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

Abstract

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.

Optimal Investment, Derivative Demand and Arbitrage Under Price Impact

“…The price p ∞ (0) is the limit of the arbitrage-free prices E 0,n [h] in the sense of Definition 4.2 (as well as Definition 4.3 for u fixed as n → ∞), so p = p ∞ (0) corresponds to the large investor obtaining a very advantageous price as the market makers approach risk neutrality. Lastly, we remark that [5,Corollary 4.6] verifies (4.9) for any sequence of traded prices {p n } n∈N ⊂ (h, h) provided p n → p = p ∞ (0). There is no need for p to be fixed across all n. We end this section with an observation.…”

“…Recall that 1/β = 1/α + 1/γ is the combined risk tolerance of the investor and market maker. Using the results of Section 3, together with the general theory developed in [5], we now show that optimal demands are on the order of 1/β, and hence large positions arise endogenously as either the investor's or market maker's risk aversion vanishes. To this latter point, the two situations consistent with γ → 0 are when the number of market makers increases (as this implies growth in the aggregate risk tolerance 1/γ) and as an approximation to market maker risk-neutrality.…”

“…We now make the above precise, considering a sequence β n → 0. Let Assumption 4.1 hold, and to be consistent with [5], set…”

“…There is no need for p to be fixed across all n. We end this section with an observation. [5] makes a general connection between endogenously rising large positions and asymptotic market completeness (c.f. Section 6.2 therein).…”

“…Since we are assuming K o t = R d , for any level of market maker risk aversion, the market with price impact is complete. Therefore, the "asymptotic market completeness" of [5] corresponds, not to the vanishing inability to hedge claims in the price impact model, but rather to "asymptotic market maker risk neutrality" in the price impact model. However, it is possible to associate asymptotic market maker risk neutrality with asymptotic completeness, and this is done through the lens of basis risk models (c.f.…”