Abstract
In this paper, we deal with the pricing of European options in an incomplete market. We use the common risk measures Value-at-Risk and Expected Shortfall to define good-deals on a financial market with log-normally distributed rate of returns. We show that the pricing bounds obtained from the Value-at-Risk admit a non-smooth behavior under parameter changes. Additionally, we find situations in which the seller’s bound for a call option is smaller than the buyer’s bound. We identify the missing convexity of the Value-at-Risk as main reason for this behavior. Due to the strong connection between good-deal bounds and the theory of risk measures, we further obtain new insights in the finiteness and the continuity of risk measures based on multiple eligible assets in our setting.
Highlights
Pricing in incomplete financial markets is a demanding task for which various approaches are considered in the finance literature
We study good-deal bounds based on Value-at-Risk and Expected Shortfall acceptance sets
Based on a benchmark example we demonstrate that the Value-at-Risk good-deal bounds for a European type call option behave non-smooth under varying the underlying stock price
Summary
Pricing in incomplete financial markets is a demanding task for which various approaches are considered in the finance literature. Working under the latter condition, the consideration of the Value-at-Risk and the Expected Shortfall allows us to demonstrate the differences between non-convex and convex pricing bounds. Our results allow to conclude that pricing w.r.t. the non-convex bounds is problematic, whereas the good-deal bounds based on Expected Shortfall provide a reasonable way for option pricing in incomplete markets.
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