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Pricing options on illiquid assets with liquid proxies using utility indifference and dynamic-static hedging

2013; Taylor & Francis; Volume: 14; Issue: 3 Linguagem: Inglês

10.1080/14697688.2013.816766

1469-7696

Igor Halperin, Andrey Itkin,

Credit Risk and Financial Regulations

Abstract This work addresses the problem of optimal pricing and hedging of a European option on an illiquid asset Z using two proxies: a liquid asset S and a liquid European option on another liquid asset Y. We assume that the S-hedge is dynamic while the Y-hedge is static. Using the indifference pricing approach, we derive a Hamilton–Jacobi–Bellman equation for the value function. We solve this equation analytically (in quadrature) using an asymptotic expansion around the limit of perfect correlation between assets Y and Z. While in this paper we apply our framework to an incomplete market version of Merton's credit-equity model, the same approach can be used for other asset classes (equity, commodity, FX, etc.), e.g. for pricing and hedging options with illiquid strikes or illiquid exotic options. Keywords: Incomplete marketsAsset pricingDerivative pricing modelsQuantitative finance techniquesHedging with utility based preferencesComputational financePricing with utility based preferencesJEL Classification: C60C63 Acknowledgments We thank Peter Carr, participants of the "Global Derivatives USA 2011" conference, and two anonymous referees for useful comments, and Nic Trainor for the editorial work. I.H. would like tothank Andrew Abrahams and Julia Chislenko for support and interest in this work. Notes Here, we refer to this instrument as an index; but, it could be any 'linear' instrument, such as stock, forward, etc. For a discussion of such scenarios in commodities markets, see Trolle and Schwartz (Citation2009). Most of the formulae below, excluding those that use specific forms of pay-offs, are general and applicable for other similar settings. The stock is equivalent to our index in the setting of the Merton's optimal investment problem. In case of equity options, these are strikes, while in the credit world they represent bond notionals. As an example, we mention the case of equity options referencing the same underlying, i.e. , but . We may want to hedge an illiquid option with strike (say, deep OTM) with a liquid option on the same underlying, but with a different strike . Under this set-up, we have , i.e. a prefect correlation case. See, e.g. Landau and Lifshitz (Citation1988). While the parameter is always and positive, the parameter could be positive or negative for typical values of correlations. For example, if , then , while for , we have . The cosine law can also be used to find proper values of correlation parameters in the limit . To this end, we first use equation (Equation20) to convert the estimated triplet into a triplet of independent variables and then take the limit while keeping and constant. In other words, write , where . As , this is a regular map. From theory, we know that the asymptotic solution converges to the true solution as . However, due to our definition of via , this is not an interesting case. Compare with equation (Equation31).