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This paper studies the impact of dry markets for underlying assets on the optimal stopping time and optimal exercise policy of American derivatives. We consider that the underlying is transacted at all points in time except for a subset of dates, for which there is an exogenous probability that trading may exist. Using superreplicating strategies, we derive expectation representations for the range of arbitrage-free values of the derivatives. For arbitrary probability, an enlarged filtration jointly induced by the price process and the market existence process makes ordinary stopping times sufficient to characterize such representation. For the deterministic case where the probability is zero, randomized stopping times are required. Several comparisons of the ranges obtained with the two market restrictions are performed. Finally, we conclude that market incompleteness caused by dryness may delay the optimal exercise of American derivatives.

Derivatives were originally priced by assuming that markets were complete, computing the value of a derivative as the value of a self-financing, and replicating portfolio on the underlying risky asset and risk-free bond. Such portfolio could be rebalanced by continuously trading the underlying asset and bonds. The value of the initial portfolio could be shown to be the no-arbitrage price of the derivative. In the case of American derivatives, Karatzas [

In this paper we assume that an American derivative and its respective underlying asset may not be traded at some points in time, generating incomplete markets, and then study the impact of this constraint on its optimal stopping time. In particular, we show that this incompleteness may delay the optimal exercise of American derivatives as compared to the case of complete markets. We are also able to write upper and lower bounds for their possible non-arbitrage values in terms of both randomized and ordinary stopping times.

The market for an asset is said to be liquid when there are numerous buyers and sellers, making a quick trade easy to be executed. In such circumstances, variations in supply and demand should not have a significant impact on the transaction price. An illiquid market implies that it may not be that easy to sell or to buy the asset. The fact that the underlying asset can be traded only at some points in time can be described as a particular lack of liquidity of the market, as in [

Markets’ dryness implies that markets become incomplete in the sense that perfect hedging of the derivative in all states of nature may no longer be possible. However, for any given derivative, portfolios can be found that have the same payoff as the derivative in some states of nature and higher payoffs in the other states. Such portfolios are said to be superreplicating (or superhedging). Holding one such portfolio should be worth more than the derivative itself and therefore, the value of the cheapest of such portfolios should be seen as a bound on the value of the derivative. The nature of the superreplicating bounds for European derivatives was well characterized in the context of incomplete markets by [

There has been a relatively extensive literature characterizing in varying degrees of generality the superreplicating bounds of American derivatives in incomplete markets such as in [

The results of these authors were related to the fact that^{2}, under incomplete markets, the choice of exercise policy may influence the characterization of the marketed subspace, and therefore influence the pricing of securities. A rational exercise policy may even not be well defined if the state-price deflator depends on the exercise policy. This is their argument for using optimal randomized stopping times when characterizing the superreplication bounds of American derivatives under proportional transaction costs.

We contribute to this literature by showing that in the case where market incompleteness is generated by dryness, the supremum may be taken over usual deterministic stopping times, as opposed to the randomized stopping times in [

Furthermore, under some regularity conditions, in dry markets the choice of exercise policy does not affect the characterization of the marketed subspace. Actually, we are able to show that this type of market incompleteness may delay the optimal exercise of American derivatives as compared to the case where the underlying asset may be traded at every point in time.

Our work is organized as follows. Section 2 presents the model and relevant probabilistic concepts, after which Section 3 presents the results for the upper and lower superreplicating bounds of American derivatives. In Section 4 these different bounds are compared. The exercise policy in dry markets is discussed in section 5. Finally, in Section 6 we conclude. Our main technical proofs are presented in the Appendix.

In this paper we shall follow the notation used in [

A path on the event tree is a set of nodes

in the union satisfies

Dry markets are characterized by the fact that transactions are possible with probability one only at some points in time t in a set

We can think of the existence (or not) of trade at time t as the realization of a random variable

and the enlarged filtration

with

When^{4}. In that case, consider at any node

Under the absence of arbitrage, the initial value of the derivative coincides with

When markets are dry the number of traded securities at some points in time may not be sufficient to allow the construction of a self-financing replicating portfolio. Markets are then said to be incomplete. In that case, there is not a unique arbitrage-free value for the American derivative. By replacing the notion of replicating strategy by the notion of superreplication strategy it is possible however to derive an arbitrage-free range of values for the American derivative.

We now focus on the construction of the admissible superreplicating strategies for probabilistic dry markets. At any point in time, the number of shares and the amount invested in the risk-free asset will depend on the existence, or inexistence, of trade at the previous moments in time. However, these values will not depend on the future existence of trade.

Let

since the portfolio cannot be rebalanced at time t. In that sense the admissible trading strategies will depend on the last point in time where trade occured. For that reason we shall use the notation

Just as in the deterministic case, let

Hence,

In an analogous way to the deterministic case, the definition of self-financed strategy and superreplicating strategy is dependent on whether one is in a short or in a long position in the derivative.

In what follows we characterize the upper and lower arbitrage-free bounds for the value of the American derivatives in the described framework. In order to do that, we first define some necessary mathematical objects in the following section.

In this section we adapt to the case of dry markets some definitions and results from [

Notice that

Definition 1. A node probability measure is a nonnegative function

Let

Definition 2. A node probability measure

The martingale term is justified in the sense that under that measure, for given

Definition 3. A node probability measure on the event tree is said to be y-simple if, for each

The following theorem is analogous to Theorem 6.7 in [

Theorem 1 (Chalasani and Jha) The extreme points of the set

Proof. Theorem 6.7 in [

Definition 4. An adjusted probability measure is a nonnegative function

with

Let the set of all such probability measures be denoted by

Consider

Definition 5. A y-simple node probability measure is said to be associated with a given stopping time if

Let the set of all node probability measures with this property be denoted by

Definition 6. For any probability measure

The set of all

Definition 7. A measure-strategy pair

We can now enunciate the following result, adapted from [

Theorem 2. Consider a node probability measure

is strictly positive. Conversely, if

Proof. Consider that the auxiliary node probability measures,

and

The mathematical definitions provided above are dependent of y. In what follows we show that it is possible to define an adjusted probability and a randomized stopping time in the original tree related with the concepts just presented.

Definition 8. An adjusted probability measure

The set of all such probability measures

A randomized stopping time is a nonnegative

where

Definition 9. For a given randomized stopping time

for any

Let

Theorem 3. For any given stopping time

with

If

The randomized stopping time

If

Otherwise,

Proof. See Appendix.

The upper bound for the value of an American derivative (also called its upper hedging price) is the maximum value for which the derivative would be traded without allowing for arbitrage opportunities. Such upper bound coincides with the value of the cheapest self-financed portfolio that the seller of the derivative can acquire in order to be completely hedged against any possibility of early exercise of the American derivative.

A strategy is said to be self-financed if, for any given

with

A sequence of portfolios

superreplicating strategy if the value of each portfolio is higher than or equal to the payoff of the derivative at any node in the next transaction time. In other words, for any trading dates t_{1} and t_{2} such that _{1}, _{2} a value

Since the upper bound

where the decision variables are the

More formally, for any given _{2} as

subject to the superreplicating constraints:

and subject to the self-financing constraints:

for any

Using results from linear programming the upper arbitrage-free bound of the American derivative can be written as follows.

Theorem 4. The upper hedging price of an American derivative in a probabilistic dry market can be written as

Proof. The proof follows from characterizing the dual of the linear programming problem above characterizing

The upper bound solving the problem above can also be seen to be the solution of a more intuitive problem. In fact, it can be shown that this upper bound maximizes over all possible stopping times the expected discounted payoff, when the expectation is optimized among all adjusted probability measures. In other words,

Theorem 5. The upper hedging price of an American derivative in a probabilistic dry market can be written as

where

Proof. Using Theorem 1 and Theorem 2 the conclusion is straightforward.

Note that this result is the same that would be obtained if the filtration describing the risky asset price is augmented, with no uncertainty about the existence of trade and no trade in some nodes. Using this filtration, ordinary stopping times are sufficient to write the upper bound as an expectation.

However, the upper bound on the value of an American derivative can also be written using randomized stopping times under an adjusted probability measure. The adjusted probability measure must be decomposed in such a way that, if an augmented filtration is considered, the risky asset price is a martingale. Under the original filtration

Theorem 6. The upper hedging price of an American derivative in a probabilistic dry market can be written as

with

Proof. In order to prove that the optimum value determined by the optimization problems in Theorem 6 coincides with the one presented in Theorem 5, we notice that

We then establish a relation between the two sets over which the optimization problems presented in both theorems are performed. We begin by considering an element

for any element

By the definition of a

The point that explains the difference between our result and that of Chalasani and Jha in [

In the setting of [

In our case it is not possible to exercise the derivative when the underlying asset is not traded, and hence there is no need to hedge for exercise at those points where it is not possible to rebalance the portfolio. In particular, in the case of probabilistic dry markets, our representation of the superreplicating bounds with deterministic stopping times is strongly driven by the fact that we consider the enlarged filtration resulting from the price process and the trade-existence process. This enlarged filtration allows for at most one node, per path, such that the value of the superhedging portfolio fully replicates the derivative’s payoff, avoiding this way the randomized stopping times. If that were not the case, the resulting stopping times could also be randomized. In fact, had we considered only the filtration generated by the price process, for any given price path it could be optimal fully replicate the derivative’s payoff at different moments in time.

The lower bound for the value of an American derivative (also called its lower hedging price) is the minimum value for which the derivative would be traded without allowing for arbitrage opportunities. Such lower bound coincides with the value of the most expensive self-financed portfolio that the buyer of the American derivative can sell in order to be completely hedged.

For any given stopping time

For any given stopping time

The value process of the portfolio

For each

each node that is a predecessor of the node where

The set of these set of portfolios, one for each

For a long position in the derivative, a set of portfolios that belongs to

with

A set of portfolios that belongs to

For each

More formally, the lower bound for the value of the American derivative can thus be seen as the solution of the following problem:

subject to the superreplicating constraint

if

for any

Additionally, for any node

for any

Let us consider the worse scenario in what concerns the existence of trade, that is, the situation that corresponds to the

Theorem 7. The lower hedging price of an American derivative in a dry market can be written as

with

Proof. Notice that the solution of the deterministic case

The lower bound solving the problem above can also be seen as the solution of a more intuitive problem, maximizing the expected discounted payoff, of the derivative over all possible stopping times, when the expectation is minimized among all adjusted probability measures. In other words,

Theorem 8. The lower hedging price of an American derivative in a dry market can be written as

with

where

Proof. Follows from the result of the last theorem together with Theorem 2.

This result has already been conjectured as an extension in [

In this section we will compare the arbitrage-free bounds of an American derivative in a deterministic dry market (when

The upper bound in a probabilistic dry market is higher than or equal to the upper bound if the market is dry in the deterministic sense. Moreover, it is also equal to or higher than the upper bound if transactions were possible at all points in time (

The lower bound in a probabilistic dry market is equal to the lower bound if the market is dry in the deterministic sense. Moreover, it is also lower than or equal to the lower bound if transactions were possible at all points in time (

If the market is incomplete, even with the existence of transactions at all points in time, it is in general not possible to find a unique arbitrage free value for the American derivative. However, it is also possible to establish an arbitrage free range of variation for the value of the American derivative. This range will be a subset of the arbitrage free range of variation for the value of the American derivative in the case of probabilistic dryness, but may not be a subset of the arbitrage free range of variation in the deterministic case.

In this paper we have considered that the only source of market incompleteness would be the non-existence of underlying trade at some points in time. If transactions were possible at all points in time, markets would be complete and there would be a unique arbitrage-free value for any American derivative. We found out that this unique arbitrage-free value for each American derivative belongs to the arbitrage-free range of variation for its value under a probabilistic dry market. However, it may not belong to the arbitrage-free range if a deterministic dry market is considered.

In order to understand the optimal exercise policy, we start presenting the case of a complete market.

In the case of complete markets, the value of an American derivative is given by

where

Definition 10. A stopping time frontier is the set of nodes (i, t) such that

Recalling that there is an optimal stopping node for each possible path5, we define the interior of the stopping time frontier as follows.

Definition 11. The interior of the stopping time frontier is the set of predecessors of the stopping time frontier.

It follows that no rational agent exercises the American derivative at a node inside the stopping time frontier, because at such nodes, the American derivative is worth more than the corresponding exercise. A rational agent would exercise the American derivative whenever the stopping time frontier is reached. This happens because the derivative’s payoff at that point is larger than the cost of a replicating portfolio, guaranteeing the derivative’s payoff in the future.

If the solution is not unique, there may be indeterminacy, even in this case of complete markets. An example illustrates this point. Consider the non-terminal node

We also assume that, at node

Moreover, let

In this case, the value of the portfolio, at node

Since the replicating portfolio satisfies (12) and (13), the solution of the dual problem is not unique. There are several node probability measures q solving the maximization problem that characterizes the value of the derivative. Let

such that for any

If the maximization problem characterizing the value is not uniquely solved by a node probability measure q, then the stopping time and the adjusted probability measure are also not uniquely defined. For instance, the value can be written as

with

As the stopping time is not unique, there are several stopping time frontiers, each one associated with a different stopping time. For any node inside all possible stopping time frontiers, the argument of the unique case solution applies and the agent does not have any incentive to exercise the American derivative. However, when the first stopping time frontier is reached, namely node

In the case where several stopping time frontiers coexist in this complete market setting, the exercise at any stopping time frontier before the last frontier provides a payoff equal to the value of the derivative. Also note that the last stopping time frontier is reduced to the role of a unique stopping time frontier, if the American derivative is not exercised at the previous frontiers.

With incomplete dry markets the problem is more complex. In order to characterize the optimal exercise policy, we use the stopping time

First, if the reduced filtration

Second, if we consider the enlarged filtration

Definition 12. A stopping time frontier is a pair

Remark 1. Notice that complete markets corresponds to the case where there is only one vector

Remark 2. Note that for two different sets

Just as in the complete markets’ case, the interior of the stopping time frontier is defined as the set of predecessors of the stopping time frontier.

Even for price paths with a strictly positive q, the optimal stopping exercise is not uniquely defined using pure arbitrage arguments. For a given realization

a rational holder who wants to guarantee a given amount at

However, if the American derivative is not exercised at any predecessor of the stopping time frontier, it will be exercised when the stopping time frontier is reached. The reason is that, at the frontier, the payoff is higher than the value of its replicating portfolio. However, if in a given path there is no node with a strictly positive q, the optimal stopping time can be such that

A third and final point, is that the situation occurring in the complete market case leading to a non-unique solution of the dual problem, may also happen when markets are incomplete.

In this section we establish the following result.

Proposition 9. For every path such that the stopping time is unique, the stopping time frontier under complete markets is contained in the union of the stopping time frontier under dry markets and its interior.

Proof. Consider a given

Hence, nodes in the interior of the stopping time frontier under complete markets are also in the interior of the stopping time frontier under incomplete markets. On the other hand

We now turn to the case where there is not a unique stopping time. In that case, for each path w, pick the node

Let the set

Proposition 10. The envelope of the stopping time frontier under complete markets is contained in the union of the envelope of the stopping time frontiers under dry markets and its interior.

Proof. Analogous to the proof above.

We now arrive to a quite significant corollary of this exercise in incomplete dry markets.

Corollary 1. Under uniqueness of the stopping time, rational exercise of American options under incomplete markets driven by dryness may occur later than it would occur under complete markets.

Remark 3. Notice that, under dry markets, American options may be exercised before their optimal stopping time, only under the condition specified by Equation (14), i.e., only under indifference. If not for that case, market incompletenes may only affect optimal exercise of an American option by delaying it.

An analysis of the optimal exercise of American options in a context where market’s incompleteness is generated by transaction costs leads to the conclusion that the optimal exercise is characterized by randomized stopping times. Our paper analyzes the same problem when market incompleteness is generated by dry markets, making clear that the need for randomized stopping times depends on the way market incompleteness is generated. We conclude that in the case of deterministic dry markets, the bounds for the values of American derivatives are the supremum of the implied European derivatives (i.e., European derivatives with the same underlying traded in the same dry market), this supremum being taken over deterministic stopping times. In the general probabilistic dry markets’ case, there is an additional source of uncertainty, namely the existence or not of trade at given points in time, which can be interpreted as the realization of an additional stochastic process. If an enlarged filtration, resulting from the price process and the trade existence process is considered, only ordinary stopping times are required to describe the upper and lower bounds. However, if the enlarged filtration was not considered, and the stopping times were defined by using only the filtration induced by the price process, then they could be randomized. This last conclusion recovers the results for the transaction cost model.

In a complete market the arbitrage free value of the derivative is unique and equal to the value of the replicating portfolio. However, in our incomplete market framework driven by dry markets, that fact no longer holds true. Ruling out arbitrage opportunities identifies a range of variation for the value of the derivative. The arbitrage-free value intervals for the deterministic case, for the probabilistic case and for the liquid case are compared. We find out that the range in the probabilistic case includes the range of the deterministic case and also the unique value of the liquid case. Moreover, the lower bound in the probabilistic case coincides with the one in the deterministic case. However, a relation cannot be established between the arbitrage-free range in the deterministic case and the one in the liquid case when transactions are possible at all points in time.

Moreover, when a complete market is considered, the optimal exercise policy corresponds to the stopping time that is the supremum of the implied European derivatives. Consistently with the absence of a unique arbitrage-free price, if American options are considered in this dry markets setting, the optimal exercise policy is also not well defined. The reason is that there are paths where the stopping time is not uniquely defined and in addition, the fact that if the filtration induced by the price process is considered, randomized stopping time must be used. However, we were able to show that incompleteness generated by dry markets may in general delay the optimal exercise of American derivatives.

This work was funded by National Funds through FCT―Fundação para a Ciência e Tecnologia under the project Ref. UID/ECO/00124/2013 and by POR Lisboa under the project LISBOA-01-0145-FEDER-007722. This support is greatly appreciated.

de Matos, J.M. and Lacerda, A. (2016) Randomized Stopping Times and Early Exercise for American Derivatives in Dry Markets. Journal of Mathematical Finance, 6, 842-865. http://dx.doi.org/10.4236/jmf.2016.65057

Proof. The proof that

For any given node

Consider a given path such that at the terminal node

Consider a given path such that at the node

Let

Moreover, as

However, if there are not a successor

and the proof is complete.

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