Suppose that a pricing ruleΠis given on the market. Consider now a set ofdbenchmark price processesS1,ã ã ã,Sd(the so-calledunderlyings).
We will now show how to recover the local-martingale or σ−martingale properties forS=(S1,ã ã ã,Sd) (see e.g. [4, 13, 10]) within our framework.
In the usual approach, to build gain processes of trading strategies as stochastic integrals one requires that S is an Rd-valued semimartingale with respect to the reference probabilityP. In our model-free context the natural counterpart is the assumption thatSis aQ-semimartingale. One can then introduce stochastic integrals with respect toSand define a notion of replicating strategy:
Definition 4.1. Given a pricing ruleΠ on the market, represented by a martingale measure Q and an Rd-valuedQ-semimartingale S, a payoff H ∈ L0 is said to beS-replicable if there exist ax ∈ Rand a predictable
process (strategy)ϕsuch that:
1. ϕisS-integrable underQ.
2. Q(Πt(H)=x+t
0 ϕdS)=1.
Remark 4.1. In the above definition and in what follows probabilistic no- tions are induced by the pricing rule through its representingQ.
Delbaen and Schachermayer [10] linked the No Free Lunch with Vanishing Risk property underPwith the existence of a probability measure equiv- alent toPunder whichSis aσ-martingale, a notion introduced in [5]. We will now show how theσ-martingale property appears in our context.
Let us recall the following result from Emery [11]:
Proposition 4.1. [11] Let S be a d-dimensional semimartingale on (Ω,F,(Ft)t∈[0,T],Q)and denote by L(S)(Q)the set of predictable and S-integrable processes under the probability measureQ. The following assertions are equiva- lent:
1. there exist a d-dimensionalQ- martingale N and a positive (scalar) process ψ∈ ∩1≤i≤dL(Ni)(Q)such that Si=
ψdNi;
2. there exists a countable predictable partition (Bn)n of Ω×R+ such that IBndSiis aQ-martingale for every i,n;
3. there exist (scalar) processesηi with paths thatQ−a.s.never touch zero, such thatηi∈L(Si)(Q)and
ηidSiis aQ-local martingale.
Definition 4.2. We say thatSis aσ-martingale underQif it satisfies any of the equivalent conditions of the above Proposition.
The above equivalences illustrate that theσ-martingale property is a gen- eralization of the local martingale property.
Remark 4.2. Whenever the the (Bn)ncan be written as stochastic intervals ]Tn,Tn+1] whereTnis a sequence of stopping times increasing to+∞, then the previous definition coincides with that of local martingale.
If for someithe processSiis not the market price of a traded asset (but, for instance, a non-traded risk factor such as an instantaneous forward rate or instantaneous volatility process) thenSi isnot necessarily a martingale.
However, the result by Emery allows us to recover theσ-martingale fea- tures ofSunderQfrom a straightforward analysis ofthe market spanned by S.Roughly speaking, there must be a traded derivativeHwith underlying S, which isS-replicable via a hedging strategy that is always non zero:
Proposition 4.2. Suppose that for all i there exists an Si-replicable derivative Hitraded in the market with a strategy(ϕit)t∈[0,T] thatQ-a.s. never touches zero.
Then S is aσ-martingale underQ.
Proof. SinceHiis traded with market priceVi= Π(Hi), our Theorem 3.1 implies that the gain
ϕidSiis aQ-martingale. Then, given the assumption on theϕis,Sis aσ-martingale underQfrom a direct application of item 3, Proposition 4.1.
Remark 4.3. (The ’No Free Lunch with Vanishing Risk’ property) IfS is indeed a σ−martingale underQ, then the market spanned bySsatisfies the NFLVR condition with respect to Q(and henceforth with respect to anyP∼Q). In fact, consider theQ-admissible strategiesϕi.e. whose gain processes areQ-almost surely bounded from below:
∃c>0,Q(
ϕdS≥ −c)=1
IfSis aσ-martingale underQ, such strategies give rise to gain processes which areQ-supermartingales (see e.g. [10]). Hence absence of arbitrage obviously holds, sinceEQ[T
0 ϕdS]≤0. An application of Fatou’s Lemma then shows that NFLVR also holds.
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A Class of Financial Products and Models Where Super-replication Prices are Explicit
L. Carassus1, E. Gobet2, E. Temam1
1Laboratoire de Probabilit´es et Mod`eles Al´eatoires - Universit´e Paris 7 - Denis Diderot - Case 7012 - 2, place Jussieu -
75251 Paris cedex 05 - France
2Laboratoire Jean Kuntzman - ENSIMAG INP Grenoble - BP 53 - 38041 Grenoble cedex 09 - France
We consider a multidimensionalfinancial model with mild con- ditions on the underlying asset price process. The trading is only allowed at somefixed discrete times and the strategy is constrained to lie in a closed convex cone. We show how the minimal cost of a super hedging strategy can be easily computed by a backward recursive scheme. As an application, when the underlying as- set follows a stochastic differential equation including stochastic volatility or Poisson jumps, we compute those super-replication prices for a range of European and American style options, includ- ing Asian, Lookback or Barrier Options. We also perform some multidimensional computations.
Key words:Closed formula for Super-replication cost; convex cone constraints on portfolio; exotic European and American options.
AMS Classification:90A09, 60G42, 26B25.
1. Introduction
We consider afinancial market consisting of d risky assets with dis- counted price process denoted byS, and one risk-less bond: the trading is allowed only atfixed discrete times. We assume that the trading strategies are also subject to portfolio constraints. Namely, given a closed convex coneK with vertex in 0, the vector of number of shares invested in the risky assets is constrained to lie inK. Such formalization includes in par- ticular incomplete markets and markets with short-selling constraints. It is well-known that in those contexts, it is not possible to define an unique
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fair price, i.e. the initial cost of a strategy replicating a given contingent claim, as in the context of complete markets. A possible way of defining a price is to consider the minimal initial wealth needed to hedge with- out risk the contingent claim. This is called the super-replication cost and has been introduced in the binomial setup for transaction costs by Bensaid-Lesne-Pag`es-Scheinkman [1], in aL2-setup for transaction costs and short-sales constraints by Jouini-Kallal [14, 15] and in the diffusion setup for incomplete markets by El Karoui-Quenez [9]. In the context of convex constraints, this notion has been studied among others by Cvitani´c- Karatzas [4], Karatzas-Kou [17], Broadie-Cvitani´c-Soner [3] and in a great generality by F¨ollmer-Kramkov [11]. In those papers a dual formulation is given. Namely, the super-replication cost of an European contingent claim,H, is essentially the supremum over a given set of probability of the expectation ofH(or a modification ofH). Nevertheless this dual formula- tion does not enable in general to effectively compute the super-replication price.
Here we combine primal and dual formulations, in order to provide a closed formula for European and American style options under general assumptions on the underlyingS(namely, an usual non degeneracy condi- tion), and also to give the hedging strategy. In the case of European vanilla options,finding the super-replication price reduces to compute some con- cave envelop of the payofffunction. For more general options, it involves recursive computations using again kind of concave envelops. The coeffi- cients of the affine function which appears in the concave envelop give the hedging strategy. The application of this algorithm turns to be simple to derive the super-replication prices of all usual options. Patry [20] obtains a similar formula, in the Black-Scholes case, for an European vanilla option.
Our effective computation shows that, when the asset prices can heavily fluctuate, the super-replication prices are trivial in the sense that they correspond to basic strategies such as ”Buy and Hold”. In particular, for an European call option, the super-replication price is equal to the initial price of the underlying: this result has been already obtained in the context of transaction costs by Cvitani´c-Shreve-Soner [7] and Cvitani´c-Pham-Touzi [6], and for a continuous time stochastic volatility model by Cvitani´c-Pham- Touzi [5]. Our approach emphasizes the fact that the super-replication price depends on the law of the underlying asset price process only through its null sets. These results are presumably not surprising, even if, up to our knowledge, only specific one dimensional cases have been handled in the literature. Finally, this work provides a relatively complete answer to the problem of how to explicitly compute super-replication prices in a general discrete time strategies framework.
The paper is organized as follows. In Section 2, we describe thefinancial
model and give the notation of the paper. Then, we recall the notion of No Arbitrage and state a dual formulation for the super-replication problem: while this result is standard in the European case (see Kabanov- R`asonyi-Stricker [16] and Schachermayer [23]), it has not been yet stated in the American context (this is Theorem 2.2, which proof is postponed in Appendix). Section 3 is devoted to the closed formulae for the super- replication prices, and their proofs. In Section 4, we effectively compute the super-replication price for European and American style exotic options (including Asian, Lookback or Barrier options), when there is only one risky asset (See table 1 for those explicit computations). We also handle some multidimensional examples. These results hold true if the underlying asset law admits a positive density w.r.t. the Lebesgue measure: it includes for example Black-Scholes model, general stochastic differential equations, stochastic volatility models, or models governed by Brownian motion and Poisson process. We will also see that increasing the number of hedging dates does not modify the super-replication prices.