  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
problem 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
the case of 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.
$\epsilon$-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.
