Lookback option pricing

A lookback option offers the holder the right to buy a certain asset at the lowest price realized during a certain period. Therefore, thus called lookback option. In case of a put, it offers the holder to sell a certain asset at the highest price realized during a certain period. Due to the path dependent nature, the most straightforward way to price lookback options is through on Monte Carlo simulations. Important is that, lookback options have a floating strike price and as a result, always end up in the money. Therefore, lookback options tend to be more expensive.

payoff_{lookback call} = max(S_T - min(S_t), 0)

payoff_{lookback put} = max(max(S_t) - S_T, 0)

Lookback option pricing simulation ingredients

Due to the odd payoff of lookback option, it can not be valued through an analytical formula. However they can be valued through a simulation experiment. In many simulation exercises, the geometric Brownian motion, as shown below, can be used to model the underlying stock behaviour. In this formula S equals the price of the stock, μ equals the stock’s return, σ equals the stock’s volatility and Δt equals 1 time step.

    $$ \Delta S = \mu \cdot S \cdot \Delta t + \sigma \cdot S \cdot \epsilon \cdot \sqrt{\Delta t} $$

In order to implement the stock price evolution in Excel this has to be restated as follows:

    $$ S_{t+1} = S_{t} \cdot e^{(\mu - \frac{\sigma^{2}}{2})\Delta t + \sigma \cdot \epsilon \sqrt{\Delta t}} $$

With an uncertainty parameter ε generated by a certain distribution, often just a normal distribution.

\epsilon \sim N(0,1)

geometric brownian motion

Lookback option pricing simulation implementation

The value of a lookback option can in practice be determined based on the following method:

Step 1: Determine the return μ, the volatility σ, the risk free rate r, the time horizon T and the time step Δt

Step 2: Generate using the formula a price sequence

Step 3: Calculate the payoff of the lookback call and, or put and store it

Step 4: Apply step 2 and 3 N times (e.g. 10000)

Step 5: Calculate the average of all the stored payoffs

Step 6: Discount this value back to today

Summary

Lookback options of the right to buy or sell an asset at its most favorable realized price. These exotic options are more expensive and always end up in the money. Monte Carlo simulations support the lookback option pricing process.

Lookback option pricing Excel implementation

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