Jan Vecer

A new PDE approach for pricing arithmetic average Asian options

In this paper, arithmetic average Asian options are studied. It is observed that the Asian option is a special case of the option on a traded account. The price of the Asian option is characterized by a simple one-dimensional partial differential equation which could be applied to both continuous and discrete average Asian option. The article also provides numerical implementation of the pricing equation. The implementation is fast and accurate even for low volatility and/or short maturity cases.

Pricing Asian Options in a Semimartingale Model

In this paper we studyy arithmetic Asian options when the underlying stock is driven by special semimartingale processes. We show that the inherently path dependent problem of pricing Asian options can be transformed into a problem without path dependence in the payoff function. We also show that the price is driven by a process with independent increments, Levy processes being a special case. This approach applies for both discretely or continuously options.

Options on a traded account: Vacation calls, vacation puts and passport options

Abstract. In this article we study options on a traded account. In terms of the actions available to the buyer, the options we study are more general than a class of options known as {\em passport options}; in terms of the model of the underlying asset they are more restrictive. Using probabilistic techniques, we find the value of these options, the optimal strategy of the buyer, and the hedging strategy the seller should use in response to a (not necessarily optimal) strategy by the buyer.

Portfolio sensitivity to changes in the maximum and the maximum drawdown

In this article, we define new ‘Greeks’ for financial derivatives: sensitivities to the running maximum and the running maximum drawdown of an underlying asset. Some types of portfolios, such as the net asset value of a hedge fund or performance fees, are sensitive to these parameters. In order to illustrate the concept of the new ‘Greeks’, we derive probabilistic representations of sensitivities for two classes of financial contracts: forwards on the maximum drawdown and lookback options. These results allow us to interpret the delta-hedge of the contracts in a novel way.

Estimating the Effect of the Red Card in Soccer: When to Commit an Offense in Exchange for Preventing a Goal Opportunity

We study the effect of the red card in a soccer game. A red card is given by a referee to signify that a player has been sent off following serious misconduct. The player who has been sent off must leave the game immediately and cannot be replaced during the game. His team must continue the game with one player fewer. We estimate the effect of the red card from betting data on the FIFA World Cup 2006 and Euro 2008, showing that the scoring intensity of the penalized team drops significantly, while the scoring intensity of the opposing team increases slightly. We show that a red card typically leads to a smaller number of goals scored during the game when a stronger team is penalized, but it can lead to an increased number of goals when a weaker team is punished. We also show when it is better to commit a red card offense in exchange for the prevention of a goal opportunity.

PDE methods for maximum drawdown

Maximum drawdown is a risk measure that plays an important role in portfolio management. In this paper, we address the question of computing the expected value of the maximum drawdown using a partial dierential equation (PDE) approach. First, we derive a two-dimensional convection-diusi on pricing equation for the maximum drawdown in the Black-Scholes framework. Due to the properties of the maximum drawdown, this equation has a nonstandard boundary condition. We apply an alternating direction implicit method to solve the equation numerically. We also discuss stability and convergence of the numerical method.

On Probabilistic Excitement of Sports Games

In this paper we introduce a quantitative measure of the excitement of sports games. This measure can be thought of as the variability of the expectancy of winning as a game progresses. We illustrate the concept of excitement at soccer games for which the theoretical win expectancy can be well approximated from a Poisson model of scoring. We show that in the Poisson model, higher scoring rates lead to increased expected excitement. Given a particular strength of a team, the most exciting games are expected with opponents who are slightly stronger. We apply this theory to the FIFA World Cup 2006 games, where the winning expectancy was independently estimated by betting markets. Thus, it was possible to compute the expected and the realized excitement of each given game from the trading data.

VALUING CALLABLE AND PUTABLE REVENUE-PERFORMANCE-LINKED PROJECT BACKED SECURITIES

Public owners face a constant demand for developing new projects and for funding the renewal, maintenance and operation of existing infrastructure projects. One way to raise capitals to provide new financial resources to constrained budgets is to securitize a stream of revenue cash flows from a portfolio of mature infrastructure projects. We present a new type of PBS, the revenue performance-linked project backed securities (PBS), with embedded call and put options. In this new PBS setting, risks for issuers and buyers can be confined within a cut-off area. This risk hedging feature is expected to facilitate the trading of such products.