Look Back Option

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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.


Image:Flbsc1.svg

ThetaScript 

%% The model LookBack computes the price "P" of 
%% a Look-Back Option   
Model LookBack        
    import S    "Underlying stock"            
    import EUR  "Numeraire" 
    export P    "Option value"
 
    P = E(V!)
 
    n = 500 % number of observations
    T = 1   % maturity time
 
    % Set start value of minimum
    s_min= S
 
    loop n                 	        
        Theta T/n
        s_min = min( S, s_min)
    end 
 
    V= max(S - s_min, 0) * EUR
end

Black-Scholes Price

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


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 σ
Interest Rate r



Thetagram

Image:Lookback.png

Numerical Example

A graph of the convergence of the Floating Strike Look-Back option to the Black-Scholes price versus the number of time-steps is shown below. 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 σ 40%
Interest Rate r 5%
Maturity T 1 year
Numeraire EUR 1



Image:Flbsc3.svg

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