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# Look Back Option

## Contents

## Contract Description

There are several possible definitions of look-back options. A common one is the Floating Strike Look-Back Call option which we will focus on here. This option gives the owner the right to buy the underlying security `S` at the lowest price of the underlying that was observed during the option's lifetime `T`. The floating strike look-back put gives the owner the right to sell the underlying at the highest price.

## ThetaML implementation of a look back option

The following ThetaML model `LookBack` computes the price `P` of a Look-Back 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 LookBack import S "Underlying stock prices" import EUR "Numeraire" export P "Lookback 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' allows the values of 'V' at to be determined at a later instance %when 'V' is assigned some values P = E(V!) n = 500 % number of observations T = 1 % maturity time % Set start value of minimum s_min= S loop n %the ThetaML command 'Theta' advances time by 'T/n' years Theta T/n s_min = min( S, s_min) end %at option maturity time T, set the European put option payoffs; %the option payoffs are discounted to time 0 by the discount numeraire 'EUR' V = max(S - s_min, 0) * EUR end

The payoff description language ThetaML treats the valuation problem of the lookback option as a set of operator objects:

- the economic state is described by the underlying stock process
`S`and the numeraire`EUR`, where`S`and`EUR`are modeled by some stochastic models and are subsequently imported as processes into the structural model`LookBack`. The structural model`LookBack`is defined by the model body in the above source code and lives on a model time axis. The model time axis is a time grid that combines all the model time points in the state variables. Time passing along the model time axis is enabled by the ThetaML operator`Theta`that describes time-determined behavior of the lookback option.

- the valuation function
`V`computes the payoffs`V = max(S - s_min, 0) * EUR<tt> with the economic state variables <tt>S`and`EUR`as function arguments

- the operator
`E`applied to the valuation function`V`returns the lookback option value

## Black-Scholes price for floating strike look back option

An analytic price formula for a Floating Strike Look-Back option can be found in the Black-Scholes Model. It is expressed as

$ C_{float} = Se^{-DT}N(a_2)+Se^{-rt} \frac{\sigma^2}{2(r-D)} \left [ \left ( \frac{S}{S_{min}} \right ) N \left ( -a_1+\frac{2(r-D)}{\sigma} \right )-e^{DT}N(-a_1)\right ] $

where

$ a_1 = \frac {\ln \left (\frac{S}{S_{min}} \right )+(r-D+0.5\sigma^2)T}{\sigma\sqrt{T}} $

and

$ a_2=a_1-\sigma\sqrt T $ </center>

Option | Look Back Option |

Underlying | Common stock |

Underlying price | S |

Start Date | 0 |

Maturity Date | T |

Call | True |

Payoff | C |

Divident yield | D |

Cumulative Normal Distribution | N(.) |

Volatility | $ \sigma $ |

Interest Rate | $ r $ |

## Thetagram graphic illustration of modelling look back option

## Numerical example of pricing floating strike look back option

The graph below gives the convergence of the Floating Strike Look-Back option price to the Black-Scholes price as the number of time-steps is increased. Note that a large number of time-steps (>500) are required for accurate estimates.

Number of Monte Carlo simulations: 1000 random_seed: varied from 1..100 Black-Scholes Price: P=29.9573

Parameter | Symbol | Value |

Underlying price | S | 100 |

Volatility | $ \sigma $ | 40% |

Interest Rate | r | 5% |

Maturity | T | 1 year |

Numeraire | EUR | 1 |