No-arbitrage and hedging with liquid American options

No-arbitrage and hedging with liquid American options

Abstract.

Since most of the traded options on individual stocks is of American type it is of interest to generalize the results obtained in semi-static trading to the case when one is allowed to statically trade American options. However, this problem has proved to be elusive so far because of the asymmetric nature of the positions of holding versus shorting such options. Here we provide a unified framework and generalize the fundamental theorem of asset pricing (FTAP) and hedging dualities in [5] to the case where the investor can also short American options. Following [5], we assume that the longed American options are divisible. As for the shorted American options, we show that the divisibility plays no role regarding arbitrage property and hedging prices. Then using the method of enlarging probability spaces proposed in [12], we convert the shorted American options to European options, and establish the FTAP and sub- and super-hedging dualities in the enlarged space both with and without model uncertainty.

Key words and phrases:
semi-static trading strategies, Liquid American options, Fundamental theorem of asset pricing, sub/super hedging dualities
E. Bayraktar is supported in part by the National Science Foundation under grant DMS-1613170 and by the Susan M. Smith Professorship.

1. Introduction

Recently there has been some fundamental work on no-arbitrage and hedging in a financial market where stocks are traded dynamically and liquid options are traded statically (semi-static strategies), see e.g., [11, 7, 8, 1] and the references therein. Even though, [3, 16, 15, 6, 12], consider the problem of hedging American options in this framework, it is worth noting that in all the above papers the liquid options are restricted to be European-style. But since most options on individual stocks are of American type, it is of practical interest to consider these problems when one can use the American options for hedging purposes. So far only three papers considered American options as hedging devices: [9] studies the completeness of the market where American put options of all the strike prices are available for semi-static trading, [10] studies the no arbitrage conditions on the price function of American put options where European and American put options are available, [5] considers FTAP and hedging duality with liquid American options.

The difficulty of using American options in semi-static trading lies in the asymmetric nature of positions of holding versus shorting this option. Our starting point in this paper is [5], where we assume that the liquid American options can only be bought, but not sold, and only the sub-hedging price but not the super-hedging price of the hedged American option is considered. The reason is that, if liquid American options are sold, or if the super-hedging is considered, then the investor needs to use a trading strategy that is adapted to the stopping strategy used by the holders of the American options. From this point of view, the problem becomes more complicated. In this paper, we resolve these difficulties and generalize the FTAP and hedging results to the case where shorting American options is also allowed, hence creating a unified treatment of the problem. In particular, we assume that there are liquid American options that can be sold, and we also consider the super-hedging price of an American option.

We assume that the longed American options are divisible, and as demonstrated by [5, Section 2], this is a crucial assumption for obtaining the FTAP and subhedging duality. We show in this paper that the divisibility plays no role for the shorted American options regarding arbitrage and hedging. Then using the method of enlarging probability spaces proposed in [12] (also see [2] for its second version), we convert the shorted American options to European options. We establish the FTAP and sub- and super-hedging dualities for American options for models with and without model uncertainty.

A main contribution of this paper is that it provides a unified framework for the FTAP and hedging dualities when American options are available for both static buying and selling. There is extensive literature on FTAP and hedging duality with liquid European options. On the other hand, due to the flexibility of American options, conceptually and technically it is much more difficult to take liquid American options into consideration. Yet most of options in the market are of American style. As far as we know, before this paper there are only three papers [9, 10, 5] consider American options being liquid options, yet none of them provides such a general framework, not to mention the FTAP and hedging duality results in this framework.

We assume options are bounded. This is because our proof crucially relies on the compactness of liquidating strategies under the weak star topology (or Baxter-Chacon topology, see e.g., [13]). The boundedness of American options is needed in order to use this weak star topology to prove the FTAP and hedging results without model uncertainty, as well as apply some minimax argument for the proof of hedging dualities with model uncertainty. One novelty of this paper lies in the incorporation of liquid American options, and liquidating strategies for American options is crucial and also practical for the FTAP and hedging results. We think such boundedness assumption is not restrictive. First of all, what we have in mind for liquid options are put options, which are bounded (e.g., [10] considers American put options). Second, we are in a finite discrete time set-up, and within finitely many time steps, even the stock prices could be reasonably assumed to be bounded, not to mention the option payoffs.

Another assumption is the continuity of options in the case of model uncertainty. Such assumption can be expected. First, a discretization argument is used for the proof of hedging dualities, and continuity is essential for the discretization. Second, for each (longed) American option we have infinitely many possible payoffs associated with this American option, due to infinitely many possible liquidating strategies. In this sense, we can think of one American option as infinitely many European options (but with merely one price). For the existing literature on FTAP and hedging with infinitely many European options in a model-free or model uncertainty setup, some continuity assumption of options is often imposed. See e.g., [12, 1, 7]. In fact, our continuity assumption is weaker in the sense that even though we assume American options at each period are continuous in , their payoffs may still be discontinuous in because liquidating strategies may not be continuous. We again think such continuity assumption is not restrictive, since in practice most of options are (semi-)continuous.

The rest of the paper is organized as follows. In the next section, we first show that it makes no difference whether the shorted American options are divisible or not for the definitions of no-arbitrage and hedging prices. Then we work on an enlarged space and establish the FTAP and hedging dualities for a given model. In Section 3, we extend the FTAP and hedging dualities to the case of model uncertainty.

2. No-arbitrage and hedging without model ambiguity

In this section, we first describe the setup of our financial model without model ambiguity. We show that it makes no difference for arbitrage and hedging prices whether the shorted (not longed) American options are divisible or not. Then we reformulate the problems of arbitrage and hedging in an enlarged probability space, and establish FTAP and hedging dualities. Theorems 2.1-2.4 are the main results of this section.

2.1. Original probability space

Let be a filtered probability space, where is assumed to be separable, and represents the time horizon in discrete time. Let be an adapted process taking values in which represents the stock prices. Let , , be -measurable, representing the payoffs of European options. Let , ; , , be -adapted processes, representing the payoff processes of American options. We assume that we can only buy but not sell each and at time with price and respectively, and we can only sell but not buy each at time with price . Denote , , and for scalar . Similarly we will use and for denoting vectors (of processes and prices). For simplicity, we assume that are bounded. Let be -adapted, representing the payoff of the American options whose sub/superhedging prices we are interested in calculating by semi-statically trading in . For simplicity, we assume that is bounded. We call and the sub-hedged longed American options, and and the super-hedged shorted American options.

Remark 2.1.

Here may represent the options whose trade is quoted with bid-ask spreads. For example, for an American option with bid price and ask price , we can treat it as two American options, one that can only be bought at price , and the other that can only be sold at price .

We assume that the European options are only available to buy. This is in fact without loss of generality, because to short a European option is equivalent to long a European option . On the other hand, unlike European options, the treatments of American options that are bought and sold are very different, and that is the reason we separate American options into and .

We will often consider two cases, , where represents the number of shorted American options. To be more specific, for , the FTAP or sub-hedging will be considered, and there are shorted American options involved including . For , the super-hedging of an American option will be considered, and there are shorted American options including and the super-hedged American option.

If no American options are shorted (i.e., and we consider the sub-hedging ), then the only information the investor can observe is , and hence she will use a dynamic trading strategy in the stocks that is -adapted. Moreover, motivated by [5, Section 2], we assume that the American options and (for sub-hedging) are divisible. That is, the investor can break each unit American options into pieces, and exercise each piece separately. We use the phrase liquidating strategy to describe this type of exercise policy. it is more precisely defined as follows:

Definition 2.1.

An -adapted process is said to be an -liquidating strategy, if for , and . Denote as the set of all -liquidating strategies.

Due to the existence of shorted American options and (recall that is shorted when we consider the problem of super-hedging), the investor’s dynamic trading strategy for stock and liquidating strategy for longed American option should also be adapted to the stopping/liquidating strategies chosen by the holders of and . Moreover, since options and sub-hedged are assumed to be divisible, it is natural to assume that options (and super-hedged ) are also divisible. This assumption makes practical sense: the investor may sell to several agents, or sell to the same agent a large shares of . However, as we will demonstrate, in terms of no-arbitrage and hedging prices (which we will define in the next section), it is in fact sufficient to let all shares of (and super-hedged ) be exercised once (i.e., holder of uses stopping times).

2.2. Discussion of divisibility for shorted American options for no-arbitrage and hedging

In this sub-section we show that whether the shorted American options are divisible or not, the definitions of no-arbitrage and hedging prices coincide.

Definitions of no arbitrage and hedging prices with divisibility

Let be the set of -adapted processes taking values in . Let

 V:={(v0,…,vT)∈RT+1+: T∑t=0vt=1},

which represents the space of liquidating strategies for each shorted American option. Let

 ^Hn:={^H(⋅):Vn↦H: ^Hr(v1,…,vn)=^Hr(u1,…,un), if vkt=ukt  for  t=0,…,r; k=1,…,n},

and

 ^Ln:={^η(⋅):Vn↦L:^ηr(v1,…,vn)=^ηr(u1,…,un), if vkt=ukt  for t=0,…,r; k=1,…,n},

where for , and is the -fold Cartesian product of . Denote the set of semi-static trading strategies

 ^An:={(^H,a,b,^μ,c): (a,b,c)∈RL+×RM+×RN+, ^H∈^Hn, ^μ∈(^Ln)M}.

For , denote the payoff using semi-static trading strategy w.r.t. the prices , with the realization for shorted American options,

 ^Φnα,β,γ(^H,a,b,^μ,c)(v):=^H(v)⋅S+a(f−α)+b((^μ(v))(g)−β)−N∑k=1ck(vk(hk)−γk), (2.1)

where for ,

 H⋅S:=T−1∑t=0Ht(St+1−St),

and for ,

 μ(g):=(μ1(g1),…,μM(gM))

with

 μj(gj):=T∑t=0gjtμjt.

In the above equations we used denoted the inner product of vectors, say and , by . Here at the right-hand-side of (2.1), the first term represents the payoff from trading stocks, second term the payoff from trading European options, third term the payoff from trading longed American options. and last term the payment for shorted American options.

Definition 2.2 (No arbitrage).

We say NA holds w.r.t. the prices , if for any

 ^ΦNα,β,γ(^H,a,b,^μ,c)(v)≥0,P-a.s. for any v∈VN,

implies

 ^ΦNα,β,γ(^H,a,b,^μ,c)(v)=0,P-a.s. for any v∈VN.

We say SNA (SNA stands for “strict no arbitrage”) holds, if there exists such that NA holds w.r.t. the prices .

Definition 2.3 (Hedging prices).

We define the sub-hedging price of by

 π––1(ϕ) := sup{x∈R: ∃(^H,a,b,^μ,c)∈^AN and ^η∈^LN, s.t. ^ΦNα,β,γ(^H,a,b,^μ,c)(v)+(^η(v))(ϕ)≥x P-a.s.% , ∀v∈VN},

and its super-hedging price by

 ¯¯¯π1(ϕ) := inf{x∈R: ∃(^H,a,b,^μ,c)∈^AN+1, s.t. x+^ΦN+1α,β,γ(^H,a,b,^μ,c)(v)≥vN+1(ϕ) P-a.s., ∀v=(v1,…,vN+1)∈VN+1},

For a European option , define its sub-hedging price as

 π––1e(ψ):=sup{x∈R: ∃(^H,a,b,^μ,c)∈^AN, s.t. ^ΦNα,β,γ(^H,a,b,^μ,c)(v)+ψ≥x P-a.s% ., ∀v∈VN}.

Let us clarify that should be applied before in the above definition.

Definitions of no arbitrage and hedging prices without divisibility

For , let

 ~Hn:={~H(⋅):Tn↦H: ~Hr(t1,…,tn)=~Hr(s1,…,sn) % for r

and

 ~Ln:={~η(⋅):Tn↦L:~ηr(t1,…,tn)=~ηr(s1,…,sn) for r

where , and

 r∗=infk∈I(sk∧tk)withI={i∈{1,…,n}: si≠ti}.

Denote the set of semi-static trading strategies by

 ~An:={(~H,a,b,~μ,c): (a,b,c)∈RL+×RM+×RN+, ~H∈~Hn, ~μ∈(~Ln)M}.

and the payoff using semi-static trading strategy w.r.t. the prices , with the realization for shorted American options by

 ~Φnα,β,γ(~H,a,b,~μ,c)(t):=~H(t)⋅S+a(f−α)+b((~μ(t))(g)−β)−N∑k=1ck(hktk−γk).
Definition 2.4 (No-arbitrage).

We say NA holds w.r.t. the prices , if for any

 ~ΦNα,β,γ(~H,a,b,~μ,c)(t)≥0,P-a.s. for any t∈TN,

implies

 ~ΦNα,β,γ(~H,a,b,~μ,c)(t)=0,P-a.s. for any t∈TN.

We say SNA holds, if there exists such that NA holds w.r.t. the prices .

Definition 2.5 (Hedging prices).

We define the sub-hedging price of as

 π––2(ϕ) := sup{x∈R: ∃(~H,a,b,~μ,c)∈~AN and ~η∈~LN, s.t. ~ΦNα,β,γ(~H,a,b,~μ,c)(t)+(~η(t))(ϕ)≥x P% -a.s., ∀t∈TN},

and its super-hedging price as

 ¯¯¯π2(ϕ) := inf{x∈R: ∃(~H,a,b,~μ,c)∈~AN+1, s.t. x+~ΦN+1α,β,γ(~H,a,b,~μ,c)(t)≥ϕtN+1 P-a.s., ∀t=(t1,…,tN+1)∈TN+1}.

For a European option , define its sub-hedging price as

 π––2e(ψ):=sup{x∈R: ∃(~H,a,b,~μ,c)∈~AN, s.t. ~ΦNα,β,γ(~H,a,b,~μ,c)(t)+ψ≥x P-a.s., ∀t∈TN}.

The Equivalence of the no-arbitrage definitions and the hedging prices

We have and and

Proof.

For the simplicity of presentation, we will only show for and . The proof can be very easily adapted for the more general case.

Since is clear, we focus on the reverse inequality. Let , then there exists such that for any ,

 ~H(t1,t2)⋅S−c1(h1t1−γ1)−c2(h2t2−γ2)+ψ≥x,P-a.s..

Define ,

 ^Hr(u,v)=T∑s=0T∑t=0usvt~Hr(s,t),u=(u0,…,uT),v=(v0,…,vT)∈V.

For any , if for , and , then

 ^Ht∗(u,v)=T∑s=0T∑t=0usvt~Ht∗(s,t) =t∗∑s=0t∗∑t=0usvt~Ht∗(s,t)+T∑s=t∗+1t∗∑t=0usvt~Ht∗(s,t)+t∗∑s=0T∑t=t∗+1usvt~Ht∗(s,t)+T∑s=t∗+1T∑t=t∗+1usvt~Ht∗(s,t) =t∗∑s=0t∗∑t=0usvt~Ht∗(s,t)+T∑s=t∗+1ust∗∑t=0vt~Ht∗(T,t)+T∑t=t∗+1vtt∗∑s=0us~Ht∗(s,T)+T∑s=t∗+1usT∑t=t∗+1vt~Ht∗(T,T) =t∗∑s=0t∗∑t=0u′sv′t~Ht∗(s,t)+T∑s=t∗+1u′st∗∑t=0v′t~Ht∗(T,t)+T∑t=t∗+1v′tt∗∑s=0u′s~Ht∗(s,T)+T∑s=t∗+1u′sT∑t=t∗+1v′t~Ht∗(T,T) =t∗∑s=0t∗∑t=0u′sv′t~Ht∗(s,t)+T∑s=t∗+1t∗∑t=0u′sv′t~Ht∗(s,t)+t∗∑s=0T∑t=t∗+1u′sv′t~Ht∗(s,t)+T∑s=t∗+1T∑t=t∗+1u′sv′t~Ht∗(s,t) =T∑s=0T∑t=0u′sv′t~Ht∗(s,t)=^Ht∗(u′,v′),

where for the third and fifth equalities we use the non-anticipativity of (see the definition of ). This implies . Now for any ,

 ^H(u,v)⋅S−c1(u(h1)−γ1)−c2(v(h2)−γ2)+ψ =T−1∑r=0T∑s=0T∑t=0usvt~Hr(s,t)(Sr+1−Sr)−c1T∑s=0us(h1s−γ1)−c2T∑t=0vt(h2t−γ2)+ψ =T∑s=0T∑t=0usvt[~H(s,t)⋅S−c1(h1s−γ1)−c2(h2t−γ2)+ψ] ≥x,P-a.s..

This implies that . By the arbitrariness of , we have . ∎

Theorem 2.2.

SNA and SNA are equivalent.

Proof.

Let SNA hold. Denote the sub-hedging price of each (resp. sub-hedging price of each , super-hedging price of each ) using stock and other liquid options by (resp. , ). We will not differentiate the hedging prices for type 1 and type 2 since by Theorem 2.1 they are the same. By SNA, there exists , such that NA holds w.r.t. the prices . This implies

 αi−ε≥π––′e(fi),βj−ε≥π––′(gj),γk+ε≤¯¯¯π′(hk). (2.2)

Suppose SNA fails, then for any there would exist such that

 ^ΦNα−1m,β−1m,γ+1m(^Hm,am,bm,^μm,cm)(v)≥0,P-a% .s. ∀v∈VN, (2.3)

and such that

If , then we would have that

 ^H(vm)⋅S≥0, P-a.s.,andP{^H(vm)⋅S>0}>0. (2.4)

SNA implies that for any , if -a.s. then -a.s.. This contradicts (2.4). Therefore, at least one of is not zero. Denote

 dm:=max{ami,bmj,cmk, i=1,…,L, j=1,…,M, k=1,…,N}>0.

By (2.3),

 1dm^ΦNα,β,γ(^Hm,am,bm,^μm,cm)(v)+L+M+Nm≥0,P-a.s.%  ∀v∈VN. (2.5)

For each at least one of is equal to . Without loss of generality, (up to a sub-sequence) assume . Then by (2.5),

 π––′e(f1)≥α1−L+M+Nm.

Since is arbitrary we have that , which contradicts (2.2). This shows that SNA implies SNA.

We can show that SNA implies SNA using a similar argument. ∎

Remark 2.2.

In terms of arbitrage and hedging prices, divisibility are essential for and the sub-hedged as indicated by [5, Section 2], but not essential for and the super-hedged as indicated by Theorems 2.1 and 2.2. Therefore in the rest the paper we will assume that the shorted American options and super-hedged are not divisible.

2.3. Enlarged probability space

We will follow the method in [12] to reformulate the problems of arbitrage and hedging in an enlarged space. The advantage for working on the enlarged space is that the shorted American options become European options.

We will again let or . The case when is for FTAP and sub-hedging , i.e., either is not involved or the investor longs . The case for is for super hedging . Let . Here (resp. ) represents the space of the exercise times for shorted American options (resp. and for super-hedging). For , let ,

 θk(¯¯¯ωn)=tk,¯¯¯ωn=(ω,t1,…,tn)∈¯¯¯¯Ωn. (2.6)

We extend from to , i.e., for . We similarly extend and (for sub-hedging, i.e., when ) and we denote the extensions as , respectively. For , we extend from to ,

 ¯¯¯¯¯hkn(¯¯¯ωn):=hktk(ω),¯¯¯ωn=(ω,t1,…,tn)∈¯¯¯¯Ωn.

Similarly, (for super-hedging) we extend from to ,

 ¯¯¯ϕN+1(¯¯¯ωN+1):=ϕtN+1(ω),¯¯¯ωN+1=(ω,t1,…,tN+1)∈¯ΩN+1.
Remark 2.3.

The extensions and serve different roles: is an American option, which will be considered in the sub-hedging problem, while is a European option, which will be considered in the super-hedging problem.

Next, let us define the enlarged filtration. For ,

 ¯¯¯¯¯Fnt:=σ(Ft×Tn,{θk≤s},s=0,…,t, k=1,…,n),

where . Denote .

Finally, for , let be any probability measure on 1 with full support. That is, for any

 Pn({t1,…,tn})>0.

Let . will serve as the enlarged filtered probability space.

Let (resp. ) be the set of -adapted (resp. -adapted) processes, representing the set of dynamic trading strategies for the stock based on information as well as the exercise times of (resp. and for super-hedging). Similarly, let be the set of -liquidating strategies. Below we give the definition of semi-static trading strategies in the enlarged filtered probability space.

Definition 2.6.

For , a quintuplet is said to be an -semi-static trading strategy, if , , and . Denote as the set of -semi-static trading strategies.

The payoff using semi-static trading strategy w.r.t. the prices is given by

 ¯¯¯¯Φnα,β,γ(¯¯¯¯¯H,a,b,¯¯¯μ,c):=¯¯¯¯¯H⋅¯¯¯¯Sn+a(¯¯¯fn−α)+b(¯¯¯μ(¯¯¯gn)−β)−c(¯¯¯hn−γ).

2.4. FTAP and hedging dualities

Definition 2.7 (No arbitrage).

For , we say no arbitrage (NA) holds in