Optimal hedging and parameter uncertainty

We explore the impact of drift parameter uncertainty in a basis risk model, an incomplete market in which a claim on a non-traded asset is optimally hedged using a correlated traded stock. Using analytic expansions for indifference prices and hedging strategies, we develop an efficient procedure to generate terminal hedging error distributions when the hedger has erroneous estimates of the drift parameters. These show that the effect of parameter uncertainty is occasionally benign, but often very destructive. In light of this, we develop a filtering approach in which the hedger updates her parameter estimates from observations of the asset prices, and we find an analytic soultion to the hedger's combined filtering and control problem in the case that the drift of the traded asset is known with certainty

Optimal investment and hedging under partial and inside information

This article concerns optimal investment and hedging for agents who must use trading strategies which are adapted to the filtration generated by asset prices, possibly augmented with some inside information related to the future evolution of an asset price. The price evolution and observations are taken to be continuous, so the partial (and, when applicable, inside) information scenario is characterised by asset price processes with an unknown drift parameter, which is to be filtered from price observations. We first give an exposition of filtering theory, leading to the Kalman-Bucy filter. We outline the dual approach to portfolio optimisation, which is then applied to the Merton optimal investment problem when the agent does not know the drift parameter of the underlying stock. This is taken to be a random variable with a Gaussian prior distribution, which is updated via the Kalman filter. This results in a model with a stochastic drift process adapted to the observation filtration, and which can be treated as a full information problem, and an explicit solution to the optimal investment problem is possible. We also consider the same problem when the agent has noisy knowledge at time $0$ of the terminal value of the Brownian motion driving the stock. Using techniques of enlargement of filtration to accommodate the insider's additional knowledge, followed by filtering the asset price drift, we are again able to obtain an explicit solution. Finally we treat an incomplete market hedging problem. A claim on a non-traded asset is hedged using a correlated traded asset. We summarise the full information case, then treat the partial information scenario in which the hedger is uncertain of the true values of the asset price drifts. After filtering, the resulting problem with random drifts is solved in the case that each asset's prior distribution has the same variance, resulting in analytic approximations for the optimal hedging strategy

Malliavin calculus method for asymptotic expansion of dual control problems

We develop a technique based on Malliavin-Bismut calculus ideas, for asymptotic expansion of dual control problems arising in connection with exponential indifference valuation of claims, and with minimisation of relative entropy, in incomplete markets. The problems involve optimisation of a functional of Brownian paths on Wiener space, with the paths perturbed by a drift involving the control. In addition there is a penalty term in which the control features quadratically. The drift perturbation is interpreted as a measure change using the Girsanov theorem, leading to a form of the integration by parts formula in which a directional derivative on Wiener space is computed. This allows for asymptotic analysis of the control problem. Applications to incomplete It\^o process markets are given, in which indifference prices are approximated in the low risk aversion limit. We also give an application to identifying the minimal entropy martingale measure as a perturbation to the minimal martingale measure in stochastic volatility models

Utility indifference pricing with market incompleteness

Utility indifference pricing and hedging theory is presented, showing how it leads to linear or to non-linear pricing rules for contingent claims. Convex duality is first used to derive probabilistic representations for exponential utility-based prices, in a general setting with locally bounded semi-martingale price processes. The indifference price for a finite number of claims gives a non-linear pricing rule, which reduces to a linear pricing rule as the number of claims tends to zero, resulting in the so-called marginal utility-based price of the claim. Applications to basis risk models with lognormal price processes, under full and partial information scenarios are then worked out in detail. In the full information case, a claim on a non-traded asset is priced and hedged using a correlated traded asset. The resulting hedge requires knowledge of the drift parameters of the asset price processes, which are very difficult to estimate with any precision. This leads naturally to a further application, a partial information problem, with the drift parameters assumed to be random variables whose values are revealed to the hedger in a Bayesian fashion via a filtering algorithm. The indifference price is given by the solution to a non-linear PDE, reducing to a linear PDE for the marginal price when the number of claims becomes infinitesimally small

The minimal entropy measure and an Esscher transform in an incomplete market model

We consider an incomplete market model with one traded stock and two correlated Brownian motions $W$,$\widetilde{W}$. The Brownian motion $W$ drives the stock price, whose volatility and Sharpe ratio are adapted to the filtration $\mathbb{F} := (\widetilde{\mathcal{F}}_{t})_{0 \le t \le T}$ generated by $\widetilde{W}$. We show that the projections of the minimal entropy and minimal martingale measures onto $\widetilde{\mathcal{F}}_{T}$ are related by an Esscher transform involving the correlation between $W$,$\widetilde{W}$, and the mean-variance trade-off process. The result leads to a new formula for the marginal exponential utility-based price of an $\widetilde{\mathcal{F}}_{T}$-measurable European claim

Optimal exercise of an executive stock option by an insider

We consider an optimal stopping problem arising in connection with the exercise of an executive stock option by an agent with inside information. The agent is assumed to have noisy information on the terminal value of the stock, does not trade the stock or outside securities, and maximises the expected discounted payoff over all stopping times with regard to an enlarged filtration which includes the inside information. This leads to a stopping problem governed by a time-inhomogeneous diffusion and a call-type reward. We establish conditions under which the option value exhibits time decay, and derive the smooth fit condition for the solution to the free boundary problem governing the maximum expected reward, and derive the early exercise decomposition of the value function. The resulting integral equation for the unknown exercise boundary is solved numerically and this shows that the insider may exercise the option before maturity, in situations when an agent without the privileged information may not. Hence we show that early exercise may arise due to the agent having inside information on the future stock price

Efficient option pricing with transaction costs

A fast numerical algorithm is developed to price European options with proportional transaction costs using the utility-maximization framework of Davis (1997). This approach allows option prices to be computed by solving the investor’s basic portfolio selection problem without insertion of the option payoff into the terminal value function. The properties of the value function can then be used to drastically reduce the number of operations needed to locate the boundaries of the no-transaction region, which leads to very efficient option valuation. The optimization problem is solved numerically for the case of exponential utility, and comparisons with approximately replicating strategies reveal tight bounds for option prices even as transaction costs become large. The computational technique involves a discrete-time Markov chain approximation to a continuous-time singular stochastic optimal control problem. A general definition of an option hedging strategy in this framework is developed. This involves calculating the perturbation to the optimal portfolio strategy when an option trade is executed

Preserving unitarity in a novel perturbative technique for solving quantum field theory

A new perturbative technique for solving a scalar φ2 P theory consists of expanding a φ2(1+δ) interaction in powers of δ. The Green functions are computed as a power series in δ by applying a linear differential operator to the Green functions of a specially constructed intermediate Lagrangian. We confront this linear procedure with the quadratic requirement of perturbative unitarity. We verify up to order δ3 that unitarity is indeed satisfied, by virtue of the precise structure of the intermediate Lagrangian. Unitarity gives constraints on that structure, but does not fix it uniquely. © 1989 Springer-Verlag

The minimal entropy measure and an Esscher transform in an incomplete market model

We consider an incomplete market model with one traded stock and two correlated Brownian motions $W$,$\widetilde{W}$. The Brownian motion $W$ drives the stock price, whose volatility and Sharpe ratio are adapted to the filtration $\mathbb{F} := (\widetilde{\mathcal{F}}_{t})_{0 \le t \le T}$ generated by $\widetilde{W}$. We show that the projections of the minimal entropy and minimal martingale measures onto $\widetilde{\mathcal{F}}_{T}$ are related by an Esscher transform involving the correlation between $W$,$\widetilde{W}$, and the mean-variance trade-off process. The result leads to a new formula for the marginal exponential utility-based price of an $\widetilde{\mathcal{F}}_{T}$-measurable European claim

Utility-based valuation and hedging of basis risk with partial information

We analyse the valuation and hedging of a claim on a non-traded asset using a correlated traded asset under a partial information scenario, when the asset drifts are unknown constants. Using a Kalman filter and a Gaussian prior distribution for the unknown parameters, a full information model with random drifts is obtained. This is subjected to exponential indifference valuation. An expression for the optimal hedging strategy is derived. An asymptotic expansion for small values of risk aversion is obtained via PDE methods, following on from payoff decompositions and a price representation equation. Analytic and semi-analytic formulae for the terms in the expansion are obtained when the minimal entropy measure coincides with the minimal martingale measure. Simulation experiments are carried out which indicate that the filtering procedure can be beneficial in hedging, but sometimes needs to be augmented with the increased option premium, that takes into account parameter uncertainty, in order to be effective. Empirical examples are presented which conform to these conclusions