Cliquet option

Cliquet is a redirect to this article. For the French resistance fighter, see Charles Cliquet .

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As Cliquet option is called in the financial world, a path-dependent exotic option whose payoff is determined by several intermediate observations at different, predetermined times. It consists of several at-the-money options, whereby the first option with a base price of 100% of the current rate becomes active when the contract is concluded. As soon as this expires, the next one is activated again with a base price of the current rate. This process is repeated until the option expires.

definition

A cliquet option on an underlying is determined by the following parameters : ${\displaystyle (S_{t}),\;t\geq 0}$

• the number of observations ${\displaystyle n\in \mathbb {N} }$
• the times of the observations (T corresponds to the expiry date) ${\displaystyle 0=t_{0}<t_{1}<t_{2}\ldots <t_{n}=T}$
• the local barriers ${\displaystyle floor_{l}\leq 0,cap_{l}\geq 0}$
• the global barriers . ${\displaystyle floor_{g}\leq 0,cap_{g}\geq 0}$

Over time, the interim returns are now observed. At the end time T, the following amount is paid out: ${\displaystyle P_{i}:={\frac {S_{t_{i+1}}}{S_{t_{i}}}}-1}$

${\displaystyle X=min(max(floor_{g},[\sum _{i=0}^{n-1}min(max(floor_{l},P_{i}),cap_{l})]),cap_{g})}$

First, the individual returns are limited upwards and downwards (by the local barriers), then added up and then limited again (by the global barriers).

If both the local and the global floor are negative, the payout amount can also be negative. In this case, the cliquet option is no longer an option in the narrower sense, but only a contingent claim .

rating

As with most path-dependent options, the assessment of a cliquet option can only be carried out analytically in special cases. For example , there is a simple solution in the Black-Scholes model , especially when the global barriers are not active. Then the following applies to the price of the option:

${\displaystyle P=E(e^{-rT}X)=e^{-rT}\sum _{i=0}^{n-1}E[min(cap_{l},max(floor_{l},P_{i}))]}$,

Where r denotes the risk-free interest rate and the expected value is calculated in terms of a martingale measure. If the observation times are also equidistant , the expression is simplified to

: ${\displaystyle P=n\cdot e^{-rT}E[min(cap_{l},max(floor_{l},P_{0}))]}$.

However, if more complicated capital market models are used (e.g. general Lévy processes or models with stochastic volatility ), the only option often left is to estimate the option price using a Monte Carlo simulation .