International Journal of Theoretical and Applied Finance

Real option valuation has traditionally been concerned with investment under project value uncertainty while assuming that the agent has perfect confidence in a specific model. However, agents do not generally have perfect confidence in their model and this ambiguity may affect their decisions. In addition, the value of real investments is not typically fully spanned by tradable assets because markets are incomplete as is typically the case in energy and commodities. In this paper, we account for the agent’s aversion to model ambiguity and address market incompleteness through the notion of robust indifference prices. We derive analytical results for the perpetual option to invest and the linear complementarity problem that the finite-time version of this problem satisfies. Ambiguity aversion has a number of effects on decision making some of which cannot be explained by altering the agent’s risk aversion. For example, ambiguity averse agents are found to exercise real options both earlier and later than their ambiguity neutral counterparts, depending on whether ambiguity stems from uncertainty in the dynamics of the project value or the dynamics of a hedging asset.

We consider robust pre-commitment and time-consistent mean-variance optimal asset allocation strategies, that are required to perform well also in a worst-case scenario regarding the development of the asset price. We show that worst-case scenarios for both strategies can be found by solving a specific equation each time step. In the unconstrained asset allocation case, the robust pre-commitment as well as the time-consistent strategy are identical to the corresponding robust myopic strategies, by which investors perform robust portfolio control only for one time step and conduct a risk-free strategy afterwards. In the experiments, the robustness of pre-commitment and time-consistent strategies is studied in detail. Our analysis and numerical results indicate that the time-consistent allocation strategy is more stable when possible incorrect assumptions regarding the future asset development are modeled and taken into account. In some situations, the time-consistent strategy can even generate higher efficient frontiers than the pre-commitment strategy (which is counter-intuitive), because the time-consistency restriction appears to protect an investor in such a situation.

In this paper, we study the financing of high uncertainty projects. High uncertainty is defined as the lack of knowledge of whether growth options exist. In this paper, we will describe this uncertainty by a probability distribution which describes the arrival of a growth option at a deterministic time. Once the option arrives, an additional uncertainty exists since it is not certain that it is profitable to exercise it. We value the corporate securities with contingent claims valuation both for a whole equity-financed firm and a debt-equity-financed firm. Unlike traditional capital structure models, we find nonconvex value functions for the firm vis-a-vis the debt coupon under specific parametrizations. High and low leverage can yield similar firm value maximizing policies.

Capital allocations have been studied in conjunction with static risk measures in various papers. The dynamic case has been studied only in a discrete-time setting. We address the problem of allocating risk capital to subportfolios for the first time in a continuous-time dynamic context. For this purpose, we introduce a differentiability result for backward stochastic Volterra integral equations and apply this result to derive continuous-time dynamic capital allocations. Moreover, we study a dynamic capital allocation principle that is based on backward stochastic differential equations and derive the dynamic gradient allocation for the dynamic entropic risk measure.

Unlike conventional convertible bonds, contingent convertible (CoCo) bonds are converted into equity shares of the issuing bank subject to certain trigger mechanisms (accounting and/or regulatory trigger) when the issuing bank is under financial nonviable state. We consider pricing of these CoCo bonds using the contingent claims approach, where the state variables are the stock price and Tier 1 capital ratio. We use the Parisian feature to model the regulatory trigger where equity conversion is triggered when the capital ratio stays under the nonviable state consecutively for a certain period of time. The accounting trigger is modeled using the one-touch barrier feature associated with the capital ratio. The Parisian trigger feature adds one extra path dependent state variable in the pricing model of a CoCo bond. We design effective numerical algorithms for pricing the CoCo bonds using the extended Fortet method that avoid adding one extra state variable for the Parisian feature of regulatory trigger. Pricing properties of the CoCo bonds under both regulatory trigger and accounting trigger are explored.

We show that an explicit approximate implied volatility formula can be obtained from a Black–Scholes formula approximation that is 2% accurate. The relative error of the approximate implied volatility is uniformly bounded for options with any moneyness and with arbitrary large or small option maturities and volatilities, including for long dated options and options on highly volatile underlying assets. For options within a large trading range, such as options with maturity less than five years and implied volatility less than 150%, the error of the approximate implied volatility relative to the Black–Scholes implied volatility is less than 10% points.

We consider two classes of wrong-way risk models in the context of CVA: static (resampling) and dynamic (reduced form). Although both potentially suffer from arbitrage problems, their tractability makes them appealing to the industry and therefore deserve additional study. For example, Gaussian copula-based resampling and reduced-form with “Hull–White intensities” yield analytical expected positive exposure (EPE) profiles when the portfolio price process (i.e. exposure process) is Gaussian. However, the first approach disregards credit volatility whilst the second can provide default probabilities larger than 1. We therefore enlarge the study by introducing a new dynamic approach for credit risk, consisting in the straight modeling of the survival (Azéma supermartingale) process using the $\mathrm{\Phi }$-martingale. This method is appealing in that it helps fixing some drawbacks of the above models. Indeed, it is a dynamic method (it disentangles correlation and credit volatility) that preserves probabilities in $[0,1]$ without affecting the analytical tractability of the model. In particular, calibration to any valid default probability curve is automatic and the closed-form expression for the EPE profiles remains available under Gaussian exposures. For each approach, we derive analytically the EPE profiles (conditional upon default) associated to prototypical exposure processes of Forward Rate Agreement (FRA) and Interest Rate Swap (IRS) in all cases and provide a comparison and discuss the implied Credit Valuation Adjustment (CVA) figures.