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A cliquet or ratchet option is a series of at the money options, with periodic settlement, resetting the strike value at the then current price level, at which time, the option locks in the difference between the old and new strike and pays that out as the profit. The profit can be accumulated until final maturity, or paid out at each reset date.
Introduction || Ratchet Types || Monte Carlo Simulation || Quasi-Monte Carlo Simulation || Finite Differences Method || Known Names / Variants || References
A cliquet or ratchet option is a series of at the money options, with periodic settlement, resetting the strike value at the then current price level, at which time, the option locks in the difference between the old and new strike and pays that out as the profit. The profit can be accumulated until final maturity, or paid out at each reset date.
A ladder option is similar to a cliquet or ratchet option, with the exception that the gains are only locked in when the asset breaks through certain barrier levels, making a ladder option essentially a cliquet knock-in barrier. We will consider the pricing of ladder options in a separate page.
The regular ratchet option is simply the cliquet / ratchet described in the introduction of this page.
Compound Ratchet Options Compound ratchet options are similar to standard ratchet options, with the exception that each return gained at reset dates is carried forward, increasing the notional value. No payouts are made at each reset date and the accumulated amount is given only at maturity.
With standard vanilla type options, a higher volatility generally gives a higher option price due to compensation for so-called volatility risk. In the case with cliquets, this is not so. Cliquets are highly sensitive to changes in volatility and therefore have a high vega; furthermore, low volatility levels give rise to higher option prices, making this option more expensive than standard vanilla types.
Pricing: The value of cliquet options can be considered by thinking of them as forward-start options. Rubinstein (1990) presents analytical closed form formulae for forward-start options within a Black-Scholes framework which we discuss in a separate page, but reiterate here for reference purposes:
Where the variable
Where T is the time to maturity and
Therefore, to price out cliquets, the pricing formula is simply the sum of the forward start calls, given as:
Where
As per usual, one of the most reliable methods to price path-dependent options is through the use of simulation methods. By making use of antithetic variables or control variate techniques, one can price cliquets with little computational time and usable results.
There has been extensive research into the use of Quasi-MCS employing low discrepancy sequences in order to price path-dependent options. Winiarski (1998) show that the use of low-discrepancy sequences, coupled with a Brownian bridge or an incremental approach method produces cliquet values with reduced error levels (relative to standard MCS or QMCS without the mentioned methods). It is shown that accurate pricing of cliquets / ratchets can be found with approximately 2,000 iterations.
We recommend the use of QMCS for accurate results out of all mentioned methods, particularly when there are numerous reset dates. Using a low-discrepancy sequence and the inversion method given by Moro or Acklam, convergence of the option value is quicker and faster than other methods.
Governing the price of the cliquet / ratchet is a partial differential equation. A way in which we can determine the price of any option which has it's value given by a PDE, we can solve the equation by use of finite differences by determining the number of dimensions in the pricing problem and then solving.
In the case of this option, the problem lies in four dimensions (see Wilmott (2002). The PDE which governs the ratchet is given as:
Where V is a function given below, t is time,
Which we can see is identical to the standard Black-Scholes vanilla option PDE. The reason for this is because the cliquet / ratchet is a series of forward start European options, and hence can be valued in a similar fashion by transposing the number of reset periods with individual options.
We can solve this using a finite difference method, but considering that the term V is now dependent on four variables:
1) The current value of S
By use of similarity reduction, we can reduce the four-dimensional problem, down to three. By this, we mean that the current and last value of S can be reduced to 1 variable due to it's 'similarity'. We now have 3-dimensions as follows:
1) Where:
With this adjustment, we can show that the PDE changes from:
to:
For more on the aforementioned, see Wilmott (2002).
We can solve the above PDE using finite difference methods and consideration of the boundary conditions. For detailed analysis into finite difference techniques, see Tavella & Randall (2000).
Binary Cliquet
Additional/Useful List of resources Papers:
Acklam, P., Online Documentation, 2002 |
is set to 1 as cliquets are set to being at the money. The variables 
and 
are given as:
is the time to forward start.
denote the ith maturity time and ith time to forward respectively. For ratchet puts, we can find a similar relation with forward start puts.
is the volatility measure, r is the risk free interest rate and S is the asset price. We use the same notation given in Wilmott's paper.
