Black-Scholes Setting Research Articles

Crypto quanto and inverse options

AbstractOver 90% of exchange trading on crypto options has always been on the Deribit platform. This centralized crypto exchange only lists inverse products because they do not accept fiat currency. Likewise, other major crypto options platforms only list crypto–stablecoin trading pairs in so‐called direct options, which are similar to the standard crypto options listed by the CME except the US dollar is replaced by a stablecoin version. Until now a clear mathematical exposition of these products has been lacking. We discuss the sources of market incompleteness in direct and inverse options and compare their pricing and hedging characteristics. Then we discuss the useful applications of currency protected “quanto” direct and inverse options for fiat‐based traders and describe their pricing and hedging characteristics, all in the Black–Scholes setting.

The Convergence Rate of Option Prices in Trinomial Trees

We study the convergence of the binomial, trinomial, and more generally m-nomial tree schemes when evaluating certain European path-independent options in the Black–Scholes setting. To our knowledge, the results here are the first for trinomial trees. Our main result provides formulae for the coefficients of 1/n and 1/n in the expansion of the error for digital and standard put and call options. This result is obtained from an Edgeworth series in the form of Kolassa–McCullagh, which we derive from a recently established Edgeworth series in the form of Esseen/Bhattacharya and Rao for triangular arrays of random variables. We apply our result to the most popular trinomial trees and provide numerical illustrations.

Time-Discrete Hedging of Down-and-Out Puts with Overnight Trading Gaps

Hedging down-and-out puts (and up-and-out calls), where the maximum payoff is reached just before a barrier is hit that would render the claim worthless afterwards, is challenging. All hedging methods potentially lead to large errors when the underlying is already close to the barrier and the hedge portfolio can only be adjusted in discrete time intervals. In this paper, we analyze this hedging situation, especially the case of overnight trading gaps. We show how a position in a short-term vanilla call option can be used for efficient hedging. Using a mean-variance hedging approach, we calculate optimal hedge ratios for both the underlying and call options as hedge instruments. We derive semi-analytical formulas for optimal hedge ratios in a Black–Scholes setting for continuous trading (as a benchmark) and in the case of trading gaps. For more complex models, we show in a numerical study that the semi-analytical formulas can be used as a sufficient approximation, even when stochastic volatility and jumps are present.

A computational approach to hedging Credit Valuation Adjustment in a jump-diffusion setting

This study contributes to understanding Valuation Adjustments (xVA) by focussing on the dynamic hedging of Credit Valuation Adjustment (CVA), corresponding Profit & Loss (P&L) and the P&L explain. This is done in a Monte Carlo simulation setting, based on a theoretical hedging framework discussed in existing literature. We look at hedging CVA market risk for a portfolio with European options on a stock, first in a Black-Scholes setting, then in a Merton jump-diffusion setting. Furthermore, we analyze the trading business at a bank after including xVAs in pricing. We provide insights into the hedging of derivatives and their xVAs by analyzing and visualizing the cash-flows of a portfolio from a desk structure perspective. The case study shows that not charging CVA at trade inception results in an expected loss. Furthermore, hedging CVA market risk is crucial to end up with a stable trading strategy. In the Black-Scholes setting this can be done using the underlying stock, whereas in the Merton jump-diffusion setting we need to add extra options to the hedge portfolio to properly hedge the jump risk. In addition to the simulation, we derive analytical results that explain our observations from the numerical experiments. Understanding the hedging of CVA helps to deal with xVAs in a practical setting.

THE EFFECT OF THE ASSUMED INTEREST RATE AND SMOOTHING ON VARIABLE ANNUITIES

AbstractIn this paper, we consider the risk–return trade-off for variable annuities in a Black–Scholes setting. Our analysis is based on a novel explicit allocation of initial wealth over the payments at various horizons. We investigate the relationship between the optimal consumption problem and the design of variable annuities by deriving the optimal so-called assumed interest rate for an investor with constant relative risk aversion preferences. We investigate the utility loss due to deviations from this. Finally, we show analytically how habit-formation-type smoothing of financial market shocks over the remaining lifetime leads to smaller year-to-year volatility in pension payouts, but to increases in the longer-term volatility.

PRIIP-KID: providing retail investors with inappropriate product information?

Providers of so-called packaged-retail and insurance-based investment products (“PRIIPs”) have to draw up a standardized key-information document (“KID”) since 1st of January 2018 when they offer such products in the European Union. In addition to some standard information on the product and its provider, this key information document discloses the product’s riskiness by means of a summary risk indicator, its performance potential by means of so-called performance scenarios and its included costs by means of a summary cost indicator. The European Commission has issued regulatory technical standards stating how the risk indicator, the performance scenarios and the cost indicator shall be calculated. This paper analyzes these “calculation recipes”, focusing on the risk indicator and the performance scenarios. Since, the European Commission issues these formulae without providing the assumed methodologies, our analyses on the one side shed light on the (presumed) underlying ideas and on the other side detect methodical and technical errors. We show that the risk indicator’s formula can be derived in a Black–Scholes setting considering a single premium investment. Since insurance companies are generally required to produce key information documents for regular premium payments as well, we show that an application of this formula to regular premium payments overestimates the products’ “true” risk. Therefore, we propose amended formulae for the risk indicator for regular premium payments which perform much better than the current specification. Further, we identify methodical and technical errors prevailing in the requirement regarding the (presumed) performance scenarios’ calculation. Taking into account the revision of the PRIIPs-directive at the end of 2018, this paper provides a good starting point for fixing current methodical and technical issues when risk indicator and performance scenarios are assessed.

Asian option pricing with orthogonal polynomials

In this paper we derive a series expansion for the price of a continuously sampled arithmetic Asian option in the Black–Scholes setting. The expansion is based on polynomials that are orthogonal with respect to the log-normal distribution. All terms in the series are fully explicit and no numerical integration nor any special functions are involved. We provide sufficient conditions to guarantee convergence of the series. The moment indeterminacy of the log-normal distribution introduces an asymptotic bias in the series, however we show numerically that the bias can safely be ignored in practice.

Asian Option Pricing with Orthogonal Polynomials

In this paper we derive a series expansion for the price of a continuously sampled arithmetic Asian option in the Black-Scholes setting. The expansion is based on polynomials that are orthogonal with respect to the log-normal distribution. All terms in the series are fully explicit and no numerical integration nor any special functions are involved. We provide sufficient conditions to guarantee convergence of the series. The moment indeterminacy of the log-normal distribution introduces an asymptotic bias in the series, however we show numerically that the bias can safely be ignored in practice.

Machine Learning for Quantitative Finance: Fast Derivative Pricing, Hedging and Fitting

In this paper, we show how we can deploy machine learning techniques in the context of traditional quant problems. We illustrate that for many classical problems, we can arrive to speed-ups of several orders of magnitude by deploying machine learning techniques based on Gaussian process regression. The price we have to pay for this extra speed is some loss of accuracy. However, we show that this reduced accuracy is often well within reasonable limits and hence very acceptable from a practical point of view. The concrete examples concern fitting and estimation. In the fitting context, we fit sophisticated Greek profiles and summarize implied volatility surfaces. In the estimation context, we reduce computation times for the calculation of vanilla option values under advanced models, the pricing of American options and the pricing of exotic options under models beyond the Black-Scholes setting.

Optimal strategy for a fund manager with option compensation

I consider the problem of portfolio optimization for a manager whose compensation is given by the sum of a constant and a variable term. The constant term is a fixed percentage of the managed funds that is payed to the manager independently of his performance. The variable term is a premium that is proportional to the profit earned by the manager over a benchmark at a certain evaluation date. I find the optimal strategy and the optimal portfolio value in the Black–Scholes setting when the benchmark is a linear combination of the risky asset and the money market account.

Risk Minimization and Portfolio Diversification

We consider a capital at risk (CaR) minimization problem in an incomplete market Black-Scholes setting. The optimization problem is studied, given the possibility that a correlation constraint between the wealth process and a financial index is imposed. The optimal portfolio is not unique and it is analytically characterized. In the special case of complete market, the optimal portfolio is unique and is obtained in closed form. The effect of the correlation constraint is also explored; it turns out that this constraint leads to a more diversified portfolio.

CoCo bonds and implied CET1 volatility

s at AFMathConf 2016 CoCo Bonds and Implied CET1 Volatility Ine Marquet KU Leuven Dep. of Mathematics, Celestijnenlaan 200B, 3001 Heverlee Belgium ine.marquet@wis.kuleuven.be Joint work with: J. De Spiegeleer, W. Schoutens In this talk we introduce the notion of implied Common Equity Tier 1 volatility and the concept of a risk-adjusted distance to trigger. Using a derivatives-based valuation approach, we are able to derive the implied CET1 volatility from the market price of a CoCo bond in a Black-Scholes setting. The numerical results show how different contingent convertibles issued by the same bank and sharing a similar contractual CET1 trigger, have almost identical implied CET1 volatility levels. The same results confirm the difference in market risk between Tier 2 and Additional Tier 1 CoCo bonds. The ability to obtain an implied level for the CET1 volatility offers other interesting results. We are able to determine the implied CET1 level corresponding to a coupon cancellation. It further allowed us for example to look at the severity of one of the stress tests imposed by the ECB on European banks in November 2014. In that perspective we were also able to derive implied PONV CET1 levels.

SIMPLIFIED HEDGE FOR PATH-DEPENDENT DERIVATIVES

Path-dependent derivatives are typically difficult to hedge. Traditional dynamic delta hedging does not perform well because of the difficulty to evaluate the Greeks and the high cost of constantly rebalancing. We propose to price and hedge path-dependent derivatives by constructing simplified alternatives that preserve certain distributional properties of their terminal payoffs, and that can be hedged by semi-static replication. The method is illustrated by a geometric Asian option and by a lookback option in the Black–Scholes setting, for which explicit forms of the simplified alternatives exist. Extensions to a Lévy market and to a Heston stochastic volatility model are discussed as well.

Valuation of power option for uncertain financial market

Power option is such an option whose payoff is based on the price of the underlying asset raised to some power. Unlike Black–Scholes setting, we investigate the valuation of power options under the assumption that the underlying stock price is assumed to follow an uncertain differential equation, and derive the pricing formulas of power options for Liu’s uncertain stock model with the method of uncertain calculus based on uncertainty theory. Some numerical examples are given to illustrate the pricing formulas.

Reconstructing the Joint Probability Distribution From Basket Prices: A Mildly Ill-Posed Problem

In this thesis, we study basket options which are contracts on N assets. It is known that the risk neutral joint probability density of the assets is uniquely defined by the prices of basket options with positive weights. However, the proof of this fact requires a step involving the inversion of a Laplace transform. It is known that inversion of a Laplace transform is a severely ill-posed problem. The purpose of this thesis is to understand whether inversion of the basket transform is ill-posed or not.In our dissertation, we propose three methods to determine and prove the order of ill-posedness for the case when the basket contains only one asset and then map the problem to Radon transform to address two or higher dimensional case. Our proofs show that that problem is mildly ill-posed of order 1/2 (N -1) 2 given N dimensions.We also conduct numerical experiments under the Black-Scholes setting. First, using the reconstruction method from Cohen and Pagliacci, our results show that this method functions well in one-dimension but it becomes numerically intensive in our two-dimensional implementation, and we were unable to achieve convergence. Alternatively, we have looked at using an inverse Radon transform method to reconstruct the joint density numerically. As this requires knowledge of basket prices with negative weights, we have tried extending the basket price in this region as an odd function. For comparison, we compared this against the reconstruction from approximate basket prices calculated directly in the negative weight region. It is fascinating that these two approaches give similar results. This may indicate that the errors are due to the two-moment matching approximation rather than the odd extension. This intriguing possibility would certainly merit further investigation if more time were available.We believe that our mathematical particularly, but also our numerical results indicate there is cause for optimism that a stable numerical algorithm to reconstruct the joint density function from basket prices will be found in the future.

CVaR hedging under stochastic interest rate

In this paper we assess the partial hedging problems by formulating hedging strategies that minimize conditional value-at-risk (CVaR) of the portfolio loss under stochastic interest rate environment. The combination of stochastic interest and CVaR hedging method makes the valuing approach more complex than the existing model with constant interest rate. We take up two issues in searching the optimal CVaR hedging strategy: given the initial capital constraint we minimize the CVaR of the portfolio loss; by prescribing a bound on the risk, we also minimize the hedging cost. As an illustration of this hedging technique we derive hedging strategies for a European call option with the Black Scholes setting under HJM framework; explicit formulas are presented. We also investigate CVaR hedging problems by using the real financial data.

Implied CET1 Volatility

In this work we introduce the notion of implied Core Equity Tier 1 volatility and the concept of a risk-adjusted distance to trigger. Using a derivatives-based valuation approach, we are able to derive the implied CET1 volatility from the market price of a CoCo bond in a Black-Scholes setting. The numerical results in this paper show how different contingent convertibles issued by the same bank and sharing a similar contractual CET1 trigger, have almost identical implied CET1 volatility levels. The same results confirm the difference in market risk between Tier 2 and Additional Tier 1 CoCo bonds. The ability to obtain an implied level for the CET1 volatility offers other interesting results. We are able to determine the implied CET1 level corresponding to a coupon cancellation. It further allowed us for example to look at the severity of one of the stress tests imposed by the ECB on European banks in November 2014. In that perspective we were also able to derive implied PONV CET1 levels.

BOUNDARY EVOLUTION EQUATIONS FOR AMERICAN OPTIONS

We consider the problem of finding optimal exercise policies for American options, both under constant and stochastic volatility settings. Rather than work with the usual equations that characterize the price exclusively, we derive and use boundary evolution equations that characterize the evolution of the optimal exercise boundary. Using these boundary evolution equations we show how one can construct very efficient computational methods for pricing American options that avoid common sources of error. First, we detail a methodology for standard static grids and then describe an improvement that defines a grid that evolves dynamically while solving the problem. When integral representations are available, as in the Black–Scholes setting, we also describe a modified integral method that leverages on the representation to solve the boundary evolution equations. Finally we compare runtime and accuracy to other popular numerical methods. The ideas and methodology presented herein can easily be extended to other optimal stopping problems.

Boundary Evolution Equations for American Options

We consider the problem of finding optimal exercise policies for American options, both under constant and stochastic volatility settings. Rather than work with the usual equations that characterize the price exclusively, we derive and use boundary evolution equations that characterize the evolution of the optimal exercise boundary. Using these boundary evolution equations we show how one can construct very efficient computational methods for pricing American options that avoid common sources of error. First, we detail a methodology for standard static grids and then describe an improvement that defines a grid that evolves dynamically while solving the problem. When integral representations are available, as in the Black–Scholes setting, we also describe a modified integral method that leverages on the representation to solve the boundary evolution equations. Finally we compare runtime and accuracy to other popular numerical methods. The ideas and methodology presented herein can easily be extended to other optimal stopping problems.

Explicit Representation of Cost-Efficient Strategies

In this paper, we give an explicit representation of the lowest cost strategy (or cost-efficient strategy) to achieve a given payoff distribution. For any inefficient strategy, we are able to construct financial derivatives which dominate in the sense of first-order or second-order stochastic dominance. We highlight the connections between cost-efficiency and dependence (copulas). This allows us to extend the theory to deal with state-dependent constraints to better reflect real world preferences. We show in particular that path-dependent strategies (although inefficient in the Black Scholes setting) may become optimal in the presence of state-dependent constraints.