The system [of finance] is too complex to be run on error-strewn hunches and gut feelings, but current mathematical models don’t represent reality adequately. The entire system is poorly understood and dangerously unstable. The world economy desperately needs a radical overhaul and that requires more mathematics, not less.

This article in the Guardian is a little late to the party and has an intentionally misleading headline, but brings up some points that are usually too esoteric to survive in print:

Any mathematical model of reality relies on simplifications and assumptions. The Black-Scholes equation was based on arbitrage pricing theory, in which both drift and volatility are constant. This assumption is common in financial theory, but it is often false for real markets. The equation also assumes that there are no transaction costs, no limits on short-selling and that money can always be lent and borrowed at a known, fixed, risk-free interest rate. Again, reality is often very different.

There are more false assumptions like Gaussianity of log-returns, complete markets, martingale price paths, etc., but these are merely technical complaints, which can be patched (as many are doing). The real issue is, as the author notes, “… instability is common in economic models … mainly because of the poor design of the financial system.” Namely, there is a lack of accounting for behavioral effects that result in feedback, which give rise to rather more fundamental issues that would require the “radical overhaul” alluded to in the opening quotation to resolve. There are some problems that could be tackled in this area.

• Damping: As posed here and here, could damping of price and flow mechanisms in markets create more stability? And at what expense to liquidity or other notions of efficiency? Are there fundamental trade-offs between liquidity and stability — e.g., something like the Gibbs phenomenon in filter design?
• Redundant beliefs: As explained here, risk sharing (e.g. via CDS) stops working when the network of institutions becomes densely connected (and therefore causes short risk feedback cycles.) Is there a way to calculate the correct network risk in a distributed way, by something like iterative message passing, and share that information in a way that does not reveal proprietary information?
• Impact: More generally, price of risk products is set based on some static model of markets, e.g., statistics and predictions assuming things remain the same. However, when risk is mined and sold, it changes the decisions of participants (viz. allowing hedging makes it possible to take larger risks) in a way that could invalidate the data on which pricing was based. Is there a relatively simple, separable model for decision propagation (e.g. a correction term, or an amplification term or filter that modifies a parameter on which the price is based)?

Beyond feedback issues, there are many other questions, such as:

• Heterogeneity: What is the real effect of heterogeneous participants, especially those with different time horizons or risk appetites for asset purchases, on price discovery? What, if any, notion of fairness can we guarantee in these cases? Or, what notion of fairness is actually being computed by standard markets?
• Large systems: How do we pass from small-scale game-theoretic models to large-scale system models? For example, in thermodynamics, we can identify macroscopic variables like temperature with microscopic variables like kinetic energy. Can we identify market variables like volatility with any underlying behavioral variables? What are the “right” (e.g. fundamental, minimal) variables to completely describe a market?
• Instrument design: Are expiring instruments like bonds fundamentally better than infinite-horizons ones like stocks? Can we design market instruments to have certain guarantees on things like volatility? Can we design atomic risk instruments that have payoffs in easy-to-understand terms?

Maybe this list will inspire the field to advance past its current medieval state.