Multi-asset Derivatives Research Articles

On the Optimal Choice of Strike Conventions in Exchange Option Pricing

An important but rarely-addressed option pricing question is how to choose appropriate strikes for implied volatility inputs when pricing more exotic multi-asset derivatives. By means of Malliavin calculus, we construct an asymptotically optimal log-linear strike convention for exchange options under stochastic volatility models. This novel approach allows us to minimize the difference between the corresponding Margrabe computed price and the true option price. We show that this optimal convention does not depend on the specific stochastic volatility model chosen and, furthermore, that parameter estimation can be dramatically simplified by using market observables as inputs. Numerical examples are given that provide strong support for the new methodology.

Improved robust price bounds for multi-asset derivatives under market-implied dependence information

Improved robust price bounds for multi-asset derivatives under market-implied dependence information

Total Value Adjustment of Multi-Asset Derivatives under Multivariate CGMY Processes

Counterparty credit risk (CCR) is a significant risk factor that financial institutions have to consider in today’s context, and the COVID-19 pandemic and military conflicts worldwide have heightened concerns about potential default risk. In this work, we investigate the changes in the value of financial derivatives due to counterparty default risk, i.e., total value adjustment (XVA). We perform the XVA for multi-asset option based on the multivariate Carr–Geman–Madan–Yor (CGMY) processes, which can be applied to a wider range of financial derivatives, such as basket options, rainbow options, and index options. For the numerical methods, we use the Monte Carlo method in combination with the alternating direction implicit method (MC-ADI) and the two-dimensional Fourier cosine expansion method (MC-CC) to find the risk exposure and make value adjustments for multi-asset derivatives.

Pricing Multiasset Derivatives by Variational Quantum Algorithms

Pricing Multiasset Derivatives by Variational Quantum Algorithms

Model-Free Bounds for Multi-Asset Options Using Option-Implied Information and Their Exact Computation

We consider derivatives written on multiple underlyings in a one-period financial market, and we are interested in the computation of model-free upper and lower bounds for their arbitrage-free prices. We work in a completely realistic setting, in that we only assume the knowledge of traded prices for other single- and multi-asset derivatives and even allow for the presence of bid–ask spread in these prices. We provide a fundamental theorem of asset pricing for this market model, as well as a superhedging duality result, that allows to transform the abstract maximization problem over probability measures into a more tractable minimization problem over vectors, subject to certain constraints. Then, we recast this problem into a linear semi-infinite optimization problem and provide two algorithms for its solution. These algorithms provide upper and lower bounds for the prices that are ε-optimal, as well as a characterization of the optimal pricing measures. These algorithms are efficient and allow the computation of bounds in high-dimensional scenarios (e.g., when d = 60). Moreover, these algorithms can be used to detect arbitrage opportunities and identify the corresponding arbitrage strategies. Numerical experiments using both synthetic and real market data showcase the efficiency of these algorithms, and they also allow understanding of the reduction of model risk by including additional information in the form of known derivative prices. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported by the Nanyang Technological University [NAP Grant] and the Hellenic Foundation for Research and Innovation [Grant HFRI-FM17-2152]. Supplemental Material: The data files and online appendices are available at https://doi.org/10.1287/mnsc.2022.4456 .

Derivatives-based portfolio decisions: an expected utility insight

This paper challenges the use of stocks in portfolio construction, instead we demonstrate that Asian derivatives, straddles, or baskets could be more convenient substitutes. Our results are obtained under the assumptions of the Black–Scholes–Merton setting, uncovering a hidden benefit of derivatives that complements their well-known gains for hedging, risk management, and to increase utility in market incompleteness. The new insights are also transferable to more advanced stochastic settings. The analysis relies on the infinite number of optimal choices of derivatives for a maximized expected utility theory agent; we propose risk exposure minimization as an additional optimization criterion inspired by regulations. Working with two assets, for simplicity, we demonstrate that only two derivatives are needed to maximize utility while minimizing risky exposure. In a comparison among one-asset options, e.g. American, European, Asian, Calls and Puts, we demonstrate that the deepest out-of-the-money Asian products available are the best choices to minimize exposure. We also explore optimal selections among straddles, which are better practical choice than out-of-the-money Calls and Puts due to liquidity and rebalancing needs. The optimality of multi-asset derivatives is also considered, establishing that a basket option could be a better choice than one-asset Asian call/put in many realistic situations.

Pricing with À La Carte Joint Distributions: Multi-Asset Derivatives Via Copulas and the Collocating Local Volatility Model

Pricing with À La Carte Joint Distributions: Multi-Asset Derivatives Via Copulas and the Collocating Local Volatility Model

Pricing Multi-Asset Derivatives by Finite-Difference Method on a Quantum Computer

Following the recent great advance of quantum computing technology, there are growing interests in its applications to industries, including finance. In this article, we focus on derivative pricing based on solving the Black–Scholes partial differential equation by the finite-difference method (FDM), which is a suitable approach for some types of derivatives but suffers from the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">curse of dimensionality</i> , that is, exponential growth of complexity in the case of multiple underlying assets. We propose a quantum algorithm for FDM-based pricing of multi-asset derivative with exponential speedup with respect to dimensionality compared with classical algorithms. The proposed algorithm utilizes the quantum algorithm for solving differential equations, which is based on quantum linear system algorithms. Addressing the specific issue in derivative pricing, that is, extracting the derivative price for the present underlying asset prices from the output state of the quantum algorithm, we present the whole of the calculation process and estimate its complexity. We believe that the proposed method opens the new possibility of accurate and high-speed derivative pricing by quantum computers.

Detection of arbitrage opportunities in multi-asset derivatives markets

Abstract We are interested in the existence of equivalent martingale measures and the detection of arbitrage opportunities in markets where several multi-asset derivatives are traded simultaneously. More specifically, we consider a financial market with multiple traded assets whose marginal risk-neutral distributions are known, and assume that several derivatives written on these assets are traded simultaneously. In this setting, there is a bijection between the existence of an equivalent martingale measure and the existence of a copula that couples these marginals. Using this bijection and recent results on improved Fréchet–Hoeffding bounds in the presence of additional information on functionals of a copula by [18], we can extend the results of [33] on the detection of arbitrage opportunities to the general multi-dimensional case. More specifically, we derive sufficient conditions for the absence of arbitrage and formulate an optimization problem for the detection of a possible arbitrage opportunity. This problem can be solved efficiently using numerical optimization routines. The most interesting practical outcome is the following: we can construct a financial market where each multi-asset derivative is traded within its own no-arbitrage interval, and yet when considered together an arbitrage opportunity may arise.

BOUNDS ON MULTI-ASSET DERIVATIVES VIA NEURAL NETWORKS

Using neural networks, we compute bounds on the prices of multi-asset derivatives given information on prices of related payoffs. As a main example, we focus on European basket options and include information on the prices of other similar options, such as spread options and/or basket options on subindices. We show that, in most cases, adding further constraints gives rise to bounds that are considerably tighter. Our approach follows the literature on constrained optimal transport and, in particular, builds on the work of Eckstein &amp; Kupper (2018) [Computation of optimal transport and related hedging problems via penalization and neural networks, Appl. Math. Optimiz. 1–29].

A reduced PDE method for European option pricing under multi-scale, multi-factor stochastic volatility

The number of tailor-made hybrid structured products has risen more prominently to fit each investor’s preferences and requirements as they become more diversified. The structured products entail synthetic derivatives such as combinations of bonds and/or stocks conditional on how they are backed up by underlying securities, stochastic volatility, stochastic interest rates or exchanges rates. The complexity of these multi-asset structures yields lots of difficulties of pricing the products. Because of the complexity, Monte-Carlo simulation is a possible choice to price them but it may not produce stable Greeks leading to a trouble in hedging against risks. In this light, it is desirable to use partial differential equations with relevant analytic and numerical techniques. Even if the partial differential equation method would generate stable security prices and Greeks for single asset options, however, it may result in the curse of dimensionality when pricing multi-asset derivatives. In this study, we make the best use of multi-scale nature of stochastic volatility to lift the curse of dimensionality for up to three asset cases. Also, we present a transformation formula by which the pricing group parameters required for the multi-asset options in illiquid market can be calculated from the underlying market parameters.

Calibration of local correlation models to basket smiles

Allowing correlation to be local, i.e., state-dependent, in multi-asset models allows better hedging by incorporating correlation moves in the Delta. When options on a basket - be it a stock index, a cross-foreign exchange rate or an interest rate spread - are liquidly traded, one may want to calibrate a local correlation to these option prices. Only two particular solutions have been suggested so far in the literature. Both impose a particular dependency of the correlation matrix on the asset values that one has no reason to undergo. They may also fail to be admissible, i.e., positive semi-definite. We explain how, by combining the particle method presented in smile calibration problem solved by Guyon and Henry-Labordere (2011) with a simple affine transform, we can build all the calibrated local correlation models. The two existing models appear as special cases (if admissible). For the first time, one can now choose a calibrated local correlation in order to fit a view on the correlation skew, or reproduce historical correlation, or match some exotic option prices, thus improving the pricing, hedging and risk-management of multi-asset derivatives. This technique is generalized to calibrate models that combine stochastic interest rates, stochastic dividend yield, local stochastic volatility and local correlation. Numerical results show the wide variety of calibrated local correlations and give insight into a difficult (still unsolved) problem: finding lower bounds/upper bounds on general multi-asset option prices given the whole surfaces of implied volatilities of a basket and its constituents.

Pricing and hedging of derivatives in contagious markets

It is well documented that stock markets are contagious. A negative shock to one market increases the probability of adverse shocks to other markets. We model this contagion effect by including mutually exciting jump processes in the dynamics of the indexes’ log-returns. On top of this we add a stochastic volatility component to the dynamics. It is important to take the contagion effect into account if derivatives written on a basket of assets are to be priced or hedged. Due to the affine model specification the joint characteristic function of the log-returns is known analytically, and for two specifications we detail how the model can be calibrated efficiently to option prices. In total we calibrate over an extended period of time the specifications to options data on four US stock indexes, and show how the models achieve satisfactory pricing errors. We study the effect of contagion on multi-asset derivatives prices and show how for certain derivatives the impact is heavy. Moreover, we derive hedge ratios for European put and call options and perform a numerical experiment, which illustrates the impact of contagious jumps on option prices and hedge ratios. Mutually exciting processes have been analyzed for multivariate intensity modeling for the purpose of credit derivatives pricing, but have not been used for pricing/hedging options on equity indexes.

Is the Information Obtained from European Options on Equally Weighted Baskets Enough to Determine the Prices of Exotic Derivatives Such as Worst of Options?

In recent years there has been a remarkable growth of multi-asset options. These options exhibit sensitivity to the volatility of the underlying assets, as well as to their correlations. The call versus call is a product commonly used to trade correlation within the inter-dealer broker markets. The buyer of correlation buys a European call on the equally weighted basket option and sells a weighted average of European calls on each asset. In this case, the following important question arises: Is the information provided by equally weighted basket options enough to price other European multi-asset exotic derivatives such as worst-of or outperformance options? This article investigates this issue under a stochastic correlation framework. Importantly, this article shows that, when pricing multi-asset exotic derivatives, matching the prices of European equally weighted basket options, quoted in the market, does not guaranty the absence of model risk even in the case where the exotic payoff is observed only at maturity.

Detection of arbitrage in a market with multi-asset derivatives and known risk-neutral marginals

In this paper we study the existence of arbitrage opportunities in a multi-asset market when risk-neutral marginal distributions of asset prices are known. We first propose an intuitive characterization of the absence of arbitrage opportunities in terms of copula functions. We then address the problem of detecting the presence of arbitrage by formalizing its resolution in two distinct ways that are both suitable for the use of optimization algorithms. The first method is valid in the general multivariate case and is based on Bernstein copulas that are dense in the set of all copula functions. The second one is easier to work with but is only valid in the bivariate case. It relies on results about improved Fréchet–Hoeffding bounds in presence of additional information. For both methods, details of implementation steps and empirical applications are provided.

Pricing and Hedging of Derivatives in Contagious Markets

It is well documented that stock markets are contagious. A negative shock to one market increases the probability of adverse shocks to other markets. We model this contagion effect by including mutually exciting jump processes in the dynamics of the index's log-returns, so that a jump in one market increases the intensities of more jumps in the same market and in other markets. Between jumps the intensities revert to their long-run means. On top of this we add a stochastic volatility component to the dynamics. It is important to take the contagion effect into account if derivatives written on a basket of assets are to be priced or hedged. Due to the affine model specification the joint characteristic function of the log-returns is known analytically, and for two specifications we detail how the model can be calibrated efficiently to option prices using Fourier transform methods. In total we calibrate over an extended period of time the specifications to options data on four US stock indexes; the Amex Biotechnology Index, the Morgan Stanley Technology index, the Securities Broker/Dealer index, and the Natural Gas index, and investigate the effect of contagion on multi-asset derivatives prices. Moreover, we compute the model hedge ratios for put and call options and investigate the historical hedging performances of the contagion specifications. Mutually exciting processes have been analyzed for multivariate intensity modeling for the purpose of credit derivatives pricing, but have not been used for pricing/hedging options on equity indexes.

Skewing Up Correlation: Understanding Correlation Skew in Equity Derivatives

The goal of this paper is to fill in the gap between the multi-asset equity derivatives market and the mathematical models developed to treat the correlation skew problem. We analyze the structure of the equity correlation market and explain which parts are affected by correlation skew. Using a Jacobi instantaneous correlation model as a vehicle to gain intuition, we approach the issue from both the perspective of historical realized correlation as well as from the market-implied one. We study the effect of correlation skew on the typical structures traded in the correlation market. The main result is the calibration procedure for such a model. Once calibrated, the model provides a link between various correlation derivatives, most importantly between the basket dispersion trades and Worst-Of structures, the two main vehicles of correlation business.

PRICING MULTI-ASSET DERIVATIVES WITH REGIME-SWITCHING VOLATILITIES

In this paper we provide a valuation method for multi-asset derivatives with single jump regime-switching volatilities. We suppse that volatilities of assets are affected by an n-dimensional independent Markov regime-switching process.

Stochastic Correlation and Volatility Mean-reversion – Empirical Motivation and Derivatives Pricing via Perturbation Theory

The dependence structure is crucial when modelling several assets simultaneously. We show for a real-data example that the correlation structure between assets is not constant over time but rather changes stochastically, and we propose a multidimensional asset model which fits the patterns found in the empirical data. The model is applied to price multi-asset derivatives by means of perturbation theory. It turns out that the leading term of the approximation corresponds to the Black–Scholes derivative price with correction terms adjusting for stochastic volatility and stochastic correlation effects. The practicability of the presented method is illustrated by some numerical implementations. Furthermore, we propose a calibration methodology for the considered model.

A Jump-Diffusion Approach for Pricing Exotic Multi-Asset Derivatives

We propose a simple multidimensional jump-diffusion process for pricing exotic derivatives with multiple underlyings. This process ensures the possibility of sudden drops in asset prices, fits several well-known empirical properties of asset returns, and incorporates dependencies between diffusions, jump intensities, and jump sizes. We show by pricing barrier reverse convertibles that neglecting jumps and dependencies can result in underestimation of the spreads associated with these investments. This feature is likely to have consecutive effects for studies on the overpricing of exotic multi-asset financial derivatives.