This paper introduces a parameterization of the normal mixture diffusion (NMD) local volatility model that captures only a short-term smile effect, and then extends the model so that it also captures a long-term smile effect. We focus on the ‘binomial’ NMD parameterization, so-called because it is based on simple and intuitive assumptions that imply the mixing law for the normal mixture log price density is binomial. With more than two possible states for volatility, the general parameterization is related to the multinomial mixing law. In this parsimonious class of complete market models, option pricing and hedging is straightforward since model prices and deltas are simple weighted averages of Black–Scholes prices and deltas. But they only capture a short-term smile effect, where leptokurtosis in the log price density decreases with term, in accordance with the ‘stylised facts’ of econometric analysis on ex-post returns of different frequencies and the central limit theorem. However, the last part of the paper shows that longer term smile effects that arise from uncertainty in the local volatility surface can be modeled by a natural extension of the binomial NMD parameterization. Results are illustrated by calibrating the model to several Euro–US dollar currency option smile surfaces.

Bitcoin prices are driven by upward as well as downward jumps and so the bitcoin implied volatility surface behaves differently from those of established options markets. We analyze tick-level Deribit option price data, demonstrating increasing support for the limits-to-arbitrage hypothesis. Hence market makers are managing order imbalance and inventory more effectively as Deribit bitcoin options trading volumes increases. On the demand side, volatility traders drive both at-the-money and out-of-the-money option prices, the latter also being driven by directional traders. Directional effects were most pronounced during the price bubble of 2021. Further refinements of our tests assess time-to-maturity and time-of-day effects.

A price process is scale-invariant if and only if the returns distribution is independent of the price measurement scale. We show that most stochastic processes used for pricing options on financial assets have this property and that many models not previously recognised as scale-invariant are indeed so. We also prove that price hedge ratios for a wide class of contingent claims under a wide class of pricing models are model-free. In particular, previous results on model-free price hedge ratios of vanilla options based on scale-invariant models are extended to any contingent claim with homogeneous pay-off, including complex, path-dependent options. However, model-free hedge ratios only have the minimum variance property in scale-invariant stochastic volatility models when price–volatility correlation is zero. In other stochastic volatility models and in scale-invariant local volatility models, model-free hedge ratios are not minimum variance ratios and our empirical results demonstrate that they are less efficient than minimum variance hedge ratios.

We analyse minute-level multi-dimensional information flows within and between bitcoin spot and derivatives. We show that perpetual swaps and futures traded on the unregulated exchanges Huobi, OKEx and BitMEX are much the strongest instruments for bitcoin price discovery and we examine potential determinants of their leadership strength. Prices on the regulated CME bitcoin futures and the US-based spot exchanges react to, rather than lead, price movements on the unregulated exchanges and they may do so relatively slowly. In a multi-dimensional setting including the main price leaders within futures, perpetuals and spot markets, the CME futures have a very minor effect on price discovery, even less than the spot exchanges Bitfinex, Bitstamp and Coinbase. Our findings highlight the persistent problems stemming from inconsistent regulation in bitcoin spot and derivatives markets, including insufficient price stability and lack of resistance to manipulative trading. We conclude that the SEC are correct to maintain such issues as their main concern for bitcoin ETF applications.

We apply Markov chain Monte Carlo methods to time series data on S&P 500 index returns, and to its option prices via a term structure of VIX indices, to estimate 18 different affine and non-affine stochastic volatility models with one or two variance factors, and where jumps are allowed in both the price and the instantaneous volatility. The in-sample fit to the VIX term structure shows that the second (stochastic long-term volatility) factor is required to fit the VIX term structure. Out-of-sample tests on the fit to individual option prices, as well as in-sample tests, show that the inclusion of jumps is less important than allowing for non-affine dynamics. The estimation and testing periods together cover more than 21 years of daily data.

The Basel 2 Accord requires regulatory capital to cover stress tests, yet no coherent and objective framework for stress testing portfolios exists. We propose a new methodology for stress testing in the context of market risk models that can incorporate both volatility clustering and heavy tails. Empirical results compare the performance of eight risk models with four possible conditional and unconditional return distributions over different rolling estimation periods. When applied to major currency pairs using daily data spanning more than 20 years we find that stress test results should have little impact on current levels of foreign exchange regulatory capital.

Credit default swap (CDS) spreads display pronounced regime specific behaviour. A Markov switching model of the determinants of changes in the iTraxx Europe indices demonstrates that they are extremely sensitive to stock volatility during periods of CDS market turbulence. But in ordinary market circumstances CDS spreads are more sensitive to stock returns than they are to stock volatility. Equity hedge ratios are three or four times larger during the turbulent period, which explains why previous research on single-regime models finds stock positions to be ineffective hedges for default swaps. Interest rate movements do not affect the financial sector iTraxx indices and they only have a significant effect on the other indices when the spreads are not excessively volatile. Raising interest rates may decrease the probability of credit spreads entering a volatile period.

We test the joint efficiency of the bitcoin and ether options and perpetual futures markets and identify the determinants of arbitrage opportunities. Our novel fiat-currency-free put–call parity relationship motivates new arbitrage tests for options-only and option–perpetual cross-markets. Bitcoin and ether derivatives markets are becoming more efficient, especially for options of maturity ≥ 15 days. Bitcoin derivative markets are generally more efficient than ether derivative markets, but arbitrage strategies can still be highly profitable even under conservative transaction cost scenarios, which include slippage for large orders, especially during periods of high trading volumes or when the blockchain traffic becomes more congested.