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This paper shows that standard disaster risk models are inconsistent with movements in stock market volatility and credit spreads during disasters. We resolve this shortcoming by incorporating persistent macroeconomic crises into a structural credit risk model. The model successfully captures the joint dynamics of aggregate consumption, financial leverage, and asset market risks, both unconditionally and during crises. Leverage strongly amplifies fundamental shocks by continuing to rise while crises endure. We structurally estimate the model and show that it replicates the firm-level implied volatility curve and its cross-sectional relation with observable proxies of default risk.

As in the previous chapters, we consider a market model consisting in two assets: one non-risky (bond), the other risky (stock). While before we focused on discrete-time market models, here we introduce the so-called Black-Scholes model : a well-known example of continuous-time market model.

In the following, we consider a market model where a non-risky asset (called bond) and a risky asset (called stock) are available. The bond price is denoted by B, while the stock price is denoted by S.

Chapter 8 focuses on the valuation of options along with their applications to arbitrage and hedging. After a general introduction to derivatives markets and hedging, put-call parity is explored, with its application to collars as a hedging technique. One, two, and n-time period binomial pricing models are derived from the binomial hedge ratio. The Black-Scholes model is discussed along with its "Greeks." Formation of delta and gamma neutral portfolios is explained. The Black-Scholes model is extended to FX options, which are applied to the management of exchange exposure. The Black-Scholes model is derived in the chapter appendix.

In the following, we recall the notions of arbitrage, completeness and option pricing in quite general one-period (or multi-period) market models, but always based on a finite sample space.

It is widely accepted to use conditional value-at-risk for risk management needs and option pricing. As a rule, there are difficulties in exact calculations of conditional value-at-risk. In the paper, we use the conditional value-at-risk methodology to price spread options, extending some approximation approaches for these needs. Our results we illustrate by numerical calculations which demonstrate their effectiveness. We also show how conditional value-at-risk pricing can help with regulatory needs inspired by the Basel Accords.

By employing a randomisation procedure on the variance parameter of the standard geometric Brownian motion (GBM) model, we construct new families of analytically tractable asset pricing models. In particular, we develop two explicit families of processes that are respectively referred to as the randomised gamma (G) and randomised inverse gamma (IG) models, both characterised by a shape and scale parameter. Both models admit relatively simple closed-form analytical expressions for the transition density and the no-arbitrage prices of standard European-style options whose Black-Scholes implied volatilities exhibit symmetric smiles in the log-forward moneyness. Surprisingly, for integer-valued shape parameter and arbitrary positive real scale parameter, the analytical option pricing formulas involve only elementary functions and are even more straightforward than the standard (constant volatility) Black-Scholes (GBM) pricing formulas. Moreover, we show some interesting characteristics of the risk-neutral transition densities of the randomised G and IG models, both exhibiting fat tails. In fact, the randomised IG density only has finite moments of the order less than or equal to one. In contrast, the randomised G density has a finite first moment with finite higher moments depending on the time-to-maturity and its scale parameter. We show how the randomised G and IG models are efficiently and accurately calibrated to market equity option data, having pronounced implied volatility smiles across several strikes and maturities. We also calibrate the same option data to the wellknown SABR (Stochastic Alpha Beta Rho) model.

Modern volatility managed indexes employ antiquated close-to-close volatility models to scale exposure. This has material consequences on the option pricing dynamics for structured investments, many of which reference a volatility managed index because of option pricing benefits associated with the constant volatility. The author tests two daily close-to-close models and three intraday “realized volatility” models that leverage the rich information dynamics found in intraday data. The results, across three highly liquid US large-cap indexes, suggest that the latter three models produce more consistent volatility profiles, or alternatively, a lower volatility of volatility, without sacrificing performance (i.e., the Sharpe ratio). All else being equal, this lower volatility of volatility should be preferred by product issuers and investors because of the resulting option pricing benefits.

The aims of the current article are threefold. First, to investigate the power of deterministic option-implied trees, constructed either by forward or by backward induction, in pricing European opt...

This study aims to identify a theoretical fair value of the Sharpe ratio. For this purpose, we propose a formula with a closed-form solution to estimate a fair, expected, or ex ante value for the Sharpe ratio by framing it in the option pricing model. We find that firm value, growth rate, return volatility, risk-free rate, and the maturity and face value of the debt contract are determinants of the Sharpe ratio. In particular, a firm’s investing and financing policies can affect the firm stock’s Sharpe ratios. Contrary to what might reasonably be expected, our results show that a firm with a higher equity value tends to have a lower Sharpe ratio value. The results show the potential and feasibility of integrating three fields in future research: performance evaluations, option pricing, and corporate finance.

Stephen A. Ross was one of the most influential scholars in the field of financial economics in the late twentieth century. Ross's work was central to several novel domains of economic inquiry. His contributions included the arbitrage pricing theory (APT), the risk-neutral pricing of contingent claims, the binomial option pricing model, a theory of the term structure of interest rates, a seminal contribution to the economic theory of agency, and insights about conditioning biases in ex post performance measurement. In this article, we discuss his seminal papers and the broad scope of his curiosity within the arc of a remarkably productive and influential career that spanned five decades and yet ended sooner than most who knew him expected.

Owing to the ability to discern the future and the past in the over-the-counter market, the lookback option is regarded as one of the well-known path-dependent financial derivatives. Firstly, because the historical actual financial data is usually unreliable, scarce and unavailable, the underlying asset of the lookback option is regarded as an uncertain variable. Fractional-order derivative p rather than the integer derivative is applied and a more fine-grained portrayal of the real economic market is obtained based upon the uncertain fractional-order differential equation (UFDE). Next, through the existing extreme value theorems of the UFDE, European lookback (containing call and put cases) option pricing formulas are obtained for the uncertain fractional-order stock model (UFSM) and uncertain fractional-order mean-reverting models (UFMM), respectively. At last, the predictor-corrector method (PCM) is used to design numerical algorithms for calculating European lookback option price. Moveover, some numerical example are experienced to demonstrate the reasonability of our models.

In this paper, we combine modern portfolio theory and option pricing theory so that a trader who takes a position in a European option contract and the underlying assets can construct an optimal portfolio such that at the moment of the contract’s maturity the contract is perfectly hedged. We derive both the optimal holdings in the underlying assets for the trader’s optimal mean-variance portfolio and the amount of unhedged risk prior to maturity. Solutions assuming the cases where the price dynamics in the underlying assets follows discrete binomial price dynamics, continuous diffusions, stochastic volatility, volatility-of-volatility, and Merton-jump diffusion are derived.

We show how and why to use a financially meaningful differential regularization method when pricing options by Monte Carlo simulation, be that in polynomial regression or neural network context.

We present an analytical option pricing formula for the European options, in which the price dynamics of a risky asset follows a mean-reverting process with a time-dependent parameter. The process can be adapted to describe a seasonal variation in price such as in agricultural commodity markets. An analytical solution is derived based on the solution of a partial differential equation, which shows that a European option price can be decomposed into two terms: the payoff of the option at the initial time and the time-integral over the lifetime of the option driven by a time-dependent parameter. Finally, results obtained from the formula have been compared with Monte Carlo simulations and a Black–Scholes-type formula under various kinds of long-run mean functions, and some examples of option price behaviours have been provided.

In this work we propose a simple and efficient algorithm to numerically approximate the time-dependent implied volatility for jump–diffusion models in option pricing that generalize the Black–Scholes equation. Here we use implicit–explicit difference schemes to compute the derivative part with fully implicit method and the integral term — in an explicit way. An average in time linearization of the diffusion term is applied, followed by a special decomposition of the unknown volatility function, which enables us to derive the implied volatility in an explicit form. Furthermore, the correctness of the algorithms is established. The presented numerical simulations demonstrate the capabilities of the current approach and confirm the robustness of the proposed methodology.

Levy process models can capture the large price changes on sudden exogenous events and can better demonstrate the high peak and heavy tail characteristics of financial data. The Fourier transformation method is famous for pricing derivatives under the Levy processes beause of its efficiency, separation models from payoff function, and handling models with characteristic functions, but it is criticized for its restriction on path dependency. In this article, we propose a unified cosine willow tree method, which inherits the merits of the transformation method but overcomes the shortcomings. Moreover, the hedging Greeks can be obtained as a by-product from the tree structure with minor extra cost. Some popular variance derivatives are also discussed to demonstrate the flexibility of the proposed method in handling path dependency. Finally, the theoretical convergence is analyzed for various Levy process models. Key Findings ▪ Propsoe a recombining willow tree framework for stochastic processes with explicit characteristic functions on option pricing and hedging. ▪ Analyze the convergence of the willow tree structure on option pricing under L’evy processes. ▪ Propose efficient pricing algorithms for volatility option, variance swap with corridor and timer option.

It has been found that the surface of implied volatility has appeared in financial market embrace volatility “Smile” and volatility “Smirk” through the long-term observation. Compared to the conventional Black-Scholes option pricing models, it has been proved to provide more accurate results by stochastic volatility model in terms of the implied volatility, while the classic stochastic volatility model fails to capture the term structure phenomenon of volatility “Smirk.” More attempts have been made to correct for American put option price with incorporating a fast-scale stochastic volatility and a slow-scale stochastic volatility in this paper. Given that the combination in the process of multiscale volatility may lead to a high-dimensional differential equation, an asymptotic approximation method is employed to reduce the dimension in this paper. The numerical results of finite difference show that the multiscale volatility model can offer accurate explanations of the behavior of American put option price.

The goal of this paper is to investigate the method outlined by one of us (PR) in Cherubini et al. (2009) to compute option prices. We name it the SINC approach. While the COS method by Fang and Osterlee (2009) leverages the Fourier-cosine expansion of truncated densities, the SINC approach builds on the Shannon Sampling Theorem revisited for functions with bounded support. We provide several results which were missing in the early derivation: i) a rigorous proof of the convergence of the SINC formula to the correct option price when the support grows and the number of Fourier frequencies increases; ii) ready to implement formulas for put, Cash-or-Nothing, and Asset-or-Nothing options; iii) a systematic comparison with the COS formula for several log-price models; iv) a numerical challenge against alternative Fast Fourier specifications, such as Carr and Madan (1999) and Lewis (2000); v) an extensive pricing exercise under the rough Heston model of Jaisson and Rosenbaum (2015); vi) formulas to evaluate numerically the moments of a truncated density. The advantages of the SINC approach are numerous. When compared to benchmark methodologies, SINC provides the most accurate and fast pricing computation. The method naturally lends itself to price all options in a smile concurrently by means of Fast Fourier techniques, boosting fast calibration. Pricing requires to resort only to odd moments in the Fourier space. A previous version of this manuscript circulated with the title `Rough Heston: The SINC way'., Comment: 49 pages, 4 figures, 13 tables

Many studies find that traditional option pricing models fail to work in practice. The implied volatility smile is one example. In this study, we examine deviations of spot prices from prices impli...

This paper proposes a nonparametric estimator for the state price density implied by a single cross-section of European options with different strikes and the same maturity. The proposed estimator has two distinctive features. First, it extracts information from both call and put options, as opposed to only call options. Second, it does not require estimating any second-order derivative; instead, it solves a constrained and penalized linear regression. The asymptotic analysis faces two challenges because the state price density is defined by a Fredholm integral equation of the first kind with an unbounded support, and the kernel function is unbounded and non-differentiable. We address these challenges by exploiting the structure of the option pricing problem. After establishing the estimator’s consistency and convergence rate, we apply it to estimate the state price densities implied by the S&P500 index options and those by the VIX options. The sample period includes the recent financial crisis and the Great Recession, during which the turbulent market conditions imposed substantial challenges on the estimation. We show that the estimator can work with both daily and high-frequency observations. We also study whether the various features of this density can predict future asset returns and obtain positive findings. Finally, we apply the method to examine the causal effects of monetary policy announcements on the financial market, using high-frequency data.

Aims: The main purpose of this study is to understand whether Logistic regression has certain benefits in the evaluation of American options. As far as the Monte Carlo method is concerned, the least square method is traditionally used to evaluate American options, but in fact, Logistic regression is generally quite good in classification performance. Therefore, this study wants to know if Logistic regression can improve the accuracy of evaluation in American options. Study Design: The selection of options parameters required in the simulation process mainly considers the average level of actual market conditions in the past few years in terms of dividend yield and risk-free interest rate. The part of the stock price and the strike price mainly considers three different situations: in-the-money, out-of-the-money and at the money. Methodology: This study applied the Logistic regression in Monte Carlo method for the pricing of American. Uses the ability of logistic regression to help determine whether the American option should be exercised early for each stock price path. The validity of the proposed method is supported by some vanilla put cases testing. The parameters used in all cases tested are considered the current state of the market. Conclusion: This study demonstrates the effectiveness of the proposed approach using numerical examples, revealing significant improvements in numerical efficiency and accuracy. Several test cases showed that the relative error of all tests are below 1%.

In this paper, we start from the no-arbitrage constraints in option pricing and develop a novel hybrid gated neural network (hGNN) based option valuation model. We adopt a multiplicative structure of hidden layers to ensure model differentiability. We also select the slope and weights of input layers to satisfy the no-arbitrage constraints. Meanwhile, a separate neural network is constructed for predicting option-implied volatilities. Using S&P 500 options, our empirical analyses show that the hGNN model substantially outperforms well-established alternative models in the out-of-sample forecasting and hedging exercises. The superior prediction performance stems from our model’s ability in describing options on the boundary, and in offering analytical expressions for option Greeks which generate better hedging results.

In this paper, we propose two new representation formulas for the conditional marginal probability density of the multi-factor Heston model. The two formulas express the marginal density as a convolution with suitable Gaussian kernels whose variances are related to the conditional moments of price returns. Via asymptotic expansion of the non-Gaussian function in the convolutions, we derive explicit formulas for European-style option prices and implied volatility. The European option prices can be expressed as Black–Scholes style terms plus corrections at higher orders in the volatilities of volatilities, given by the Black–Scholes Greeks. The explicit formula for the implied volatility clearly identifies the effect of the higher moments of the price on the implied volatility surface. Further, we derive the relationship between the VIX index and the variances of the two Gaussian kernels. As a byproduct, we provide an explanation of the bias between the VIX and the volatility of total returns, which offers support to recently proposed methods to compute the variance risk premium. Via a series of numerical exercises, we analyse the accuracy of our pricing formula under different parameter settings for the one- and two-factor models applied to index options on the S&P500 and show that our approximation substantially reduces the computational time compared to that of alternative closed-form solution methods. In addition, we propose a simple approach to calibrate the parameters of the multi-factor Heston model based on the VIX index, and we apply the approach to the double and triple Heston models.

This paper analyzes the effects of the lower bound for interest rates on the distributions of inflation and interest rates. We study a stylized New Keynesian model where the policy instrument is subject to a lower bound to motivate the empirical analysis. Two equilibria emerge: In the “target equilibrium,” policy is unconstrained most or all of the time, whereas in the “liquidity trap equilibrium,” policy is mostly or always constrained. We use options data on future interest rates and inflation to study whether the decrease in the natural real rate of interest leads to forecast densities consistent with the theoretical model. Qualitatively, we find that the evidence is consistent with the theoretical predictions in the target equilibrium and find no evidence in favor of the liquidity trap equilibrium. Quantitatively, while the lower bound has a sizable effect on the distribution of future interest rates, its impact on forecast densities for inflation is relatively modest. We develop a lower bound indicator that captures the effects of the lower bound on the distribution of interest rates.

In this paper, we present a numerical technique for solving the time-fractional Black-Scholes (TFBS) equation describing European options. The time-fractional derivative is described by means of Caputo and a compact finite difference method is employed for discretization of space derivative. Stability and convergence of the fully discrete scheme are studied. Two test problems with the known analytical solutions are considered to demonstrate the efficiency and accuracy of the method and validate the theoretical result. Further, in order to demonstrate the practicability of the method, the method is applied to three European options pricing problems governed by the TFBS equations, where analytical solutions to these problems are not known. We study the effect of fractional order derivative on the solution profile corresponding to option price.

In this study, using the method of discounting the terminal expectation value into its initial value, the pricing formulas for European options are obtained under the assumptions that the financial market is risk-aversive, the risk measure is standard deviation, and the price process of underlying asset follows a geometric Brownian motion. In particular, assuming the option writer does not need the risk compensation in a risk-neutral market, then the obtained results are degenerated into the famous Black–Scholes model (1973); furthermore, the obtained results need much weaker conditions than those of the Black–Scholes model. As a by-product, the obtained results show that the value of European option depends on the drift coefficient μ of its underlying asset, which does not display in the Black–Scholes model only because μ = r in a risk-neutral market according to the no-arbitrage opportunity principle. At last, empirical analyses on Shanghai 50 ETF options and S&P 500 options show that the fitting effect of obtained pricing formulas is superior to that of the Black–Scholes model.

We develop a structural model of the firm that allows after-interest cash to be directed to dividends, buybacks or some combination thereof. We study the effects of dilutions and buybacks on the value of the firm's claimants and on options written on the stock. We distinguish between options on equity and options on individual shares and value both types. We examine employee stock options and establish how their value changes in accordance with changing payout policies. Our model allows us to quantify and analyze the wealth transfer that occurs from shareholders to holders of employee stock-based contingent claims.

In this study, we consider option pricing under a Markov regime-switching GARCH-jump (RS-GARCH-jump) model. More specifically, we derive the risk neutral dynamics and propose a lattice algorithm to price European and American options in this framework. We also provide a method of parameter estimation in our RS-GARCH-jump setting using historical data on the underlying time series. To measure the pricing performance of the proposed algorithm, we investigate the convergence of the tree-based results to the true option values and show that this algorithm exhibits good convergence. By comparing the pricing results of RS-GARCH-jump model with regime-switching GARCH (RS-GARCH) model, GARCH-jump model, GARCH model, Black–Scholes (BS) model, and Regime-Switching (RS) model, we show that accommodating jump effect and regime switching substantially changes the option prices. The empirical results also show that the RS-GARCH-jump model performs well in explaining option prices and confirm the importance of allowing for both jump components and regime switching.

Black-Scholes option pricing model is used to decide theoretical price of different Options contracts in many stock markets in the world. In can find many generalizations of BS model by modifying some assumptions of classical BS model. In this paper we compared two such modified Black-Scholes models with classical Black-Scholes model only for Indian option contracts. We have selected stock options form 5 different sectors of Indian stock market. Then we have found call and put option prices for 22 stocks listed on National Stock Exchange by all three option pricing models. Finally, we have compared option prices for all three models and decided the best model for Indian Options. Motivation/Background: In 1973, two economists, Fischer Black, Myron and Robert Merton derived a closed form formula for finding value of financial options. For this discovery, they got a Nobel prize in Economic science in 1997. Afterwards, many researchers have found some limitations of Black-Scholes model. To overcome these limitations, there are many generalizations of Black-Scholes model available in literature. Also, there are very limited study available for comparison of generalized Black-Scholes models in context of Indian stock market. For these reasons we have done this study of comparison of two generalized BS models with classical BS model for Indian Stock market. Method: First, we have selected top 5 sectors of Indian stock market. Then from these sectors, we have picked total 22 stocks for which we want to compare three option pricing models. Then we have collected essential data like, current stock price, strike price, expiration time, rate of interest, etc. for computing the theoretical price of options by using three different option pricing formulas. After finding price of options by using all three models, finally we compared these theoretical option price with market price of respected stock options and decided that which theoretical price has less RMSE error among all three model prices. Result: After going through the method described above, we found that the generalized Black-Scholes model with modified distribution has minimum RMSE errors than other two models, one is classical Black-Scholes model and other is Generalized Black-Scholes model with modified interest rate.

Unpredictable nature of financial derivatives market has always been a challenge for investors to profit from these markets. It is known that computing volatility of underlying assets is a source of this challenge. In this work, we present an application of our data-driven neuro arch (DDNA) volatility model that is driven by the market data of the stock prices to price options. That is, we use the volatility forecast from our DDNA model along with the Monte Carlo (MC) simulations to compute option prices. Since the MC method requires a large number of simulations for better precision, we implement the proposed model on two cloud resources (Amazon’s EMR and Google’s Cloud DataProc) using the Hadoop MapReduce paradigm. Also, since the MC strategy is prone to errors due to uncertainties and random numbers, we propose to generate a fuzzified range of option prices instead of a single crisp option value to minimize these errors. Our experimental configuration consists of c3.xlarge instances and n1-standard-4 instances, both having 4 vCores, for EMR and GDP respectively. For a largest number of 10 million simulations on 40 VMs, the option pricing results (computed in 39 and 33 s respectively on EMR and GDR) are close to the last traded option value found on option chain tables with an error of 0.00101 (0.1%). The proposed DDNA model for forecasting volatility together with MC option pricing model implemented on MapReduce outperforms the existing option pricing models in terms of efficiency and accuracy. This proposed strategy could be used by investors for computing option prices precisely with relative ease, allowing them to value the numerous available option contracts for their investment decisions.

This study investigates valuation of guaranteed minimum maturity benefits (GMMB) in variable annuity contract in the case where the guarantees can be surrendered at any time prior to the maturity. In the event of the option being exercised early, early surrender charges will be applied. We model the underlying mutual fund dynamics under regime-switching volatility. The valuation problem can be reduced to an American option pricing problem, which is essentially an optimal stopping problem. Then, we obtain the pricing partial differential equation by a standard Markovian argument. A detailed discussion shows that the solution of the problem involves an optimal surrender boundary. The properties of the optimal surrender boundary are given. The regime-switching Volterra-type integral equation of the optimal surrender boundary is derived by probabilistic methods. Furthermore, a sensitivity analysis is performed for the optimal surrender decision. In the end, we adopt the trinomial tree method to determine the optimal strategy.

Our main objective of this paper is to introduce four individual techniques of pricing options; the techniques are Binomial method, Trinomial method, Monte Carlo simulation and Black-Scholes-Merton model. Because they play a significant role in option valuation of stock price dynamics, risk managements as well as stock market. In this paper, we briefly discuss all these four methods with their properties and behavior. We also focused on numerical technique for the higher accuracy of option pricing and compare them graphically. We use the Computer Algebra System (CAS) Python (Edition 2019.3.1) for this purpose. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 7, Dec 2020 P 1-7

In this paper, we present a double exponential jump-diffusion option pricing model with stochastic interest rates, stochastic volatility, and stochastic jump intensity. In addition, Markov regime-s...

Motivated by the theory of demand-based option pricing in imperfect markets, we examine the relation between short-sale constraints and equity option returns, conditional on the level of mispricing in the underlying stock. We report a monotonic relation between various measures of short-sale constraints and delta-hedged returns of put options on overpriced stocks. This relation is robust to controls for firm attributes and limits to arbitrage proxies. Our findings suggest that while investors drive up the demand for these put options, dealers command a high premium as compensation for the increased market making risk. We do not find a robust relation for either put options on underpriced stocks or call options.

Liu process is a basic process in fuzzy environment. As an extension of Liu process, fractional Liu process has attracted the attention of many scholars. In this paper, a fuzzy stock model driven by fractional Liu process is established, and its European, American, Asian, power options pricing formulas are given. In order to better understand these formulas, we give a few numerical examples to illustrate the changes of European option price with different parameters when time is fixed. However, these examples are not based on real-life data since the lack of parameter estimation method for fuzzy differential equation driven by Liu process. Then the changes of American option price are given when time and parameters are both changed. At the same time, we study the parameter interval where the option price fluctuates greatly. Finally, the fuzzy stock model is extended to the generalized case, and the stock price is given.

We provide a lean, non-technical exposition on the pricing of path-dependent and European-style derivatives in the Cox–Ross–Rubinstein (CRR) pricing model. The main tool used in this paper for simplifying the reasoning is applying static hedging arguments. In applying the static hedging principle, we consider Arrow–Debreu securities and digital options, or backward random processes. In the last case, the CRR model is extended to an infinite state space which leads to an interesting new phenomenon not present in the classical CRR model. At the end, we discuss the paradox involving the drift parameter $$\mu $$ μ in the Black–Scholes–Merton model pricing. We provide sensitivity analysis and an approximation of the speed of convergence for the asymptotically vanishing effect of drift in prices.

Asian rainbow option is option on the minimum or the maximum of several average prices. In modern financial market, Asian rainbow option is an effective instrument for asset allocation and risk management. The investor with Asian rainbow option enjoys an entitlement to select a max or min from multiple assets with an exercise price at maturity date. The investor has to defray fee to acquire this right, which raises the option pricing issue. This paper mainly explores the pricing of Asian rainbow option in the uncertain financial environment, in which the underlying assets prices are treated as uncertain processes. Here, the pricing formulas of Asian rainbow option are derived under the condition that stock prices obey uncertain differential equations driven by independent Liu processes. Furthermore, some numerical examples are designed to compute the prices of these options.

[EN] A new efficient numerical method is proposed for valuation of American option on zero-coupon bond using Hull and White model. By applying the front-fixing transformation suggested by Holmes and Yang, the original free boundary problem is transformed into a new fixed boundary partial differential equation (PDE) problem, where the optimal stopping boundary is one of the unknowns of the problem. The numerical finite difference scheme for the transformed problem is constructed. Stability and convergence rate is studied empirically. Numerical simulation of the computation of both the option price and the optimal stopping boundary are illustrated with examples and the comparison with the Hull and White tree method., Ministerio de Ciencia e Innovacion, Grant/Award Number: MTM2017-89664-P

The remarkable performance of deep learning is based on its ability to learn high-level features by processing large amounts of data. This exceptionally superior performance has attracted the attention of researchers studying option pricing. However, option data are more expensive and less accessible than other types of data and are imbalanced because of the liquidity of options. This motivated us to propose a new option pricing and delta-hedging framework called DeepOption. This framework, which is based on deep learning, can improve the performance even when applying imbalanced real option data. In particular, the framework fuses simulated big data, known as distilled data, obtained using various traditional parametric methods. The proposed model employs the following three-stage training approach: Our model is pre-trained using big distilled data after it is fine-tuned using real option data through transfer learning. Finally, a delta branch is added to the model and trained. We experimentally evaluated the proposed method using three sets of real option data, namely S&P 500 European call options, EuroStoxx50 call options, and Hang Seng Index put options. Our experimental results on option pricing demonstrate that our proposed model outperforms parametric methods and other machine learning methods. Specifically, our model, which uses pre-training with distilled data, reduces the overall mean absolute percentage error (MAPE) by more than 50%, compared with that of a deep learning model using only real option data without pre-training.

We examine the performance of the Black-Scholes (B-S) formulas around (i.e., before, during and after) two periods of market stress: the subprime crisis (October, 2008) and the onset of the Covid-19 pandemic (March, 2020). This is interesting for two reasons. First, most studies regarding the accuracy of the B-S formulas have been done under regular or normal market conditions (granted, a fuzzy concept), not under conditions of severe market stress. And second, these two crises are quite different in nature. The subprime crisis was generated within a specific segment of the capital markets and affected only some sectors of the economy;the Covid-19 crisis was due to an external shock and still continues to have a much wider impact. We find, in agreement with previous studies under different circumstances, that the accuracy of these formulas is very poor. Moreover, we find that the degree of mispricing, which is significant and in many cases exceed 100%, is manifest in both crises, and in all scenarios analyzed.