Derivatives Pricing

A Path Integral Approach for Time-Dependent Hamiltonians with Applications to Derivatives Pricing ArXiv ID: 2408.02064 “View on arXiv” Authors: Unknown Abstract We generalize a semi-classical path integral approach originally introduced by Giachetti and Tognetti [“Phys. Rev. Lett. 55, 912 (1985)”] and Feynman and Kleinert [“Phys. Rev. A 34, 5080 (1986)”] to time-dependent Hamiltonians, thus extending the scope of the method to the pricing of financial derivatives. We illustrate the accuracy of the approach by presenting results for the well-known, but analytically intractable, Black-Karasinski model for the dynamics of interest rates. The accuracy and computational efficiency of this path integral approach makes it a viable alternative to fully-numerical schemes for a variety of applications in derivatives pricing. ...

Basket Options with Volatility Skew: Calibrating a Local Volatility Model by Sample Rearrangement ArXiv ID: 2407.02901 “View on arXiv” Authors: Unknown Abstract The pricing of derivatives tied to baskets of assets demands a sophisticated framework that aligns with the available market information to capture the intricate non-linear dependency structure among the assets. We describe the dynamics of the multivariate process of constituents with a copula model and propose an efficient method to extract the dependency structure from the market. The proposed method generates coherent sets of samples of the constituents process through systematic sampling rearrangement. These samples are then utilized to calibrate a local volatility model (LVM) of the basket process, which is used to price basket derivatives. We show that the method is capable of efficiently pricing basket options based on a large number of basket constituents, accomplishing the calibration process within a matter of seconds, and achieving near-perfect calibration to the index options of the market. ...

Gas Fees on the Ethereum Blockchain: From Foundations to Derivatives Valuations ArXiv ID: 2406.06524 “View on arXiv” Authors: Unknown Abstract The gas fee, paid for inclusion in the blockchain, is analyzed in two parts. First, we consider how effort in terms of resources required to process and store a transaction turns into a gas limit, which, through a fee, comprised of the base and priority fee in the current version of Ethereum, is converted into the cost paid by the user. We adhere closely to the Ethereum protocol to simplify the analysis and to constrain the design choices when considering multidimensional gas. Second, we assume that the gas price is given deus ex machina by a fractional Ornstein-Uhlenbeck process and evaluate various derivatives. These contracts can, for example, mitigate gas cost volatility. The ability to price and trade forwards besides the existing spot inclusion into the blockchain could enable users to hedge against future cost fluctuations. Overall, this paper offers a comprehensive analysis of gas fee dynamics on the Ethereum blockchain, integrating supply-side constraints with demand-side modelling to enhance the predictability and stability of transaction costs. ...

A weighted multilevel Monte Carlo method ArXiv ID: 2405.03453 “View on arXiv” Authors: Unknown Abstract The Multilevel Monte Carlo (MLMC) method has been applied successfully in a wide range of settings since its first introduction by Giles (2008). When using only two levels, the method can be viewed as a kind of control-variate approach to reduce variance, as earlier proposed by Kebaier (2005). We introduce a generalization of the MLMC formulation by extending this control variate approach to any number of levels and deriving a recursive formula for computing the weights associated with the control variates and the optimal numbers of samples at the various levels. We also show how the generalisation can also be applied to the \emph{“multi-index”} MLMC method of Haji-Ali, Nobile, Tempone (2015), at the cost of solving a $(2^d-1)$-dimensional minimisation problem at each node when $d$ index dimensions are used. The comparative performance of the weighted MLMC method is illustrated in a range of numerical settings. While the addition of weights does not change the \emph{“asymptotic”} complexity of the method, the results show that significant efficiency improvements over the standard MLMC formulation are possible, particularly when the coarse level approximations are poorly correlated. ...

A Unifying Approach for the Pricing of Debt Securities ArXiv ID: 2403.06303 “View on arXiv” Authors: Unknown Abstract We propose a unifying framework for the pricing of debt securities under general time-inhomogeneous short-rate diffusion processes. The pricing of bonds, bond options, callable/putable bonds, and convertible bonds (CBs) is covered. Using continuous-time Markov chain (CTMC) approximations, we obtain closed-form matrix expressions to approximate the price of bonds and bond options under general one-dimensional short-rate processes. A simple and efficient algorithm is also developed to price callable/putable debt. The availability of a closed-form expression for the price of zero-coupon bonds allows for the perfect fit of the approximated model to the current market term structure of interest rates, regardless of the complexity of the underlying diffusion process selected. We further consider the pricing of CBs under general bi-dimensional time-inhomogeneous diffusion processes to model equity and short-rate dynamics. Credit risk is also incorporated into the model using the approach of Tsiveriotis and Fernandes (1998). Based on a two-layer CTMC method, an efficient algorithm is developed to approximate the price of convertible bonds. When conversion is only allowed at maturity, a closed-form matrix expression is obtained. Numerical experiments show the accuracy and efficiency of the method across a wide range of model parameters and short-rate models. ...

Quasi-Monte Carlo with Domain Transformation for Efficient Fourier Pricing of Multi-Asset Options ArXiv ID: 2403.02832 “View on arXiv” Authors: Unknown Abstract Efficiently pricing multi-asset options poses a significant challenge in quantitative finance. Fourier methods leverage the regularity properties of the integrand in the Fourier domain to accurately and rapidly value options that typically lack regularity in the physical domain. However, most of the existing Fourier approaches face hurdles in high-dimensional settings due to the tensor product (TP) structure of the commonly employed numerical quadrature techniques. To overcome this difficulty, this work advocates using the randomized quasi-MC (RQMC) quadrature to improve the scalability of Fourier methods with high dimensions. The RQMC technique benefits from the smoothness of the integrand and alleviates the curse of dimensionality while providing practical error estimates. Nonetheless, the applicability of RQMC on the unbounded domain, $\mathbb{“R”}^d$, requires a domain transformation to $[“0,1”]^d$, which may result in singularities of the transformed integrand at the corners of the hypercube, and hence deteriorate the performance of RQMC. To circumvent this difficulty, we design an efficient domain transformation procedure based on boundary growth conditions on the transformed integrand. The proposed transformation preserves sufficient regularity of the original integrand for fast convergence of the RQMC method. To validate our analysis, we demonstrate the efficiency of employing RQMC with an appropriate transformation to evaluate options in the Fourier space for various pricing models, payoffs, and dimensions. Finally, we highlight the computational advantage of applying RQMC over MC or TP in the Fourier domain, and over MC in the physical domain for options with up to 15 assets. ...

Risk-Neutral Probabilities Explained ArXiv ID: ssrn-1395390 “View on arXiv” Authors: Unknown Abstract All too often, the concept of risk-neutral probabilities in mathematical finance is poorly explained, and misleading statements are made. The aim of this paper Keywords: risk-neutral probabilities, martingales, stochastic calculus, derivatives pricing, Quantitative Finance Complexity vs Empirical Score Math Complexity: 7.0/10 Empirical Rigor: 2.0/10 Quadrant: Lab Rats Why: The paper focuses on theoretical foundations, including continuous-time stochastic processes like geometric Brownian motion and martingales, but lacks any empirical backtesting, data, or implementation details. flowchart TD A["Research Goal: Explain Risk-Neutral Probabilities clearly"] --> B["Methodology: Critical Review of Stochastic Calculus"] B --> C["Input: Misleading Statements in Texts"] C --> D["Computational Process: Martingale Measure Derivation"] B --> E["Input: Derivatives Pricing Models"] E --> D D --> F["Key Finding: Q-Measure vs. P-Measure"] D --> G["Key Finding: No-Arbitrage Pricing Framework"]

Discrete TimeFinance ArXiv ID: ssrn-976589 “View on arXiv” Authors: Unknown Abstract These are my Lecture Notes for a course in Discrete Time Finance which I taught in the Winter term 2005 at the University of Leeds. I am aware that the notes ar Keywords: Discrete Time Finance, Derivatives Pricing, Risk Management, Stochastic Calculus, Derivatives Complexity vs Empirical Score Math Complexity: 8.5/10 Empirical Rigor: 1.0/10 Quadrant: Lab Rats Why: The content is heavily theoretical, focused on rigorous mathematical derivations and proofs common in academic finance courses, while there is no mention of data, backtests, or practical implementation. flowchart TD A["Research Goal: Pricing & Hedging in<br>Discrete Time Models"] --> B["Key Inputs: Probability Space,<br>Adapted Processes, Filtration"] B --> C["Methodology: Dynamic Programming<br>& Martingale Representation"] C --> D["Computational Process:<br>Recursive Pricing Algorithms"] D --> E["Key Outcome 1: Fundamental<br>Theorem of Asset Pricing"] D --> F["Key Outcome 2: Optimal<br>Discrete Hedging Strategies"]