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 tn+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 t1 and the expiry tn+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 t1,...,tn.
The payoff at expiry tn+1 of the passport option is thus Vn+1=N max(0,gn+1) where gn+1 denotes the terminal gain per unit of notional N at time tn+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 (ti,ti+1):
where Ii xN is the notional of S that (H) has chosen at time ti to hold over the period (ti,ti+1). Note that each Ii 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.
When the terminal gain includes interest, it usually is defined as follows:
where Bi is the money market account at time ti usually defined using a money market rate.
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 ti,ti+1 depends on the strategy Ii chosen by (H) at time ti, on St and on the gain gt = gi+Ii (St - Si). The value of the passport option is thus exposed to change in S via its dependency with both S and g. Hence the option delta is sV + Ii gV and the option gamma is s2V + Ii2 g2V +2IigsV.
INSIGHT ON VALUATION AND HEDGING
The payoff of the passport option is V(tn+1) = N max(In(Sn+1 - Sn) + gn,0). At time tn+ just after (H) has decided on the last position In, the passport option can be replicated using a call if In is positive and a put if In is negative with expiry tn+1, strike (Sg - gn/In) and notional N*In*. The value V(tn+) of the passport option at time tn+ is thus the value of this call or put.
Just before time tn, the gain gn is known while the strategy In is still to be chosen by (H) between -1 and 1. The function V(tn+,) is convex with respect to In and therefore can have local maxima only at the extremities I = -1,1. It follows that
Between tn+,-1 and tn-, the holder has chosen the position In-1 and the gain gn-1 is known . The passport option can thus be regarded as an instrument that pays at time tn-, the quantity V(tn-,gn-1+In-1 (Sn-Sn-1),Sn).
At time tn--1 just before (H) chooses the position In-1 , the value of the passport option is since V(tn+,-1) is convex with respect to In-1:
Applying this simple argument to each trading period together with the Black-Scholes assumptions and Ito's Lemma to the function V(t,Ii-1,gi-1+Ii-1(S - Si-1),S), we derive the following equation on (ti-1,ti):
where rt is the short rate, *t is the local volatility of S and µt is the risk-neutral drift of S. The above equation must be solved numerically on each period (ti-1,ti) for I= -1 and I= 1 because of the convexity of V(ti+-1) with respect to Ii-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 byPhilippe Balland, v.p., fixed income atMerrill Lynch International, in London.