When investors buy fixed income securities and financial derivatives, how can they estimate the “expected returns” they hope to earn over short periods like one month or six months? How can they measu

When investors buy fixed income securities and financial derivatives, how can they estimate the “expected returns” they hope to earn over short periods like one month or six months? How can they measure the riskiness of these returns? With so many Wall Street wizards with doctorates in mathematics and nuclear physics, one would think that answers to such questions should be relatively straightforward.

“It’s not so easy,” says Isenberg Finance Professor Sanjay Nawalkha, whose most recent paper—published in the October issue of The Journal of Finance—provides the first rigorous theoretical framework for the derivation of formulas for the expected returns and risk measures for fixed income securities and financial derivatives, which are worth more than $132 trillion in combined valuation globally. This amount exceeds the valuation of the total global equity market by at least $30 trillion.

His work—coauthored with Xiaoyang Zhuo, at the Beijing Institute of Technology—provides a significant generalization of the Nobel Prize–winning options formulas of Black and Scholes (1973) and Merton (1973), both of which are jointly referred to as the Black-Merton-Scholes (BMS) model. The BMS model allows the valuation of an option, a type of investment that allows the investor to purchase an asset for a fixed price at some future date (for example, stock options can be offered as part of an employment compensation package). Nawalkha and Zhuo’s broader framework generalizes not only these two widely used option formulas, but also the valuation formulas of a vast number of fixed income securities and financial derivatives in order to compute their expected returns and higher-order moments, such as variance, skewness, etc.

The questions regarding valuation, expected return, and risk are the most important ones in finance. The 5 Nobel Prizes in Economic Sciences that have gone to finance researchers since 1990 have been credited to 14 path-breaking papers, and all except one out of these address valuation, risk-return analysis, and market efficiency issues related to financial securities.

Nawalkha and Zhuo’s innovation has the potential to create a new investment machinery that will help many Wall Street practitioners and academic researchers estimate the risk and returns of a plethora of financial securities. “For example, if this machinery was available in, say, 2005, there is some chance that collateralized debt obligation (CDO) fund managers would have been aware of the highly negative ‘expected returns’ in this market, which could have slowed down the bubble created in the housing market in 2008,” says Nawalkha.

A New Hybrid Probability Measure

“This was not an easy paper to write since we were introducing not just another fancy valuation model in finance but a new probability measure that applies to hundreds of existing valuation models in finance,” says Nawalkha. Until Nawalkha and Zhuo’s discovery, every financial model used either the real-world probability measure or the risk-neutral probability measure Q ​(and its variants) derived by Black-Merton-Scholes in 1973. After almost 50 years, Nawalkha and Zhuo have created a new hybrid probability measure R ​, which together with its variants, subsumes both the physical measure and the risk-neutral measure (and its variants). Using their new framework, one can derive not only the valuation formulas but also the formulas for expected returns and higher-order moments over an arbitrary holding period for virtually all financial securities.

“The task seemed daunting when we first discovered the new R ​ measure in the spring of 2017,” Nawalkha says. “I had to meditate deeply on how to write a paper that would be a significant extension of literally hundreds of very widely cited papers, as a single all-encompassing paper.” He took a sabbatical in spring 2018 to focus his full attention on the paper. Months and years flew by as Nawalkha and his collaborator Zhuo painstakingly derived expected return formulas for a wide range of financial securities.

“Often the formulas were so complex that we needed to test the final solutions using numerical programming to ensure that there were no typos or mistakes,” said Zhuo.  

Nawalkha and Zhuo’s probability measure R ​, with its variants, applies to all fixed income securities and financial derivatives, including:

  • Treasury bonds, corporate bonds, mortgage bonds, and all other types of fixed income securities;
  • All types of insurance claims;
  • All financial derivatives including equity options, bond options, credit default swaps (CDSs) and CDOs, commodity options, volatility options, exchange rate options, interest rate swaps, interest rate caps and swaptions, currency swaps, futures contracts, and other financially innovative and exotic derivative products that are priced using the risk-neutral measure.

Response from Peers

When Nawalkha and Zhuo finished the manuscript in 2020, they sent it to Darrell Duffie, the Adams Distinguished Professor of Management and Professor of Finance at Stanford, who is one of the most cited professors in finance with more than 50,000 Google Scholar citations. Duffie reacted positively to the work.

That fall, Nawalkha presented the paper at the annual conference of the Center for International Securities and Derivatives Markets (CISDM), which took place virtually. Duffie, who also presented a paper (on the unpredictability of treasury bonds during the pandemic), shared his insights on Nawalkha and Zhuo’s paper with the conference participants, saying, “This paper was a breath of fresh air when Sanjay sent it to me a few months ago. I was kind of blown away by the fact that I hadn’t realized that I could do this for so many years, when I was doing all these kinds of calculations manually, kind of plugging coefficients in the underlying processes to compute everything. And now all I have to do is change to a nice measure and derive the behavior of the stochastic processes of interest under that new measure, and then just go for it.” Duffie ended his laudatory comments by saying, “I wanted to make it crystal clear that this is a big machine. It’s just waiting for applications.”

Another CISDM panelist, Peter Carr, a world-renowned expert on financial derivatives, who was at the time the chair of the Finance and Risk Engineering Department at NYU’s Tandon School of Engineering, also offered comments on the paper at the conference, discussing precedents to the new model and suggesting new uses for it. A few weeks after the CISDM conference, Carr—who passed away in 2022—hosted Nawalkha as part the Tandon School’s Brooklyn Quant Experience Lecture Series.

Other experts in the field have also praised the work. Sanjiv Das, the William and Janice Terry Professor of Finance and Business Analytics at the Santa Clara University Leavey School of Business, who wrote a book on derivatives, emailed Nawalkha to say that the paper “looks amazing.” George Constantinides, the Leo Melamed Professor of Finance at the University of Chicago Booth School of Business and a former president of the American Finance Association and of the Society for Financial Studies, emailed Nawalkha to say, “Great paper.”

Mila Getmansky Sherman, Nawalkha’s colleague at Isenberg, where she is a finance professor and the Judith Wilkinson O’Connell Faculty Fellow, wrote, “Congratulations on making a fundamental contribution to the finance theory. That only happens once in several decades, and this paper will be definitely instrumental for future research.”

The paper was accepted very quickly by the Journal of Finance, with only one revision. “I think the fast acceptance was because of Darrell’s very positive review of the paper at the 2020 CISDM conference, in which he served as the main discussant of the paper,” Nawalkha says.  

Nawalkha hopes that his work will affect not only academic research in the field, but also investment decisions at major financial institutions (mutual funds, insurance companies, pension funds, and investment banks) related to their portfolios of fixed income securities and financial derivatives.





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