In this paper we study the existence of an optimal hedging strategy for the shortfall risk measure in the game options setup. We consider the continuous time Black–Scholes (BS) model. Our first result says that in the case where the game contingent claim (GCC) can be exercised only on a finite set of times, there exists an optimal strategy. Our second and main result is an example which demonstrates that for the case where the GCC can be stopped on the whole time interval, optimal portfolio strategies need not always exist.

A game contingent claim (GCC) or game option, which was introduced in [

A hedge (for the seller) against a GCC is defined here as a pair

In real market conditions an investor (seller) may not be willing for various reasons to tie in a hedging portfolio the full initial capital required for a perfect hedge. In this case the seller is ready to accept a risk that his portfolio value at an exercise time may be less than his obligation to pay and he will need additional funds to fulfill the contract.

We consider the shortfall risk measure which is given by (see [

A natural question to ask, is whether for a given initial capital there exists a hedging strategy which minimizes the shortfall risk (an optimal hedge). For American options the existence of an optimal hedging strategy is proved by applying the Komlós lemma and relies heavily on the fact that the shortfall risk measure is a convex functional of the wealth process (see [

In this paper we treat the simplest complete, continuous time model, namely the Black–Scholes (BS) model. Our first result (Theorem

In Section

Consider a complete probability space

Define the exponential martingale

Next, let

A portfolio strategy with an initial capital

Let us recall some elementary properties that will be used in the sequel (for details see Chapters IV-V in [

For any

The shortfall risk measure is given by

We emphasis that in contrast to previous work on game options (see [

We start with some preparations. Let

For any

The following auxiliary result is an extension of Theorem 5.1 in [

Since

By applying the dominated convergence theorem, the inequality

Next, introduce the normal random variable

We arrive to the final step of the proof. Introduce the normal random variable

We arrive at the following Corollary.

(i). The result follows immediately by applying Lemma

(ii). The convexity of

Now we are ready to prove Theorem

Let

Moreover, define the random variables

Thus, in order to conclude the proof we need to show that there exists

Next, from Corollary

Finally, the completeness of the BS model implies that there exists

By combining (

We observe that the proof of Theorem

In this section we consider a game option which can be exercised at any time in the interval

Denote by

For any

We arrive at the main result.

If

Let us notice that for a given stopping time

Before we prove Theorem

The proof will be done by approximating

It remains to argue that for any

Next, we observe that for any stopping time

(i). Let

(ii). In view of (i), it is sufficient to show that

While Lemmas

(i). Let

(ii). Choose

Now, we have all the ingredients for the proof of Theorem

From Lemma

The first inequality is trivial,

Next, let

We conclude that the only remaining possibility is

We end this section with the following two remarks.

The message of Theorem

One can ask, what if we require that both of the payoff processes

The answer is yes. Let us apply Theorem

Consider a simple BS financial market with time horizon

Indeed, let

From Theorem

Theorem

By applying the weak convergence theory we can show that

An interesting question which is left for the future, is whether by allowing the investor to randomize from the start (in the spirit of [

I would like to thank Yuri Kifer for introducing me to the problems which are treated in this paper and also for related fruitful discussions. I also would like to thank Walter Schachermayer for sharing some ideas a while ago which turned out to be helpful for proving Theorem