Exotic Derivatives Research Articles

Numerical Method for Model-free Pricing of Exotic Derivatives in Discrete Time Using Rough Path Signatures

ABSTRACT We estimate prices of exotic options in a discrete-time model-free setting when the trader has access to market prices of a rich enough class of exotic and vanilla options. This is achieved by estimating an unobservable quantity called ‘implied expected signature’ from such market prices, which are used to price other exotic derivatives. The implied expected signature is an object that characterizes the market dynamics.

Gradient Boosting for Quantitative Finance

In this paper, we discuss how tree-based machine learning techniques can be used in the context of derivatives pricing. Gradient boosted regression trees are employed to learn the pricing map for a couple of classical, time-consuming problems in quantitative finance. In particular, we illustrate this methodology by reducing computation times for pricing exotic derivatives products and American options. Once the gradient boosting model is trained, it is used to make fast predictions of new prices. We show that this approach leads to speed-ups of several orders of magnitude, while the loss of accuracy is very acceptable from a practical point of view. In addition to the predictive performance of these methods, we acknowledge the importance of interpretability of pricing models. For both applications, we therefore look under the hood of the gradient boosting model and elaborate on how the price is constructed and interpreted.

Tightening robust price bounds for exotic derivatives

We investigate the optimal martingale transport problem under additional constraints and its application to robust price bounds for financial derivatives. More specifically, we derive improved price bounds by taking into account supplementary information about the variance of the returns on the underlying security. Such information can be extracted from market data and our theoretical and numerical results indeed show a significant tightening of price bounds. In this respect, our results have important implications for the practical applicability and relevance of robust price bounds.

The collocating local volatility framework – a fresh look at efficient pricing with smile

ABSTRACTIt is a market practice to price exotic derivatives, like callable basket options, with the local volatility model [B. Dupire, Pricing with a smile, Risk 7 (1994), pp. 18–20; E. Derman and I. Kani, Stochastic implied trees: Arbitrage pricing with stochastic term and strike structure of volatility, Int. J. Theor. Appl. Finance 1 (1998), pp. 61–110.] which can, contrary to stochastic volatility frameworks, handle multi-dimensionality easily. On the other hand, a well-known limitation of the nonparametric local volatility model is the necessity of a short-stepping simulation, which, in high dimensions, is computationally expensive. In this article, we propose a new local volatility framework called the collocating local volatility (CLV) model which allows for large Monte Carlo steps and therefore it is computationally efficient. The CLV model is by its construction guaranteed to be almost perfectly calibrated to implied volatility smiles/skews at a given set of expiries. Additionally, the framework allows to control forward volatilities without affecting the fit to plain vanillas. The model requires only a fraction of a second for complete calibration to simple vanilla products.

Time-Discrete Markov Martingale Transport

We study the optimal martingale transport problem under an additional constraint imposing the underlying process to be Markovian. This formulation results in a modified transportation problem in which the solutions correspond to robust price bounds for exotic derivatives within the class of calibrated martingale models exhibiting the Markov property. We investigate the arising consequences which comprise a dual perspective of the transport problem in terms of liquid replication strategies. Eventually an empirical investigation illustrates the influence of the Markov property on robust price bounds for financial derivatives.

Tightening Robust Price Bounds for Exotic Derivatives

We investigate the optimal martingale transport problem under additional constraints and its application to robust price bounds for financial derivatives. More specifically, we derive improved price bounds by taking into account supplementary information about the variance of the returns on the underlying security. Such information can be extracted from market data and our theoretical and numerical results indeed show a significant tightening of price bounds. In this respect, our results have important implications for the practical applicability and relevance of robust price bounds.

Modeling and Pricing of Energy Derivative Market

Energy derivatives are an energy instrument whose value be determined by on or derived from the values, more basic, a fundamental energy asset, such as crude oil, electricity, or natural gas. Energy derivatives are nonstandard products that have been generated by financial engineers (I. e exotic derivatives) and include exchange-traded contracts such as options and futures. In energy industries, the risk management and pricing model are important because the volatility of pricing in energy products. The price of the volatility can decrease the income of business strategies and its affects the consumer´s buying and selling decisions. For this reason, we have to manage the pricing risk and it became a pressure in the energy industries to continue the profitability and to avoid competitive disadvantages. The main goal of this study is to construct the option-pricing model for energy derivative markets.

PRICING INDEX OPTIONS BY STATIC HEDGING UNDER FINITE LIQUIDITY

We develop a model for indifference pricing in derivatives markets, where price quotes have bid–ask spreads and finite quantities. The model quantifies the dependence of the prices and hedging portfolios on an investor’s views, risk preferences and financial position as well as on the price quotes. Computational techniques of convex optimization allow for fast computation of the hedging portfolios and prices as well as sensitivities with respect to various model parameters. We illustrate the techniques by pricing and hedging of exotic derivatives on S&P index using call and put options, forward contracts and cash as the hedging instruments. The optimized static hedges provide good approximations of the options payouts and the spreads between indifference selling and buying prices are quite narrow as compared with the spread between superhedging and subhedging prices.

Integrated Commodity Inventory Management and Financial Hedging: A Dynamic Mean‐Variance Analysis

We consider a firm purchasing a storable raw material commodity from a spot market with volatile commodity prices and the access to an associated financial derivatives market. The purchased commodity is processed into an end product with uncertain demand and lost sales. The firm aims to integrate the inventory replenishment and financial hedging decisions to maximize the mean‐variance of terminal wealth over a finite horizon. Recognizing time‐inconsistency of mean‐variance criteria, we employ the dynamic programming approach to obtain a time‐consistent policy. Assuming no arbitrage in financial market, we show that the mean‐variance utility functions under the time‐consistent policy have a recursive representation which enables us to readily characterize the structure of the time‐consistent policy. We analyze two types of hedging instruments, vanilla hedges and exotic hedges, and show that inventory and financial hedging decisions can be separated in the presence of forward contracts and a myopic state‐dependent base stock policy is optimal. The optimal hedging policy can be obtained by minimizing the variance of the hedging portfolio, the value of excess inventory and the profit‐to‐go as a function of future price. In the presence of a continuum of option strikes, we show how to construct custom exotic derivatives using forwards and options of all strikes to replicate the profit‐to‐go function. We then show the optimality of the time‐consistent policy under exotic hedge for the initial mean‐variance objective. We further investigate the dynamic interplay of inventories and financial hedge and show that they can be substitutes in a dynamic environment. Finally, we compare the performances in different hedging environments to discuss how financial hedges add value and provide a numerical study.

Path Dependent Optimal Transport and Model Calibration on Exotic Derivatives

In this paper, we introduce and develop the theory of semimartingale optimal transport in a path dependent setting. Instead of the classical constraints on marginal distributions, we consider a general framework of path dependent constraints. Duality results are established, representing the solution in terms of path dependent partial differential equations (PPDEs). Moreover, we provide a dimension reduction result based on the new notion of "semifiltrations", which identifies appropriate Markovian state variables based on the constraints and the cost function. Our technique is then applied to the exact calibration of volatility models to the prices of general path dependent derivatives.

Pricing multivariate barrier reverse convertibles with factor-based subordinators

In this paper we study factor-based subordinated Levy processes in their VG and NIG specications, and focus on their ability to price multivariate exotic derivatives. Dierent model specications, calibrated to a dataset of multivariate Barrier Reverse Convertibles listed at the Swiss market, show diverse ability in capturing smile patterns and recovering empirical correlations. We show how the range of the correlation spanned by the model is linked to the process marginal distributions. Our analysis nds that there exists a tradeo between marginal and correlation fit. A sensitivity analysis is performed, showing how the product's characteristics and the model's features aect Multi Barrier Reverse Convertible prices. Market and model prices are analyzed, highlighting and explaining discrepancies

Pathwise superreplication via Vovk\u2019s outer measure

Since Hobson’s seminal paper (Hobson in Finance Stoch. 2:329–347, 1998), the connection between model-independent pricing and the Skorokhod embedding problem has been a driving force in robust finance. We establish a general pricing–hedging duality for financial derivatives which are susceptible to the Skorokhod approach.Using Vovk’s approach to mathematical finance, we derive a model-independent superreplication theorem in continuous time, given information on finitely many marginals. Our result covers a broad range of exotic derivatives, including lookback options, discretely monitored Asian options, and options on realized variance.

Development of the Vietnam Derivative Stock Market

Stock Market is the most important parts of the financial markets; it is providing long-term investment for the economy. Financial markets are very sensitive, just one loss in a market division may spread to financial markets large and very easily lead to a financial crisis, causing damage to the economy. Thereby posing a requirement is to stabilize the financial markets. Therefore, during the past decade with the growth of the exotic derivatives market, one of the most important developments of the financial markets, the derivative financial instruments are increasingly widespread use, flexibility in financing activities. These are new tools and complex, so want to use it effectively Vietnam need to study and learn it. Derivative securities are derivative financial instruments derived from stocks and have a relationship with the stock closely origin. These are financial instruments versatile and are important tools used in a flexible manner to help enterprises and investors on stock market risk treatment on stock prices and help speculators looking to profit. But derivatives to create ceramic molding from investors, and very complicated only for professional investors and stock market development. In Vietnam, the stock market just put into operation since 2000, also in phase initially built up the problem applies derivative securities is inadequate, but as the market develops, it becomes necessary. After 15 years of establishment and development of Vietnam's stock market, dated 5/5/2015, the Prime Minister issued Decree No. 42/2015/ND-CP on derivative securities and stock market Derivative. This is considered a prerequisite for the operation and development of the stock market derivative Vietnam in June 2017; contribute to improve the market structure of modern finance, next to the stock market and bond market government listed.

Vulnerable Exotic Derivatives

Understanding and managing counterparty credit risk exposure in derivatives contracts has become a crucial element of real-world trading as well as theoretical modeling. But existing models are limited in the number of securities involved and in the assumed dynamics of the underlying asset returns processes. In this article, the authors present a framework with two counterparties who enter into a derivatives contract in which either of them may default, and the derivative’s payoff depends on the joint distribution of n different assets. Three specific examples illustrate application of the approach: an option on a security in which both counterparties are subject to default risk; a vulnerable spread option, in which one risky counterparty issues an option tied to the price spread between two other assets; and a defaultable swap with one underlying and two risky counterparties who commit to a series of future cash flows. The resulting pricing formulas are mathematically complicated, but as closed-form solutions (with the caveat that integral expressions are approximated using finite step size and number of terms in an infinite series), they are much more efficient than Monte Carlo simulation in reaching a given level of accuracy.

Vulnerable Exotic Derivatives

Vulnerable Exotic Derivatives

Swing Option Pricing by Dynamic Programming with B-spline Density Projection

Swing options are a type of exotic financial derivative which generalize American options to allow for multiple early-exercise actions during the contract period. These contracts are widely traded in commodity and energy markets, but are often difficult to value using standard techniques due to their complexity and strong path-dependency. There are numerous interesting varieties of swing options, which differ in terms of their intermediate cash flows, and the constraints (both local and global) which they impose on early-exercise (swing) decisions. We introduce an efficient and general purpose transform-based method for pricing discrete and continuously monitored swing options under exponential L\'evy models, which applies to contracts with fixed rights clauses, as well as recovery time delays between exercise. The approach combines dynamic programming with an efficient method for calculating the continuation value between monitoring dates, and applies generally to multiple early-exercise contracts, providing a unified framework for pricing a large class of exotic derivatives. Efficiency and accuracy of the method is supported by a series of numerical experiments which further provide benchmark prices for future research.

Fair value accounting in the absence of prudence in accounting standards: an illustration with exotic derivatives

ABSTRACTThe aim of this paper is to contribute to the current discussion about fair value accounting (FVA). We discuss the problem surrounding FVA by relying on the role played by prudence, its meaning, and how the treatment of prudence has changed in the accounting framework of standard setters due to its ‘apparent’ inconsistency with neutrality. To highlight the relevance of this issue, we provide (1) a brief analysis of the high impact that Level 2 fair value estimates have on large US and European banks’ financial positions; (2) a ‘case study’ by pricing two common exotic derivatives and comparing the valuation results of two different assumptions of volatility (local vs. stochastic); and (3) a discussion of potential solutions to the problem surrounding FVA. Our findings are consistent with the argument that neutrality is supported by the exercise of prudence in achieving a faithful representation, since a non-conservative use of FVA can lead bank managers towards model misspecification error in the valuation of complex financial instruments. We conclude by arguing that the problem surrounding FVA can be mitigated if prudence is reinstated by standards setters.

Tightness and duality of martingale transport on the Skorokhod space

The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the continuous-time martingale transport on the Skorokhod space of càdlàg paths. Similar to the classical setting of optimal transport, we introduce different dual problems and establish the corresponding dualities by a crucial use of the S-topology and the dynamic programming principle.

The CLV Framework - A Fresh Look at Efficient Pricing with Smile

It is a market practice to price exotic derivatives, like callable basket options, with the local volatility model (B. Dupire, 1994, E. Derman & I. Kani, 1994) which can, contrary to stochastic volatility frameworks, handle multi-dimensionality easily. On the other hand a well-known limitation of the nonparametric local volatility model is the necessity of a short-stepping simulation, which, in high dimensions, is computationally expensive. In this article we propose a new local volatility framework called the Collocating Local Volatility (CLV) model which allows for large Monte Carlo steps and therefore it is computationally efficient. The CLV model is by its construction guaranteed to be almost perfectly calibrated to implied volatility smiles/skews at a given set of expiries. Additionally, the framework allows to control forward volatilities without affecting the fit to plain vanillas. The model requires only a fraction of a second for complete calibration to simple vanilla products and, finally, it allows for calculation of the Greeks without re-simulating of the Monte Carlo paths. We will apply the CLV model for pricing FX barrier options and also show that under this new framework we can easily price Bermudan options on 100-dimensional baskets of correlated underlyings within just a few seconds. Illustrative examples will be also presented.

Valuation of CMS range notes in a multifactor LIBOR market model

In the framework of a multifactor LIBOR market model (LMM), this paper presents a new approach for finding an approximate distribution of constant maturity swap (CMS) rates under the terminal martingale measure. With this approach, we derive an analytical pricing formula for CMS range notes, which is both intuitive and tractable. Many exotic CMS rate derivatives are widely traded in the marketplace or embedded in structure notes.