In this paper, we provide new and concrete, elements for pricing and hedging Basis Risk. We consider the problem of an agent paying a derivative written on a risky asset V on which trading is not possible, not allowed or costly. For example, for liquidity reasons, an investor can sell an option on a stock and prefer to hedge with the associated index, or in the commodities market hedge with Fioul Oil 1% an option on Fioul Oil Straight Run 0,5%. In all these cases, one considers a more liquid asset S which is highly correlated with V and then price and hedge investing in S and cash only.
This is a typical incomplete market and the natural extension of No Arbitrage pricing, i.e. replication, is the super-replication concept. But, in the Black Scholes diffusion world, it is well known that this lead to unreasonably high values. For example, the super replication price of a call option on a non-tradable asset is equal to the initial value of this asset provided that it is possible to buy it at the beginning of the trading period.
Another method has been introduced by Cochrane and Saa-Requejo (2001): the No Good Deal (NGD) pricing. The idea is to exclude from admissible strategies, portfolios which have too high “Sharpe Ratio” because, similarly to arbitrage opportunities, good deals would quickly disappear as investors would immediately grab them. How should we define Sharpe ratio? In economic theory, the Sharpe ratio of a claim measures the degree to which the expected return of the claim exceeds the risk free rate, as a proportion of the standard deviation of this claim. For dynamic strategy, the definition of Sharpe ratio is not so clear and there exist different versions in the literature. We refer to Cochrane and Saa-Requejo (2001), Björk and Slinko (2006), Bayraktar and Young (2008) or Klöppel and Schweizer (2007) among others.
Klöppel and Schweizer (2007) define Sharpe Ratio globally and find that this NGD constraint, i.e. imposing a bound on the Sharpe ratio of any portfolio based on exchangeable claims, is equivalent to a bound on the variance of the density of the pricing measures. Note that this definition of Sharpe Ratio and No Good Deal price is linked to the notion of coherent risk measure and coherent NGD utility function of Cherny (2008).
Cochrane and Saa-Requejo (2001) and Björk and Slinko (2006) use an instantaneous notion of Sharpe Ratio and the authors assert that the NGD constraint leads to a bound on the market risk premium (considering both coverable and uncoverable risks). We remark that only a bound on the coverable risk premium naturally appears and consequently, it seems that their notion of NGD price is not directly related to instantaneous definition of Sharpe Ratio. We also show that it is also not related to the global Sharpe Ratio.
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Pricing and Hedging Basis Risk under No Good Deal Assumption
