Goethe Center for Scientific Computing (G-CSC)

Goethe University Frankfurt

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Valuation of Performance-Dependent Options

Project description

Valuation of Performance-Dependent Options

T. GerstnerPerformance-Race

Companies make big efforts to bind their staff to them for long periods of time in order to prevent a permanent change of executives in important positions. Besides high wages, such efforts use long-term incentives and bonus schemes. One widespread form of such schemes consists in giving the participants a conditional award of shares. If the participant stays with the company for at least a prescribed time period he will receive a certain number of company shares at the end of the period.

Typically, the exact amount of shares is determined by a performance criterion such as the company’s gain over the period or its ranking among comparable firms (the peer group). This way, such bonus schemes induce uncertain future costs for the company. For the corporate management and especially for the shareholders, the actual value of such bonus programmes is quite interesting. One way to determine an upper bound on this value is to take the price of vanilla call options on the maximum number of possibly needed shares. This upper bound, however, often significantly overestimates the true value of the bonus programme since its specific structure is not respected.

Contingent claim theory states that the accurate value of such bonus programmes is given by the fair price of options which include the used performance criteria in their payoff. Such options are called performance-dependent options. Their payoff yields exactly the required shares at the end of the bonus scheme. This way, performance-dependent options minimize the amount of money the company would need to hedge future payments arising from the bonus scheme.

Similar performance comparison criteria are currently used in various financial products, for example many hedge funds are employing so-called portable alpha strategies. Recently, pure performance-based derivatives have entered the market in the form of so-called alpha certificates. Here, typically the relative performance of a basket of stocks is compared to the relative performance of a stock index. Such products are either used for risk diversification or for pure performance speculation purposes.

In this project, we develop a framework for the efficient valuation of fairly general performance-dependent options. Thereby, we assume that the performance of an asset is determined by the relative increase of the asset price over the considered period of time. This performance is then compared to the performances of a set of benchmark assets. For each possible outcome of this comparison, a different payoff can be realized.

Performance-MappingWe use multidimensional stochastic models for the temporal development of all asset prices required for the performance ranking. The martingale approach then yields a fair price of the performance-dependent option as a multidimensional integral whose dimension is the number of stochastic processes used in the model. In so-called full models the number of stochastic processes equals the number of assets. In reduced models, the number of processes is smaller. Unfortunately, in neither case direct closed-form solution for these integrals are availabla. Moreover, the integrand is typically discontinuous which makes accurate numerical solutions difficult to achieve. In this project we develop numerical methods to solve these integration problems. For reduced models, tools from computational geometry are developed, such as the fast enumeration of the cells of a hyperplane arrangement and the determination of its orthant decomposition.


  1. T. Gerstner and M. Griebel:
    Sparse grids.
    In Encyclopedia of Quantitative Finance, J. Wiley & Sons, 2009.

  2. T. Gerstner and M. Holtz:
    Valuation of performance-dependent options.
    Applied Mathematical Finance, 15(1):1-20, 2008.

  3. T. Gerstner and M. Holtz:
    Geometric tools for the valuation of performance-dependent options.
    In Computational Finance and its Application II, pp. 161-170, WIT Press, 2006.

  4. T. Gerstner, M. Holtz and R. Korn:
    Valuation of performance-dependent options in a Black-Scholes framework.
    In Numerical Methods for Finance, pp. 203-214. Chapman & Hall/CRC, 2007