Frontiers in Quantitative Finance: Volatility and Credit Risk Modeling
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Market completeness is thennaturally formulated in terms of moment determinacy.
This allows us toderive equivalent conditions for the absence of arbitrage between generalpayoffs not limited to single-asset call options. We also focus on the par-ticular case of basket calls or European call options on a basket of assets. Basket calls appear in equity markets as index options and in interest ratederivatives market as spread options or swaptions, and are key recipients ofmarket information on correlation. The paper is organized as follows. We begin by describing the oneperiod market and illustrate our approach on a simple example, intro-ducing the payoff semigroup formed by the market securities and theirproducts.
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Section 2 starts with a brief primer on harmonic analysis onsemigroups after which we describe the general no-arbitrage conditions onthe payoff semigroup. We also show how the products in this semigroupcomplete the market. A Moment Approach to Static Arbitrage Fi-nally, there is a riskless asset with payoff 1 at maturity and price 1 todayand we assume, without loss of generality here, that interest rates are equalto zero we work in the forward market.
We look for conditions on p pre-cluding arbitrage in this market, that is, buy and hold portfolios formed atno cost today which guarantee a strictly positive payoff at maturity. We want to answer the following simple question: Given the marketprice vector p, is there an arbitrage opportunity a buy-and-hold arbitrage inthe continuous market terminology between the assets xi and the securitiess j x? In fact, if we simply discretize theproblem on a uniform grid with L steps along each axis, this problem is stillequivalent to an exponentially large linear program of size O Ln. Here, welook for a discretization that does not involve the state price measure butinstead formulates the no arbitrage conditions directly on the market pricevector p.
Inthis case, conditions 1.
Werecognize 1. A Moment Approach to Static Arbitrage 7 In the next section, we will show that the no-arbitrage conditions 1. We also detail under which technical conditions the securitiesin S make the one-period market complete. In all the results that follow, wewill assume that the asset distribution has compact support.
Verysimilar but much more technical results hold in the noncompact case, asdetailed in the preprint . Testing for no arbitrage is then equivalentto testing for the existence of a moment function f on S that matches themarket prices in 1. Theorem 1. Then, [23, Th. Linearity of f simply fol-lows from the linearity of semicharacters on the market semigroup in 1. However, payoffs doplay a role through the semigroup structure.
Credit risk: modeling, valuation, and hedging
When no such relationships exist however, the conditions inTheorem 1. We illustrate this pointfurther in Section 1. Here, we suppose that there is no arbitrage in the one period market. In fact, we show below that when asset payoffshave compact support, this pricing measure is unique.
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This result shows that the securities in S make the market complete inthe compact case. In practice, wecan get a reduced set of conditions by only considering elements of S up toa certain even degree 2d: Sd 1, x1,. The exponential growth of Nd withn and m means that only small problem instances can be solved using cur-rent numerical software.
Finally, as wewill see in Section 1. A Moment Approach to Static Arbitrage 13 see . While the set of nonnegative portfolios is intractable, the set ofportfolios that are sums of squares of payoffs in S hence nonnegative canbe represented using linear matrix inequalities. Using theconic duality in 1. The key difference between this pro-gram and 1.
Weknow from  and  that the absence of arbitrage in this dynamicmarket is equivalent to the existence of a martingale measure on the as-sets x1,. A partial answer is given by thefollowing majorization result, which can be traced to Blackwell, Stein, Sher-man, Cartier, Meyer, and Strassen. Finding tractable conditions for the existence of a martingale measurewith given marginals, outside of the particular case of vanilla Europeancall options considered in  or for the density families discussed in ,remains however an open problem. In practice however, problemsize and conditioning issues still make problems such as 1.
This is partly due to the fact that these packages do not explicitly ex-ploit the group structure of the problems derived here to reduce numericalcomplexity. By interpreting theno-arbitrage conditions as a moment problem, we have derived equivalent.
Frontiers in Quantitative Finance: Volatility and Credit Risk Modeling by Rama Cont
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