International Journal of Financial Engineering

This paper provides an in-depth analysis of the structuring and the pricing of an innovative financial market product. This instrument is called a contingent conversion convertible bond or “CoCoCo”. This hybrid bond is itself a combination of two other hybrid instruments: a contingent convertible (“CoCo”) and a convertible bond. This combination introduces more complexity in the structure but it also allows investors to profit from strong share price performances. This upside potential is added on top of the normal contingent convertible mechanics of CoCos, which expose the investors to mainly downside risk. First, we explain how the features of the contingent convertible bonds on one side and the features of the standard convertible bonds on the other side are combined. Thereafter, we propose a pricing approach which moves away from the standard Black$\&$Scholes setting. The CoCoCos are evaluated using the Heston process to which a Hull-White interest rate process has been added. We demonstrate the importance of using a stochastic interest rate when modeling this instrument. Finally we quantify the loss absorbing capacity of this instrument.

This paper introduces a new efficient and practical weak approximation for option price under local stochastic volatility model as marginal expectation of stochastic differential equation, using iterative asymptotic expansion with Malliavin weights. The explicit Malliavin weights for SABR model are shown. Numerical experiments confirm the validity of our discretization with a few time steps.

This paper proposes a trading strategy that dynamically rebalances static super-replicating portfolios, which is very useful for both investment and hedging strategies. In order to investigate general properties of the strategy, we derive the Doob–Meyer decomposition for the value process without any specifications of models under the continuous processes of the underlying variables. In particular, we find that the increasing part of the decomposition characterizes the performance of the strategy. Also, we obtain more concrete features for cross-currency and one-touch options based on our general framework. Moreover, numerical examples for cross-currency options demonstrate the effectiveness of our strategy for investment and hedging.

In this paper, it has been shown that the combined use of exponential operators and special functions provides a powerful tool to solve certain class of generalized space fractional Laguerre heat equation. It is shown that exponential operators are powerful and effective method for solving certain singular integral equations and space fractional Black–Scholes equation.

The pricing equations of the average options with jump diffusion processes can be formulated as two-dimensional partial integro-differential equations (PIDEs). In the uncertain volatility model, for options with non-convex and non-concave payoffs, such as the butterfly spread, the PIDEs are nonlinear. We use the semi-Lagrangian method to reduce the two-dimensional nonlinear PIDE to a one-dimensional nonlinear PIDE along the trajectory of the average price, and use a Newton-type iteration to guarantee the convergence of the discrete solution to the viscosity solution. Monotonicity and stability as well as the convergence results are derived. Numerical tests of convergence for a variety of cases, including average butterfly spread and ordinary butterfly spread, are presented.

In this paper, a new approach is proposed to improve forecasting performances. We analyze the co-movement of precious metals (daily data of gold, silver and platinum starting from July, 2011) using multiple wavelet coherence and determine the movement dependencies on frequency–time space. The data is split into frequencies using scale by scale continuous wavelet transform. All three time series retaining the same frequency scale are (i) selected, (ii) inversed and (ii) forecasted using multivariate model, Vector Auto Regressive Moving Average (VARMA). We conclude that the efficiency of VARMA forecasting is substantially increased because of same frequency highly correlated time series obtained by using scale by scale wavelet transform. Moreover, the direction of price shift (increasing/decreasing trend) is prospected to an adequately distinguishable degree.

The two-additive-factor Gaussian model G2$++$ is a famous stochastic model for the instantaneous short rate. It has functional qualities required in various practical purposes, as in Asset Liability Management and in Trading of interest rate derivatives. Though closed formulas for the prices of various main interest-rate instruments are known and used under the G2$++$ model, it seems that references for the corresponding sensitivities are not clearly presented over the financial literature. To fill this gap is one of our purposes in the present work. We derive here analytic expressions for the sensitivities of zero-coupon bond, coupon-bearing bonds, portfolio of coupon bearing bonds. The sensitivities under consideration here are those with respect to the shocks linked to the unobservable two-uncertainty shock risk/opportunity factors underlying the G2$++$ model. As a such, the hedging of a position sensitive to the interest rate by means of a portfolio (in accordance with the market participants practice) becomes easily transparent as just resulting from the balance between the various involved sensitivities.

An equity-linked note with complex payoff and a path dependent variable maturity is analyzed analytically and numerically. This note pays up to one coupon, the size of which depends on the period count in which it is paid and with initial investment partially protected. Valuation of this instrument is achieved in closed form in terms of multivariate normal distribution function, but simple, fast and accurate numerical solution in terms of Chapman–Kolmogorov equations is preferred. The note has a number of unique features, some of which expose the investors to types of risk not usually encountered in either equity or fixed income world.

Optimal dividend strategy in dual risk model is well studied in the literatures. But to the best of our knowledge, all the previous works assumes deterministic interest rate. In this paper, we study the optimal dividends strategy in dual risk model, under a stochastic interest rate, assuming the discounting factor follows a geometric Brownian motion or exponential Lévy process. We will show that closed form solutions can be obtained.

In this paper, we present a pricing model for European options in a mixed fractional Hull–White interest rate model. By using the variable transform techniques and mathematical physics methods, we derive closed-form pricing formulas for this pricing problem, which are the main contribution of this paper and expand the relevant literature’s conclusions. Moreover, we provide numerical examples to illustrate the effects of main parameters of the mixed fractional interest rate model on the option price. Numerical results show that the long memory property of interest rates plays an important role in determining the option price and cannot be neglected in option pricing.

This paper presents a Heston-based pricing model for contingent convertible bonds (CoCos). The main finding is that skew in the implied volatility surface has a significant impact on the CoCo price. Hence stochastic volatility models, like the Heston model, which incorporate smile and skew are appropriate in the context of pricing CoCos. The financial crisis of 2007–2008 triggered an avalanche of financial worries for financial institutions around the globe. After the collapse of Lehman Brothers, governments intervened and bailed out banks using tax-payer’s money. Preventing such bail-outs in the future and designing a more stable banking sector in general, requires both higher capital levels and regulatory capital of a higher quality. Bank debt needed therefore to be made absorbing. This is where CoCos come in. The Lloyds Banking Group introduced the first CoCo bonds as early as December 2009. Since the first issuance of a CoCo bond more than five years ago, the market has continued to grow and has now reached a size over €120bn. CoCos are hybrid financial instruments that convert into equity or suffer a write-down of the face value upon the appearance of a trigger event, often in terms of the bank’s CET1 level in combination with a regulatory trigger. The valuation of CoCos boils down to the quantification of the trigger probability and the expected loss suffered by the investors if such a trigger event eventually takes place. There are at least two schools of thought regarding valuation of CoCos. Structural models can be put at work or investors can rely on market implied models. The latter category uses market data (share prices, CDS levels and implied volatility, etc.) in order to calculate the theoretical price of a CoCo bond. In De Spiegeleer and Schoutens The Journal of Derivatives, the pricing of CoCo notes has been worked out in a market implied Black–Scholes context. In this paper we move away from the assumption of a constant volatility which is the back-bone of Black–Scholes based valuation and put the Heston model at work and study CoCos in a stochastic volatility context. The existence of a semi closed-form formula for European options pricing under the Heston model allows for a fast calibration of the model. In our approach we combined market quotes of listed option prices with CDS data. As a case study, the procedure was applied on the Tier 2 10NC $7\frac{5}{8}\%$ CoCo issued by Barclays in 2012.

We examine currency options in the double exponential jump-diffusion version of the Heston stochastic volatility model for the exchange rate. We assume, in addition, that the domestic and foreign stochastic interest rates are governed by the CIR dynamics. The instantaneous volatility is correlated with the dynamics of the exchange rate return, whereas the domestic and foreign short-term rates are assumed to be independent of the dynamics of the exchange rate and its volatility. The main result furnishes a semi-analytical formula for the price of the European currency call option in the hybrid foreign exchange/interest rates model.

This paper studies the effect of fractional volatility on path-dependent options, which are highly sensitive to the volatility structure of a targeted underlying asset process. To this end, we propose an approximation formula for average and barrier options when volatility follows a fractional Brownian motion. Furthermore, using the analytical formula, we investigate the impact of the Hurst index on option prices. Overall, our important finding is that when the maturity is short and speed of mean-reversion is slow, the impact of the Hurst index strongly influences the option prices and that is non-negligible. This is an important lesson for practitioners who uses standard Brownian motion.

The estimation of partial derivatives of the price in respect to the main financial variables, called Greeks, is an essential task for a trader in order to understand the sensitivity of a derivative to the input of pricing model. The study of the level of reactivity of the mark to market is an essential task to manage properly the market risk of a portfolio. Due to the negative interest rates in Euro Area, the pricing model of interest-rates options (cap, floor and swaption) has been changed from a log-normal to a normal framework. The aim of this paper is to investigate the effects of this model change on the calculation of option sensitivities.