Model-independent superhedging under portfolio constraints
In a discrete-time market, we study model-independent superhedging, while the semi-static superhedging portfolio consists of three parts: static positions in liquidly traded vanilla calls, static positions in other tradable, yet possibly less liquid, exotic options, and a dynamic trading strategy in risky assets under certain constraints. By considering the limit order book of each tradable exotic option and employing the Monge-Kantorovich theory of optimal transport, we establish a general superhedging duality, which admits a natural connection to convex risk measures. With the aid of this duality, we derive a model-independent version of the fundamental theorem of asset pricing. The notion “finite optimal arbitrage profit”, weaker than no-arbitrage, is also introduced. It is worth noting that our method covers a large class of Delta constraints as well as Gamma constraint.\JEL
Keywords:model-independent pricing robust superhedging limit order book fundamental theorem of asset pricing portfolio constraints Monge-Kantorovich optimal transport
To avoid model mis-specification, one may choose to consider only “must-be-true” implications from the market. The standard approach, suggested by Dupire Dupire94 (), leverages on market prices of liquidly traded vanilla call options: one does not manage to specify a proper physical measure, but considers all measures that are consistent with market prices of vanilla calls as plausible pricing measures. These measures then provide model-independent bounds for prices of illiquid exotic options, and motivate the practically useful semi-static hedging, which involves static holdings in vanilla calls and dynamic trading in risky assets. Pioneered by Hobson Hobson98 (), this thread of research has drawn substantial attention; see e.g. BHR01 (), BP02 (), HLW05-QF (), HLW05-IME (), LW05 (), CDDV08 (), CO11 (), BHP13 (), and DS14-PTRF (). In particular, Beiglböck, Henry-Labordère & Penkner establish in BHP13 () a general duality of model-independent superhedging, under a discrete-time setting where market prices of vanilla calls with maturities at or before the terminal time are all considered.
In reality, what we can rely on goes beyond vanilla calls. In the markets of commodities, for instance, Asian options and calendar spread options are largely traded, with their market or broker quotes easily accessible. In the New York Stock Exchange and the Chicago Board Options Exchange, standardized digital and barrier options have been introduced, mostly for equity indexes and Exchange Traded Funds. What we can take advantage of, as a result, includes market prices of not only vanilla calls, but also certain tradable exotic options.
In this paper, we take up the model-independent framework in BHP13 (), and intend to establish a general superhedging duality, under the consideration of additional tradable options besides vanilla calls, as well as portfolio constraints on trading strategies in risky assets. More specifically, our semi-static superhedging portfolio consists of three parts: 1. static positions in liquidly traded vanilla calls, as in the literature of robust hedging; 2. static positions in additional tradable, yet possibly less liquid, exotic options; 3. a dynamic trading strategy in risky assets under certain constraints.
While tradable, the additional exotic options may be very different from vanilla calls, in terms of liquidity. Their limit order books are usually very shallow and admit large bid-ask spreads, compared to those of the underlying assets and the associated vanilla calls. It follows that we need to take into account the whole limit order book, instead of one single market quote, of each of the additional options, in order to make trading possible. We formulate the limit order books in Section 2.1, and consider the corresponding non-constant unit price functions. On the other hand, portfolio constraints on trading strategies in risky assets have been widely studied under the model-specific case; see CK93 () and JK95 () for deterministic convex constraints, and FK97 (), CPT01 (), Napp03 (), and Rokhlin05 (), among others, for random and other more general constraints. Our goal is to place portfolio constraints under current model-independent context, and investigate its implication to semi-static superhedging.
We particularly consider a general class of constraints which enjoys adapted convexity and continuous approximation property (Definition 2.7). This already covers a large collection of Delta constraints, including adapted convex constraints; see Remark 2.9. For the simpler case where no additional tradable option exists, we derive a superhedging duality in Proposition 3.10, by using the theory of optimal transport. This in particular generalizes the duality in BHP13 () to the multi-dimensional case with portfolio constraints; see Remarks 3.12 and 3.13. Then, on strength of the convexity of the non-constant unit price functions, we are able to extend the above duality to the general case where additional tradable options exist; see Theorem 3.14. Note that Acciaio, Beiglböck, Penkner & Schachermayer ABPS13 () also applies to model-independent superhedging in the presence of tradable exotic options, while assuming implicitly that each option can be traded liquidly. Theorem 3.14 can therefore be seen as a generalization of ABPS13 () that deals with different levels of liquidity; see Remark 3.16.
The second part of the paper investigates the relation between the superhedging duality and the fundamental theorem of asset pricing (FTAP). It is well known in the classical model-specific case that the FTAP yields the superhedging duality. This relation has been carried over to the model-independent case by ABPS13 (), where an appropriate notion of model-independent arbitrage was introduced. In the same spirit as in ABPS13 (), we define model-independent arbitrage in Definition 4.1, under current setting with additional tradable options and portfolio constraints. With the aid of the superhedging duality in Theorem 3.14, we are able to derive a model-independent FTAP; see Theorem 4.8. While the theorem itself does not distinguish between arbitrage due to risky assets and arbitrage due to additional tradable options, Lemmas 4.4 and 4.6 can be used to differentiate one from the other. It is also worth noting that we derive the FTAP as a consequence of the superhedging duality. This argument was first observed in DS14 (), as opposed to the standard argument of deriving the superhedging duality as a consequence of the FTAP, used in both the model-specific case and ABPS13 ().
With the FTAP (Theorem 4.8) at hand, we observe from Theorem 3.14 and Proposition 3.17 that the problems of superhedging and risk-measuring can be well-defined even when there is model-independent arbitrage to some extent. We relate this to optimal arbitrage under the formulation of CT13 (), and show that superhedging and risk-measuring are well-defined as long as “the optimal arbitrage profit is finite”, a notion weaker than no-arbitrage; see Proposition 4.15. We also compare Theorem 4.8 with (FS-book-11, , Theorem 9.9), the classical model-specific FTAP under portfolio constraints, and observe that a closedness condition in FS-book-11 () is no longer needed under current setting. An example given in Section 4.1 indicates that availability of vanilla calls obviates the need of the closedness condition.
Finally, we extend our scope to Gamma constraint. While Gamma constraint does not satisfy adapted convexity in Definition 2.7 (ii), it admits additional boundedness property. Taking advantage of this, we are able to modify previous results to obtain the corresponding superhedging duality and FTAP in Propositions 6.3 and 6.6.
This paper is organized as follows. In Section 2, we prescribe the set-up of our studies. In Section 3, we establish the superhedging duality, and investigate its connection to other dualities in the literature and convex risk measures. In Section 4, we define model-independent arbitrage under portfolio constraints with additional tradable options, and derive the associated FTAP. The notion “finite optimal arbitrage profit”, weaker than no-arbitrage, is also introduced. Section 5 presents concrete examples of portfolio constraints and the effect of additional tradable options. Section 6 deals with constraints which do not enjoy adapted convexity, but admit some boundedness property. Appendix A contains a counter-example which emphasizes the necessity of the continuous approximation property required in Definition 2.7.
2 The set-up
We consider a discrete-time market, with a finite horizon . There are risky assets , whose initial price is given. There is also a risk-free asset which is normalized to . Specifically, we take as the canonical process on the path-space , and denote by the natural filtration generated by .
2.1 Vanilla calls and other tradable options
At time , we assume that the vanilla call option with payoff can be liquidly traded, at some price given in the market, for all , , and . The collection of pricing measures consistent with market prices of vanilla calls is therefore
where denotes the collection of all probability measures defined on .
In view of (HR12, , Proposition 2.1), for each and , as long as is nonnegative, convex and satisfies , and , the relation already prescribes the distribution of on , which will be denoted by . Thus, by setting as the law of under , we have
Given , note that for all and (which can be seen by taking in (2.1)).
We do not assume that is increasing for each fixed and . This condition, normally required in the literature (see e.g. (BHP13, , p. 481)), implies that the set of martingale measures
is non-empty, which underlies the superhedging duality in BHP13 (). In contrast, the superhedging duality in Proposition 3.10 below hinges on a different collection which contains (see Definition 3.4). Since it is possible that our duality holds while , imposing “ is increasing” is not necessary.
Besides vanilla calls, there are other options tradable, while less liquid, at time . Let be a (possibly uncountable) index set. For each , suppose that is the payoff function of an option tradable at time . Let be the number of units of being traded at time , with denoting a purchase order and a selling order. Let denote the total cost of trading units of . Throughout this paper, we impose the following condition:
We can then define the unit price for trading units of by
Condition (C) is motivated by the typical structure of a limit order book of a nonnegative option, as demonstrated in Figure 1. That is, the option can be purchased only at prices with number of units respectively, and sold only at prices with number of units respectively, where reflects the bid-ask spread and and belong to . The possibility of allows for infinitely many buy/sell prices in the order book.
We keep track of , the total number of units that can be bought at or below the price , for . Similarly, is the total number of units that can be sold at or above the price , for all . The total cost of trading units of is then given by
where we set , , , , and use the convention that and . As shown in Figure 2, satisfies (C). In particular, is linear on if and only if ; this means that can be traded liquidly at the price , which is the slope of .
Condition (C) captures two important features of the prices of : (1) the bid-ask spread, formulated as ; (2) the non-linearity, i.e. the unit price is non-constant. This setting in particular allows for zero spread when , while at the same time the limit order book may induce non-linear pricing. This happens to a highly liquid asset for which the bid-ask spread is negligible, but transaction cost becomes significant for large trading volumes. Also, is linear if and only if can be traded liquidly, with whatever units, at one single price (which is the slope of ).
Note that BZZ14 () has recently considered bid-ask spreads, but not non-linear pricing, of hedging options under model uncertainty. In a model-independent setting, while a non-linear pricing operator for hedging options has been used in DS14 (), the non-linearity does not reflect the non-constant unit price of an option in its limit order book (see (DS14, , (2.3))); instead, it captures a market where the price of a portfolio of options may be lower than the sum of the respective prices of the options (see the second line in the proof of (DS14, , Lemma 2.4)).
2.2 Constrained trading strategies
Definition 2.6 (Trading strategies)
We say is a trading strategy if is a constant and is Borel measurable for all . Moreover, the stochastic integral of with respect to will be expressed as
where in the right hand side above, , , and “ ” denotes the inner product in . We will denote by the collection of all trading strategies. Also, for any collection , we introduce the sub-collections
In this paper, we require the trading strategies to lie in a sub-collection of , prescribed as below.
Definition 2.7 (Adaptively convex portfolio constraint)
is a set of trading strategies such that
For any and adapted process with for all ,
For any , , and , there exist a closed set and such that
In Definition 2.7, (i) and (ii) are motivated by (FS-book-11, , Section 9.1), while (iii) is a technical assumption which allows us to perform continuous approximation in Lemma 3.3. This approximation in particular enables us to establish the superhedging duality in Proposition 3.10. In fact, if we only have conditions (i) and (ii), the duality in Proposition 3.10 may fail in general, as demonstrated in Appendix A.
An explained below, Definition 2.7 (iii) covers a large class of convex constraints.
Remark 2.9 (Adapted convex constraints)
Let be an adapted set-valued process such that for each , maps to a closed convex set which contains . Consider the collection of trading strategies
This is equivalent to the following condition:
To check (iii), let us fix , , and . For each , by Lusin’s theorem, there exists a closed set such that is continuous and . Under (2.5), we can apply the theory of continuous selection (see e.g. (Michael56, , Theorem 3.2)) to find a bounded continuous function such that on . Now, set , and define , which is by definition closed in . We see that , , and on for all . This already verifies Definition 2.7 (iii).
3 The superhedging duality
For a path-dependent exotic option with payoff function , we intend to construct a semi-static superhedging portfolio, which consists of three parts: static positions in vanilla calls, static positions in , and a dynamic trading strategy . More precisely, consider
We intend to find , , and such that
where we assume that . In the definition of , we specifically require to be strictly positive. This is because corresponds to trading the risky assets, which is already incorporated into and should not be treated as part of the static positions. By setting as the collection of all , we define the superhedging price of by
By introducing , we may express (3.2) as
Our goal in this section is to derive a superhedging duality associated with .
3.1 Upper variation process
In order to deal with the portfolio constraint , we introduce an auxiliary process for each , as suggested in (FS-book-11, , Section 9.2).
Given , the upper variation process for is defined by
First, Note that the conditional expectation in the definition of is well-defined, thanks to Remark 2.1. Next, since Definition 2.7 (i)-(ii) implies whenever , we may replace by in the above definition. It follows that
For any and , we have
This in particular implies that
For each ,
In view of (3.5), it suffices to show that, for each fixed , there exists such that . Take such that for all . By Definition 2.7 (iii), for any , there exist closed in and such that , on , and for all . It follows that
Thanks to Remark 2.1, the random variable is -integrable. We can then conclude from the above inequality that . ∎
Let be the collection of such that .
Given any , the process is a local -supermartingale, for all .
This result follows from the argument in (FS-book-11, , Proposition 9.18). We present the proof here for completeness. Consider the stopping time
where the conditional expectation is well-defined thanks to Remark 2.1. Given a process , let us denote by the stopped process . Observe that
Thanks again to Remark 2.1, this implies that is -integrable. Moreover,
where the inequality follows from (3.3). Since , the above inequality shows that is a -supermartingale. ∎
With some integrability at the terminal time , the local supermartingale in the above result becomes a true supermartingale.
Fix and . If -integrable random variable such that -a.s., then for all . This in particular implies that is a true -supermartingale.
Using the same notation as in the proof of Lemma 3.6, we know that there exist a sequence of stopping times such that -a.s. and the stopped process is a -supermartingale, for each . We will prove this lemma by induction. Given any such that , we obtain from the supermartingale property that
3.2 Derivation of the superhedging duality
In view of the static holdings of in (3.1), we introduce
Set . We observe that
Consider the collection of measures
Fix . For any , suppose the following two conditions hold.
for some or ,
for some or .
By the convexity of , we have . Thus, in view of (3.7), if is a finite set, and (i)-(ii) above are satisfied for all , then .
We will work on deriving a duality between defined in (3.2) and
The following minimax result, taken from (Terkelsen72, , Corollary 2), will be useful.
Let be a compact convex subset of a topological vector space, be a convex subset of a vector space, and be a function satisfying
For each , the map is convex on .
For each , the map is upper semicontinuous and concave on .
Let us first derive a superhedging duality for the case where , i.e. no option is tradable at time except vanilla calls. The pathwise relation in (3.1) reduces to