Pricing Options in Incomplete Equity Markets via the Instantaneous Sharpe Ratio

We use a continuous version of the standard deviation premium principle for pricing in incomplete equity markets by assuming that the investor issuing an unhedgeable derivative security requires compensation for this risk in the form of a pre-specified instantaneous Sharpe ratio. First, we apply our method to price options on non-traded assets for which there is a traded asset that is correlated to the non-traded asset. Our main contribution to this particular problem is to show that our seller/buyer prices are the upper/lower good deal bounds of Cochrane and Sa\'{a}-Requejo (2000) and of Bj\"{o}rk and Slinko (2006) and to determine the analytical properties of these prices. Second, we apply our method to price options in the presence of stochastic volatility. Our main contribution to this problem is to show that the instantaneous Sharpe ratio, an integral ingredient in our methodology, is the negative of the market price of volatility risk, as defined in Fouque, Papanicolaou, and Sircar (2000).Comment: Keywords: Pricing derivative securities, incomplete markets, Sharpe ratio, correlated assets, stochastic volatility, non-linear partial differential equations, good deal bound

The valuation of guaranteed lifelong withdrawal benefit options in variable annuity contracts and the impact of mortality risk

n light of the growing importance of the variable annuities market, in this paper we introduce a theoretical model for the pricing and valuation of guaranteed lifelong withdrawal benefit (GLWB) options embedded in variable annuity products. As the name suggests, this option offers a lifelong withdrawal guarantee; therefore, there is no limit on the total amount that is withdrawn over the term of the policy because if the account value becomes zero while the insured is still alive, he or she continues to receive the guaranteed amount annually until death. Any remaining account value at the time of death is paid to the beneficiary as a death benefit. We offer a specific framework to value the GLWB option in a market-consistent manner under the hypothesis of a static withdrawal strategy, according to which the withdrawal amount is always equal to the guaranteed amount. The valuation approach is based on the decomposition of the product into living and death benefits. The model makes use of the standard no-arbitrage models of mathematical finance, which extend the Black-Scholes framework to insurance contracts, assuming the fund follows a geometric Brownian motion and the insurance fee is paid, on an ongoing basis, as a proportion of the assets. We develop a sensitivity analysis, which shows how the value of the product varies with the key parameters, including the age of the policyholder at the inception of the contract, the guaranteed rate, the risk-free rate, and the fund volatility. We calculate the fair fee, using Monte Carlo simulations under different scenarios. We give special attention to the impact of mortality risk on the value of the option, using a flexible model of mortality dynamics, which allows for the possible perturbations by mortality shock of the standard mortality tables used by practitioners. Moreover, we evaluate the introduction of roll-up and step-up options and the effect of the decision to delay withdrawing. Empirical analyses are performed, and numerical results are provided

An Object-Oriented Framework For Valuing Shout Options on High-Performance Computer Architectures

A shout option is a financial contract which allows the holder to change the payoff during the lifetime of the contract. For example, the holder could have the right to set the strike price to the current value of the underlying asset. Complex versions of these options are embedded in financial products which offer various types of maturity guarantees such as segregated funds marketed by Canadian insurance companies. The value of these options can be determined by solving a collection of coupled partial differential equations (PDEs). In this work we develop an extensible, object-oriented framework for valuing these contracts which is capable of exploiting modern, high-performance supercomputing architectures. We use this framework to study and illustrate practical aspects of valuing and hedging these contracts

Pricing Methods and Hedging Strategies for Volatility Derivatives

In this paper we investigate the behaviour and hedging of discretely observed volatility derivatives. We begin by comparing the e#ects of variations in the contract design, such as the di#erences between specifying log returns or actual returns, taking into consideration the impact of possible jumps in the underlying asset. We then focus on the di#culties associated with hedging these products. Naive delta-hedging strategies are ine#ective for hedging volatility derivatives since they require very frequent rebalancing and have limited ability to protect the writer against possible jumps in the underlying asset. We investigate the performance of a hedging strategy for volatility swaps that establishes small, fixed positions in straddles and out-of-the-money strangles at each volatility observation

Numerical methods and volatility models for valuing cliquet options

Several numerical issues for valuing cliquet options using PDE methods are investigated. The use of a running sum of returns formulation is compared to an average return formulation. Methods for grid construction, interpolation of jump conditions, and application of boundary conditions are compared. The effect of various volatility modelling assumptions on the value of cliquet options is also studied. Numerical results are reported for jump diffusion models, calibrated volatility surface models, and uncertain volatility models