Option Pricing under Randomised GBM Models

By employing a randomisation procedure on the variance parameter of the standard geometric Brownian motion (GBM) model, we construct new families of analytically tractable asset pricing models. In particular, we develop two explicit families of processes that are respectively referred to as the randomised gamma (G) and randomised inverse gamma (IG) models, both characterised by a shape and scale parameter. Both models admit relatively simple closed-form analytical expressions for the transition density and the no-arbitrage prices of standard European-style options whose Black-Scholes implied volatilities exhibit symmetric smiles in the log-forward moneyness. Surprisingly, for integer-valued shape parameter and arbitrary positive real scale parameter, the analytical option pricing formulas involve only elementary functions and are even more straightforward than the standard (constant volatility) Black-Scholes (GBM) pricing formulas. Moreover, we show some interesting characteristics of the risk-neutral transition densities of the randomised G and IG models, both exhibiting fat tails. In fact, the randomised IG density only has finite moments of the order less than or equal to one. In contrast, the randomised G density has a finite first moment with finite higher moments depending on the time-to-maturity and its scale parameter. We show how the randomised G and IG models are efficiently and accurately calibrated to market equity option data, having pronounced implied volatility smiles across several strikes and maturities. We also calibrate the same option data to the wellknown SABR (Stochastic Alpha Beta Rho) model.

The arbitrage In Securities Market Model And Some There Properties

The applications of functional analysis in economics began worked out since the by presenting theoretical studies related to the development and balance of financial markets by building mathematical models with linear topological space , describing and defining the economic balance of the stock market in mathematical formulas and terms , and then using the theorems of linear topological spaces such as Han's theorems . Banach , separation theorems , open function theorem ,closed statement theorem and so on to create the necessary and sufficient condition to make the market model achieve viability , achieve no arbitrage , and not recognize No free Lunches

The Time Variation in Risk Appetite and Uncertainty

We formulate a dynamic no-arbitrage asset pricing model for equities and corporate bonds, featuring time variation in both risk aversion and economic uncertainty. The joint dynamics among cash flows, macroeconomic fundamentals, and risk aversion accommodate both heteroskedasticity and non-Gaussianity. The model delivers measures of risk aversion and uncertainty at the daily frequency. We verify that equity variance risk premiums are very informative about risk aversion, whereas credit spreads and corporate bond volatility are highly correlated with economic uncertainty. Our model-implied risk premiums outperform standard instruments for predicting asset excess returns. Risk aversion is substantially correlated with consumer confidence measures and in early 2020 reacted more strongly to new COVID cases than did an uncertainty proxy. This paper was accepted by Haoxiang Zhu, finance.

Localized radial basis functions for no-arbitrage pricing of options under stochastic alpha–beta–rho dynamics

Closed-form explicit formulas for implied Black–Scholes volatilities provide a rapid evaluation method for European options under the popular stochastic alpha–beta–rho (SABR) model. However, it is well known that computed prices using the implied volatilities are only accurate for short-term maturities, but, for longer maturities, a more accurate method is required. This work addresses this accuracy problem for long-term maturities by numerically solving the no-arbitrage partial differential equation with an absorbing boundary condition at zero. Localized radial basis functions in a finite-difference mode are employed for the development of a computational method for solving the resulting two-dimensional pricing equation. The proposed method can use either multiquadrics or inverse multiquadrics, which are shown to have comparable performances. Numerical results illustrate the accuracy of the proposed method and, more importantly, that the computed risk-neutral probability densities are nonnegative. These two key properties indicate that the method of solution using localized meshless methods is a viable and efficient means for price computations under SABR dynamics. doi:10.1017/S1446181121000237

European Option Pricing Formula in Risk-Aversive Markets

In this study, using the method of discounting the terminal expectation value into its initial value, the pricing formulas for European options are obtained under the assumptions that the financial market is risk-aversive, the risk measure is standard deviation, and the price process of underlying asset follows a geometric Brownian motion. In particular, assuming the option writer does not need the risk compensation in a risk-neutral market, then the obtained results are degenerated into the famous Black–Scholes model (1973); furthermore, the obtained results need much weaker conditions than those of the Black–Scholes model. As a by-product, the obtained results show that the value of European option depends on the drift coefficient μ of its underlying asset, which does not display in the Black–Scholes model only because μ = r in a risk-neutral market according to the no-arbitrage opportunity principle. At last, empirical analyses on Shanghai 50 ETF options and S&P 500 options show that the fitting effect of obtained pricing formulas is superior to that of the Black–Scholes model.

On the Option Pricing by the Binomial Model

In this note we study the binomial model applied to European, American and Bermudan type of derivatives. Our aim is to give the necessary and sufficient conditions under which we can define a fair value via replicating portfolios for any derivative using simple mathematical arguments and without using no arbitrage techniques. Giving suitable definitions we are able to define rigorously the fair value of any derivative without using concepts from probability theory or stochastic analysis therefore is suitable for students or young researchers. It will be clear in our analysis that if $e^{r \delta} \notin [d,u]$ then we can not define a fair value by any means for any derivative while if $d \leq e^{r \delta} \leq u$ we can. Therefore the definition of the fair value of a derivative is not so closely related with the absence of arbitrage. In the usual probabilistic point of view we assume that $d < e^{r \delta} < u$ in order to define the fair value but it is not clear what we can (or we can not) do in the cases where $e^{r \delta} \leq d$ or $e^{r \delta} \geq u$.

Circular Economy and Value Creation: Sustainable Finance with a Real Options Approach

This paper presents a methodological proposal that integrates the circular economy concept and financial valuation through real options analysis. The Value Hill model of a circular economy provides a representation of the course followed by the value of an asset. Specifically, after the primary use, the life of an asset may be extended by going through four phases: the 4R phases (Reuse, Refurbish, Remanufacture and Recycle). Financial valuation allows us to quantify value creation from firms’ asset circularity under uncertainty, modelled by binomial trees. Furthermore, the 4R phases are valued as real options by applying no-arbitrage opportunity arguments. The major contribution of this paper is to provide a quantitative approach to the value of circularity in a general context that is adaptable to firms’ specific situations. This approach is also useful for translating relevant information for stakeholders and policy makers into something with economic and financial value.

No-arbitrage concepts in topological vector lattices

We provide a general framework for no-arbitrage concepts in topological vector lattices, which covers many of the well-known no-arbitrage concepts as particular cases. The main structural condition we impose is that the outcomes of trading strategies with initial wealth zero and those with positive initial wealth have the structure of a convex cone. As one consequence of our approach, the concepts NUPBR, NAA$$_1$$ 1 and NA$$_1$$ 1 may fail to be equivalent in our general setting. Furthermore, we derive abstract versions of the fundamental theorem of asset pricing (FTAP), including an abstract FTAP on Banach function spaces, and investigate when the FTAP is warranted in its classical form with a separating measure. We also consider a financial market with semimartingales which does not need to have a numéraire, and derive results which show the links between the no-arbitrage concepts by only using the theory of topological vector lattices and well-known results from stochastic analysis in a sequence of short proofs.