International Journal of Theoretical and Applied Finance

The purpose of this paper is introducing rigorous methods and formulas for bilateral counterparty risk credit valuation adjustment (CVA) on interest-rate portfolios. In doing so, we summarize the general arbitrage-free valuation framework for counterparty risk adjustments in presence of bilateral default risk, including the default of the investor. We illustrate the symmetry in the valuation and show that the adjustment involves a long position in a put option plus a short position in a call option, both with zero strike and written on the residual net present value of the contract at the relevant default times. We allow for correlation between the default times of the investor and counterparty, and for correlation of each with the underlying risk factor, namely interest rates. We also analyze the often neglected impact of credit spread volatility. We include close-out netting rules in our examples, although other agreements, such as periodic margining or collateral posting, are left for future work.

We analyze the errors arising from discrete readjustment of the hedging portfolio when hedging options in exponential Lévy models, and establish the rate at which the expected squared error goes to zero when the readjustment frequency increases. We compare the quadratic hedging strategy with the common market practice of delta hedging, and show that for discontinuous option pay-offs the latter strategy may suffer from very large discretization errors. For options with discontinuous pay-offs, the convergence rate depends on the underlying Lévy process, and we give an explicit relation between the rate and the Blumenthal-Getoor index of the process.

We generalize the arbitrage-free valuation framework for counterparty credit risk (CCR) adjustments when credit triggers are allowed in the contract. The settlement of the deal for the investor could be either obliged or optional to execute when the counterparty hits the credit trigger before any default events from the two parties. General formulas for credit value adjustment (CVA) are given for all four cases: obliged unilateral, obliged bilateral, optional unilateral and optional bilateral. The unilateral CVA with an optional credit trigger is found to be the same as the unilateral CVA with an analogous obliged credit trigger. We show that adding credit triggers will decrease the unilateral CVA for both obliged and optional cases, which are in line with the motivation of investors to reduce CCR. However, adding credit triggers may not necessarily reduce bilateral CVA. Counter-intuitively, we show that the bilateral CVA may actually increase by adding credit triggers. Moreover, the increased amount of bilateral CVA due to credit triggers for one party is exactly the same amount of bilateral CVA reduced for the other party. The CVA calculation is subjected to large uncertainty of model risks, mostly due to the lack of data for calibrating jump-to-default probabilities. Some explicit models for obliged unilateral CVA are discussed with special caveats on the model assumptions. Numerical examples are also given to illustrate the model risk of CVA calculation due to the uncertainty of jump sizes, even though pure jump models are assumed.

This paper investigates the use of multiple directions of stratification as a variance reduction technique for Monte Carlo simulations of path-dependent options driven by Gaussian vectors. The precision of the method depends on the choice of the directions of stratification and the allocation rule within each strata. Several choices have been proposed but, even if they provide variance reduction, their implementation is computationally intensive and not applicable to realistic payoffs, in particular not to Asian options with barrier. Moreover, all these previously published methods employ orthogonal directions for multiple stratification. In this work we investigate the use of algorithms producing convenient directions, generally non-orthogonal, combining a lower computational cost with a comparable variance reduction. In addition, we study the accuracy of optimal allocation in terms of variance reduction compared to the Latin Hypercube Sampling. We consider the directions obtained by the Linear Transformation and the Principal Component Analysis. We introduce a new procedure based on the Linear Approximation of the explained variance of the payoff using the law of total variance. In addition, we exhibit a novel algorithm that permits to correctly generate normal vectors stratified along non-orthogonal directions. Finally, we illustrate the efficiency of these algorithms in the computation of the price of different path-dependent options with and without barriers in the Black-Scholes and in the Cox-Ingersoll-Ross markets.

In this paper, we quantify the impact on the representative agent's welfare of the presence of derivative products spanning covariance risk. In an asset allocation framework with stochastic (co)variances, we allow the agent to invest not only in the stocks but also in the associated variance swaps. We solve this optimal portfolio allocation program using the Wishart Affine Stochastic Correlation framework, as introduced in Da Fonseca, Grasselli and Tebaldi (2007): it shares the analytical tractability of the single-asset counterpart represented by the [36] model and it seems to be the natural framework for studying multivariate problems when volatilities as well as correlations are stochastic. What is more, this framework shows how variance swaps can implicitly span the covariance risk. We provide the explicit solution to the portfolio optimization problem and we discuss the structure of the portfolio loadings with respect to model parameters. Using real data on major indexes, we find that the impact of covariance risk on the optimal strategy is huge. It first leads to a portfolio that is mostly driven by the market price of volatility-covolatility risks. It is then strongly leveraged through variance swaps, thus leading to a much higher utility, when compared to the case when investing in such derivatives is not possible.

We propose a tractable model for the exchange rate in a target zone with realignment. The target zone exchange rate dynamics is assumed to obey a bounded regular diffusion with two-sided unattainable barriers. The realignment is modeled as a continuous-time two-state Markov chain. Under the stationary setting of the Markov chain, a general pricing formula for the derivative written on the exchange rate is derived in the presence of the realignment risk. The Jacobi diffusion model is studied as an example and numerical results are presented for illustration.

Portfolio optimization under downside risk is of crucial importance to asset managers. In this article we consider one such particular measure given by the notion of Capital at Risk (CaR), closely related to Value at Risk. We consider portfolio optimization with respect to CaR in the Black-Scholes setting with time dependent parameters and investment strategies, i.e., continuous-time portfolio optimization. We review the results from our previous work in unconstrained portfolio optimization, and then investigate and solve the corresponding problems with the additional constraint of no-short-selling. Analytical formulae are derived for the optimal strategies, and numerical examples are presented.