International Journal of Theoretical and Applied Finance

Constant proportion portfolio insurance (CPPI) strategy is a very popular investment solution which provides an investor with a capital protection as well as allows for an equity market participation. In this paper, we propose a two-step approach to the numerical optimization of the CPPI main parameter, multiplier. First, we identify an admissible range of the multiplier values by controlling the shortfall probability (chosen as a measure of the gap risk). Second, within the admissible range, we choose the optimal multiplier value with respect to the omega ratio (chosen as a performance measure). We illustrate the performance of our optimization algorithm on simulated CPPI paths in the Black–Scholes environment with discrete trading as well as on the historical S&P500 data using the block-bootstrap simulations.

We study a credit risk model of a financial market in which the dynamics of intensity rates of two default times are described by linear combinations of three independent geometric Brownian motions. The dynamics of two default-free risky asset prices are modeled by two geometric Brownian motions which are dependent of the ones describing the default intensity rates. We obtain closed form expressions for the no-arbitrage prices of both risk-free and risky credit default swaps given the reference filtration initially and progressively enlarged by the two default times. The accessible default-free reference filtration is generated by the standard Brownian motions driving the model.

It is known that an adaptation of Newton’s method allows for the computation of functional inverses of formal power series. We show that it is possible to successfully use a similar algorithm in a fairly general analytical framework. This is well suited for functions that are highly tangent to identity and that can be expanded with respect to asymptotic scales of “exp-log functions”. We next apply our algorithm to various well-known functions coming from the world of quantitative finance. In particular, we deduce asymptotic expansions for the inverses of the Gaussian and the Black–Scholes pricing functions.

Risk measurement and pricing of financial positions are based on modeling assumptions, which are common assumptions on the probability distribution of the position’s outcomes. We associate a model with a probability measure and investigate model risk by considering a model space. First, we incorporate model risk into market risk measures by introducing model weighted and superposed market risk measures. Second, we quantify model risk itself and propose axioms for model risk measures. We introduce superposed model risk measures that quantify model risk relative to a reference model, which is the financial institution’s model of choice. Several risk measures that we propose require a probability distribution on the model space, which can be obtained from data by applying Bayesian analysis. Examples and a case study illustrate our approaches.

We propose a new framework to value employee stock options (ESOs) that capture multiple exercises of different quantities over time. We also model the ESO holder’s job termination risk and incorporate its impact on the payoffs of both vested and unvested ESOs. Numerical methods based on Fourier transform and finite differences are developed and implemented to solve the associated systems of PDEs. In addition, we introduce a new valuation method based on maturity randomization that yields analytic formulae for vested and unvested ESO costs. We examine the cost impact of job termination risk, exercise intensity and various contractual features.

This paper proposes a set of desirable axioms to characterize performance measures in the context of portfolio management. A performance measure consistent with the axioms is called “ideal”. We observe that a popular performance measure, Farinelli–Tibiletti (FT) ratio [S. Farinelli & L. Tibiletti (2008) Sharpe thinking in asset ranking with one-sided measures, European Journal of Operational Research 185 (3), 1542–1547], which captures potential-seeking behavior, is not ideal. It violates a very important property of portfolio theory, the diversification. As an alternative, we propose a new ideal performance measure, upside beta ratio (UBR). To examine its performance, we rank mutual funds for UBR and other four performance measures — Sharpe, Sortino, FT and Jensen’s alpha — and then we compare the rankings of UBR with the rankings of other ratios. In addition, the performance of top-ranked funds are compared through back-testing and out-of-sample analysis. Our findings reveal that the UBR performs significantly better than the other ratios in most scenarios. Finally, in order to check robustness of the new measure, a parameter sensitivity analysis is presented.

This work builds on the work by Joslin et al. [(2011) A new perspective on Gaussian dynamic term structure Models, The Review of Financial Studies 24, 926–970] on the affine dynamics of portfolios of yields and addresses the unresolved issues of ‘internal consistency’ mentioned in the same paper. It shows the unexpected constraints that have to be satisfied by the $\mathbb{Q}$-measure evolution of the yield curve if the portfolio of yields has to be interpreted as their principal components. This choice of state variables is common in the recent literature and so our findings are intrinsically interesting. However, we show that our results also extend to a wide class of choices for state variables, when these are chosen as linear combinations of yields. We show that these constraints have important financial consequences, which, to our knowledge, have not been appreciated. In particular, this paper highlights some puzzling issues of compatibility between the $\mathbb{Q}$- and $\mathbb{P}$-measure dynamics of Gaussian dynamic term structure models when principal components are chosen as state variables, once the constraints we derive are taken into account.