Research


All AMMs are CFMMs.  All DeFi markets have invariants.  A DeFi market is arbitrage-free if and only if it has an increasing invariant 

October 2023 version.


EMA-type Trading Strategies Maximize Utility under Partial Information

Frontiers of Mathematical Finance (2023) vol 2 issue 1, 124 - 140. 

Joint with X Chen. 


Semi-robust Replication of Barrier-style Claims on Price and Volatility

Applied Mathematical Finance (2022) vol 28 issue 6, 534-559.

Joint with P Carr and M Lorig.


Robust Replication of Volatility and Hybrid Derivatives on Jump-Diffusions

Mathematical Finance (2021) vol 31 issue 4, 1394-1422.

Joint with P Carr and M Lorig.


Variance Swaps on Time-Changed Markov Processes

SIAM Journal on Financial Mathematics (2021) vol 12 issue 2. 

Joint with P Carr and M Lorig.


How Leverage Shifts and Scales a Volatility Skew: Asymptotics for Continuous and Jump Dynamics

Joint with R Wang.


Asymptotics of Implied Volatility to Arbitrary Order

Finance and Stochastics (2014) vol 18 issue 2, 349-392.

Joint with K Gao.


Variation and Weighted Variation Swaps on Time-Changed Levy Processes

Finance and Stochastics (2013) vol 17 issue 4, 685-716.

Joint with P Carr.


The Small-Time Smile and Term Structure of Implied Volatility Under the Heston Model

SIAM Journal on Financial Mathematics (2012) vol 3, 690-708.

Joint with M Forde and A Jacquier.


Variance Swaps on Time-Changed Levy Processes

Finance and Stochastics (2012) vol 16 issue 2, 335-355.

Joint with P Carr and L Wu.


Displaced Lognormal Volatility Skews: Analysis and Applications to Stochastic Volatility Simulations

Annals of Finance, vol 8 issue 2 (2012), 159-181.

Joint with D Wang.


Volatility Derivatives

Annual Review of Financial Economics, vol 1 (2009), 319-339.

Joint with P Carr.


Weighted Variance Swap, Corridor Variance Swap, Gamma Swap, Realized Volatility Options

in Encyclopedia of Quantitative Finance (2010).


Hedging Variance Options on Continuous Semimartingales

Finance and Stochastics, vol 14 issue 2 (2010), 179-207.

Joint with P Carr.


Put-Call Symmetry: Extensions and Applications

Mathematical Finance, vol 19 issue 4 (2009), 523-560.

Joint with P Carr.


Robust Replication of Volatility Derivatives

PRMIA award for Best Paper in Derivatives, MFA 2008 Annual Meeting.

Joint with P Carr.


Realized Volatility and Variance: Options via Swaps including unpublished appendices

RISK, vol 20 issue 5 (2007), 76-83.

Joint with P Carr.


The Moment Formula for Implied Volatility at Extreme Strikes

Mathematical Finance, vol 14 issue 3 (July 2004), 469-480.


Option Pricing by Transform Methods: Extensions, Unification, and Error Control (Feb 05 version)

Journal of Computational Finance, vol 7 issue 3 (Spring 2004), 51-86.


Implied Volatility: Statics, Dynamics, and Probabilistic Interpretation

In Recent Advances in Applied Probability, Springer (2005).


Implied and Local Volatilities under Stochastic Volatility

International Journal of Theoretical and Applied Finance, vol 4 issue 1 (2001) 45-89.