Lookback option: Lookback Option with Fix Strike

In this article I have explained you about the lookback option with fixed strike.

Lookback Option with Fix Strike

The option’s strike price is fixed as for the standard European options.It differs in a way that at maturity the option is not exercised at the price: the maximum difference between the optimal underlying asset price and the strike is the payoff.

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For the call option, the holder opt to exercise at the point when the underlying asset price is at its highest level whereas if we talk about put option the holder opt to exercise at the underlying asset’s lowest price. Respectively for the Lookback call and the Lookback put, the payoff functions are given by:

$LC_{fix}=max(S_{max}-K,0), ~~ text{and} ~~ LP_{fix}=max(K-S_{min},0),$

where the maximum asset’s price is denoted by S_max during the life of the option, minimum asset’s price is denoted by S_min during the life of the option, and strike price is denoted by K.

Arbitrage Free Price of Lookback Options with Fix Strike

We can also price the European Lookback options with fix strike by using the Black-Scholes Model, and its notations, here you should assume the same as for the Lookback option with floating strike and here also the same notations are used, at time t<T the price of the European Lookback call option having fix strike is given by the following statement.

If M < K,

$LC_t^K = SPhi(a_1(S,K)) - Ke^{-rtau}Phi(a_2(S,K)) + frac{Ssigma^2}{2r} ( Phi(a_1(S,K)) - e^{-rtau}(K/S)^{frac{2r}{sigma^{2}}}Phi(a_3(S,K))),$

and if M > K,

$LC_t^K = (M-K)e^{-rtau} + SPhi(a_1(S,M)) - Me^{-rtau}Phi(a_2(S,M)) + frac{Ssigma^2}{2r} ( Phi(a_1(S,M)) - e^{-rtau}(M/S)^{frac{2r}{sigma^{2}}}Phi(a_3(S,M))).$

In fact, there are two different prices that depends on whether M is greater than K or not, and it can be explained by the following. The event {S_max > K} is always true, if M > K. This implies that at time t the expected payoff function is then S_max − K instead of max(S_max − K,0).

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Similarly, the price of the European Lookback put option at time t<T with fix strike is given by the following statement.

If m > K,

$LP_t^K = -SPhi(-a_1(S,K)) + Ke^{-rtau}Phi(-a_2(S,K)) - frac{Ssigma^2}{2r} ( Phi(-a_1(S,K)) - e^{-rtau}(K/S)^{frac{2r}{sigma^{2}}}Phi(-a_3(S,K))),$

and if m < K,

$LP_t^K = (K-m)e^{-rtau} - SPhi(-a_1(S,m)) + me^{-rtau}Phi(-a_2(S,m)) - frac{Ssigma^2}{2r} ( Phi(-a_1(S,m)) - e^{-rtau}(m/S)^{frac{2r}{sigma^{2}}}Phi(-a_3(S,m))).$

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