Pricing Index Options by Static Hedging Under Finite Liquidity

Abstract

We develop a model for indifference pricing in derivatives markets where price quotes have bid-ask spreads and finite quantities. The model quantifies the dependence of the prices and hedging portfolios on an investors beliefs, risk preferences and financial position as well as on the price quotes. Computational techniques of convex optimisation allow for fast computation of the hedging portfolios and prices as well as sensitivities with respect to various model parameters. We illustrate the techniques by pricing and hedging of exotic derivatives on S&P index using call and put options, forward contracts and cash as the hedging instruments. The optimized static hedges provide good approximations of the options payouts and the spreads between indifference selling and buying prices are quite narrow as compared with the spread between super-and subhedging prices.

Coherent risk measures alone are ineffective in constraining portfolio losses

“…We may then view the option payoffs as random variables in this probability model, and this will describe the market in full. The idea of studying this market is taken from Armstrong et al (2017).…”

Optimal capital allocation in correlated mutual funds under exponential loss utility

“…In this study, we model our portfolio optimization problem by minimizing an agent's risk preference described by an exponential utility function based on Armstrong, Pennanen and Rakwongwan's work in 2018. 11 The problem is numerically solved using Python. By considering six mutual funds, we compute optimal allocations of an initial capital under different constraints.…”

“…In this work, we use Monte Carlo technique for the approximation. However, as opposed to that of, 11 this problem has correlated multi random factors. Instead of simulating each price independently using Monte Carlo technique, to have all prices correlated with the given correlation matrix, we need to make an adjustment to the Brownian motion simulation.…”

Optimal semi-static hedging in illiquid markets

“…The hedging strategies involve buy-and-hold positions in the derivatives while the underlying and cash are traded dynamically. We use a Galerkin method to approximate the hedging problem by a finite-dimensional convex optimization problem which is then numerically solved by an interior point method much like in [APR18] in a purely static setting. The approach extends with minor modifications to situations where the dynamically traded underlying is also subject to bid ask spreads.…”