Abstract
In this paper, we consider the problem of equal risk pricing and hedging in which the fair price of an option is the price that exposes both sides of the contract to the same level of risk. Focusing for the first time on the context where risk is measured according to convex risk measures, we establish that the problem reduces to solving independently the writer and the buyer's hedging problems with zero initial capital. By further imposing that the risk measures decompose in a way that satisfies a Markovian property, we provide dynamic programming equations that can be used to solve the hedging problems for both European and American options. All of our results are general enough to accommodate situations where the risk is measured according to a worst-case risk measure, as is typically done in robust optimization. Our numerical study illustrates the advantages of equal risk pricing over schemes that only account for a single party, pricing based on quadratic hedging (i.e. ϵ-arbitrage pricing), or pricing based on a fixed equivalent martingale measure (i.e. Black–Scholes pricing). In particular, the numerical results confirm that when employing an equal risk price both the writer and the buyer end up being exposed to risks that are more similar and on average smaller than what they would experience with the other approaches.
Highlights
One of the main challenges in pricing and hedging financial derivatives is that the market is often incomplete and there exists unhedgeable risk that needs to be further accounted for in pricing
In the case of discrete time hedging, we show how the boundaries of such a fair price interval can be obtained by using dynamic programming for both European and American options as long as the convex risk measures employed by the two parties are one-step decomposable and satisfy a Markovian property
In the context where the underlying asset follows a geometric Brownian motion, we show for the first time how robust optimization can motivate the use of worst-case risk measures that only consider a subset of the outcome space
Summary
One of the main challenges in pricing and hedging financial derivatives is that the market is often incomplete and there exists unhedgeable risk that needs to be further accounted for in pricing. In the case of discrete time hedging, we show how the boundaries of such a fair price interval can be obtained by using dynamic programming for both European and American options as long as the convex risk measures employed by the two parties are one-step decomposable and satisfy a Markovian property (see Section 2.1 for proper definitions). Our numerical experiments indicate that the fair price interval might converge, as the number of rebalancing periods increases, to the Black- Scholes price when an uncertainty set inspired by the work of [4] is properly calibrated in a market driven by a geometric Brownian motion If supported theoretically, such a property would close the gap between risk neutral pricing and risk averse discrete-time hedging using worst-case risk measures. We further refer the reader to an extensive set of Appendices describing detailed arguments supporting all propositions and lemmas presented in this article
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