Option valuation as an expectation in the complex domain: The Black-Scholes case

It is very well known that the first succesful valuation of a stock option was done by solving a deterministic partial differential equation (PDE) of the parabolic type with some complementary conditions specific for the option. In this approach, the randomness in the option value process is eliminated through a no-arbitrage argument. An alternative approach is to construct a replicating portfolio for the option. From this viewpoint the payoff function for the option is a random process which, under a new probabilistic measure, turns out to be of a special type, a martingale. Accordingly, the value of the replicating portfolio (equivalently, of the option) is calculated as an expectation, with respect to this new measure, of the discounted value of the payoff function. Since the expectation is, by definition, an integral, its calculation can be made simpler by resorting to powerful methods already available in the theory of analytic functions. In this paper we use precisely two of those techniques to find the well-known value of a European call

Option valuation as an expectation in the complex domain: The Black-Scholes case

It is very well known that the first succesful valuation of a stock option was done by solving a deterministic partial differential equation (PDE) of the parabolic type with some complementary conditions specific for the option. In this approach, the randomness in the option value process is eliminated through a no-arbitrage argument. An alternative approach is to construct a replicating portfolio for the option. From this viewpoint the payoff function for the option is a random process which, under a new probabilistic measure, turns out to be of a special type, a martingale. Accordingly, the value of the replicating portfolio (equivalently, of the option) is calculated as an expectation, with respect to this new measure, of the discounted value of the payoff function. Since the expectation is, by definition, an integral, its calculation can be made simpler by resorting to powerful methods already available in the theory of analytic functions. In this paper we use precisely two of those techniques to find the well-known value of a European call

Endogenous spillovers, increased competition and re-organization waves

We consider an entrepreneur that is the sole producer of a costreducing skill, but the entrepreneur that hires a team to usethe skill cannot prevent collusive trade for the innovation related knowledge between employees and competitors. We showthat there are two types of diffusion avoiding strategies forthe entrepreneur to preempt collusive communication i) settingup a large productive capacity (the traditional firm) and ii)keeping a small team (the lean firm). The traditional firm ischaracterized by its many "marginal" employees that work shortdays, receive flat wages and are incompletely informed about the innovation. The lean firm is small in number of employees,engages in complete information sharing among members, that are paid with stock option schemes. We find that the lean firm is superior to the traditional firm when technological entry costsare low and when the sector is immature.

How does ownership structure invfluence bank risk? Analyzing the role of managerial incentives

This paper analyzes how ownership concentration and managerial incentives influences bank risk for a large sample of US banks over the period 1997-2007. Using 2SLS simultaneous equations models, we show that ownership concentration has a positive total effect on bank risk. This is the result of a positive direct effect, which reflects monitoring and opportunistic behavior, and a negative indirect effect, which works through the design of managerial incentive contracts and reflects shareholder preferences toward risk. Large shareholders reduce bank risk by reducing the sensitivity of CEO wealth to stock volatility (Vega) and by increasing the CEO pay-performance sensitivity (Delta). In addition, we show that the direct and indirect effect of ownership concentration on bank risk depends on the type of the largest shareholder (a family, a bank, a corporation or an institutional investor), as well as, on the total shareholding held by each type as a group. Our results suggest that the positive relation between ownership concentration and risk is not the result of preferences towards more risk. Rather, they point at opportunistic behavior of large shareholders.

Option pricing under stochastic volatility and stochastic interest rate in the Spanish case

Among the underlying assumptions of the Black-Scholes option pricingmodel, those of a fixed volatility of the underlying asset and of aconstantshort-term riskless interest rate, cause the largest empirical biases. Onlyrecently has attention been paid to the simultaneous effects of thestochasticnature of both variables on the pricing of options. This paper has tried toestimate the effects of a stochastic volatility and a stochastic interestrate inthe Spanish option market. A discrete approach was used. Symmetricand asymmetricGARCH models were tried. The presence of in-the-mean and seasonalityeffectswas allowed. The stochastic processes of the MIBOR90, a Spanishshort-terminterest rate, from March 19, 1990 to May 31, 1994 and of the volatilityofthe returns of the most important Spanish stock index (IBEX-35) fromOctober1, 1987 to January 20, 1994, were estimated. These estimators wereused onpricing Call options on the stock index, from November 30, 1993 to May30, 1994.Hull-White and Amin-Ng pricing formulas were used. These prices werecomparedwith actual prices and with those derived from the Black-Scholesformula,trying to detect the biases reported previously in the literature. Whereasthe conditional variance of the MIBOR90 interest rate seemed to be freeofARCH effects, an asymmetric GARCH with in-the-mean and seasonalityeffectsand some evidence of persistence in variance (IEGARCH(1,2)-M-S) wasfoundto be the model that best represent the behavior of the stochasticvolatilityof the IBEX-35 stock returns. All the biases reported previously in theliterature were found. All the formulas overpriced the options inNear-the-Moneycase and underpriced the options otherwise. Furthermore, in most optiontrading, Black-Scholes overpriced the options and, because of thetime-to-maturityeffect, implied volatility computed from the Black-Scholes formula,underestimatedthe actual volatility.

A comparison of stock market mechanisms

This paper studies the relationship between the amount of publicinformation that stock market prices incorporate and the equilibriumbehavior of market participants. The analysis is framed in a static, NREEsetup where traders exchange vectors of assets accessing multidimensionalinformation under two alternative market structures. In the first(the unrestricted system), both informed and uninformed speculators cancondition their demands for each traded asset on all equilibrium prices;in the second (the restricted system), they are restricted to conditiontheir demand on the price of the asset they want to trade. I show thatinformed traders incentives to exploit multidimensional privateinformation depend on the number of prices they can condition upon whensubmitting their demand schedules, and on the specific price formationprocess one considers. Building on this insight, I then give conditionsunder which the restricted system is more efficient than the unrestrictedsystem.

A generalization of Hull and White formula and applications to option pricing approximation

By means of Malliavin Calculus we see that the classical Hull and White formulafor option pricing can be extended to the case where the noise driving thevolatility process is correlated with the noise driving the stock prices. Thisextension will allow us to construct option pricing approximation formulas.Numerical examples are presented.

On the valuation and incentive effects of executive cash bonus contracts

Executive compensation packages are often valued in an inconsistent manner: while employee stock options (ESOs) are typically valued ex-ante, cash bonuses are valued ex-post. This renders the existing valuation models of employee compensation packages theoretically unsatisfactory and, potentially, empirically distortive. In this paper, we propose an option-based framework for ex-ante valuation of cash bonus contracts. After obtaining closed-form expressions for ex-ante values of several frequently used types of bonus contracts, we utilize them to explore the e¤ects that the shape of a bonus contract has on the executive s attitude toward risk-taking. We, also, study pay-performance sensitivity of such contracts. We show that the terms of a bonus contract can dramatically impact both risk-taking behavior as well as pay-performance incentives. Several testable predictions are made, and venues of future research outlined.

Asymptotic analysis of stock price densities and implied volatilities in mixed stochastic models

In this paper, we obtain sharp asymptotic formulas with error estimates for the Mellin con- volution of functions de ned on (0;1), and use these formulas to characterize the asymptotic behavior of marginal distribution densities of stock price processes in mixed stochastic models. Special examples of mixed models are jump-di usion models and stochastic volatility models with jumps. We apply our general results to the Heston model with double exponential jumps, and make a detailed analysis of the asymptotic behavior of the stock price density, the call option pricing function, and the implied volatility in this model. We also obtain similar results for the Heston model with jumps distributed according to the NIG law.

The private and fiscal returns to schooling and the effect of public policies on private incentives to invest in education: a general framework and some results for the EU

This paper develops a comprehensive framework for the quantitative analysis of the private and fiscal returns to schooling and of the effect of public policies on private incentives to invest in education. This framework is applied to 14 member states of the European Union. For each of these countries, we construct estimates of the private return to an additional year of schooling for an individual of average attainment, taking into account the effects of education on wages and employment probabilities after allowing for academic failure rates, the direct and opportunity costs of schooling, and the impact of personal taxes, social security contributions and unemployment and pension benefits on net incomes. We also construct a set of effective tax and subsidy rates that measure the effects of different public policies on the private returns to education, and measures of the fiscal returns to schooling that capture the long-term effects of a marginal increase in attainment on public finances under c

Inequity version and team incentives

We study optimal contracts in a simple model where employees are averse to inequity as modelled by Fehr and Schmidt (1999). A "selfish" employer can profitably exploit such preferences among its employees by offering contracts which create inequity off-equilibrium and thus, they would leave employees feeling envy or guilt when they do not meet the employer's demands. Such contracts resemble team and relative performance contracts, and thus we derive conditions under which it may be beneficial to form work teams of employees with distributional concerns who were previously working individually. Similar results are obtained for status-seeking and efficiency concerns preferences.

Market Competition and Lower Tier Incentives

The relationship between competition and performance-related pay has been analysed in single-principal-single-agent models. While this approach yields good predictions for managerial pay schemes, the predictions fail to apply for employees at lower tiers of a firm's hierarchy. In this paper, a principal-multi-agent model of incentive pay is developed which makes it possible to analyze the effect of changes in the competitiveness of markets on lower tier incentive payment schemes. The results explain why the payment schemes of agents located at low and mid tiers are less sensitive to changes in competition when aggregated firm data is used. JEL classification numbers: D82, J21, L13, L22. Keywords: Cournot competition, Contract delegation, Moral hazard, Entry, Market size, Wage cost.

Financial incentives in academia: research versus development

This paper investigates the effects of monetary rewards on the pattern of research. We build a simple repeated model of a researcher capable to obtain innovative ideas. We analyse how the legal environment affects the allocation of researcher's time between research and development. Although technology transfer objectives reduce the time spent in research, they might also induce researchers to conduct research that is more basic in nature, contrary to what the skewing problem would presage. We also show that our results hold even if development delays publication.

Does affirmative action reduce effort incentives? A contest game analysis

In this paper a contest game with heterogeneous players is analyzed in which heterogeneity could be the consequence of past discrimination. Based on the normative perception of the heterogeneity there are two policy options to tackle this heterogeneity: either it is ignored and the contestants are treated equally, or affirmative action is implemented which compensates discriminated players. The consequences of these two policy options are analyzed for a simple two-person contest game and it is shown that the frequently criticized trade-off between affirmative action and total effort does not exist: Instead, affirmative action fosters effort incentives. A generalization to the n-person case and to a case with a partially informed contest designer yields the same result if the participation level is similar under each policy.