It Takes Two to Tango: Estimation of the Zero-Risk Premium Strike of a Call Option via Joint Physical and Pricing Density Modeling

It is generally said that out-of-the-money call options are expensive and one can ask the question from which moneyness level this is the case. Expensive actually means that the price one pays for the option is more than the discounted average payoff one receives. If so, the option bears a negative risk premium. The objective of this paper is to investigate the zero-risk premium moneyness level of a European call option, i.e., the strike where expectations on the option’s payoff in both the P- and Q-world are equal. To fully exploit the insights of the option market we deploy the Tilted Bilateral Gamma pricing model to jointly estimate the physical and pricing measure from option prices. We illustrate the proposed pricing strategy on the option surface of stock indices, assessing the stability and position of the zero-risk premium strike of a European call option. With small fluctuations around a slightly in-the-money level, on average, the zero-risk premium strike appears to follow a rather stable pattern over time.

CONIC CVA AND DVA FOR OPTION PORTFOLIOS

In this paper, we propose a framework for credit and debit valuation adjustments (CVA and DVA, respectively) for options and option portfolios which is based on conic finance, that is, where the positions are valued at their bid or ask prices depending on whether they are assets or liabilities. This can be achieved by transforming the pricing measure via appropriate distortion functions, depending on (at least) one parameter. We apply our methodology, which is based on the Wang transform, to portfolios of European commodity futures options, and we show that both CVA and DVA are significantly impacted by bid-ask spreads, when compared to their traditional risk-neutral counterparts. In particular, we show that DVA decreases when computed under conic finance settings, which is in line with the regulatory efforts to rein in DVA gains for financial institutions resulting from their own credit quality deterioration. Finally, we investigate the robustness of our approach with respect to the calibrated parameters, and we show that the calibrated distortion parameter is an excellent explanatory variable for the observed bid-ask spreads.

Pricing of Commodity Derivatives on Processes with Memory

Spot option prices, forwards and options on forwards relevant for the commodity markets are computed when the underlying process S is modelled as an exponential of a process ξ with memory as, e.g., a Volterra equation driven by a Lévy process. Moreover, the interest rate and a risk premium ρ representing storage costs, illiquidity, convenience yield or insurance costs, are assumed to be stochastic. When the interest rate is deterministic and the risk premium is explicitly modelled as an Ornstein-Uhlenbeck type of dynamics with a mean level that depends on the same memory term as the commodity, the process ( ξ ; ρ ) has an affine structure under the pricing measure Q and an explicit expression for the option price is derived in terms of the Fourier transform of the payoff function.

CAT BOND PRICING UNDER A PRODUCT PROBABILITY MEASURE WITH POT RISK CHARACTERIZATION

Frequent large losses from recent catastrophes have caused great concerns among insurers/reinsurers, who then turn to seek mitigations of such catastrophe risks by issuing catastrophe (CAT) bonds and thereby transferring the risks to the bond market. Whereas, the pricing of CAT bonds remains a challenging task, mainly due to the facts that the CAT bond market is incomplete and that the pricing usually requires knowledge about the tail of the risks. In this paper, we propose a general pricing framework based on a product pricing measure, which combines a distorted probability measure that prices the catastrophe risks underlying the CAT bond with a risk-neutral probability measure that prices interest rate risk. We also demonstrate the use of the peaks over threshold (POT) method to uncover the tail risk. Finally, we conduct case studies using Mexico and California earthquake data to demonstrate the applicability of our pricing framework.

EFFICIENT LONG-DATED SWAPTION VOLATILITY APPROXIMATION IN THE FORWARD-LIBOR MODEL

We provide efficient swaption volatility approximations for longer maturities and tenors under the lognormal forward-LIBOR model (LFM). In particular, we approximate the swaption volatility with a mean update of the spanning forward rates. Since the joint distribution of the forward rates is not known under a typical pricing measure, we resort to numerical discretization techniques. More specifically, we approximate the mean forward rates with a multi-dimensional weak order 2.0 Itō–Taylor scheme. The higher-order terms allow us to more accurately capture the state dependence in the drift terms and compute conditional expectations with second-order accuracy. We test our approximations for longer maturities and tenors using a quasi-Monte Carlo (QMC) study and find them to be substantially more effective when compared to the existing approximations, particularly for calibration purposes.

Maximum Market Price of Longevity Risk Under Solvency Regimes: The Case Of Solvency II

Longevity risk constitutes an important risk factor for life insurance companies and it can be managed through longevity-linked securities. The market of longevity-linked securities is at present far from being complete and does not allow to find a unique pricing measure. We propose a method to estimate the maximum market price of longevity risk depending on the risk margin implicit within the calculation of the technical provisions as defined by Solvency II. The maximum price of longevity risk is determined for a survivor forward (S-forward), an agreement between two counterparties to exchange at maturity a fixed survival-dependent payment for a payment depending on the realized survival of a given cohort of individuals. The maximum prices determined for the S-forwards can be used to price other longevity-linked securities, such as q-forwards. The Cairns-Blake-Dowd model is used to represent the evolution of mortality over time, that combined with the information on the risk margin, enables us to calculate upper limits for the risk-adjusted survival probabilities, the market price of longevity risk and the S-forward prices. Numerical results can be extended for the pricing of other longevity-linked securities.

The CJEU Assesses Another Minimum Pricing Measure Without Properly Contextualising it

Case C-221/15 Criminal proceedings against Etablissements Fr. Colruyt NV [2016] ECLI:EU:C: 2016:704

A CHANGE OF MEASURE PRESERVING THE AFFINE STRUCTURE IN THE BARNDORFF-NIELSEN AND SHEPHARD MODEL FOR COMMODITY MARKETS

For a commodity spot price dynamics given by an Ornstein–Uhlenbeck (OU) process with Barndorff-Nielsen and Shephard stochastic volatility, we price forwards using a class of pricing measures that simultaneously allow for change of level and speed in the mean reversion of both the price and the volatility. The risk premium is derived in the case of arithmetic and geometric spot price processes, and it is demonstrated that we can provide flexible shapes that are typically observed in energy markets. In particular, our pricing measure preserves the affine model structure and decomposes into a price and volatility risk premium. In the geometric spot price model, we need to resort to a detailed analysis of a system of Riccati equations, for which we show existence and uniqueness of solution and asymptotic properties that explain the possible risk premium profiles. Among the typical shapes, the risk premium allows for a stochastic change of sign, and can attain positive values in the short end of the forward market and negative in the long end.

FVA and CVA under margining

Purpose – The purpose of this paper is to provide a framework of replication pricing of derivatives and identify funding valuation adjustment (FVA) and credit valuation adjustments (CVA) as price components. Design/methodology/approach – The authors propose the notion of bilateral replication pricing. In the absence of funding cost, it reduces to unilateral replication pricing. The absence of funding costs, it introduces bid–ask spreads. Findings – The valuation of CVA can be separated from that of FVA, so-called split up. There may be interdependence between FVA and the derivatives value, which then requires a recursive procedure for their numerical solution. Research limitations/implications – The authors have assume deterministic interest rates, constant CDS rates and loss rates for the CDS. The authors have also not dealt with re-hypothecation risks. Practical implications – The results of this paper allow user to identify CVA and FVA, and mark to market their derivatives trades according to the recent market standards. Originality/value – For the first time, a line between the risk-neutral pricing measure and the funding risk premiums is drawn. Also, the notion of bilateral replication pricing extends the unilateral replication pricing.