# PASSPORT OPTION WITH DISCRETE UPDATE

European options on an underlying S were initially introduced as a mean of protection against a rise or fall of S between today and the expiry of the option.

European options on an underlying
*S*
were initially introduced as a mean of protection against a rise or fall of
*S*
between today and the expiry of the option. Bermudan options were introduced to allow the option holder to exercise prior to expiry. Passport or trader options were recently introduced to give even more flexibility to the option holder as they are calls on the gain realized in trading
*S*
according to a strategy chosen by the option holder.

TERMS OF THE PASSPORT OPTION

At the expiry date
*t _{n}*

_{+1}, the writer (W) of a passport option pays the holder (H) of the option, the maximum between zero and the gain realized by (H) in trading the underlying

*S*between the start

*t*

_{1}and the expiry

*t*

_{n}_{+1}according to a strategy of his/her choice. The chosen strategy must have positions that lie between

*-N*and

*N*and that are updated at some agreed dates

*t*

_{1},...,

*t*.

_{n}The payoff at expiry
*t _{n}*

_{+1}of the passport option is thus

*V*

_{n}_{+1}=

*N*max(0,

*g*

_{n}_{+1}) where

*g*

_{n}_{+1}denotes the terminal gain per unit of notional

*N*at time

*t*

_{n}_{+1}for the trading strategy chosen by (H) among all strategies that are allowed by the contract. The precise definition of the terminal gain varies from contract to contract depending on whether the gain includes interest on trading gain or not.

When the terminal gain does not include interest, it is usually defined as the sum of the trading gains realized over each trading period
(*t _{i}*,

*t*

_{i+}_{1}):

where
*I _{i}*
x

*N*is the notional of

*S*that (H) has chosen at time

*t*to hold over the period (

_{i}*t*,

_{i}*t*

_{i+}_{1}). Note that each

*I*must lie between -1 and 1 for the strategy to be acceptable. We give some insight in the next section, on the valuation and hedging of the passport option in this case.

_{i}When the terminal gain includes interest, it usually is defined as follows:

where
*B _{i}*
is the money market account at time

*t*usually defined using a money market rate.

_{i}A Bermudan option is priced and hedged assuming that the buyer of the option will optimize the time of his exercise. The price of the Bermudan option is then the supremum over all adapted exercise dates
*
of the price of a European option with random maturity
*. If the holder of this Bermudan option is not optimal in his exercise then (H) will loose in carry. The passport option is similar. Since the strategy
*I*
that (H) will choose is unknown, the price of the trader option is the supremum of the replication cost given a random strategy over all adapted permissible strategies. Like Bermudan options, if the strategy chosen by (H) is not the most expensive to replicate then the writer of the passport option (W) will gain in carry.

The value of the passport option between two successive updates
*t _{i}*,

*t*

_{i+}_{1}depends on the strategy

*I*chosen by (H) at time

_{i}*t*, on

_{i}*S*and on the gain

_{t}*g*=

_{t}*g*+

_{i}*I*(

_{i}*S*-

_{t}*S*). The value of the passport option is thus exposed to change in

_{i}*S*via its dependency with both

*S*and

*g*. Hence the option delta is

*+*

_{s}V*I*

_{i}*and the option gamma is *

_{g}V

_{s}^{2}

*V*+

*I*

_{i}^{2}

_{g}^{2}

*V +2I*

_{i}**.

_{g}_{s}*V*INSIGHT ON VALUATION AND HEDGING

The payoff of the passport option is
*V(t _{n+}*

_{1}) =

*N*max(

*I*(

_{n}*S*

_{n}_{+1}- S

*) +*

_{n}*g*,0). At time

_{n}*t*

_{n}*just after (H) has decided on the last position*

^{+}*I*, the passport option can be replicated using a call if

_{n}*I*is positive and a put if

_{n}*I*is negative with expiry

_{n}*t*

_{n+}_{1}, strike (S

*-*

_{g}*g*

_{n}*/*

*I*

*) and notional*

_{n}*N**

*I**. The value

_{n}*V*(

*t*

_{n}*) of the passport option at time*

^{+}*t*

_{n}*is thus the value of this call or put.*

^{+}Just before time
*t _{n}*, the gain

*g*is known while the strategy

_{n}*I*is still to be chosen by (H) between -1 and 1. The function

_{n}*V*(

*t*

_{n}*,) is convex with respect to*

^{+}*I*and therefore can have local maxima only at the extremities

_{n}*I*= -1,1. It follows that

Between
*t _{n}*

*,*

^{+}_{-1}and

*t*

_{n}*, the holder has chosen the position*

^{-}*I*

_{n}_{-1}and the gain

*g*

_{n}_{-1}is known . The passport option can thus be regarded as an instrument that pays at time

*t*

_{n}*, the quantity*

^{-}*V*(

*t*

_{n}*,*

^{-}*g*

_{n}_{-1}+

*I*

_{n}_{-1}(

*S*-

_{n}*S*

_{n}_{-1}),

*S*)

_{n}

_{.}At time
*t _{n}*

^{-}

*just before (H) chooses the position*

_{-1}*I*

_{n}_{-1}, the value of the passport option is since

*V*(

*t*

_{n}*,*

^{+}_{-1}) is convex with respect to

*I*

_{n}_{-1}:

Applying this simple argument to each trading period together with the Black-Scholes assumptions and Ito's Lemma to the function
*V*(*t*,*I _{i}*

_{-1}

*,g*

_{i}_{-1}+

*I*

_{i}_{-1}(

*S - S*

_{i}_{-1}),

*S*), we derive the following equation on (

*t*

_{i}_{-1},

*t*):

_{i}where
*r _{t}*
is the short rate,
*

*is the local volatility of*

_{t}*S*and µ

*is the risk-neutral drift of*

_{t}*S*. The above equation must be solved numerically on each period (

*t*

_{i}_{-1},

*t*) for

_{i}*I= -1*and

*I= 1*because of the convexity of

*V*(

*t*

_{i}

^{+}_{-1}) with respect to

*I*

_{i}_{-1}.

This equation can be adapted to cover the passport option with barrier on the gain (corresponding to a stop loss strategy) or on the spot or the case of stochastic volatility.

*This week's Learning Curve was written by**Philippe Balland**, v.p., fixed income at**Merrill Lynch International**, in London.*