In the face of severe climate change, researchers have looked for assistance from financial instruments. They have examined how to hedge the risks of these instruments created by market fluctuations through various green financial derivatives, including green bonds (i.e., fixed-income financial instruments designed to support an environmental goal). In this study, we designed a green bond index option contract. First, we combined an autoregressive moving-average model (AMRA) with a generalized autoregressive conditional heteroskedasticity model (GARCH) to predict the green bond index. Next, we established a fractional Brownian motion option pricing model with temporally variable volatility. We used this approach to predict the closing price of the China Bond–Green Bond Index from 3 January 2017 to 30 December 2021 as an empirical analysis. The trend of the index predicted by the ARMA–GARCH model was consistent with the actual trend and predictions of actual prices were highly accurate. The modified fractional Brownian motion option pricing model improved the pricing accuracy. Our results provide a policy reference for the development of a green financial derivatives market, and can accelerate the transformation of markets towards a more sustainable economic development model.

This article proposes a semi-martingale approximation to a fractional Lévy process that is capable of capturing long and short memory in the stochastic process together with fat tails. The authors use the semi-martingale process in option pricing and empirically compare its performance to other option pricing models, including a stochastic volatility Lévy process. They contribute to the empirical literature by being the first to report the implied Hurst index computed from observed option prices using the Lévy process model. Calibrating the implied Hurst index of S&P 500 option prices in a period that covers the 2008 financial crisis, they find that the risk-neutral measure is characterized by a short memory in turbulent markets and a long memory in calm markets.

Basket options have long been an important structured product. Although basket options have been extensively studied in the literature, there are few published papers that deal with the pricing of basket options with stochastic interest rates. This study presents two novel basket option pricing models that permit the interest rates to be random. The paper presents a powerful calculation technique for the problem when underlying stock returns are continuous. Finally, we use a regular grid method to the calculation of the formula of two-asset basket option when underlying stock returns are continuous and a mixture of both the regular grid method and a Monte Carlo method to the one when underlying stock returns are discontinuous, and sensitivity analyses are presented.

By testing properties implied by one-dimensional diffusion option pricing models, we find that call (put) prices in the Chinese 50ETF option market move in opposite (same) direction with the underlying between 13.39% and 27.89% (between 12.45% and 33.98%) of the time for 5-minute and 1-day sampling intervals respectively. Given fundamental different investor structures in U.S. and China option markets, we also observe some important unique features in the 50ETF option price dynamics. More importantly, we demonstrate that these striking violations reduce substantially in 2016 compared with those in 2015, indicating that Chinese stock option market becomes more efficient.

In this paper, we derive the corresponding implied VIX formulas under the locally riskneutral valuation relationship proposed by Duan (1995) when various forms of GARCH model are proposed for S&P 500 index. The empirical study shows that the GARCH implied VIX is consistently and significantly lower than the CBOE VIX for all kinds of GARCH model investigated. Moreover, the magnitude of the difference suggests that the GARCH option pricing model is not capable of capturing the variance premium, which indicates the incompleteness of the GARCH option pricing under the locally risk-neutral valuation relationship. The source of this kind of incompleteness is then theoretically analyzed. It is shown that the framework of GARCH option pricing model fails to incorporate the price of volatility risk or variance premium.

Volatility long memory is a stylized fact that has been documented for a long time. Existing literature have two ways to model volatility long memory: component volatility models and fractionally integrated volatility models. This paper develops a new fractionally integrated GARCH model, and investigates its performance by using the Standard and Poor’s 500 index returns and cross-sectional European option data. The fractionally integrated GARCH model signi?cantly outperforms the simple GARCH(1, 1) model by generating 37% less option pricing errors. With stronger volatility persistence, it also dominates a component volatility model, who has enjoyed a reputation for its outstanding option pricing performance, by generating 15% less option pricing errors. We also con?rm the fractionally integrated GARCH model’s robustness with the latest option prices. This paper indicates that capturing volatility persistence represents a very promising direction for future study.

China launched her warrant market in August 2005 in the split share structure reform of listed companies. As up to now, equity trading on margin and short-sale of any form are still prohibited in China. This warrant market enables investors to trade on information that otherwise might be prohibitively expensive to trade on. The Chinese warrant market created top trading volume and turnover with only a handful of different warrants traded. This paper first studies the Chinese warrant market. Empirical evidence shows that the market prices of warrants are much higher systematically than the Black-Scholes prices with historical volatility. Moreover, the paper documents ample evidence that the one-dimensional diffusion model does not apply well in the Chinese warrant market. The prices of a warrant and its underlying asset do not support the monotonicity, perfect correlation and option redundancy properties. The paper also studies the cumulated gains of a delta-hedged warrant portfolio. In the Chinese warrant market, the cumulated delta-hedged gains for almost all expired warrants are negative. The negative gains are mainly driven by the volatility risk, and the trading values of the warrants for puts and the market risk for calls. The investors are trading some other risks in addition to the underlying risk.

This is the first study that uses Merton’s (1974) option pricing model to compute default measures for individual firms and assess the effect of default risk on equity returns. The size effect is a default effect, and this is also largely true for the book-to-market (BM) effect. Both exist only in segments of the market with high default risk. Default risk is systematic risk. The Fama-French (FF) factors SMB and HML contain some default-related information, but this is not the main reason that the FF model can explain the cross-section of equity returns.

This paper deals with the option-pricing problem. In the first part of the paper we study in more details the discrete setting of the option-pricing problem usually referred to as the binomial scheme. We highlight basic differences between the old and the new approaches. The main qualitative distinction of the new pricing approach from either binomial or Black Scholes’s is that it represents the option price as a stochastic process. This stochastic interpretation can not give straightforward advantage for an investor due to stochastic setting of the pricing problem. The new approach explicitly states that the options price is more risky than represented by binomial scheme or Black Scholes theory. Continuous setting will be considered in the second part of the paper following [1]. One significant conclusion follows from the new model. It states that there is no sense in using either neutral probabilities or ‘neutral world’ applications for options valuation either theoretically or numerically. Recall that after the Black Scholes’ publication [2] the ‘simplified’ approach named later binomial scheme was introduced in [3]. In this paper referring to the historical tradition we first represent discrete scheme. In several examples we discuss two-period plain vanilla option valuation. Then we extend the discrete scheme applications to an exotic option-pricing referred to as a compound option. The compound option in Black Scholes setting was first studied in [4] and then in [5,6]. To highlight the difference between stochastic and deterministic option price definitions note that if a deterministic value is interpreted as a perfect or fair price we can comment that the stochastic interpretation provides this number or any other with the probability that real world option value at maturity will be bellow chosen number. This probability is a pricing risk of the option. Thus with an investor’s motivation of the option pricing the stochastic approach gives information about the risk taking. The investor analyzing option price and corresponding risk makes a decision to purchase the option or not. As far as this paper presents alternative point on option pricing it might be useful to present a short history of this development. Recall that according the US law institutions must provide clients by the risk information regarding client’s prospective on their investments. This circumstance implies importance new approach measuring risk of investments. Different parts of this paper were submitted and sent to journals, conferences, and prominent professors. The third part of the paper was sent to Federal Reserve from the Congressman office and simple examples showing drawbacks of the benchmark option valuation method were submitted to SEC in August 2002.