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In this article, the multitouch option, also called the n-touch option (or the “baseball” option when n=3) is analyzed and valued in closed form. This is a kind of barrier option that has been traded for a long time on the markets, but that does not yet admit a known valuation formula. The multitouch option sets a gradual knock-out/knock-in mechanism based on the number of times the underlying asset has crossed a predefined barrier in various time intervals before expiry. The higher the number of predefined time intervals during which the barrier has been touched, the lower the value of a knock-out contract at expiry, and conversely for a knock-in one. Multitouch options can be viewed as an extension of step barrier options, preserving the ability of the latter to adjust the exposure to risk over time, while eliminating the notorious danger of “sudden death” that holders of step barrier options are faced with. They are thus less risky and more flexible than step barrier options, and all the more so when compared to standard barrier options. This article also provides closed-form valuation of multitouch options with nonstandard features such as an outside barrier or a barrier defined as a continuous function of time.

International audience; This paper deals with a subset of lookback options known as cliquet options. The latter lock in the best underlying asset price over a number of prespecified dates during the option life. The specific uses of these contracts are analyzed, as well as two different hedging techniques. Closed form valuation formulae are provided under standard assumptions. They are easy to implement, very efficient and accurate compared to Monte Carlo simulation approximations.

All the explicit formulae for the valuation of lookback and barrier options available in the financial literature assume continuous monitoring of the underlying asset. In practice, however, monitoring is always discrete, and the gap between continuously and discretely monitored option values can be very large. In this paper, we provide explicit formulae for discretely monitored lookback and barrier options. They allow for non-constant volatility, interest rate, dividend rate and barrier parameters that vary as step functions of time. They can deal with any number and spacing of monitoring dates. They are not restricted to particular payoffs or strike price specifications. We also provide a simple rule for the numerical integration of these high-dimensional formulae, as well as an efficient interpolation method.

International audience; In this paper, we show how to obtain explicit formulae for a variety of popular path-dependent contracts with complex payoffs involving joint distributions of several extrema. More specifically, we give formulae for standard step-up and step-down barrier options, as well as partial and outside step-up and step-down barrier options, involving multiple integrals of dimensions ranging between three and five. Our method can be extended to other exotic path-dependent payoffs as well as to higher dimensions. Numerical results show that the quasi random integration of these formulae involving multivariate distributions of correlated Gaussian random variables provides option values more quickly and more accurately than Monte Carlo simulation.

International audience; This paper provides new explicit results for some boundary crossing distributions in a multi-dimensional geometric Brownian motion framework when the boundary is a piecewise constant function of time. Among their various possible applications, they enable accurate and efficient analytical valuation of a large number of option contracts traded in the financial markets belonging to the classes of barrier and lookback options.

International audience; Double barrier options have been traded for a long time in the markets and they are embedded in a variety of popular structured products. However, in their standard form, they lack flexibility inasmuch as they feature a constant barrier level during the entire option life. Step double barrier options overcome this limitation by allowing investors to set the knock-out or knock-in levels they want at the time of their choosing rather than by imposing an arbitrary mathematical form upon the time-varying double barrier. Although step double barrier options are actively traded, no analytical formula is known for their valuation and hedging. In this paper, not only regular step double barrier options are analytically valued, but also more exotic contracts combining periods with and without active barriers as well as step double barrier options written on several assets. Financial properties of these instruments are discussed in comparison with other existing contracts.

International audience; This paper extends the analytical valuation of options on the maximum or the minimum of several risky assets in several directions. The first extension consists in including more assets in the payoff and making the latter more flexible by adding knock-in and knock-out provisions. The second extension consists in pricing these contracts in a multivariate jump-diffusion framework allowing for a stochastic two-factor term structure of interest rates. In both cases, explicit formulae are provided which yield prices quasi instantaneously and with utmost precision. Hedge ratios can be easily and accurately derived from these formulae.