Robust replication of volatility and hybrid derivatives on jump diffusions

Comparison of Model for Pricing Volatility Swaps

In this chapter, we give an overview on applications of change of time methods considered in this book in Chapters 4–8 These applications include yet another (among many) derivation of the Black-Scholes formula; the derivation of option pricing formula for a mean-reverting asset in energy finance; pricing of variance, volatility, covariance, and correlation swaps for the classical Heston model; pricing of variance and volatility swaps in energy markets; pricing of financial and energy derivatives with multifactor Levy models; and pricing of variance and volatility swaps and hedging of volatility swaps for the delayed Heston model. This chapter not only describes the applications of the change of time method but also constitutes the ultimate difference between Barndorff-Nielsen-Shiryaev’s book (2010) and present book.

Analytically pricing volatility options and capped/floored volatility swaps with nonlinear payoffs in discrete observation case under the Merton jump-diffusion model driven by a nonhomogeneous Poisson process

Pricing volatility derivatives under the modified constant elasticity of variance model

Analytically pricing variance and volatility swaps with stochastic volatility, stochastic equilibrium level and regime switching

Moment Methods for Exotic Volatility Derivatives

By this study, we want to provide tools for an economic agent who wants to maintain her maximized intertemporal consumption rate over time period regularly by trading activity in capital market. She would use the diversification principle to compose portfolio that is consisted of risky and risk free assets, with initial wealth by using principle of dynamic programming in stochastic control problem. While our target function is to maximize the discounted utility function over consumption rate, the agent may use this portion of wealth for hedonistic lifestyle without thinking the bequest aim. Motivated by observation that seminal work of Samuelson (1969)’s optimal investment on risky asset was independent of time and wealth process, we were replacing the source of uncertainty in the risky asset by binomial process in order to improve Samuelson's solution. The risky asset price evolution is determined to move in two possible directions of up and down in the next time instantly with certain probability, following binomial trees scheme. Working within this setting, we manage to obtain trading strategies as explicit solution for the optimal investment and consumption decisions which were being expressed as time-dependent functions as well as wealth process feedback form. This solution goes along with inter-temporal concept that our consumption and investment strategies tomorrow will be affected with our wealth today. We exemplify the results by numerical examples and perform Monte Carlo simulation.

In this paper, we consider volatility swap and variance swap when the underlying asset is described by a process with multiple stochastic volatility models. The model considered in this paper is the multi-factor Heston stochastic volatility model. We obtain pricing formulas for the weighted variance swap and approximate expressions for the weighted volatility swap. The bounds of the arbitrage-free variance swap price are also found. The proposed pricing formulas are easy to compute in real time and can be applied efficiently for practical applications. We study the problem of hedging volatility swap with variance swap. We also determined the optimal amount of the underlying asset that has to be held for minimizing the hedging error by taking positions in options and weighted variance swap. From the numerical analysis, a couple of important features of the usefulness of the multi-factor Heston stochastic volatility model are discussed.

This paper presents a robust methodology for the valuation of options on variance swaps and volatility swaps. While existing literature has often focused on the pricing of the swaps themselves under stochastic volatility models, the valuation of options on these second-order derivative products remains a challenge, particularly within a framework that captures empirical market properties such as volatility clustering and long memory. Our approach addresses this problem by relying on the precise calculation of the optimal exercise prices (or strike rates) for variance and volatility swaps. To achieve this, we introduce and utilize a multifractional extension of the classic SABR model. This enhanced version of the model incorporates a deterministic and time-varying Hurst function, h(t), which allows for the modeling of a richer and more flexible volatility dynamics, capable of replicating the varying persistence phenomena observed in real markets. The proposed methodology is detailed comprehensively. We derive the theoretical formulas for the swap prices, which form the basis for calculating the optimal exercise prices. The valuation of options (call and put) on these swaps is then performed by employing advanced numerical techniques, adapted to the complexity of the multifractional process.

In the existing literature, the representativeness heuristic is commonly viewed as a cause of asset price overreaction to new information. This article proves that the representativeness heuristic can cause asset price underreaction to new information in a competitive securities market. Specifically, there is one risk-free asset and one risky asset. Both rational and heuristic traders trade can against each other or against noise traders, whose demand is random. The payoff of the risky asset is unknown but all traders receive an informational signal about the risky asset’s payoff before any trading takes place. Due to the representativeness heuristic, the updated mean of the risky asset’s payoff for heuristic traders is higher (lower) than that for rational traders when the realization of the informational signal is above (below) the expected payoff of the risky asset. The results of this study suggest that regardless of noise traders being net buyers or sellers, the representativeness heuristic causes the asset price to overreact to new information close to the expected payoff of the risky asset and causes the asset price to underreact to new information far above or below the expected payoff of the risky asset. <b>TOPICS:</b>Statistical methods, risk management, in markets

SummaryThis paper proposes an empirical asset pricing test based on the homogeneity of the factor risk premia across risky assets. Factor loadings are considered to be dynamic and estimated from data at higher frequencies. The factor risk premia are obtained as estimates from time series regressions applied to each risky asset. We propose Swamy‐type tests robust to the presence of generated regressors and dependence between the pricing errors to assess the homogeneity of the factor risk premia and the zero intercept hypothesis. An application to US industry portfolios shows overwhelming evidence rejecting the capital asset pricing model, and the three and five factor models developed by Fama and French (Journal of Financial Economics, 1993, 33, 3–56; Journal of Financial Economics, 2015, 116, 1–22). In particular, we reject the null hypotheses of a zero intercept, homogeneous factor risk premia across risky assets, and the joint test involving both hypotheses.

This paper presents an adaptive management formulation for determining the timing and intensity of thinning and the time of clearcut for stands with any age structure and with stochastic log prices. In contrast to previous timber management studies with stochastic prices, the formulation allows the determination of the best time to switch from forestry to another land use or from selection harvesting to plantation forestry. Harvests are determined with a piecewise linear function that relates harvest intensity and stand value. The parameters of the function include its slope, location, and length. Optimal parameter values are determined using stochastic simulation. Delivered log price is a stationary process that is the sum of a deterministic constant and a normally distributed random error. Numerical results are presented for young and mature stands of California white fir [Abies concolor (Gord. &amp; Glend.) Lindl. (Iowiana [Gord.])]. The results include when to terminate forestry and switch land use. The expected time of clearcut is inversely related to the bare land value. The effect of price variation on the time of clearcut depends on land value. For low land values, increasing price variation decreases the expected time of clearcut; for high land values, the effect of price variation is reversed. The results include estimates of the relative economic efficiency of plantation versus uneven-age forestry. The size of trees in the initial stand rather than price variation phys the critical role in determining the relative economic returns. Because the results are dependent on the assumption that the price process is stationary, caution should be used in extending the results beyond the case presented here. For. Sci. 37(5):1224-1238.

This paper is dedicated to studying the optimal investment proportions of three types of assets with symmetry, namely, risky assets, risk-free assets, and wealth management products, when the stochastic expenditure process follows a jump-diffusion model. The stochastic expenditure process is treated as an exogenous cash flow and is assumed to follow a stochastic differential process with jumps. Under the Cox–Ingersoll–Ross interest rate term structure, it is presumed that the prices of multiple risky assets evolve according to a multi-dimensional geometric Brownian motion. By employing stochastic control theory, the Hamilton–Jacobi–Bellman (HJB) equation for the household portfolio problem is formulated. Considering various risk-preference functions, particularly the Hyperbolic Absolute Risk Aversion (HARA) function, and given the algebraic form of the objective function through the terminal-value maximization condition, an explicit solution for the optimal investment strategy is derived. The findings indicate that when household investment behavior is characterized by random expenditures and symmetry, as the risk-free interest rate rises, the optimal proportion of investment in wealth-management products also increases, whereas the proportion of investment in risky assets continually declines. As the expected future expenditure increases, households will decrease their acquisition of risky assets, and the proportion of risky-asset purchases is sensitive to changes in the expectation of unexpected expenditures.

Worst-Case Portfolio Optimization with Proportional Transaction Costs

This paper investigates an optimal investment problem for an open-end fund with dynamic flows. The accumulated fund inflow and outflow processes are modeled by two independent compound Poisson processes, respectively. The fund manager allocates the fund capital between a risk-free asset and a risky asset. The correlation between the fund inflow process and the risky asset's price process is under consideration. Inspired by Berk and Van Binsbergen [(2015). Measuring skill in the mutual fund industry. Journal of Financial Economics, 118(1), 1–20], we first introduce the accumulated value added process to measure the fund performance. The manager aims to maximize the expected utility of the fund's accumulated value added. We derive the optimal investment strategy explicitly via dynamic programming approach. Furthermore, a verification theorem is provided and some properties of the optimal strategy are analyzed. Finally, sensitivity analysis is given to illustrate the effects of model parameters on the optimal strategy.