International Journal of Theoretical and Applied Finance

We revisit the "Smile Dynamics" problem, which consists in relating the implied leverage (i.e. the correlation of the at-the-money volatility with the returns of the underlying) and the skew of the option smile. The ratio between these two quantities, called "Skew-Stickiness Ratio" (SSR) by Bergomi (2009), saturates to the value 2 for linear models in the limit of small maturities, and converges to 1 for long maturities. We show that for more general, non-linear models (such as the asymmetric GARCH model), Bergomi's result must be modified, and can be larger than 2 for small maturities. The discrepancy comes from the fact that the volatility skew is, in general, different from the skewness of the underlying. We compare our theory with empirical results, using data both from option markets and from the underlying price series, for the S&P 500 and the DAX. We find, among other things, that although both the implied leverage and the skew appear to be too strong on option markets, their ratio is well explained by the theory. We observe that the SSR indeed becomes larger than 2 for small maturities, signalling the presence of non-linear effects.

A forward equation, which is also called the Dupire formula, is obtained for European call options when the price dynamics of the underlying risky assets are assumed to follow a regime-switching local volatility model. Using a regime-switching version of the adjoint formula, a system of coupled forward equations is derived for the price of the European call over different states of the economy.

We consider the problem of hedging a European-type option in a market where asset prices have jump-diffusion dynamics. It is known that markets with jumps are incomplete and that there are several risk-neutral measures one can use to price and hedge options. In order to address these issues, we approximate such a market by discretizing the jumps in an averaged sense, and complete it by including traded options in the model and hedge portfolio. Under suitable conditions, we get a unique risk-neutral measure, which is used to determine the option price integro-partial differential equation, along with the asset positions that will replicate the option payoff. Upon implementation on a particular set of stock and option prices, our approximate complete market hedge yields easily computable asset positions that equal those of the minimal variance hedge, while at the same time offers protection against upward jumps and higher profit compared to delta hedging.

We show that a necessary and sufficient condition for the explosion of implied volatility near expiry in exponential Lévy models is the existence of jumps towards the strike price in the underlying process. When such jumps do not exist, the implied volatility converges to the volatility of the Gaussian component of the underlying Lévy process as the time to maturity tends to zero. These results are proved by comparing the short-time asymptotics of the Black–Scholes price with explicit formulas for upper and lower bounds of option prices in exponential Lévy models.

This paper extends the Wiener–Itô chaos expansion approach proposed by Funahashi & Kijima (2015) to an equity-interest-rate hybrid model for the pricing of European contingent claims with special emphasis on calibration to the option markets. Our model can capture the volatility skew and smile of option markets, as well as the stochastic nature of interest rates. Further, the proposed method is applicable to widely used option pricing models such as local volatility models (LVM), stochastic volatility models (SVM), and their combinations with the stochastic nature of interest rates; hence, it is suitable for practical purposes. Through numerical examples, we show that our approximation is quite accurate even for long-maturity and/or high-volatility cases.

The development of the theory of the expansion of filtrations took place mostly three decades ago, in the 1980s. Researchers developed two types of expansions: Initial expansions, where one adds information to the σ algebra ℱ₀, and progressive expansions, where information is added dynamically to turn a positive random variable (such as a last exit time) into a stopping time. The goal was to preserve the semimartingale property in the enlarged filtration. In this paper, we propose a new type of expansion, that of expansion with a stochastic process. This has antecedents in the work of Jeulin, Kohatsu-Higa, and a few others, but the theoretical and systematic approach given here is new. We begin by showing one can enlarge a filtration with a point process rather easily, and that semimartingales remain semimartingales under the expansions, if done correctly. Since one can approximate most stochastic processes with marked point processes, we then prove convergence theorems of the sequence of point processes together with their corresponding enlarged filtrations. To do this, we rely on a theory of the convergence of filtrations. We next need to give conditions such that semimartingales remain semimartingales in the limit. This is delicate, and we obtain partial results, but they are sufficient for our needs; this involves a kind of continual initial expansion procedure. One drawback of this procedure is that we do not always obtain the semimartingale decompositions for the enlarged filtrations of the limiting case, except in a class of examples that are in a Brownian motion paradigm. Recently, there have been attempts to model mathematically insider trading via a theory of enlarged filtrations. There are many technical issues involved, and creating a model with the presence of arbitrage is a constant problem, a situation one wishes to avoid. We show how one can use this procedure to create models of insider trading that do not have arbitrage opportunities, but under which the risk neutral measure for the insider is different than is the risk neutral measure for the rest of the market.

Different authors find optically very different patterns ("tents" and "bats") when excess returns from US Treasuries are regressed against forward rates. A separate source of disagreement is whether the recent tent-shaped factor found by Cochrane & Piazzesi (2004, 2005, 2008) is fundamentally different from the slope (in yields) explanatory factor for the time-dependent term premia that has been identified since the early 1990s. We show that "tent" and "bat" patterns produce predictions of excess returns that are economically indistinguishable, both locally and globally. We also argue that, if the slope (in yield space) is indeed the most important explanatory factor for excess returns, then a simple transformation of the loadings of the forwards rates on the slope factor naturally recovers an approximate tent shape in forward-rate space. However, a transformation from a slope factor in yields to loadings for forward rates does not easily produce a bat pattern. This may provide indirect corroboration to Cochrane & Piazzesi's (2004) claim that the bat factor is a result of near colinearity for smoothed data, and that the "tent" solution is intrinsically different from the "bat" solution.