Liquidity-free implied volatilities: An approach using conic finance

In this paper, we consider the problem of calculating risk-neutral implied volatilities of European options without relying on option mid prices but solely on bid and ask prices. We provide an approach, based on the conic finance paradigm, that allows to uniquely strip risk-neutral implied volatilities from bid and ask quotes, and that does not require restrictive assumptions. Our methodology also allows to jointly calculate the implied liquidity of the market. The idea outlined in this paper can be applied to calculate other implied parameters from bid and ask security prices as soon as their theoretical risk-neutral counterparts are strictly increasing with respect to the former.

Estimation of the Bid-Ask Prices for the European Discrete Geometric Average and Arithmetic Average Asian Options

Conic finance is a new and exciting development in quantitative finance, which is widely applied to several topics in finance. The theory of conic finance extends the law of one price to the law of two prices, which yields closed forms for bid-ask prices of European options. In this paper, within the framework of conic finance, we derive effective, explicit, approximate formulas to estimate the bid-ask prices for the European discrete geometric average and arithmetic average Asian options. Finally, we give two examples to demonstrate and validate that the approximate closed-form solutions are efficient and accurate.

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.

Estimation of Ask and Bid Prices for Geometric Asian Options

Traditional derivative pricing theories usually focus on the risk-neutral price or the equilibrium price. However, in highly competitive financial markets, we observed two prices which are called bid and ask prices; then the unique risk-neutral price fails to hold. In this paper, within the framework of conic finance, we provide a useful approach to evaluate the ask and bid prices of geometric Asian options and obtain the explicit formulas for the ask and bid prices. Finally, numerical examples show that the higher the market liquidity parameter γ, the wider the spread and hence the less the liquidity.

CONIC CPPIs

Constant proportion portfolio insurance (CPPI) is a structured product created on the basis of a trading strategy. The idea of the strategy is to have an exposure to the upside potential of a risky asset while providing a capital guarantee against downside risk with the additional feature that in case the product has since initiation performed well more risk is taken while if the product has suffered mark-to-market losses, the risk is reduced. In a standard CPPI contract, a fraction of the initial capital is guaranteed at maturity. This payment is assured by investing part of the fund in a riskless manner. The other part of the fund’s value is invested in a risky asset to offer the upside potential. We refer to the floor as the discounted guaranteed amount at maturity. The percentage allocated to the risky asset is typically defined as a constant multiplier of the fund value above the floor. The remaining part of the fund is invested in a riskless manner. In this paper, we combine conic trading in the above described CPPIs. Conic trading strategies explore particular sophisticated trading strategies founded by the conic finance theory i.e. they are valued using nonlinear conditional expectations with respect to nonadditive probabilities. The main idea of this paper is that the multiplier is taken now to be state dependent. In case the algorithm sees value in the underlying asset the multiplier is increased, whereas if the assets is situated in a state with low value or opportunities, the multiplier is reduced. In addition, the direction of the trade, i.e. going long or short the underlying asset, is also decided on the basis of the policy function derived by employing the conic finance algorithm. Since nonadditive probabilities attain conservatism by exaggerating upwards tail loss events and exaggerating downwards tail gain events, the new Conic CPPI strategies can be seen on the one hand to be more conservative and on the other hand better in exploiting trading opportunities.