Path that dips then recovers
Start 100, minimum 90, finish 110. The call buys the 20-point rise from the trough. The put is worthless because the finish is also the high.
- Call, undiscounted
- 20
- Discounted call (1%, 1y)
- 19.801
- Put
- 0.000
Lookback options (Monte-Carlo)
Monte-Carlo pricing for floating-strike lookback calls and puts, with in-the-money statistics and the extrema seen in the run.
Each path records its own minimum and maximum. The call pays off the rise from the trough; the put pays off the fall from the peak. Random draws mean two runs will not match exactly. Figures are theoretical and for information only.
How the lookback Monte-Carlo price is built
A floating-strike lookback lets you buy at the path minimum or sell at the path maximum. This page estimates that by simulating discrete paths and averaging the discounted payoffs.
Call = e−rT × max(ST − m, 0)
- m — minimum spot on that path (including the start)
- M — maximum spot on that path
- Put = e−rT × max(M − ST, 0)
- The printed price is the mean over all paths, rounded to three decimals
Step volatility uses the same 252-day scaling as the Asian calculator. Because a lookback is almost always in the money on a path that moved at all, ITM rates are often close to 100% once you have more than a few steps.
Worked examples
The first two examples fix a single path and apply the same discount and three-decimal rounding as a finished price. The third compares the default market with vanilla Black-Scholes. Simulated prices will differ from run to run.
Path that rallies then gives back
Start 100, maximum 120, finish 100. The put collects the 20-point drop from the peak. The call is worthless because the finish equals the start, which was also the minimum.
- Put, undiscounted
- 20
- Discounted put
- 19.801
- Call
- 0.000
Default market vs vanilla
Spot 100, 30% vol, 1% rate, 365 days. Vanilla ATM call / put are 12.368 and 11.373. A lookback should print above those, with noise from 500 paths of 5 steps.
- Vanilla ATM call
- 12.368
- Vanilla ATM put
- 11.373
- Lookback (this page)
- varies by run
Related calculators on this site
Lookbacks use path extrema; Asians use the path average. Vanilla Black-Scholes remains the closed-form European, and VaR turns a volatility number into a loss estimate instead of a path-dependent price.
Lookback option FAQ
A floating-strike lookback. The call pays the final spot minus the minimum spot seen on that path; the put pays the maximum minus the final spot. There is no separate strike input.
The same discrete Monte-Carlo step as the Asian calculator: a normal shock scaled to the step, a drift from the risk-free rate, and a floor at zero. Each path tracks its own min and max. The discounted payoff is averaged over all paths and rounded to three decimals.
The holder effectively buys at the path minimum or sells at the path maximum. That optionality is worth more than a vanilla struck at today's spot, so lookback prices typically sit above the 12.368 / 11.373 vanilla ATM pair on the option calculator default market.
They are the highest maximum and lowest minimum observed across every path in that run, not the extrema of a typical path. They will jump around from one submission to the next.
The random draws are not seeded. Defaults of 500 simulations and 5 steps leave Monte-Carlo noise. More paths reduce it; more steps refine the recorded min and max along each path.
Continuous lookbacks have known formulas under Black-Scholes assumptions. This page is a discrete Monte-Carlo approximation of the floating-strike contract, so it will not match a continuous closed form, especially with only a few steps.
Disclaimer: the contents of this website are for informational purposes only and do not constitute any investment recommendation. The visitor acts at his own risk.