A note on utility based pricing and asymptotic risk diversification

Bouchard, Bruno; Élie, Romuald; Moreau, Ludovic

doi:10.1007/s11579-011-0055-0

Cited by 5 publications

(8 citation statements)

References 18 publications

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“…Starting at least from [22], utility indifference pricing has attracted a lot of attention, see for example [9] for detailed overview. Recently, indifference pricing for large position sizes has been studied in [8,33,34]. In [34] the authors consider a sequence of a particular semi-complete market indexed by n that becomes complete as n → ∞ and, assuming the unhedgeable component of the non-traded asset vanishes in accordance to a Large Deviation Principle (LDP), it is shown that optimal purchase quantities become large at precisely the Large Deviations scaling.…”

Section: Introductionmentioning

confidence: 99%

The pricing of contingent claims and optimal positions in asymptotically complete markets

Anthropelos¹,

Robertson²,

Spiliopoulos³

2017

Ann. Appl. Probab.

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We study utility indifference prices and optimal purchasing quantities for a contingent claim, in an incomplete semi-martingale market, in the presence of vanishing hedging errors and/or risk aversion.Assuming that the average indifference price converges to a well defined limit, we prove that optimally taken positions become large in absolute value at a specific rate. We draw motivation from and make connections to Large Deviations theory, and in particular, the celebrated Gärtner-Ellis theorem. We analyze a series of well studied examples where this limiting behavior occurs, such as fixed markets with vanishing risk aversion, the basis risk model with high correlation, models of large markets with vanishing trading restrictions and the Black-Scholes-Merton model with either vanishing default probabilities or vanishing transaction costs. Lastly, we show that the large claim regime could naturally arise in partial equilibrium models.

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Section: Introductionmentioning

confidence: 99%

The pricing of contingent claims and optimal positions in asymptotically complete markets

Anthropelos¹,

Robertson²,

Spiliopoulos³

2017

Ann. Appl. Probab.

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“…As mentioned in the Introduction, these types of models typically appear in the insurance industry; see Bouchard et al. (), De Donno et al. (), Brennan and Schwartz (), Milevsky and Promislow (), and De Donno and Pratelli ().…”

Section: A “Large Market” Examplementioning

confidence: 99%

“…This form encompasses the case when B is a suitably weighted sum of component claims, or an aggregated claim; see, for example, Bouchard et al. (), De Donno et al. (), and Robertson ().…”

Section: A “Large Market” Examplementioning

confidence: 99%

“…The notion of large markets was introduced in Kabanov and Kramkov () and since then, several papers have studied theoretical questions related to asymptotic arbitrage and extending the fundamental theorems of asset pricing; see Kabanov and Kramkov (), Klein and Schachermayer (), and Klein (). In applications, these types of models frequently appear in the insurance industry; see Bouchard, Elie, and Moreau (), De Donno, Guasoni, and Pratelli (), Brennan and Schwartz (), Milevsky and Promislow (), De Donno and Pratelli (), among many others. Therefore, after introducing the abstract semicomplete setting in Section , in Section we present in detail a large market example used in the insurance industry, where assets are geometric Brownian motions (with arbitrary correlations) and where the claim is the sum of independent components.…”

Section: Introductionmentioning

confidence: 99%

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Indifference Pricing for Contingent Claims: Large Deviations Effects

Robertson

Spiliopoulos

2016

Mathematical Finance

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Abstract. We study utility indifference prices and optimal purchasing quantities for a non-traded contingent claim in an incomplete semi-martingale market with vanishing hedging errors. We make connections with the theory of large deviations. We concentrate on sequences of semi-complete markets where in the n th market, the claim Bn admits the decomposition Bn = Dn + Yn. Here, Dn is replicable by trading in the underlying assets Sn, but Yn is independent of Sn. Under broad conditions, we may assume that Yn vanishes in accordance with a large deviations principle as n grows. In this setting, for an exponential investor, we identify the limit of the average indifference price pn(qn), for qn units of Bn, as n → ∞. We show that if |qn| → ∞, the limiting price typically differs from the price obtained by assuming bounded positions sup n |qn| < ∞, and the difference is explicitly identifiable using large deviations theory. Furthermore, we show that optimal purchase quantities occur at the large deviations scaling, and hence large positions arise endogenously in this setting.

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“…In addition to the over‐the‐counter derivatives market, large claim limit pricing has applications to the insurance industry. Bouchard, Elie, and Moreau () consider large positions pricing for liabilities with both financial and insurancial risks, such as revenue insurance contracts or mortality derivatives. Here, the claim is actually the sum of the contracts.…”

Section: Introductionmentioning

confidence: 99%

Pricing for Large Positions in Contingent Claims

Robertson

2015

Mathematical Finance

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ABSTRACT. Approximations to utility indifference prices are provided for a contingent claim in the large position size limit. Results are valid for general utility functions on the real line and semi-martingale models.It is shown that as the position size approaches infinity, the utility function's decay rate for large negative wealths is the primary driver of prices. For utilities with exponential decay, one may price like an exponential investor. For utilities with a power decay, one may price like a power investor after a suitable adjustment to the rate at which the position size becomes large. In a sizable class of diffusion models, limiting indifference prices are explicitly computed for an exponential investor. Furthermore, the large claim limit is seen to endogenously arise as the hedging error for the claim vanishes.

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A note on utility based pricing and asymptotic risk diversification

Cited by 5 publications

References 18 publications

The pricing of contingent claims and optimal positions in asymptotically complete markets

The pricing of contingent claims and optimal positions in asymptotically complete markets

Indifference Pricing for Contingent Claims: Large Deviations Effects

Pricing for Large Positions in Contingent Claims

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