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Risks 2014, 2(4), 425-433; doi:10.3390/risks2040425
Abstract: We show that the recent results on the Fundamental Theorem of Asset Pricing and the super-hedging theorem in the context of model uncertainty can be extended to the case in which the options available for static hedging (hedging options) are quoted with bid-ask spreads. In this set-up, we need to work with the notion of robust no-arbitrage which turns out to be equivalent to no-arbitrage under the additional assumption that hedging options with non-zero spread are non-redundant. A key result is the closedness of the set of attainable claims, which requires a new proof in our setting.
We consider a discrete time financial market in which stocks are traded dynamically and options are available for static hedging. We assume that the dynamically traded asset is liquid and trading in them does not incur transaction costs, but that the options are less liquid and their prices are quoted with a bid-ask spread. (The more difficult problem with transaction costs on a dynamically traded asset is analyzed in  and .) As in  we do not assume that there is a single model describing the asset price behavior but rather a collection of models described by the convex collection of probability measures, which does not necessarily admit a dominating measure. One should think of as being obtained from calibration to the market data. We have a collection rather than a single model because generally we do not have point estimates but a confidence intervals for the parameters of our models. Our first goal is to obtain a criteria for deciding whether the collection of models represented by is viable or not. Given that is viable we would like to obtain the range of prices for other options written on the dynamically traded assets. The dual elements in these result are martingale measures that price the hedging options correctly (i.e., consistent with the quoted prices). As in classical transaction costs literature, we need to replace the no-arbitrage condition by the stronger robust no-arbitrage condition, as we shall see in Section 2. In Section 3 we will make the additional assumption that the hedging options with non-zero spread are non-redundant (see Definition 3.1). We will see that, under this assumption, no-arbitrage and robust no-arbitrage are equivalent. Our main results are Theorems 2.1 and 3.1.
2. Fundamental Theorem with Robust No Arbitrage
Let be the prices of d traded stocks at time and be the set of all predictable -valued processes, which will serve as our trading strategies. Let be the payoff of e options that can be traded only at time zero with bid price and ask price , with (the inequality holds component-wise). We assume and g are Borel measurable, and there are no transaction costs in the trading of stocks.
(No-arbitrage and robust no-arbitrage). We say that condition NA() holds if for all ,
We say that condition holds if there exists such that and NA() holds if g has bid-ask prices .3
(Super-hedging price). For a given a random variable f, its super-hedging price is defined as
 Let and be the super-hedging prices of and , where the hedging is done using stocks and options excluding . NAr implies either
 Clearly implies NA, but the converse is not true. For example, assume in the market there is no stock, and there are only two options: . Let be the set of probability measures on Ω, , and . Then NA holds while fails.
For , let
Let be a random variable such that . The following statements hold:
(Fundamental Theorem of Asset Pricing): The following statements are equivalent
There exists such that , such that .
(Super-hedging) Suppose holds. Let be Borel measurable such that . The super-hedging price is given by
It is easy to show in (a) implies that NA() holds for the market with bid-ask prices , Hence holds for the original market. The rest of our proof consists two parts as follows.
and the existence of an optimal hedging strategy in (b). Once we show that the set
Write , where consists of the hedging options without bid-ask spread, i.e, for , and consists of those with spread, i.e., for , for some . Denote and similarly for and . Define
Thus is bounded, and has a limit along some subsequence . Since by (2.4)
in part (a) and (3.3) in part (b). We will prove the results by an induction on the number of hedging options, as in (, [Theorem 5.1]). Suppose the results hold for the market with options . We now introduce an additional option with , available at bid-ask prices at time zero. (When the bid and ask prices are the same for f, then the proof is identical to .)
in (a): Let be the super-hedging price when stocks and are available for trading. By and (3.3) in part (b) of the induction hypothesis, we have
Now, let . (a) of induction hypothesis implies the existence of a satisfying . Define
(3.3) in (b): Let ξ be a Borel measurable function such that . Write for its super-hedging price when stocks and are traded, and . We want to show
First, we show . It suffices to show the existence of a sequence such that and . (See page 30 of  for why this is sufficient.) In other words, we want to show that
It follows from (2.10) that . Therefore, we must have that , otherwise (since the super-hedging price is positively homogenous). Recall that we have proved in part (a) that . Let . The part of (2.10) after the equality implies that . Since , we get . Since , . But by (2.8), , which is a contradiction.
Next, we show . It suffices to show for any and every , we can find such that . To this end, let which is nonempty by part (a). Define
Finally, when ξ is not bounded from above, we can apply the previous result to , and then let and use the closedness of in (2.3) to show that (3.3) holds. The argument would be the same as the last paragraph in the proof of [3, Thoerem 3.4] and we omit it here. ☐
3. A Sharper Fundamental Theorem with the Non-Redundancy Assumption
We now introduce the concept of non-redundancy. With this additional assumption we will sharpen our result.
(Non-redundancy). A hedging option is said to be non-redundant if it is not perfectly replicable by stocks and other hedging options, i.e., there does not exist and a semi-static hedging strategy such that
Suppose all hedging options with non-zero spread are non-redundant. Then NA implies .
Let , where consists of the hedging options without bid-ask spread, i.e, for , and consists of those with bid-ask spread, i.e., for . We shall prove the result by induction on s. Obviously the result holds when . Suppose the result holds for . Then for , denote , and . Denote .
By the induction hypothesis, there exists be such that NA holds in the market with stocks, options u and options v with any bid-ask prices b and a satisfying . Let , , and , such that , , and . We shall show that for some n, NA holds with stocks, options u, options v with bid-ask prices and , option f with bid-ask prices and . We will show it by contradiction.
If not, then for each n, there exists such that
We have the following FTAP and super-hedging result in terms of NA instead of , when we additionally assume the non-redundancy of g.
Suppose all hedging options with non-zero spread are non-redundant. Let be a random variable such that . The following statements hold:
(Fundamental Theorem of Asset Pricing): The following statements are equivalent
, such that .
(Super-hedging) Suppose NA holds. Let be Borel measurable such that . The super-hedging price is given by
(a’)(ii)⇒(a’)(i) is trivial. Now if (a’)(i) holds, then by Lemma 3.1, (a)(i) in Theorem 2.1 holds, which implies (a)(ii) holds, and thus (a’)(ii) holds. Finally, (b’) is implied by Lemma 3.1 and Theorem 2.1(b). ☐
Theorem 3.1 generalizes the results of  to the case when the option prices are quoted with bid-ask spreads. When is the set of all probability measures and the given options are all call options written on the dynamically traded assets, a result with option bid-ask spreads similar to Theorem 3.1-(a) had been obtained by ; see Proposition 4.1 therein, although the non-redundancy condition did not actually appear. (The objective of  was to obtain relationships between the option prices which are necessary and sufficient to rule out semi-static arbitrage and the proof relies on determining the correct set of relationships and then identifying a martingale measure.)
However, the no arbitrage concept used in  is different: the author there assumes that there is no weak arbitrage in the sense of ; see also  and .5 (Recall that a market is said to have weak arbitrage if for any given model (probability measure) there is an arbitrage strategy which is an arbitrage in the classical sense.) The arbitrage concept used here and in  is weaker, in that we say that a non-negative wealth (-q.s.) is an arbitrage even if there is a single P under which the wealth process is a classical arbitrage. Hence our no-arbitrage condition is stronger than the one used in . But what we get out from a stronger assumption is the existence of a martingale measure for each . Whereas  only guarantees the existence of only one martingale measure which prices the hedging options correctly.
This research is supported by the National Science Foundation under grant DMS-0955463.
Conflicts of Interest
The authors declare no conflict of interest.
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- 1A set is -polar if it is -null for all P ∈ . A property is said to hold -q.s. if it holds outside a -polar set.
- 2When we multiply two vectors, we mean their inner product.
- 3“ri” stands for relative interior. means component-wise inclusion.
- 4 means for all .
- 5The no-arbitrage assumption in  is the model independent arbitrage of . However that paper rules out the model dependent arbitrage by assuming that a superlinearly growing option can be bought for static hedging.
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