A framework is developed for option positioning and trading based on Financial Finance Valuation (FFV), a methodology that defines value as the infimum of expected future cash flows under different test measures plus a rebate for the less likely outcomes. Positions are designed by solving a max-min optimization problem where the inner minimization evaluates a given position and the outer maximization selects the most valuable configuration. The optimization problem is solved efficiently using Disciplined Saddle Programming (DSP). The framework is implemented assuming a base measure in which return dynamics follow a Sato process with bilateral gamma laws at unit time. Backtests on SPY, AAPL and AMZN options from January 2017 to December 2024 show consistent compounding returns and stable risk-adjusted performance. The results demonstrate that FFV-based positioning can generate practical, implementable option trading strategies that adapt to changing market conditions while maintaining rigorous risk controls.

Neural net multidimensional nonlinear prediction technologies are applied to forecasting financial return distributions. This paper reports on neural net prediction technologies applied to data on bilateral gamma distributional parameters obtained from time series and option data. The exercises conducted consider predicting longer maturity risk neutral distributions from their shorter maturity counterparts, predicting tomorrow's option surface from today's as synthesized by the Sato process based on the bilateral gamma density at unit time, and physical return distribution of a stock return from the physical distribution for the exchange traded fund for the stock's sector. It is observed that considerable improvements are offered in predicting longer maturity risk neutral distributions at the lower spectrum of the set of maturities. Improvements are not as pronounced for tomorrow's surface from today's. There are improvements for index component physical return density prediction from the distribution of the index.

The objective is the development of stationary Markovian trading policies to be applied to market data. For this we rely on Lévy driven OU equations with drifts that are tempered fractional Lévy processes. For n assets a 2n dimensional Markovian system is constructed. Stationarity is induced by employing an in…nite horizon. Policies are determined to maximize a nonlinear valuation de…ned in terms of risk acceptability. A solution to the investment problem involves the use of convex sets of risk acceptability attained via rebate functions that mirror penalty functions in the theory of convex risk measures. The nonlinear valuations solve Lévy driven BSDE0s. Tractability in application is engineered by forming risk evaluations based on quantiles. The resulting strategy development is illustrated with applications to four groups of seven to ten stocks traded from 2022 to 2026: Metrics on trading performance are reported. Extensions to underlying processes with in…nite variation are also addressed as is the use nonlinear valuations that are neither concave nor convex.

Recognizing that in reality price processes display intervals of no price change we construct …nite activity processes capable of synthesizing risk neutral distributions. The proposed construction combines three independent processes, X; N; and G where X and G are Lévy subordinators while N is a standard Poisson process. The …nite activity process is then de…ned as X taken at the time given by N observed at time G; which we call here the XN G process. Ten examples of processes are introduced for di¤erent choices for X and G: Analytical results for the Lévy measure of the XN G processes are presented. In addition we evaluate the quality of the XN G processes as discontinuous continuity approximators. The use of a calibrated …nite activity process gives access to the distribution of the next move. This distribution is employed to build next move hedging strategies that maximize a conservative …nancial valuation of the hedged position. With respect to continuity approximation and hedging sensitivities to the moments of the risk neutral distribution, two of the ten models may be distinguished. These are when G is a …nite actvity process and X has either …nite activity or is a gamma process.

The multi-asset investment problem is designed to maximize a conservative …nancial valuation. Observing that conservative valuations are below mean returns, that are zero in present value terms for risk neutral distributions, it is then noted that the resulting investment problem is vacuous. Modeling risk neutral asset price processes by compound Poisson processes approximing continuity, leads to possibly positive mean returns for the sum of the next n price moves. The investment problem is recast in terms of the distribution of the sum of the next n moves. Marginal distributions for such returns are obtained from option price data. The joint distributions are modeled by invoking physical return dependencies. Conservative valuations are then formulated as a Financial Finance Valuation, a specialized form of a monetary utility. The resulting investment recommendations embedded into a strategy trading ten equity assets from the technology sector over the period January 2023 through November 2025. Five dependency models are reported on with the best performance being delivered by the Clayton and Frank copula models.

The empirically supported property of absolute variations dominating quadratic variations are employed to motivate the construction of models with percentage returns bounded by unity in absolute value. Characteristic functions are developed for the log price relative for the new return models. The models are estimated on time series and option data and demonstrate improvements delivered by the inverse logistic transformation. Applications to pricing return variations and options on them are developed. Hedging strategies use Machine Learning methods on simulated sample spaces. It is observed that the log contract hedge introduces a high volatility dynamic hedge that theoretically may be compensated by the log contract leaving the variance contract as a residual. The Machine Learned hedge works with other functions estimated here by a Gaussian Process Regression that substantially reduces the volatility of the dynamic hedge and yet leaves, approximately, the variance payout as the residual.

Spot slides record the impact on the market value of existing positions in derivative markets in response to an immediate movement in the underlying asset’s spot price. The question of an optimal spot slide is then one of designing this response to be optimal among the set of possible risk exposure functions. Structurally as a nonzero constant cannot be an exposure, its role as a numeraire asset is replaced by the selection of a numeraire exposure. The design objective replaces expected utility that is not a financial objective by a monetary utility that we define to be a Financial Finance Objective (FFO). An FFO valuation, by virtue of being a market valuation, also avoids the use of physical probabilities in designing exposures. The FFO values risk by the maximal level of the numeraire exposure that may be extracted from a position subject to the resulting risk maintaining its acceptability. Risk acceptability is defined by a convex set that may or may not contain a convex cone larger the nonnegative variables. Such a larger cone is referred to as an embedded Conic Finance Cone (CFC). The absence of a CFC embedding limits risk taking and is a simpler design problem. The more general problem with a CFC embedding is numerically solved using Disciplined Saddle Point (DSP) programming. An FFO maximizes a conservative valuation obtained on minimizing over a set of admissible measure changes. Consequently the general problem is always a saddle point problem. We therefore envisage an enhanced role for DSP in finance with applications revising hedging, investment, and both active and passive wealth management strategies.

Recognizing the assumed absence of dynamic trading bene…ts as measured by instantaneous covariations between positions and prices when the former are modeled as being both predictable and of …nite variation we turn to their evaluation over discrete intervals. Over …nite time intervals the dynamic trading bene…t is di¤erentiated from the buy and hold or static trading gain. For a number of published trading strategies where we have access to both position and price information the two measures are computed and the variety in the strategies with respect to their sources of trading returns is noted. In particular, it is observed that risk averse trading generated by concave distorted expectations of probability have a poor dynamic performance.

Risk premia in markets rise with the degree of discontinuity in price motion. This discontinuity is measured by the scale and speed coefficients of price movements embedded in the bilateral gamma process, which is modeled as the difference of two independent gamma processes. The bilateral gamma process is calibrated to both the time series data of daily returns and to option price data. Return distributions inferred from option prices are heavily influenced by the presence of risk premia. Strategies are proposed to unwind these effects with the goal of constructing return distributions that are more useful for investment analysis. Downward movements exhibit lower continuity than upward movements, both physically and under the risk-neutral measures. Moreover, risk-neutral movements are substantially less continuous than their physical counterparts for both upward and downward directions. Indices of discontinuity are developed for upward and downward movements, based on asymmetric variance-minimizing step sizes used in finite difference delta calculations.

Assuming the validity of the Miller and Modigliani (1961) thesis arguing that investors can set their own dividend policies, two questions arise. The first asks, what are the levels of these investor determined dividend yields and the second asks what they should be. The dividend levels are addressed by employing put call parity relations in option markets to identify risk neutral dividend yields. Models for the reverse measure change back to the physical measure from the risk neutral one are then used to infer the physical dividend yields. Rational levels for the dividend yields are determined on demanding ex-dividend returns to be economically acceptable risky positions. Risk acceptability is defined using the principles of conic finance and in particular by measuring the degree of acceptability by the stress level of a distorted expectation. Estimates of dividend yields and their associated acceptability levels are evaluated for ten ETF′s and 35 stocks over the period 2015 through 2023.

The risk conscious investor is defined as the maximizer of a conservative valuation or a dynamic nonlinear expectation. Both the static and dynamic problems are addressed using distortions of tail probabilities or distortions of tail measures. The multivariate static problem is solved in the context of the multivariate bilateral gamma model. For the dynamic model states and their transitions are modeled in two ways. The first defines states by a parametric model for the return distribution with transitions described by a continuous-time finite state Markov chain between the distributional possibilities. In the second model states are represented by the first four power variations with transitions given by (OU) equations for modeling stochastically the first four power variations as Tempered Fractional Levy Processes (TFLP). Numerical solutions for policy functions are implemented in trading 768 equity assets over seven years ending December 2021.

Optimal equity levels are studied from the perspective of security valuation in two price markets when acceptable risks form a convex set. The valuation functions used are financial and increase the value of claims to an additional dollar by a dollar. Such valuation functions require the specification of rebate functions for the probabilities testing acceptability. A specific rebate function termed RADICAL is introduced that delivers closed form solutions for conservative valuations. Applications are first illustrated on stylized random variables. They are followed by results based on risk neutral distributions observed in options markets. Median equity levels fall from 14% to 10% for 45 underliers over the period 2015 to 2023 as the cone of acceptable risks is enlarged to promote grated risk taking. The equity levels are observed to peak in early 2020.