A new approach for option pricing under stochastic volatility

We develop a new approach for pricing European-style contingent claims written on the time T spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast to most of the stochastic volatility literature, we do not directly model the dynamics of the instantaneous volatility. Instead, taking advantage of the recent rise of the variance swap market, we directly assume continuous dynamics for the time T variance swap rate. The initial value of this variance swap rate can either be directly observed, or inferred from option prices. We make no assumption concerning the real world drift of this process. We assume that the ratio of the volatility of the variance swap rate to the instantaneous volatility of the underlying asset just depends on the variance swap rate and on the variance swap maturity. Since this ratio is assumed to be independent of calendar time, we term this key assumption the stationary volatility ratio hypothesis (SVRH). The instantaneous volatility of the futures follows an unspecified stochastic process, so both the underlying futures price and the variance swap rate have unspecified stochastic volatility. Despite this, we show that the payoff to a path-independent contingent claim can be perfectly replicated by dynamic trading in futures contracts and variance swaps of the same maturity. As a result, the contingent claim is uniquely valued relative to its underlying’s futures price and the assumed observable variance swap rate. In contrast to standard models of stochastic volatility, our approach does not require specifying the market price of volatility risk or observing the initial level of instantaneous volatility. As a consequence of our SVRH, the partial differential equation (PDE) governing the arbitrage-free value of the contingent claim just depends on two state variables rather than the usual three. We then focus on the consistency of our SVRH with the standard assumption that the risk-neutral process for the instantaneous variance is a diffusion whose coefficients are independent of the variance swap maturity. We show that the combination of this maturity independent diffusion hypothesis (MIDH) and our SVRH implies a very special form of the risk-neutral diffusion process for the instantaneous variance. Fortunately, this process is tractable, well-behaved, and enjoys empirical support. Finally, we show that our model can also be used to robustly price and hedge volatility derivatives.     

The valuation of multivariate contingent claims under transformed trinomial approaches

This study develops a transformed-trinomial approach for the valuation of contingent claims written on multiple underlying assets. Our model is characterized by an extension of the Camara and Chung (J Futur Mark 26: 759–787, 2006) transformed-binomial model for pricing options with one underlying asset, and a discrete-time version of the Schroder (J Finance 59(5): 2375–2401, 2004) model. However, unlike the Schroder model, our model can facilitate straightforward valuation of American-style multivariate contingent claims. The major advantage of our transformed-trinomial approach is that it can easily tackle the volatility skew observed within the markets. We go on to use numerical examples to demonstrate the way in which our transformed-trinomial approach can be utilized for the valuation of multivariate contingent claims, such as binary options.     

Sovereign asset values and implications for the credit market

Using the contingent claim approach and market data on sovereign credit default swaps we assess the drivers of a country's risk perception. Deriving market-based asset values for a set of advanced economies we gain insights into the capital markets' perspectives on sovereign creditworthiness. We find the market-based asset values to be positively influenced by debt and to be an early risk indicator for economic developments. In a cross-section analysis we identify drivers of the economic risk of countries. Clustering the countries according to their debt to asset value ratios provides further insights into the market perceptions of sovereign credit risk. For example we find that the asset values of countries with higher ratios react to changes in the global equity market. Countries with a lower ratio react more to the political stability within the country.     

Irreversible investment and discounting: an arbitrage pricing approach

This paper presents a unified approach to valuing investment projects under uncertainty, based on stochastic discount factors, by linking optimal stopping theory to the no-arbitrage principle in asset pricing. An investment threshold for the case where the discount factor and the project’s cash-flow both follow a geometric Brownian motion is derived. Comparative statics of the investment trigger are obtained adding to and clarifying on the uncertainty–investment debate. Finally, two different ways to obtain discount factors are illustrated: spanning assets and representative agent analysis. The link between the characteristics of these different approaches and the optimal investment policy is clarified.     

Optimal static quadratic hedging

We propose a flexible framework for hedging a contingent claim by holding static positions in vanilla European calls, puts, bonds and forwards. A model-free expression is derived for the optimal static hedging strategy that minimizes the expected squared hedging error subject to a cost constraint. The optimal hedge involves computing a number of expectations that reflect the dependence among the contingent claim and the hedging assets. We provide a general method for approximating these expectations analytically in a general Markov diffusion market. To illustrate the versatility of our approach, we present several numerical examples, including hedging path-dependent options and options written on a correlated asset.     

PDE approach to valuation and hedging of credit derivatives

This paper presents a PDE approach in a Markovian setting to hedge defaultable derivatives. The arbitrage price and the hedging strategy for an attainable contingent claim are described in terms of solutions of a pair of coupled PDEs. For some standard examples of defaultable claims, we provide explicit formulae for prices and hedging strategies.     

Asymptotic Investment Behaviors under a Jump-Diffusion Risk Process

We study an optimal investment control problem for an insurance company. The surplus process follows the Cramer-Lundberg process with perturbation of a Brownian motion. The company can invest its surplus into a risk-free asset and a Black-Scholes risky asset. The optimization objective is to minimize the probability of ruin. We show by new operators that the minimal ruin probability function is a classical solution to the corresponding HJB equation. Asymptotic behaviors of the optimal investment control policy and the minimal ruin probability function are studied for low surplus levels with a general claim size distribution. Some new asymptotic results for large surplus levels in the case with exponential claim distributions are obtained. We consider two cases of investment control: unconstrained investment and investment with a limited amount.     

Coefficients of Asymptotic Expansions of SDE with Jumps

A new methodology for the problem of contingent claim valuation is proposed by Yoshida (Journal of Japanese Statistical Society, 22(2): 139–159, 1992, Stochastic Processes Application, 107(1): 53–81, 2003), and Takahashi and Kunitomo (2003). They used the asymptotic expansion theorem of Watanabe. Their method is applicable to various problems of contingent claim valuation. The author has obtained the asymptotic expansion formula for European call option of pure jump models (2008). In this paper, we determine the coefficients of the asymptotic expansion formula in order to test this formula numerically.     

A PDE approach for risk measures for derivatives with regime switching

This paper considers a partial differential equation (PDE) approach to evaluate coherent risk measures for derivative instruments when the dynamics of the risky underlying asset are governed by a Markov-modulated geometric Brownian motion (GBM); that is, the appreciation rate and the volatility of the underlying risky asset switch over time according to the state of a continuous-time hidden Markov chain model which describes the state of an economy. The PDE approach provides market practitioners with a flexible and effective way to evaluate risk measures in the Markov-modulated Black–Scholes model. We shall derive the PDEs satisfied by the risk measures for European-style options, barrier options and American-style options.      

Pricing a contingent claim liability with transaction costs using asymptotic analysis for optimal investment

We price a contingent claim liability (claim for short) using a utility indifference argument. We consider an agent with exponential utility, who invests in a stock and a money market account with the goal of maximizing the utility of his investment at the final time T in the presence of a proportional transaction cost ε>0 in two cases: with and without a claim. Using the heuristic computations of Whalley and Wilmott (Math. Finance 7:307–324, 1997), under suitable technical conditions, we provide a rigorous derivation of the asymptotic expansion of the value function in powers of \(\varepsilon^{\frac{1}{3}}\) in both cases with and without a claim. Additionally, using the utility indifference method, we derive the price of the claim at the leading order of \(\varepsilon^{\frac{2}{3}}\) . In both cases, we also obtain a “nearly optimal” strategy, whose expected utility asymptotically matches the leading terms of the value function. We also present an example of how this methodology can be used to price more exotic barrier-type contingent claims.     

Expensing stock options: a fair-value approach

Now that companies such as General Electric and Citigroup have accepted the premise that employee stock options are an expense, the debate is shifting from whether to report options on income statements to how to report them. The authors present a new accounting mechanism that maintains the rationale underlying stock option expensing while addressing critics' concerns about measurement error and the lack of reconciliation to actual experience. A procedure they call fair-value expensing adjusts and eventually reconciles cost estimates made at grant date with subsequent changes in the value of the options, and it does so in a way that eliminates forecasting and measurement errors over time. The method captures the chief characteristic of stock option compensation--that employees receive part of their compensation in the form of a contingent claim on the value they are helping to produce. The mechanism involves creating entries on both the asset and equity sides of the balance sheet. On the asset side, companies create a prepaid-compensation account equal to the estimated cost of the options granted; on the owners'-equity side, they create a paid-in capital stock-option account for the same amount. The prepaid-compensation account is then expensed through the income statement, and the stock option account is adjusted on the balance sheet to reflect changes in the estimated fair value of the granted options. The amortization of prepaid compensation is added to the change in the option grant's value to provide the total reported expense of the options grant for the year. At the end of the vesting period, the company uses the fair value of the vested option to make a final adjustment on the income statement to reconcile any difference between that fair value and the total of the amounts already reported.