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# Cliquet Option

## Contents

## Description

A cliquet or ratchet option can be seen as a series of at-the-money options with periodic settlements, resetting the strike value at the then-current price level, at which time the option locks-in the difference between the old and new strikes and pays that out as profit. The profit can be accumulated until final maturity or paid out at each reset date. The option's lifetime is `T`. The floating strike look-back put gives the owner the right to sell the underlying at the highest price.

Name | Cliquet Option |

Underlying | Common Stock |

Underlying Price | S |

Start Date | 0 |

Maturity Date | T |

Call | True |

Notional | N |

Floor, Global | Fg |

Floor Local | Fl |

Cap, Global | Cg |

Cap, Local | Cl |

Numeraire | EUR |

## ThetaML implementation of Cliquet option

The underlying 'S' and the discount numeraire 'EUR' are processes simulated externally. For example, the process 'S' can be a stock price process that follows a Geometric Brownian Motion or a Heston Volatility process. The discount numeraire 'EUR' can be a constant discount curve as implemented in Discounting, or a stochastic process that has a dynamics as defined in the CIR model.

model cliquet import S "Underlying stock prices" import N "Notional" import EUR "Discount numeraire" import Cg "Global Cap" import Fg "Global Floor" import Cl "Local Cap" import Fl "Local Floor" export P "Option value" %at current time, set the option value to have the same expected discounted %value as the variable 'V'; the ThetaML future operator '!' accompanying the %variable 'V' acts like a function on 'V', such that the values of 'V'at %current time remain to be determined at a later instance when 'V' is assigned %some values P = E(V!) sum = 0 loop 5 R = (S!-S)/S %the ThetaML command 'Theta' passes time by '1' years Theta 1 Z = max(Fl,min(Cl,R)) sum = sum + Z end V = N * EUR * max(Fg, min(Cg, sum)) end

## Thetagram graphic illustration of modelling Cliquet option

## Numerical example for pricing Cliquet option

The following table contains numerical results calculated with ThetaML using Geometric Brownian Motion and Jump Diffusion models. It compares the results with respect to the value given in the referenced literature.

Paremeter | Symbol | Value |

Interest rate | r | 5 % |

Maturity | r | 5 years |

Numeraire | EUR | 1 |

Notional | N | 1 |

Cap. local | Cl | 0.08 |

Cap. global | Cg | Inf |

Floor local | Fl | 0.0 |

Floor global | Fg | 0.16 |

Jump Diff. const. volatility | Const. volatility 0.2359 | Const. volatility 0.3167 | |

Modeled | 0.1774 | 0.1632 | 0.1593 |

Reference | 0.17754 | 0.163259 | 0.159339 |

## References

[1] Windcliff, H.A., P.A. Forsyth, and K.R.Vetzal, 2006, Numerical Methods and Volatility Models for Valuing Cliquet Options, Applied Mathematical Finance, Volume 13, Issue 4, p353-386.