Paper presents an analytical solution to Merton Garman model using symmetries.
problem Developing an analytical solution to the Merton Garman model.
method Perturbation theory around an exact solution with Galilean symmetry.
result Perturbative solution performs well compared to Monte Carlo simulations.
Gauge symmetries explain the emergence of Merton-Garman equation from Black-Scholes in finance.
problem Understanding the emergence of Merton-Garman equation from Black-Scholes in financial markets.
method Using Hamiltonian formulation and gauge symmetry to derive the Merton-Garman equation from Black-Scholes, analyzing the role of stochastic volatility.
result Gauge symmetry explains the appearance of stochastic volatility and its massivation via the Higgs mechanism.
In the framework of path integral the evolution operator kernel for the Merton-Garman Hamiltonian is constructed. Based on this kernel option formula is obtained, which generalizes the well-known Black-Scholes result. Possible approximation numerical schemes for path integral calculations are proposed.
Local equivalence found between Black-Scholes and Merton-Garman equations.
problem Restoring local symmetry in stock prices under stochastic volatility.
method Exploring gauge field theory to show local equivalence.
result Black-Scholes and Merton-Garman equations are locally equivalent.
The generalized 5D Black-Scholes differential equation with stochastic volatility is derived. The projections of the stochastic evolutions associated with the random variables from an enlarged space or superspace onto an ordinary space can be achieved via higher-dimensional operators. The stochastic nature of the secur…
Study on spontaneous symmetry breaking in financial markets using quantum mechanics.
problem Analyzing spontaneous symmetry breaking in financial markets.
method Using Hamiltonian form of Black-Scholes and Merton-Garman equations, analyzing symmetry breaking and interpreting Nambu-Goldstone bosons.
result Interpretation of Nambu-Goldstone bosons in financial markets.
Intuitively, the default risk of a single borrower is higher when her or his assets and debt are denominated in different currencies. Additionally, the default dependence of borrowers with assets and debt in different currencies should be stronger than in the one-currency case. By combining well-known models by Merton …
Quantum mechanics models for financial Black-Scholes model.
problem Modeling financial derivatives using quantum mechanics.
method Noncommutative quantum mechanics applied to specific mechanical systems.
result Generalized noncommutative quantum mechanics of financial models.
Comparative statistical properties of Parkinson, Garman-Klass, Roger-Satchell and bridge oscillation estimators are discussed. Point and interval estimations, related with mentioned estimators are considered
The paper connects financial vacuum conditions to spontaneous symmetry breaking in quantum finance.
problem Understanding the conditions under which the martingale condition is a non-degenerate vacuum.
method Expressing financial equations in Hamiltonian form and analyzing symmetry breaking.
result Conditions for the martingale condition to be a non-degenerate vacuum are identified.
The paper challenges the notion that asset return doesn't affect Black-Scholes-Merton model.
problem The role of asset return in the Black-Scholes-Merton model.
method Refutation of the claim through simplified stochastic calculus approach.
result The expected rate of return of the underlying asset does affect the Black-Scholes-Merton model.
Study shows Merton model limits to Poisson process with log-normal intensity, improving default portfolio prediction.
problem Improving prediction of default portfolios using complex models.
method Applying Merton model with log-normal intensity function to Poisson process, discussing temporal correlation effects.
result Power decay model provides better generalization for long-term default portfolio data.
Developed Merton's model for public companies using observed liabilities.
problem Estimating default risk for public companies.
method Campbell and Shiller's approximation method for risk-neutral values and default probabilities.
result Formulas and ML estimators for public companies' default probabilities.
Researchers find a timing error in Black-Scholes-Merton option pricing model.
problem Timing error in Black-Scholes-Merton option pricing model.
method Discovered a timing mistake in Merton's 1971 model and showed misspecification in continuous and discrete time.
result Invalidates seminal contributions to the literature including Black-Scholes (1973) and Merton (1971).
Modeling exchange rates and options using entropic dynamics.
problem Modeling the dynamics of exchange rates and European options.
method Entropic Dynamics, entropic inference, scale invariance, logarithm of exchange rate.
result Derives the Geometric Brownian Motion and the Garman-Kohlhagen model for European options.
Develops Merton's model for private companies using DDM.
problem Lack of observable asset values for private companies.
method Uses dividend discount model (DDM) to develop structural model.
result Obtains closed-form formulas for equity and liability values, default probability.
Refining previously known estimates, we give large-strike asymptotics for the implied volatility of Merton's and Kou's jump diffusion models. They are deduced from call price approximations by transfer results of Gao and Lee. For the Merton model, we also analyse the density of the underlying and show that it features …
Solves Merton's investment-consumption problem with certainty equivalent approach.
problem Maximizing CRRA utility of consumption over time and investment mix.
method Identifies a certainty equivalent problem for the Merton problem, reformulates it as an SOCP, and applies it to model predictive control.
result The certainty equivalent problem can be solved as an SOCP, facilitating model predictive control.
In this article we consider affine generalizations of the Merton jump diffusion model [Merton, J. Fin. Econ., 1976] and the respective pricing of European options. On the one hand, the Brownian motion part in the Merton model may be generalized to a log-Heston model, and on the other hand, the jump part may be generali…
Unified model integrates Bachelier and Black-Scholes-Merton for asset pricing.
problem Study of asset pricing in a natural world with negative prices or riskless rates.
method Unified framework combining Bachelier and Black-Scholes-Merton models.
result Unified model shows different option pricing depending on riskless instruments used.
Paper solves Merton's portfolio problem in a non-Markovian, non-semimartingale model.
problem Merton's portfolio optimization in a fake stationary Volterra-Heston model.
method Stochastic factor solution to a Riccati BSDE, combined with martingale optimality principle.
result Derives semi-closed form optimal strategies and value function.
The study models credit risk using Merton's framework and binomial trees.
problem Credit risk pricing and implied volatility estimation.
method Calibrated using Merton's structural model, with asset volatility derived from Black-Scholes-Merton. Implied mean return and probability surfaces constructed using a recombining binomial tree.
result Established a practical method for constructing implied credit surfaces.
Optimizes dynamic investment portfolios with correlated jumps.
problem Maximizing expected terminal wealth in a multivariate Merton model with dependent jumps.
method Approximating CVaR with comonotonic bounds and maximizing expected terminal wealth.
result Improved optimization of dynamic investment portfolios.
Investigates optimal investment strategies in financial markets with jumps.
problem Optimal portfolio selection for investors in multi-asset financial markets with jumps.
method Uses martingale optimality principle and Riccati backward stochastic differential equations with jumps.
result Derives semi-closed form optimal strategies and value function for Merton's problem.
Solves the Merton investment-consumption problem using a new approach.
problem Infinite-horizon Merton investment-consumption problem in a constant-parameter Black-Scholes-Merton market.
method Simple and elegant argument involving a stochastic perturbation of the utility function.
result Overcomes complications in existing primal verification proofs.
In this letter, I consider the issue of pricing risky debt by following Merton's approach. I generalize Merton's results to the case where the interest rate is modeled by the CIR term structure. Exact closed forms are provided for the risky debt's price.
New optimal investment strategies for finance and insurance using Hawkes-based models.
problem Optimal investment strategies in finance and insurance for specific models.
method Solving Merton investment problems with Hawkes-based models.
result New optimal investment results for finance and insurance models.
This research improves option pricing models using Heston, GARCH, and jump diffusion models.
problem Inaccurate option pricing due to Black-Scholes assumptions.
method Monte Carlo simulation, GARCH model, Heston model, Merton jump-diffusion model.
result Heston model produces estimates closer to market prices, Merton model performs well for volatile assets, GARCH model improves volatility forecasts.
Data-driven RL solves Merton's expected utility problem via policy randomization.
problem Maximizing expected utility in an incomplete market with unknown primitives.
method Policy randomization in continuous-time reinforcement learning.
result RL algorithms solve Merton's problem without estimating model primitives.
Bayesian approach to portfolio selection reduces pessimism in frequent trading.
problem Tackling the challenge of estimating drift in Merton's portfolio selection model.
method Bayesian distributionally robust control with nonlinear Wasserstein projections.
result Reduced pessimism and improved performance in frequent rebalancing compared to existing methods.
We present a theory of homogeneous volatility bridge estimators for log-price stochastic processes. The main tool of our theory is the parsimonious encoding of the information contained in the open, high and low prices of incomplete bridge, corresponding to given log-price stochastic process, and in its close value, fo…
Enhances option pricing with fractional order Black-Scholes-Merton model.
problem Improving precision and authenticity of option pricing.
method Integrates fractional order Black-Scholes-Merton with neural networks.
result Improves accuracy in capturing complex diffusion dynamics and memory effects.
While defaults are rare events, losses can be substantial even for credit portfolios with a large number of contracts. Therefore, not only a good evaluation of the probability of default is crucial, but also the severity of losses needs to be estimated. The recovery rate is often modeled independently with regard to th…
Investigates how trading boundaries change with transaction costs in portfolio selection.
problem Investigates how trading boundaries vary with transaction costs in portfolio selection.
method Analyzes Merton's problem with proportional transaction costs, showing monotonicity of trading boundaries.
result Cost-adjusted trading boundaries are monotone in transaction costs, with implications for the Merton line.
Solves wealth maximization problem using variational analysis.
problem Maximizing expected utility of terminal wealth.
method Variational analysis, forward-backward stochastic differential equation (FBSDE).
result Characterization and solutions for various utility functions.
Unified approach to Merton's portfolio problem using Pontryagin's principles.
problem Optimizing consumption and investment strategies in financial portfolios.
method PG-DPO framework combining neural networks with Pontryagin's maximum principle.
result Locally optimal policies closely tied to classical stochastic control.
The study reveals traders' risk aversion and a new risk premium from market volumes.
problem Understanding traders' rationality and risk aversion from market volumes.
method Optimal Merton dynamics model to estimate average risk aversion and price of risk.
result Validation of the proposed trading strategy model on real data.
Hybrid model outperforms benchmarks in financial forecasting.
problem Robust asset price forecasting in finance.
method Combining LSTM with Neural Levy Processes using Grey Wolf Optimizer and ANN calibration.
result Hybrid model outperforms base LSTM and other models.
This paper investigates Merton's portfolio problem in a rough stochastic environment described by Volterra Heston model. The model has a non-Markovian and non-semimartingale structure. By considering an auxiliary random process, we solve the portfolio optimization problem with the martingale optimality principle. Optim…
This paper studies the properties of discrete time stochastic optimal control problems associated with portfolio selection. We investigate if optimal continuous time strategies can be used effectively for a discrete time market after a straightforward discretization. We found that Merton's strategy approximates the per…
The study uses the Merton model to estimate PD and finds a phase transition affecting convergence speed.
problem Estimating the probability of default (PD) using limited historical data.
method Adopted the Merton model and analyzed phase transitions in default correlation.
result PD estimation converges slowly when temporal correlation decays by power law less than one.
Deep learning improves option pricing in incomplete markets.
problem Optimal pricing and hedging in incomplete jump diffusion markets.
method Stackelberg game approach, deep learning (feedforward and LSTM networks).
result Deep learning algorithm outperforms traditional methods in incomplete markets.
Extends BBSM model to incorporate ESG ratings and path dynamics.
problem Price stock options considering historical market index dynamics and ESG ratings.
method Develops discrete, binary tree option pricing model under BBSM with ESG valuation.
result Model accurately fits stock price changes and European call option prices.
This paper deals with the problem of discrete-time option pricing by the mixed fractional version of Merton model with transaction costs. By a mean-self-financing delta hedging argument in a discrete-time setting, a European call option pricing formula is obtained. We also investigate the effect of the time-step δt a…
We compare the option pricing formulas of Louis Bachelier and Black-Merton-Scholes and observe -- theoretically as well as for Bachelier's original data -- that the prices coincide very well. We illustrate Louis Bachelier's efforts to obtain applicable formulas for option pricing in pre-computer time. Furthermore we ex…
Derives a dual equation for various option types, leading to new pricing and hedging insights.
problem Pricing and hedging of various option types.
method Derives a dual equation with the same form as the Black-Scholes-Merton equation, applicable to homogeneous degree one payoffs.
result Provides simple analytic formulas for delta and gamma, and reveals put-call equality for various options.
In this work, I generalize Merton's approach of pricing risky debt to the case where the interest rate risk is modeled by the CIR term structure. Closed form result for pricing the debt is given for the case where the firm value has non-zero correlation with the interest rate. This extends previous closed form pricing …
The paper extends Merton model to price equity warrants under subdiffusive fractional Brownian motion of the short rate.
problem Equity warrant pricing under subdiffusive fractional Brownian motion of the short rate.
method The paper applies subdiffusive mechanism to analyze equity warrant in a fractional Brownian motion environment, deriving a pricing formula for equity warrant.
result The paper provides a pricing formula for equity warrants under subdiffusive fractional Brownian motion model of the short rate.