Lookback Options

Overview

Lookback options offer perfect hindsight. That is, they allow the option holder the right to purchase the underlying asset at the lowest price (call option), or sell the underlying asset at the highest price (put option) over a specified period. In option terminology, this perfect timing allows the strike price for a call to be set at the minimum price of the underlying asset, or the strike price for a put to be set at the maximum price over the stated period.

Alternatively, the payout for a call can be viewed as the amount by which the spot price at expiry exceeds the minimum spot price recorded during the life of the option. A lookback put has a payout equal to the amount by which the maximum spot price recorded during the life of the option exceeds the spot price at expiry. As a consequence, a lookback option is always in the money since the strike price for a call is always less than or equal to the underlying price, and the strike price for a put is greater or equal to the underlying price.

Formulas & Technical Details

At expiration, the payoff function for a lookback option is:

For a call

For a put

where

is the sequence of observed prices of the underlying instrument

is the price at expiration

The design of this function assumes that is the price of the underlying asset at the original settlement date of the option. In this case, , the maturity date of the option.

For details about the calculation of Greeks, see the Greeks of Options on non-Interest Rate Instruments FINCAD Math Reference document.

FINCAD Functions

All of the valuations are based on the hypotheses of the Black-Scholes framework:

· There are no taxes, margins or transaction costs;

· The risk free interest rate is constant;

· The price volatility of the underlying instrument is constant; and

· The price movements of the underlying instrument follow a lognormal distribution.

aaLook (price_u, ex, d_v, d_exp, vlt, rate_ann, cost_hldg, option_type, stat)

This function calculates fair value and risk measures for European style continuously sampled lookback options on one underlying asset. Clearly, in practice, no option can be continuously monitored. However this valuation may provide an acceptable approximation and it has the advantage that in this case, there is a closed form solution.

aaLook_MC (price_u, min_max, d_v, d_exp, vlt, rate_ann, cost_hldg, sam_seq, option_type, num_rnd, table_type)

aaLook_fs_MC (price_u, min_max, d_v, d_exp, vlt, rate_ann, cost_hldg, sam_seq, option_type, num_rnd, table_type)

These functions return, by using Monte Carlo simulation, fair value and delta for a European style lookback option which is sampled at discrete dates. If the sampling dates are regular (e.g. monthly, quarterly etc.) use the function aaLook_MC(). For other cases, use aaLook_fs_MC().

Lookback option functionality is also provided for cases that involve more than one underlying asset. For more details on these options, please refer to the option models and FINCAD Math Reference documents available in the following categories: Basket Options, Multi-Asset Options, and Spread Options.

Example

An example might be a three month put option where the strike price is set at the highest daily closing price during the option’s life. Such an option would guarantee the best timing to exit the market and therefore would be priced significantly higher than for a standard fixed-strike option.

aaLook

Argument	Description	Example Data	Switch
price_u	spot price of underlying instrument	100
max_min	For a call, the minimum price the underlying instrument has reached so far during the life of the option; for a put, the maximum price the underlying instrument has reached so far during the life of the option.	103.5
d_v	value date	1-Feb-1996
d_exp	option expiry date	1-May-1996
vlt	annualized volatility of the spot price of the underlying instrument	0.2
rate_ann	risk free discount rate for the period from the value date to the expiry date(annual compounding).	.08
cost_hldg	cost of holding the option rather than the underlying instrument (annual compounding).	0
option_type	call or put	2	put
stat	statistic to be returned	1,…2	fair value, delta

Results

Statistics	Description	Value
fair value	fair value of the option.	7.61
delta	sensitivity of the option price to small changes in the spot price.	-0.16

The same option valued using the standard Black Scholes formula results in a fair value of 4.82. Therefore, you pay 2.79 to ensure perfect timing. However, as the spread between the underlying and strike increases, there is a lower probability that the strike price would ever be set lower for a call or higher for a put. As this probability decreases, the result becomes much the same as a deep in the money option valued using a normal Black Scholes type model.

References

[1] Haug, E.G., (1998), The Complete Guide to Option Pricing Formulas, McGraw-Hill.

[2] Wilmott P., (1998), Derivatives, NY, John Wiley & Sons, Inc.

Disclaimer

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