is defined by adding a negative shift to the initial SABR stochastic-volatility process. Accordingly, the shifted SABR functional form is obtained from SABR simply by shifting swap rate and strike. Another advantage of using a shifted SABR form is that a negative shift reduces the chances of arbitrage at low strikes, because it essentially moves the problematic region further down to negative strikes. The shift parameter in shifted-SABR can either be calibrated to market quotes or given exogenously.

Alternatively, one can build a swaption cube using the recent free-boundary SABR extension of Antonov, Konikov, and Spector (2015), or a simpler normal-mixture model, where swaption prices are obtained as linear convex combination of Bachelier prices.

**Positivity constraints in the codebase:** Because of negative rates, existing pricing code in a quant library may break and return errors. This can happen essentially for two reasons: i) a mathematical operation requires rates or strikes to be positive to return a real number, as is the case for the log in Black's swaption formula; ii) rates were constrained to be positive to reduce operational risk, preventing the user from entering a negative value by mistake, as could be the case for an equity option pricer. Therefore, code changes must be introduced at different levels to either replace an existing pricing function with a different one, or to remove unnecessary constraints.

**Collateral agreements: **A Credit Support Annex (CSA) is a document annexed to the ISDA agreement signed by two counterparties, which specifies the rules for collateral posting (type, currency, frequency, asymmetries, thresholds, etc). One of these rules is that the collateral posted by the party with negative NPV to the party with positive NPV must be remunerated at a rate specified by the CSA. In the case of cash collateral, the collateral rate is typically the OIS rate in the collateral currency.

If the OIS rate turns negative, then the party posting collateral will receive a negative interest rate for it, meaning that they will pay an interest rate equal to the absolute value of the OIS rate. This seems to be the prevailing agreement. However, there may be CSAs, for instance in Japan, where the collateral rate is floored at zero, so the collateral rate is equal to the positive part of the OIS rate. This introduces an extra optionality in the valuation of deals subject to that CSA. When OIS rates are high enough, the collateral option has very little value and can be neglected. However, when rates are low or even negative, the collateral option has suddenly a non-negligible value, which nonetheless may be hard to quantify.

**Stress tests: **It is a common risk-management practice, also urged by regulators, to value a bank's portfolios under stressed market conditions. This entails the definition of stress-test scenarios, which are based on large market moves of some of the underlying risk factors, including interest rates. Assuming scenarios with negative rates has become mandatory even for economies where interest rates are still positive. It has also been suggested by the FED, which asked banks to consider the possibility of negative rates happening in the US as well.

In all these cases, the question is always the same: how low can interest rates go? After breaching the zero boundary, there is no other economically-meaningful lower barrier. Historically, minus 1 percent is the lowest value ever reached. But, in theory, an *x*-month LIBOR can go as low as about minus 1200/*x* percent.

**Initial margins: **CCP's initial-margin models are based on a historical VaR approach. For interest-rate products, a history of rate returns is used for the margin calculations. Typically, CCPs used either absolute returns or log-returns. However, with the advent of negative rates, log-returns have been replaced by shifted log returns. This has the additional advantage of better capturing the historical distribution of rate returns. In fact, rate returns tend to be "more" normal when rates are low, and "more" lognormal when rates are high. The shift parameter is typically assigned exogenously, and a typical value is 4 percent.

##### Conclusions

Several quantitative methods for pricing and risk management are affected by the new negative rate environment. In this article, I present some of the modeling anomalies that arose because of negative rates, and the ways quants have adopted to address them.

________

* Global Head of Quant Analytics, Bloomberg LP. The views and opinions expressed in this article are my own and do not represent the opinions of any firm or institution.

[1] Interest-rate models had to be upgraded but for a different reason. They had to be adapted to the new multi-curve environment, which emerged after the financial crisis.

Antonov A., Konikov M. and Spector M. (2015). The free boundary SABR: Natural extension to negative rates. *Risk*, September 2015.

Hull J., White A. (1990). Pricing interest-rate-derivative securities. *Review of Financial Studies* 3(4), 573-592.

**BOOK REVIEW**

### Contract Theory in Continuous-Time Models

#### by Jakša Cvitanić and Jianfeng Zhang

#### Springer, 2013

If mathematicians were writing fairytales, this book would be the story of Principal and Agent, their numerous misunderstandings, and their strenuous and harsh journey to find common ground to work happily ever after. Unfortunately (really?) they do not, but this does not alter the fact that this essay constitutes an excellent introduction to the mathematics of contracting theory. The basic situation studied in the book corresponds to two economic agents, who do not necessarily share the same information, and who wish to find out what could be the optimal way to structure a contract between the two of them. At its heart lie therefore two fundamental and pervasive aspects in economic relationships: incentives and asymmetry of information.

Contract theory is actually an old subject in the economics literature, dating back to the 70s, and several monographs written by renowned economists already give a comprehensive account of the discrete–time and static theory. The present manuscript however, is the first covering the progress that has been achieved, notably by the two authors, between the late 80s and the beginning of the 21st century on the continuous–time version of the problem. As is usually the case, the continuous–time formulation, though losing in realism, opens the way to elegant and general formulations using the tools of stochastic calculus that should be familiar to the vast majority of the readers of this review, that is to say in a nutshell: stochastic control, dynamic programming principle, Pontryagin maximum principle, duality theory, and even the infamous backward stochastic differential equations (BSDEs).

An important point that needs to be pointed out is that despite the fundamentally technical nature of the book, the difficulty increases progressively, and the authors have made tremendous efforts to illustrate every aspect of the book with clear and interesting economic applications, often borrowed from the literature, but revisited through their own perspective. It gives, in my view, a clear picture of what the theory can potentially achieve in terms of applications, and which is: a lot!

The book, after an introductory chapter which presents the three different incarnations of the problem at hand in simple examples, goes on to study each of them separately. Part II is devoted to the so–called *first best* case, corresponding to the situation where Principal and Agent actually have access to the same information, and simply wish to share risks between the two of them. Part III is the longest and probably the most interesting of the book. It is concerned with the *second best* where the actions of Agent are hidden to Principal, leading to possible moral hazard, in the sense that Agent may take decisions which are not in the best interest of Principal. As in Part II, the authors give a general approach to tackle the problem using the Pontryagin maximum principle and the associated systems of fully coupled forward backward stochastic differential equations (FBSDEs), which is then illustrated at length in Chapters 6 and 7. Part IV would have the potential to top Part III in terms of interest, as it considers the even more realistic setting where Principal does not have access to both the actions as well as key characteristics of Agent, the so–called *third best* or hidden type. However it ends up being shorter and may leave the readers with a sensation of incompletion, as it treats a much less general setting than the two previous parts. Notwithstanding, this case, also known as the screening problem in the economics literature, is actually far harder than the previous ones, and to this date there does not exists a general theory, so that we are indeed getting the best that is currently known. Finally, Part V is somehow a technical appendix of what is needed to understand the general theory of the previous parts and could be summed up as: everything you always wanted to know about BSDEs and FBSDEs (but were afraid to ask).

Let me almost conclude this review with a word of caution. Some readers might be rebutted by the fact that the general FBSDEs derived by the authors to characterize optimal contracts are not known to have solutions in such a broad setting, and this can only be checked on a case by case basis. In my view this just means that this book should be understood as a comprehensive *modus operandi* explaining how any particular model you could come up with can indeed be tackled using the present method. And it certainly succeeds in doing so.

Overall, this manuscript is an excellent introduction to contract theory in continuous time, both for researchers and students, that can most certainly be used as a basis for advanced courses on the subject. It should definitely be the bedside book of anyone interested in these problems. Even more than that, this is the first take on this subject that has been written by mathematicians, in the language our community is the most accustomed to, thus lifting in part the veil of mystery that is usually shrouding the economics literature. I am convinced that it will help to spread this theory and the still numerous open problems associated to it among the financial mathematics academics.

**UPCOMING Conferences**

This list contains conferences related to mathematical finance that take place in the next three months. A full list is available at: http://www.bachelierfinance.org/congresses/conferences.html. Please let us know of conferences we are not aware of and include a URL for the event.

**3rd European Actuarial Journal (EAJ) Conference**

September 5–8, 2016, Lyon, France

**Enlargement of filtration and financial applications**

September 8–9, 2016, Zürich, Switzerland

**Boston University Financial Econometrics Conference**

September 10, 2016, Boston MA, USA

**Vienna Congress on Mathematical Finance - VCMF 2016**

September 12–14, 2016, Vienna, Austria

**Summer School: Statistical Modeling of Complex Systems and Processes**

September 12–16, 2016, Heidelberg, Germany

**Workshop on Extremes and Risks in Higher Dimensions**

September 12–16, 2016, Leiden, The Netherlands

**US Fundamental Review of the Trading Book Summit**

September 13–14, 2016, New York NY, USA

**Vienna Congress on Mathematical Finance - VCMF Educational Workshop**

September 15–16, 2016, Vienna, Austria

**Salzburg Workshop on Dependence Models and Copulas**

September 19–21, 2016, Salzburg, Austria

**Set Optimization for Applications**

September 19–23, 2016, Vienna, Austria

**High-Frequency Trading - Curse or Blessing?**

September 22–23, 2016, Vienna, Austria

**7th CEQURA Conference**

September 26–27, 2016, Munich, Germany

**2nd Interest Rate Risk in the Banking Book (IRRBB) Summit**

September 28, 2016, London, United Kingdom

10% Discount for members – Code: FKM63397BFS

**3rd London - Paris Bachelier Workshop on Mathematical Finance**

September 29–30, 2016, Paris, France

**UNISActuarial SCHOOL**

October 25–27, 2016, Salerno, Italy

**V-FI Americas**

October 26–28, 2016, New York NY, USA

10% Discount for members – Code: FKM63368BFS

**Risk Measures, XVA Analysis, Cost of Capital & Central Counterparties Workshop**

October 27–28, 2016, Shanghai, China

**Swissquote Conference 2016 on the Future of Banking**

November 4, 2016, Lausanne, Switzerland

**Model Uncertainty and Robust Finance**

November 10–11, 2016, Milan, Italy

**Quant Risk Management**

November 15–16, 2016, London, United Kingdom

**SIAM Conference on Financial Mathematics (FM16)**

November 17–19, 2016, Austin, Texas, USA

**Ninth Workshop on High Performance Computational Finance (WHPCF)**

November 18, 2016, Salt Lake City UT, USA