Abstract
Risk neutral measures are defined such that the basic random assets in a portfolio are martingales. Hence, when the market model is complete, valuation of other financial instruments is a relatively straightforward task when those basic random assets constitute their underlying asset. To determine the risk neutral measure, it is assumed that the current prices of the basic assets are known exactly. However, oftentimes all we know about the current price, or that of a derivative having it as underlying, is a bid-ask range. The question then arises as to how to determine the risk neutral measure from that information. We may want to determine risk neutral measures from that information to use it, for example, to price other derivatives on the same asset. In this paper we propose an extended version of the maximum entropy method to carry out that task. This approach provides a novel solution to this problem, which is computationally simple and fast.
Highlights
Idalion Capital Group, Quantitative Trading, 12 Hay Hill, London W1J 8NR, UK; University of the Andes School of Management (Uniandes), Cl. 21 #1-20, Edificio SD, Bogotá, Colombia
The way in which maxentropic methods fit in the context of a standing effort to determine risk neutral measures from option prices, especially in the context of the binomial model, was the subject of an extensive review by [19], as well as [20]
The historical antecedents of the standard method go back to [26], whose proposal is called tilting in the statistical literature, as well as [27], who extended the work by Gibbs at the turn of the 20th century on the foundations of statistical physics
Summary
Consider a one period (static) market model, in which there is a riskless asset, with prices S0 (0). The use of maximum entropy based methods to obtain risk neutral measures for asset pricing, that is, to solve inverse problems like (1) or (5) with exact data is not new. This approach has been explored in [11,12,13,14,15,16,17,18]. The way in which maxentropic methods fit in the context of a standing effort to determine risk neutral measures from option prices, especially in the context of the binomial model, was the subject of an extensive review by [19], as well as [20] The latter use maximum entropy methods to determine a pricing measure within the scope of range data.
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