Price promotions and “freemium” app monetization

The “freemium” model for digital goods involves selling a base version of the product for free, and making premium product features available to users only on payment. The success of the model is predicated on the ability to profitably convert free users to paying ones. Price promotions (or “sales”) are often used in freemium to induce this conversion. However, the causal effect of exposing consumers to such inter-temporal price variation is unclear. While sales can generate beneficial short-run conversion, they may be harmful in the long-run if consumers inter-temporally substitute purchases to periods with low prices, or use them as signals of low product quality. These long-run concerns may be accentuated in freemium apps, where the base version is sold for free, so that sales form extreme price cuts on the overall product combination. We work with the seller of a free-to-play video game to randomize entering cohorts of users into treatment and control conditions in which promotions for in-game purchases are turned on or off. We observe complete user behavior for half a year, including purchases and consumption of in-game goods, which, in contrast to much of the extant literature, enables us to assess possible substitution over time in consumption directly. We find that conversion and revenue improve in the treatment group; and detect no evidence of harmful inter-temporal substitution or negative inferences about quality from exposure to price variation, suggesting that promotions are profitable. We conjecture that the zero price of the base product that makes its consumption virtually costless, combined with the complementarity between the base product and premium features can help explain this. To the extent that this holds across freemium contexts, the positive effects of promotions documented here may hold more generally.

     

“It's MY Service,it's MY Music”: The role of psychological ownership in music streaming consumption

Music streaming services have become today's most popular way of consuming music. These services give their users access to a comprehensive music library without providing legal ownership of that music. However, recent research suggests that music streaming users still continue to experience feelings of ownership. To advance our understanding, this study investigates the role of psychological ownership in music streaming consumption. In particular, based on the theory of psychological ownership, it is analyzed how service‐ and music‐based psychological ownership emerge. The study further investigates the relationship between these two targets of ownership and whether music‐based psychological ownership is positively related to users' intention to switch from a music streaming service's free version to its paid premium version. Using structural equation modeling, the results indicate that service‐based psychological ownership, stemming from users' investment of self into the service, is positively related to music‐based psychological ownership, which is positively influenced by the sense of control over the music accessed. The results also show that music‐based psychological ownership is strongly related to users' intention to switch from free to premium, which highlights the importance of psychological ownership for providers of music streaming services, particularly those operating a feature‐limited freemium model.     

Step‐reset options: Design and valuation

This study proposes a new design of reset options in which the option's exercise price adjusts gradually, based on the amount of time the underlying spent beyond prespecified reset levels. Relative to standard reset options, a step‐reset design offers several desirable properties. First of all, it demands a lower option premium but preserves the same desirable reset attribute that appeals to market investors. Second, it overcomes the disturbing problem of delta jump as exhibited in standard reset option, and thus greatly reduces the difficulties in risk management for reset option sellers who hedge dynamically. Moreover, the step‐reset feature makes the option more robust against short‐term price movements of the underlying and removes the pressure of price manipulation often associated with standard reset options. To value this innovative option product, we develop a tree‐based valuation algorithm in this study. Specifically, we parameterize the trinomial tree model to correctly account for the discrete nature of reset monitoring. The use of lattice model gives us the flexibility to price step‐reset options with American exercise right. Finally, to accommodate the path‐dependent exercise price, we introduce a state‐to‐state recursive pricing procedure to properly capture the path‐dependent step‐reset effect and enhance computational efficiency. © 2002 John Wiley & Sons, Inc. Jrl Fut Mark 22:155–171, 2002     

Price discovery in the options markets: An application of put‐call parity

This study investigates the relative rate of price discovery in Taiwan between index futures and index options, proposing a put‐call parity (PCP) approach to recover the spot index embedded in the options premiums. The PCP approach offers the benefits of reducing model risk and alleviating the burden of volatility estimation. Consistent with the trading‐cost hypothesis, a dominant tendency is found for futures and a subordinate but non‐trivial price discovery from options. The relative weight of options price discovery is sensitive to the methodology employed as the means of inferring the option‐implicit spot price. The empirical evidence suggests that the information contained in the PCP‐implied spot encompasses that provided by the Black‐Scholes‐implied spot. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:354– 375, 2008     

Comparing stated and realized preferences for shade‐grown vs. conventionally grown coffee

We used both stated preference and revealed preference data to estimate willingness to pay (WTP) for shade‐grown coffee as compared with conventionally grown coffee. Stated preference data was collected using contingent valuation studies. Revealed preference data came from an experiment where all survey participants received a personally identifiable voucher redeemable for a free bakery item when the holder purchased a coffee. We compared estimates of mean and median WTP a price premium for shade‐grown coffee from stated preference data with similar estimates from revealed preference data. We used a logit model to evaluate the effect of explanatory variables (measures of environmental attitudes, personal norms for pro‐environmental behaviour and demographic variables) on respondents’ WTP a price premium for shade grown coffee. Model parameters were estimated using the maximum likelihood approach. Respondents with higher scores on measures of environmental attitudes and personal norms for pro‐environmental behaviour were, on average, willing to pay more for shade‐grown coffee. While this paper examined a specific case, purchase of shade‐grown coffee, our results confirmed that stated environmental concern was a good predictor of pro‐environmental behaviour. We found that mean and median WTP estimates from stated preference methods were higher, but not significantly different than mean and median WTP estimates from actual purchases, indicating convergent validity between stated and realized preference methods. The majority of individuals both stated WTP a price premium and purchased shade‐grown coffee at a price premium. We did, however notice some interesting behaviour at the individual level where stated preferences under‐predicted realized preferences at low price‐premia and over‐predicted realized preferences at high price premia.     

ANALYTIC APPROXIMATIONS FOR MULTI‐ASSET OPTION PRICING

We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option’s price as a sum of prices of various compound exchange options, each with different pairs of subordinate multi‐ or single‐asset options. The underlying asset prices are assumed to follow lognormal processes, although our results can be extended to certain other price processes for the underlying. For some multi‐asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single‐asset option under a modified measure, and in such cases an almost exact analytic price exists. More generally, approximate analytic prices for multi‐asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the subordinate basket options. The analytic formulae for multi‐asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to subordinate multi‐asset options, and the deltas of these subordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations. Finally, a calibration algorithm is proposed and illustrated.     

The valuation of American options in a multidimensional exponential Lévy model

We consider the problem of valuation of American options written on dividend‐paying assets whose price dynamics follow a multidimensional exponential Lévy model. We carefully examine the relation between the option prices, related partial integro‐differential variational inequalities, and reflected backward stochastic differential equations. In particular, we prove regularity results for the value function and obtain the early exercise premium formula for a broad class of payoff functions.     

Sales promotions for preorder products: The role of time‐of‐release

Preorder offers are increasingly common for many types of products and services. Sales promotions, such as price discounts and free gifts, are frequently used to raise offer attractiveness and elicit a stronger sales response. Through a series of experiments, we show that a preorder promotion's effectiveness depends on whether it matches the construal level associated with timing of the featured product's release. When a match occurs, it increases consumers’ positive affect leading to higher purchase intention directly or by raising the perceived certainty of new product quality. In addition, we find interesting differences related to promotion type. Specifically, we show that a larger discount promotes low‐level construals, which leads to stronger purchase intentions only when the product is scheduled for near‐future release. Since a gift may elicit either high‐ or low‐level construals, a premium offer of higher perceived value leads to stronger purchase intentions regardless of release timing.     

PRICING AND SEMIMARTINGALE REPRESENTATIONS OF VULNERABLE CONTINGENT CLAIMS IN REGIME‐SWITCHING MARKETS

Using a suitable change of probability measure, we obtain a Poisson series representation for the arbitrage‐free price process of vulnerable contingent claims in a regime‐switching market driven by an underlying continuous‐time Markov process. As a result of this representation, along with a short‐time asymptotic expansion of the claim's price process, we develop an efficient novel method for pricing claims whose payoffs may depend on the full path of the underlying Markov chain. The proposed approach is applied to price not only simple European claims such as defaultable bonds, but also a new type of path‐dependent claims that we term self‐decomposable, as well as the important class of vulnerable call and put options on a stock. We provide a detailed error analysis and illustrate the accuracy and computational complexity of our method on several market traded instruments, such as defaultable bond prices, barrier options, and vulnerable call options. Using again our Poisson series representation, we show differentiability in time of the predefault price function of European vulnerable claims, which enables us to rigorously deduce Feynman‐Ka? representations for the predefault pricing function and new semimartingale representations for the price process of the vulnerable claim under both risk‐neutral and objective probability measures.     

A STRUCTURAL RISK‐NEUTRAL MODEL FOR PRICING AND HEDGING POWER DERIVATIVES

We develop a structural risk‐neutral model for energy market modifying along several directions the approach introduced in Aïd et al. In particular, a scarcity function is introduced to allow important deviations of the spot price from the marginal fuel price, producing price spikes. We focus on pricing and hedging electricity derivatives. The hedging instruments are forward contracts on fuels and electricity. The presence of production capacities and electricity demand makes such a market incomplete. We follow a local risk minimization approach to price and hedge energy derivatives. Despite the richness of information included in the spot model, we obtain closed‐form formulae for futures prices and semiexplicit formulae for spread options and European options on electricity forward contracts. An analysis of the electricity price risk premium is provided showing the contribution of demand and capacity to the futures prices. We show that when far from delivery, electricity futures behave like a basket of futures on fuels.     

TUG‐OF‐WAR,MARKET MANIPULATION,AND OPTION PRICING

We develop an option pricing model based on a tug‐of‐war game. This two‐player zero‐sum stochastic differential game is formulated in the context of a multidimensional financial market. The issuer and the holder try to manipulate asset price processes in order to minimize and maximize the expected discounted reward. We prove that the game has a value and that the value function is the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the nonlinear and completely degenerate infinity Laplace operator.     

Knock‐in American options

A knock‐in American option under a trigger clause is an option contract in which the option holder receives an American option conditional on the underlying stock price breaching a certain trigger level (also called barrier level). We present analytic valuation formulas for knock‐in American options under the Black‐Scholes pricing framework. The price formulas possess different analytic representations, depending on the relation between the trigger stock price level and the critical stock price of the underlying American option. We also performed numerical valuation of several knock‐in American options to illustrate the efficacy of the price formulas. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:179–192, 2004     

THE WIENER–HOPF TECHNIQUE AND DISCRETELY MONITORED PATH‐DEPENDENT OPTION PRICING

Fusai, Abrahams, and Sgarra (2006) employed the Wiener–Hopf technique to obtain an exact analytic expression for discretely monitored barrier option prices as the solution to the Black–Scholes partial differential equation. The present work reformulates this in the language of random walks and extends it to price a variety of other discretely monitored path‐dependent options. Analytic arguments familiar in the applied mathematics literature are used to obtain fluctuation identities. This includes casting the famous identities of Baxter and Spitzer in a form convenient to price barrier, first‐touch, and hindsight options. Analyzing random walks killed by two absorbing barriers with a modified Wiener–Hopf technique yields a novel formula for double‐barrier option prices. Continuum limits and continuity correction approximations are considered. Numerically, efficient results are obtained by implementing Padé approximation. A Gaussian Black–Scholes framework is used as a simple model to exemplify the techniques, but the analysis applies to Lévy processes generally.     

On American VIX options under the generalized 3/2 and 1/2 models

In this paper, we extend the 3/2 model for VIX studied by Goard and Mazur and introduce the generalized 3/2 and 1/2 classes of volatility processes. Under these models, we study the pricing of European and American VIX options, and for the latter, we obtain an early exercise premium representation using a free‐boundary approach and local time‐space calculus. The optimal exercise boundary for the volatility is obtained as the unique solution to an integral equation of Volterra type. We also consider a model mixing these two classes and formulate the corresponding optimal stopping problem in terms of the observed factor process. The price of an American VIX call is then represented by an early exercise premium formula. We show the existence of a pair of optimal exercise boundaries for the factor process and characterize them as the unique solution to a system of integral equations.     

Disagreement in a Multi‐Asset Market

This paper provides a simple framework to study the effect of disagreement in a multi‐asset market equilibrium by considering two agents who disagree about expected returns, variances, and correlation of returns of two risky assets. When agents' subjective beliefs are characterized by mean preserving spreads of a benchmark homogeneous belief, we show that the effect of the disagreement does not cancel out in general and the effect in a multi‐asset market can be very different from a single asset market. In particular, the market risk premium can increase and the risk‐free rate can decrease significantly even when the market is overoptimistic and overconfident.     

LIQUIDITY IN A BINOMIAL MARKET

We study the binomial version of the illiquid market model introduced by Çetin, Jarrow, and Protter for continuous time and develop efficient numerical methods for its analysis. In particular, we characterize the liquidity premium that results from the model. In Çetin, Jarrow, and Protter, the arbitrage free price of a European option traded in this illiquid market is equal to the classical value. However, the corresponding hedge does not exist and the price is obtained only in L² ‐approximating sense. Çetin, Soner, and Touzi investigated the super‐replication problem using the same supply curve model but under some restrictions on the trading strategies. They showed that the super‐replicating cost differs from the Black–Scholes value of the claim, thus proving the existence of liquidity premium. In this paper, we study the super‐replication problem in discrete time but with no assumptions on the portfolio process. We recover the same liquidity premium as in the continuous‐time limit. This is an independent justification of the restrictions introduced in Çetin, Soner, and Touzi. Moreover, we also propose an algorithm to calculate the option’s price for a binomial market.     

ARROW–DEBREU EQUILIBRIA FOR RANK‐DEPENDENT UTILITIES

We provide conditions on a one‐period‐two‐date pure exchange economy with rank‐dependent utility agents under which Arrow–Debreu equilibria exist. When such an equilibrium exists, we show that the state‐price density is a weighted marginal rate of intertemporal substitution of a representative agent, where the weight depends on the differential of the probability weighting function. Based on the result, we find that asset prices depend upon agents' subjective beliefs regarding overall consumption growth, and we offer a direction for possible resolution of the equity premium puzzle.     

Valuing Seller‐Defaultable Options

This study analyzes seller‐defaultable options that allow option writers to have a free‐will right to default, along with some prespecified default mechanisms. We analytically and numerically examine the pricing, hedging, defaulting, and profitability of the seller‐defaultable options, considering three possible scenarios for seller default. Analyzing the essential implications of seller‐defaultable options, we show that the option price is positively correlated with the default fine, underlying asset price, and volatility. The seller‐defaultable option's Greeks appear more complicated than those of the plain vanilla options. The likelihood of sellers defaulting increases with the underlying asset price, interest rate, volatility, and maturity time. Subject to the default mechanism, the buyers’ trading involves a trade‐off between profits and costs. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:129–157, 2013     

How to protect your premium product from low-price competitors: Price,quality, or portfolio adjustment?

In a game-theoretic framework, I analyze how a brand manufacturer can thwart new entrants into its market. Three strategic options are considered, a price adjustment of the premium product, a quality adjustment of the premium product and a portfolio adjustment of adding a fighter brand. In a basic setup, I show that the incumbent's best response to entry is to choose a portfolio adjustment. If, however, the incumbent is uncertain about whether the rival firm will enter the market, a price adjustment of the premium product might be the better alternative if launching the fighter brand is associated with costs. Moreover, if technological progress improves the efficiency of product development, a combined quality and portfolio adjustment might be the best alternative for the incumbent.     

Pricing American options with stochastic volatility: Evidence from S&P 500 futures options

This article is the first attempt to test empirically a numerical solution to price American options under stochastic volatility. The model allows for a mean‐reverting stochastic‐volatility process with non‐zero risk premium for the volatility risk and correlation with the underlying process. A general solution of risk‐neutral probabilities and price movements is derived, which avoids the common negative‐probability problem in numerical‐option pricing with stochastic volatility. The empirical test shows clear evidence supporting the occurrence of stochastic volatility. The stochastic‐volatility model outperforms the constant‐volatility model by producing smaller bias and better goodness of fit in both the in‐sample and out‐of‐sample test. It not only eliminates systematic moneyness bias produced by the constant‐volatility model, but also has better prediction power. In addition, both models perform well in the dynamic intraday hedging test. However, the constant‐volatility model seems to have a slightly better hedging effectiveness. The profitability test shows that the stochastic volatility is able to capture statistically significant profits while the constant volatility model produces losses. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:625–659, 2000