Optimal ordering and pricing policy for an inventory system with trial periods

It is a business practice that home shopping companies offer a free trial period for their products with a goal of increasing sales. Under this policy, if for any reason customers are not satisfied with the purchase, they can return the product for a refund within the trial period. To develop inventory strategies in such environment, home shopping companies should take the return phenomenon into account so as to increase their profit. This paper considers this phenomenon and develops a seasonal inventory model to deal with the problem. Two scenarios are analyzed. In the first scenario, demand is assumed to be linearly price-dependent while in the second one, it is assumed to be exponentially price-dependent. The purpose of this research is to maximize the total profit over a given planning period by determining the optimal ordering quantity and price. The analytical results demonstrate that the optimal ordering quantity and prices are obtained using closed-form formulas.     

Joint decision on production and pricing for online dual channel supply chain system

Online dual channel supply chain system and its joint decision on production and pricing under information asymmetry are investigated. First, optimal production and pricing strategies are depicted according to the centralized system. Next, two kinds of contracts are designed for the decentralized system to coordinate the channel system, and their production and pricing decisions are depicted using a principle-agent method for the asymmetric information on the traditional channel. Finally, some interesting insights are found: the centralized system is not always being better than the decentralized system with a feasible contract if the traditional and professional retailer has lower selling cost. When uncertainty in the traditional channel information is higher, the manufacturer prefers a menu of contracts according to different channel settings. When uncertainty is lower, the manufacturer prefers a single contract. Furthermore, the higher the uncertainty in the traditional channel, the more the information welfare of the traditional retailer will gain. Performance with a menu of contracts cannot outperform that with a single contract integrating optimistic and pessimistic market setting well; their difference in performance is bigger when uncertainty in the traditional channel information is less.     

Wick–Itô formula for regular processes and applications to the Black and Scholes formula

We derive a Wick–Itô formula, that is, an Itô-type formula based on Wick integration. We derive it in the context of regular Gaussian processes which include Brownian motion and fractional Brownian motion with Hurst parameter greater than 1/2. We then consider applications to the Black and Scholes formula for the pricing of a European call option. It has been shown that using Wick integration in this context is problematic for economic reasons. We show that it is also problematic for mathematical reasons because the resulting Black and Scholes formula depends only on the variance of the process and not on its dependence structure.     

A note on arbitrage,approximate arbitrage and the fundamental theorem of asset pricing

We provide a critical analysis of the proof of the fundamental theorem of asset pricing given in the paper Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing by B. Wong and C.C. Heyde [Stochastics 82 (2010), pp. 189–200] in the context of incomplete Itô-process models. We show that their approach can only work in the known case of a complete financial market model and give an explicit counter example.     

A Matched Asymptotic Expansions Approach to Continuity Corrections for Discretely Sampled Options. Part 1: Barrier Options

We propose a general framework to model equity volatility for a firm financed by equity and additional non-equity sources of funds. The stochastic nature of equity volatility is endogenous, and comes from the impact of a change in the value of the firm's assets on the financial leverage. We first present the basic model, which is an extension of the Black-Scholes model, to value corporate securities. Second, we show for the first time in the option literature, that instantaneous equity volatility is a solution of a partial differential equation similar to Black-Scholes', although it is non-linear and in general does not have any analytical solution. However, analytical approximations for equity volatility are proposed for different capital structures: (1) equity and debt, (2) equity and warrants, and (3) equity, debt and warrants. They are shown to be very accurate.     

Stochastic equity volatility related to the leverage effect II: valuation of European equity options and warrants

We propose a general framework to assess the value of the financial claims issued by the firm, European equity options and warrantsin terms of the stock price. In our framework, the firm's asset is assumed to follow a standard stationary lognormal process with constant volatility. However, it is not the case for equity volatility. The stochastic nature of equity volatility is endogenous, and comes from the impact of a change in the value of the firm's assets on the financial leverage. In a previous paper we studied the stochastic process for equity volatility, and proposed analytic approximations for different capital structures. In this companion paper we derive analytic approximations for the value of European equity options and warrants for a firm financed by equity, debt and warrants. We first present the basic model, which is an extension of the Black-Scholes model, to value corporate securities either as a function of the stock price, or as a function of the firm's total assets. Since stock prices are observable, then for practical purposes, traders prefer to use the stock as the underlying instrument, we concentrate on valuation models in terms of the stock price. Second, we derive an exact solution for the valuation in terms of the stock price of (i) a European call option on the stock of a levered firm, i.e. a European compound call option on the total assets of the firm, (ii) an equity warrant for an all-equity firm, and (iii) an equity warrant for a firm financed by equity and debt. Unfortunately, to compute these solutions we need to specify the function of the stock price in terms of the firm's assets value. In general we are unable to specify this expression, but we propose tight bounds for the value of these options which can be easily computed as a function of the stock price. Our results provide useful extensions of the Black-Scholes model.     

Pricing Equity Swaps in an Economy with Jumps

Abstract

Empirical evidence confirms that asset price processes exhibit jumps and that asset returns are not Gaussian. We provide a pricing model for equity swaps including quanto equity swaps for a non-Gaussian market. The market is driven by a general marked point process as well as by a standard multidimensional Wiener process. In order to obtain closed-form solutions of the swap values, we assume that all parameters in the asset price processes are deterministic, but possibly functions of time. We derive swap prices using martingale methods rather than replicating portfolios, and we show how to calculate the convexity correction term analytically. Our results are an extension of the results of Liao and Wang (2003 Liao, M. and Wang, M. 2003. Pricing models of equity swaps. The Journal of Futures Markets, 23(8): 751–772. [Crossref], [Web of Science ®] , [Google Scholar]; Pricing models of equity swaps, The Journal of Futures Markets, 23(8), pp. 751–772). The martingale method is the key that enables the extension.     

A Family of Maximum Entropy Densities Matching Call Option Prices

Abstract

We investigate the position of the Buchen–Kelly density (Peter W. Buchen and Michael Kelly. The maximum entropy distribution of an asset inferred from option prices. Journal of Financial and Quantitative Analysis, 31(1), 143–159, March 1996.) in the family of entropy maximizing densities from Neri and Schneider (Maximum entropy distributions inferred from option portfolios on an asset. Finance and Stochastics, 16(2), 293–318, April 2012.), which all match European call option prices for a given maturity observed in the market. Using the Legendre transform, which links the entropy function and the cumulant generating function, we show that it is both the unique continuous density in this family and the one with the greatest entropy. We present a fast root-finding algorithm that can be used to calculate the Buchen–Kelly density and give upper boundaries for three different discrepancies that can be used as convergence criteria. Given the call prices, arbitrage-free digital prices at the same strikes can only move within upper and lower boundaries given by left and right call spreads. As the number of call prices increases, these bounds become tighter, and we give two examples where the densities converge to the Buchen–Kelly density in the sense of relative entropy. The method presented here can also be used to interpolate between call option prices, and we compare it to a method proposed by Kahalé (An arbitrage-free interpolation of volatilities. Risk, 17(5), 102–106, May 2004). Orozco Rodriguez and Santosa (Estimation of asset distributions from option prices: Analysis and regularization. SIAM Journal on Financial Mathematics, 3(1), 374–401, 2012.) have produced examples in which the Buchen–Kelly algorithm becomes numerically unstable, and we use these as test cases to show that the algorithm given here remains stable and leads to good results.     

Arithmetic Asian Options under Stochastic Delay Models

Abstract

Motivated by the increasing interest in past-dependent asset pricing models, shown in recent years by market practitioners and prominent authors such as Hobson and Rogers (1998 Hobson, D. and Rogers, L. C. G. 1998. Complete models with stochastic volatility. Mathematical Finance, 8(1): 27–48.  [Google Scholar], Complete models with stochastic volatility, Mathematical Finance, 8(1), pp. 27–48), we explore option pricing techniques for arithmetic Asian options under a stochastic delay differential equation approach. We obtain explicit closed-form expressions for a number of lower and upper bounds and compare their accuracy numerically.