Research desk
Derivatives & Volatility
Options, volatility and rates: pricing, hedging and the free boundary.
The desk prices and hedges claims on traded assets, from the foundations — arbitrage, risk-neutral measures, the benchmark approach — to the hard cases: path-dependent and game options such as the Russian and Israeli options, solved by HJB equations and reflected backward SDEs. Its volatility work includes Wishart stochastic volatility linked to order-book dynamics; its rates work, the Langevin–Zamrik framework for inertial interest rates, with bond pricing, HJM forward-rate dynamics and forward-rate derivatives. It also explains products whose behaviour surprises their buyers, such as leveraged ETFs.
Four Hundred Dollars or Nine Cents
A dollar in TQQQ, the 3x Nasdaq-100 fund, became about 408 dollars between 2010 and 2026 while the plain fund made 20. The same daily-reset machine run on the S&P 500 from 1998 to 2009 turned a dollar into 9 cents while the index, with…
The Man Who Could Have Won the Nobel Twice
In 1965 Paul Samuelson wrote down an option formula with two unknowns, and admitted he had no theory to pin them down. Set both to the interest rate and it is the Black–Scholes–Merton formula, term for term, eight years early. The missing piece was not…
Wishart Stochastic Volatility and the FTQR-Hawkes Bridge
We develop a Wishart stochastic volatility framework for the two-sided futures order book and establish its role as the correct continuum limit of the multivariate FTQR-Hawkes microstructure model. The variance state V(t) evolves as a 2 × 2 positive-definite matrix CIR process, capturing the full covariation v12(t) …
Kelly, the Growth-Optimal Portfolio, and the Benchmark Approach to Option Pricing
The Kelly criterion and Black-Scholes option pricing are the same mathematical object viewed from different angles. Both lead to the growth-optimal portfolio — Platen's benchmark — which prices derivatives under the real-world measure without requiring a risk-neutral one.
Langevin Dynamics and Why They Matter in Finance
The Langevin equation was written to describe a particle in a fluid. It turns out to be exactly the right language for interest rates, volatility, and forward curves.
Two Worlds, One Price: Entropy and the Risk-Neutral Measure
The change of measure at the heart of derivative pricing is an exponential tilt of the real-world probability measure — identical in structure to the Boltzmann distribution. The cost of that tilt is relative entropy.
Forward Rate Derivatives Pricing under the Langevin-Zamrik Model
We price interest rate derivatives — caplets, floorlets, bond options, and swaptions — under the Langevin-Zamrik (LZ) model, using its representation as a Gaussian Heath-Jarrow-Morton (HJM) model in forward-rate space. Since the LZ forward rate is normally distributed (not lognormal), caplets are priced exactly as…
Forward Rate Dynamics in the Langevin-Zamrik Framework: HJM Structure and Inertial SPDEs
This paper makes four contributions to the Langevin-Zamrik (LZ) interest rate model. (1) HJM identification: we show that LZ is a Gaussian Heath-Jarrow-Morton model with closed-form volatility σHJM(τ) = -(σ/m)B2'(τ), where no-arbitrage is automatic rather than imposed. (2) Three-regime volatility: the overdamped, critically damped, and underdamped…
Bond Pricing Under Inertial Interest Rate Dynamics: The Langevin-Zamrik PDE
We introduce the Langevin-Zamrik partial differential equation, a zero-coupon bond pricing equation derived from the full underdamped Langevin dynamics of the short interest rate. The state space is two-dimensional, comprising the short rate rt and its velocity vt = ṙt. For a quadratic (OU-type) potential the…
Russian Options: HJB and Reflected BSDE — Two Methods, One Price Surface
The Russian option, introduced by Shepp and Shiryaev (1993), is a perpetual American lookback contract whose payoff is the running maximum of the underlying asset. We price it via two independent frameworks: a Hamilton-Jacobi-Bellman (HJB) variational inequality and a singly reflected backward stochastic differential equation…
Robust Israeli Options with Finite Maturity: A Hamilton-Jacobi-Isaacs Approach under Drift Ambiguity
We study finite-maturity Israeli put options in the Black-Scholes framework when the writer faces Knightian uncertainty about the risk-neutral drift. Modelling this ambiguity as a bounded drift distortion controlled adversarially by the writer, the valuation problem becomes a three-player zero-sum game whose value satisfies a…
The Russian Option: Optimal Stopping on the Running Maximum and Peskir’s Maximality Principle
The Russian option, introduced by Shepp and Shiryaev (1993), is a perpetual American option whose payoff is the running maximum Mτ = sups ≤ τ Xs of the underlying stock price, discounted to the stopping time τ. We analyse the perpetual problem for a geometric Brownian motion with drift…
The Israeli Option: Doob-Meyer Decomposition and the Double Obstacle PDE
The Israeli option, introduced by Kifer (2000), is a financial contract in which both the holder and the writer possess the right to terminate the contract at any time, making it a zero-sum Dynkin game. We analyse the perpetual Israeli put on a geometric Brownian…