Ian Buckley

Motivated by the need for parametric families of rich and yet tractable distributions in financial mathematics, both in pricing and risk management settings, but also considering wider statistical applications, we investigate a novel technique for introducing skewness or kurtosis into a symmetric or other distribution. We use a "transmutation" map, which is the functional composition of the cumulative distribution function of one distribution with the inverse cumulative distribution (quantile)… Show more

Motivated by the need for parametric families of rich and yet tractable distributions in financial mathematics, both in pricing and risk management settings, but also considering wider statistical applications, we investigate a novel technique for introducing skewness or kurtosis into a symmetric or other distribution. We use a "transmutation" map, which is the functional composition of the cumulative distribution function of one distribution with the inverse cumulative distribution (quantile) function of another. In contrast to the Gram-Charlier approach, this is done without resorting to an asymptotic expansion, and so avoids the pathologies that are often associated with it. Examples of parametric distributions that we can generate in this way include the skew-uniform, skew-exponential, skew-normal, and skew-kurtotic-normal. Show less

The entropic calibration of the risk-neutral density function is effective in recovering the strike dependence of options, but encounters difficulties in determining the relevant greeks. By use of put-call reversal we apply the entropic method to the time reversed economy, which allows us to obtain the spot price dependence of options and the relevant greeks.

The second partial derivative of a European-style vanilla option with respect to the current price of the underlying asset—the option gamma—defines a probability density function for the current underlying price. By use of entropy maximization it is possible to obtain an option gamma, from which the associated option pricing formula can be recovered by integration. A number of pricing formulae are obtained in this manner, corresponding to different specifications of the constraints. When the… Show more

The second partial derivative of a European-style vanilla option with respect to the current price of the underlying asset—the option gamma—defines a probability density function for the current underlying price. By use of entropy maximization it is possible to obtain an option gamma, from which the associated option pricing formula can be recovered by integration. A number of pricing formulae are obtained in this manner, corresponding to different specifications of the constraints. When the available market information consists solely of a set of traded option prices, the entropic formulation leads to a model-independent calibration procedure. The result thus obtained also allows one to recover the relevant Greeks. Show less

In this paper we consider a portfolio optimization problem where the underlying asset returns are distributed as a mixture of two multivariate Gaussians; these two Gaussians may be associated with “distressed” and “tranquil” market regimes. In this context, the Sharpe ratio needs to be replaced by other non-linear objective functions which, in the case of many underlying assets, lead to optimization problems which cannot be easily solved with standard techniques. We obtain a geometric… Show more

In this paper we consider a portfolio optimization problem where the underlying asset returns are distributed as a mixture of two multivariate Gaussians; these two Gaussians may be associated with “distressed” and “tranquil” market regimes. In this context, the Sharpe ratio needs to be replaced by other non-linear objective functions which, in the case of many underlying assets, lead to optimization problems which cannot be easily solved with standard techniques. We obtain a geometric characterization of efficient portfolios, which reduces the complexity of the portfolio optimization problem. Show less

We apply impulse control techniques to a cash management problem within a mean-variance framework. We consider the strategy of an investor who is trying to minimise both fixed and proportional transaction costs, whilst minimising the tracking error with respect to an index portfolio. The cash weight is constantly fluctuating due to the stochastic inflow and outflow of dividends and liabilities. We show the existence of an optimal strategy and compute it numerically.

The convergence of the linear δ expansion is studied in the context of the integral I:=F∞−∞e−gx4dx, which corresponds to massless cphi4 theory in 0 dimensions. The method consists of rewriting the exponent as -δgx4-λ(1-δ)x2 and expanding in powers of δ. The arbitrary parameter λ is fixed by the principle of minimal sensitivity, ∂IK(λ)/∂λ=0, where IK is the expansion truncated at order K with δ set equal to 1. This has a solution λ¯K only for K odd, when it gives very good numerical results. We… Show more

The convergence of the linear δ expansion is studied in the context of the integral I:=F∞−∞e−gx4dx, which corresponds to massless cphi4 theory in 0 dimensions. The method consists of rewriting the exponent as -δgx4-λ(1-δ)x2 and expanding in powers of δ. The arbitrary parameter λ is fixed by the principle of minimal sensitivity, ∂IK(λ)/∂λ=0, where IK is the expansion truncated at order K with δ set equal to 1. This has a solution λ¯K only for K odd, when it gives very good numerical results. We are able to show analytically, using saddle-point methods, that the sequence of approximants IK(λ¯K) is convergent, the error decreasing exponentially with K, even though for fixed λ the series expansion is a divergent alternating series. Show less

We generalize to four dimensions the linear δ expansion applied to gauge theory on the lattice, with Z(2), U(1), and SU(2) as the gauge groups and an unperturbed action appropriate to the strong-coupling regime. The calculation is carried to fourth order in δ, using the machinery of character expansions. The expansion is optimized in a bid to improve upon existing strong-coupling methods and our results are compared against results from these and from Monte Carlo calculations.

Using the linear δ expansion we calculate the effective potential of the O(N)×O(N)-invariant φ2χ2 theory, in 3 dimensions, to first order in δ. The φa and χa each represent N massless fields in the vector representation of O(N). We find that the optimization equations give rise to two effective potentials. In one of these, the radiative corrections move the theory away from criticality; i.e., a mass term is induced by radiative corrections. In the other case the sole effect of the radiative… Show more

Using the linear δ expansion we calculate the effective potential of the O(N)×O(N)-invariant φ2χ2 theory, in 3 dimensions, to first order in δ. The φa and χa each represent N massless fields in the vector representation of O(N). We find that the optimization equations give rise to two effective potentials. In one of these, the radiative corrections move the theory away from criticality; i.e., a mass term is induced by radiative corrections. In the other case the sole effect of the radiative corrections is to remove the valleys in the effective potential without the introduction of any mass term. Show less

We apply the linear δ expansion to non-Abelian gauge theory on the lattice, with SU(2) as the gauge group. We establish an appropriate parametrization and evaluate the average plaquette energy EP to O(δ). As a check on our results, we recover the large-β expansion up to O(1β2), which involves some O(δ) contributions. Using these contributions we construct a variant of the 1β expansion which gives a good fit to the data down to the transition region.