The author studies T. Cover’s rebalancing option (Ordentlich and Cover 1998) under discrete hindsight optimization in continuous time. The payoff in question is equal to the final wealth that would have accrued to an initial deposit of 1 unit of the numéraire into the best of some finite set of (perhaps levered) rebalancing rules determined in hindsight. A rebalancing rule (or fixed-fraction betting scheme) amounts to fixing an asset allocation (i.e., 200% equities and −100% bonds) and then continuously executing rebalancing trades so as to counteract allocation drift. Restricting the hindsight optimization to a small number of rebalancing rules (i.e., 2) has some advantages over the pioneering approach taken by Cover & Company in their theory of universal portfolios (1986, 1991, 1996, 1998), wherein one’s trading performance is benchmarked relative to the final wealth of the best unlevered rebalancing rule (of any kind) in hindsight. Our approach lets practitioners express an a priori view that one of the favored asset allocations (“bets”) in the set {b1, …, bn} will turn out to have performed spectacularly well in hindsight. In limiting our robustness to some discrete set of asset allocations (rather than all possible asset allocations), we reduce the price of the rebalancing option and guarantee that we will achieve a correspondingly higher percentage of the hindsight-optimized wealth at the end of the planning period. A practitioner who lives to delta-hedge this variant of Cover’s rebalancing option through several decades is guaranteed to see the day that his realized compound-annual capital growth rate is very close to that of the best bi in hindsight, hence the point of the rock-bottom option price.
作者研究了连续时间离散后见之明优化下T. Cover的再平衡选择(Ordentlich and Cover 1998)。所讨论的收益等于最初存入1个单位的numsamraire的最终财富，这些财富将累积到一些有限的(可能是杠杆的)后见之明确定的再平衡规则的最佳集合中。再平衡规则(或固定分数投注方案)相当于固定资产配置(即，200%的股票和- 100%的债券)，然后不断执行再平衡交易，以抵消分配漂移。将后见之明优化限制在少数再平衡规则(即2)中，比Cover & Company在其通用投资组合理论(1986、1991、1996、1998)中采用的开创性方法有一些优势，其中一个人的交易绩效是相对于后见之明的最佳无杠杆再平衡规则(任何类型)的最终财富进行基准测试的。我们的方法让从业者表达一种先验的观点，即集合{b1，…，bn}中最受青睐的资产配置(“赌注”)之一在事后会表现得非常好。通过将我们的稳健性限制在一些离散的资产配置(而不是所有可能的资产配置)上，我们降低了再平衡选项的价格，并保证在规划期结束时，我们将获得相应更高比例的后见之明优化财富。一个实践者在几十年的时间里通过delta对冲来实现Cover的这种再平衡期权的变体，他肯定会看到有一天，他的已实现的复合年资本增长率非常接近后见之明的最佳水平，因此期权价格才会跌至谷底。
{"title":"Cover’s Rebalancing Option with Discrete Hindsight Optimization","authors":"Alex Garivaltis","doi":"10.3905/jod.2021.1.135","DOIUrl":"https://doi.org/10.3905/jod.2021.1.135","url":null,"abstract":"The author studies T. Cover’s rebalancing option (Ordentlich and Cover 1998) under discrete hindsight optimization in continuous time. The payoff in question is equal to the final wealth that would have accrued to an initial deposit of 1 unit of the numéraire into the best of some finite set of (perhaps levered) rebalancing rules determined in hindsight. A rebalancing rule (or fixed-fraction betting scheme) amounts to fixing an asset allocation (i.e., 200% equities and −100% bonds) and then continuously executing rebalancing trades so as to counteract allocation drift. Restricting the hindsight optimization to a small number of rebalancing rules (i.e., 2) has some advantages over the pioneering approach taken by Cover & Company in their theory of universal portfolios (1986, 1991, 1996, 1998), wherein one’s trading performance is benchmarked relative to the final wealth of the best unlevered rebalancing rule (of any kind) in hindsight. Our approach lets practitioners express an a priori view that one of the favored asset allocations (“bets”) in the set {b1, …, bn} will turn out to have performed spectacularly well in hindsight. In limiting our robustness to some discrete set of asset allocations (rather than all possible asset allocations), we reduce the price of the rebalancing option and guarantee that we will achieve a correspondingly higher percentage of the hindsight-optimized wealth at the end of the planning period. A practitioner who lives to delta-hedge this variant of Cover’s rebalancing option through several decades is guaranteed to see the day that his realized compound-annual capital growth rate is very close to that of the best bi in hindsight, hence the point of the rock-bottom option price.","PeriodicalId":501089,"journal":{"name":"The Journal of Derivatives","volume":"48 5","pages":"8-29"},"PeriodicalIF":0.0,"publicationDate":"2021-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495084","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The question addressed is the pricing of options on the CBOE Skew Index. The option pricing theory developed partially hedges risk by taking positions in the market for options on a related asset. The option is then priced at the cost of this hedge. The theory is applied to pricing Volatility Index (VIX) options hedged by the SPDR S&P 500 ETF Trust (SPY) options and pricing options on JPMorgan hedged by Financial Select Sector SPDR (XLF) options. The approach is then applied to illustrate the pricing of CBOE Skew Index options with a hedge in the market for SPY options. The Skew Index smile is then seen to imply the VIX and SKEW of the Skew Index itself. The pricing of VIX options with SPY as the related asset has the Gaussian copula underpricing options while the t-copula significantly overprices them. The multivariate bilateral gamma models are closer to market. The premia of cross-asset hedge prices over the market price are observed to fall with moneyness and maturity and rise with the level of the VIX. TOPICS:Derivatives, options, exchange-traded funds and applications, quantitative methods, statistical methods, performance measurement Key Findings ▪ Time series data on physical returns may be used to obtain market relevant option prices provided market-relevant hedging costs are incorporated. ▪ Options on the CBOE Skew Index are priced at the cost of an SPY option hedge portfolio. ▪ Residual risk pricing technologies may be applied more widely with market calibrated parameters if desired.
{"title":"Pricing and Hedging Options on Assets with Options on Related Assets","authors":"Dilip B. Madan,King Wang","doi":"10.3905/jod.2021.1.132","DOIUrl":"https://doi.org/10.3905/jod.2021.1.132","url":null,"abstract":"The question addressed is the pricing of options on the CBOE Skew Index. The option pricing theory developed partially hedges risk by taking positions in the market for options on a related asset. The option is then priced at the cost of this hedge. The theory is applied to pricing Volatility Index (VIX) options hedged by the SPDR S&P 500 ETF Trust (SPY) options and pricing options on JPMorgan hedged by Financial Select Sector SPDR (XLF) options. The approach is then applied to illustrate the pricing of CBOE Skew Index options with a hedge in the market for SPY options. The Skew Index smile is then seen to imply the VIX and SKEW of the Skew Index itself. The pricing of VIX options with SPY as the related asset has the Gaussian copula underpricing options while the t-copula significantly overprices them. The multivariate bilateral gamma models are closer to market. The premia of cross-asset hedge prices over the market price are observed to fall with moneyness and maturity and rise with the level of the VIX. TOPICS:Derivatives, options, exchange-traded funds and applications, quantitative methods, statistical methods, performance measurement Key Findings ▪ Time series data on physical returns may be used to obtain market relevant option prices provided market-relevant hedging costs are incorporated. ▪ Options on the CBOE Skew Index are priced at the cost of an SPY option hedge portfolio. ▪ Residual risk pricing technologies may be applied more widely with market calibrated parameters if desired.","PeriodicalId":501089,"journal":{"name":"The Journal of Derivatives","volume":"49 s18","pages":"27-47"},"PeriodicalIF":0.0,"publicationDate":"2021-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zhenyu Cui,Justin Kirkby,Duy Nguyen,Stephen Taylor
In this article, we derive a closed-form explicit model-free formula for the (Black-Scholes) implied volatility. The method is based on the novel use of the Dirac Delta function, corresponding delta families, and the change of variable technique. The formula is expressed through either a limit or as an infinite series of elementary functions, and we establish that the proposed formula converges to the true implied volatility value. In numerical experiments, we verify the convergence of the formula, and consider several benchmark cases, for which the data-generating processes are respectively the stochastic volatility inspired model, and the stochastic alpha beta rho model. We also establish an explicit formula for the implied volatility expressed directly in terms of respective model parameters, and use the Heston model to illustrate this idea. The delta family and change of variable technique that we develop are of independent interest and can be used to solve inverse problems arising in other applications. TOPIC: Derivatives Key Findings ▪ A novel closed-form representation of the Black-Scholes implied volatility is developed by utilizing a delta-family technique. ▪ Convergence and error analyses of approximate forms of this representations are presented. ▪ This technique is applied to the parametric SVI and SABR models as well as the stochastic volatility Heston model.
{"title":"A Closed-Form Model-Free Implied Volatility Formula through Delta Families","authors":"Zhenyu Cui,Justin Kirkby,Duy Nguyen,Stephen Taylor","doi":"10.3905/jod.2020.1.127","DOIUrl":"https://doi.org/10.3905/jod.2020.1.127","url":null,"abstract":"In this article, we derive a closed-form explicit model-free formula for the (Black-Scholes) implied volatility. The method is based on the novel use of the Dirac Delta function, corresponding delta families, and the change of variable technique. The formula is expressed through either a limit or as an infinite series of elementary functions, and we establish that the proposed formula converges to the true implied volatility value. In numerical experiments, we verify the convergence of the formula, and consider several benchmark cases, for which the data-generating processes are respectively the stochastic volatility inspired model, and the stochastic alpha beta rho model. We also establish an explicit formula for the implied volatility expressed directly in terms of respective model parameters, and use the Heston model to illustrate this idea. The delta family and change of variable technique that we develop are of independent interest and can be used to solve inverse problems arising in other applications. TOPIC: Derivatives Key Findings ▪ A novel closed-form representation of the Black-Scholes implied volatility is developed by utilizing a delta-family technique. ▪ Convergence and error analyses of approximate forms of this representations are presented. ▪ This technique is applied to the parametric SVI and SABR models as well as the stochastic volatility Heston model.","PeriodicalId":501089,"journal":{"name":"The Journal of Derivatives","volume":"77 1","pages":"111-127"},"PeriodicalIF":0.0,"publicationDate":"2020-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider derivatives that maximize an investor’s expected utility in the stochastic volatility model. We show that the optimal derivative that depends on the stock and its variance significantly outperforms the optimal derivative that depends on the stock only. Such derivatives yield a much higher certainty equivalent return. This result implies that investors could benefit from structured financial products constructed along these ideas. TOPICS: Derivatives, fixed income and structured finance Key Findings ▪ A derivative is optimal if it maximizes an investor’s expected utility. In the stochastic volatility model, the optimal buy-and-hold derivative with the payoff that depends on the stock price and its volatility incorporates both the market risk premium and the variance risk premium. ▪ The optimal buy-and-hold derivative with the payoff that depends on the stock price and its volatility usually outperforms significantly both the optimal buy-and-hold derivative with the payoff that depends on the stock price only and the optimal buy-and-hold portfolio made up of the stock and the risk-free bond. ▪ Investors could benefit from derivatives with payoffs that depend on the stock price and its volatility.
{"title":"Optimal Volatility Dependent Derivatives in the Stochastic Volatility Model","authors":"Artem Dyachenko,Marc Oliver Rieger","doi":"10.3905/jod.2020.1.122","DOIUrl":"https://doi.org/10.3905/jod.2020.1.122","url":null,"abstract":"We consider derivatives that maximize an investor’s expected utility in the stochastic volatility model. We show that the optimal derivative that depends on the stock and its variance significantly outperforms the optimal derivative that depends on the stock only. Such derivatives yield a much higher certainty equivalent return. This result implies that investors could benefit from structured financial products constructed along these ideas. TOPICS: Derivatives, fixed income and structured finance Key Findings ▪ A derivative is optimal if it maximizes an investor’s expected utility. In the stochastic volatility model, the optimal buy-and-hold derivative with the payoff that depends on the stock price and its volatility incorporates both the market risk premium and the variance risk premium. ▪ The optimal buy-and-hold derivative with the payoff that depends on the stock price and its volatility usually outperforms significantly both the optimal buy-and-hold derivative with the payoff that depends on the stock price only and the optimal buy-and-hold portfolio made up of the stock and the risk-free bond. ▪ Investors could benefit from derivatives with payoffs that depend on the stock price and its volatility.","PeriodicalId":501089,"journal":{"name":"The Journal of Derivatives","volume":"49 3","pages":"24-44"},"PeriodicalIF":0.0,"publicationDate":"2020-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Traditional plain vanilla options may be regarded as contingent claims whose value depends upon the simple returns of an underlying asset. These options have convex payoffs, and as a consequence of Jensen’s inequality, their prices increase as a function of maturity in the absence of interest rates. This results in long-dated call option premia being excessively expensive in relation to the fraction of a corresponding insured portfolio. We show that replacing the simple return payoff with the log return call option payoff leads to substantial premium savings while providing the similar insurance protection. Call options on log returns have favorable prices for very long maturities on the scale of decades. This property enables them to be attractive securities for long-term investors, such as pension funds. TOPICS: Options, pension funds Key Findings ▪ This article develops valuation and risk techniques for a log return payoff option under a Geometric Brownian Motion. ▪ A comparison is made between premium advantages of the log return contract to those of traditional European options. ▪ A pricing and optimal excise boundary formula for perpetual and finite maturity American log return options id derived. ▪ This article examines long-term insurance applications of the new contract that are prohibitively expensive for traditional options.
{"title":"The Premium Reduction of European, American, and Perpetual Log Return Options","authors":"Stephen Taylor,Jan Vecer","doi":"10.3905/jod.2020.1.115","DOIUrl":"https://doi.org/10.3905/jod.2020.1.115","url":null,"abstract":"Traditional plain vanilla options may be regarded as contingent claims whose value depends upon the simple returns of an underlying asset. These options have convex payoffs, and as a consequence of Jensen’s inequality, their prices increase as a function of maturity in the absence of interest rates. This results in long-dated call option premia being excessively expensive in relation to the fraction of a corresponding insured portfolio. We show that replacing the simple return payoff with the log return call option payoff leads to substantial premium savings while providing the similar insurance protection. Call options on log returns have favorable prices for very long maturities on the scale of decades. This property enables them to be attractive securities for long-term investors, such as pension funds. TOPICS: Options, pension funds Key Findings ▪ This article develops valuation and risk techniques for a log return payoff option under a Geometric Brownian Motion. ▪ A comparison is made between premium advantages of the log return contract to those of traditional European options. ▪ A pricing and optimal excise boundary formula for perpetual and finite maturity American log return options id derived. ▪ This article examines long-term insurance applications of the new contract that are prohibitively expensive for traditional options.","PeriodicalId":501089,"journal":{"name":"The Journal of Derivatives","volume":"48 8","pages":"7-23"},"PeriodicalIF":0.0,"publicationDate":"2020-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article presents a discrete-time option pricing model that is rooted in reinforcement learning (RL), and more specifically in the famous Q-Learning method of RL. We construct a risk-adjusted Markov Decision Process for a discrete-time version of the classical Black-Scholes-Merton (BSM) model, where the option price is an optimal Q-function, while the optimal hedge is a second argument of this optimal Q-function, so that both the price and hedge are parts of the same formula. Pricing is done by learning to dynamically optimize risk-adjusted returns for an option replicating portfolio, as in Markowitz portfolio theory. Using Q-Learning and related methods, once created in a parametric setting, the model can go model-free and learn to price and hedge an option directly from data, without an explicit model of the world. This suggests that RL may provide efficient data-driven and model-free methods for the optimal pricing and hedging of options. Once we depart from the academic continuous-time limit, and vice versa, option pricing methods developed in Mathematical Finance may be viewed as special cases of model-based reinforcement learning. Further, due to the simplicity and tractability of our model, which only needs basic linear algebra (plus Monte Carlo simulation, if we work with synthetic data), and its close relationship to the original BSM model, we suggest that our model could be used in the benchmarking of different RL algorithms for financial trading applications. TOPICS: Derivatives, options Key Findings • Reinforcement learning (RL) is the most natural way for pricing and hedging of options that relies directly on data and not on a specific model of asset pricing. • The discrete-time RL approach to option pricing generalizes classical continuous-time methods; enables tracking mis-hedging risk, which disappears in the formal continuous-time limit; and provides a consistent framework for using options for both hedging and speculation. • A simple quadratic reward function, which presents a minimal extension of the classical Black-Scholes framework when combined with the Q-learning method of RL, gives rise to a particularly simple computational scheme where option pricing and hedging are semianalytical, as they amount to multiple uses of a conventional least-squares regression.
{"title":"QLBS: Q-Learner in the Black-Scholes(-Merton) Worlds","authors":"Igor Halperin","doi":"10.3905/jod.2020.1.108","DOIUrl":"https://doi.org/10.3905/jod.2020.1.108","url":null,"abstract":"This article presents a discrete-time option pricing model that is rooted in reinforcement learning (RL), and more specifically in the famous Q-Learning method of RL. We construct a risk-adjusted Markov Decision Process for a discrete-time version of the classical Black-Scholes-Merton (BSM) model, where the option price is an optimal Q-function, while the optimal hedge is a second argument of this optimal Q-function, so that both the price and hedge are parts of the same formula. Pricing is done by learning to dynamically optimize risk-adjusted returns for an option replicating portfolio, as in Markowitz portfolio theory. Using Q-Learning and related methods, once created in a parametric setting, the model can go model-free and learn to price and hedge an option directly from data, without an explicit model of the world. This suggests that RL may provide efficient data-driven and model-free methods for the optimal pricing and hedging of options. Once we depart from the academic continuous-time limit, and vice versa, option pricing methods developed in Mathematical Finance may be viewed as special cases of model-based reinforcement learning. Further, due to the simplicity and tractability of our model, which only needs basic linear algebra (plus Monte Carlo simulation, if we work with synthetic data), and its close relationship to the original BSM model, we suggest that our model could be used in the benchmarking of different RL algorithms for financial trading applications. TOPICS: Derivatives, options Key Findings • Reinforcement learning (RL) is the most natural way for pricing and hedging of options that relies directly on data and not on a specific model of asset pricing. • The discrete-time RL approach to option pricing generalizes classical continuous-time methods; enables tracking mis-hedging risk, which disappears in the formal continuous-time limit; and provides a consistent framework for using options for both hedging and speculation. • A simple quadratic reward function, which presents a minimal extension of the classical Black-Scholes framework when combined with the Q-learning method of RL, gives rise to a particularly simple computational scheme where option pricing and hedging are semianalytical, as they amount to multiple uses of a conventional least-squares regression.","PeriodicalId":501089,"journal":{"name":"The Journal of Derivatives","volume":"95 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：