Abstract. An investor initially shorts a divisible American option f and dynamically trades stock S to maximize her expected utility. The investor faces the uncertainty of the exercise time of f, yet by observing the exercise time she would adjust her dynamic trading strategy accordingly. We thus investigate the robust utility maximization problem V (x) = sup(H,c) infη E[U(x+H·S−c(η(f)−p))], where H is the dynamic trading strategy for S, c represents the amount of f the investor initially shorts, η is the liquidation strategy for f, and p is the initial price of f. We mainly consider two cases: In the first case the investor shorts a fixed amount of f, i.e., w.l.o.g., c = 1 and p = 0; in the second case she statically trades f, i.e., c can be any nonnegative number. We first show that in both cases V (x) = sup(H,c) infτ E[U(x+H ·S−c(fτ −p))] = infρ sup(H,c) E[U(x + H · S − c(fρ − p))], where τ is a pure stopping time, ρ is a randomized stopping time, and H satisfies certain non-anticipation condition. Then in the first case (i.e., c = 1), we show that when U is exponential, V (x) = infτ supH E[U(x+H·S−fτ)]; for general utility this equality may fail, yet can be recovered if we in addition let τ be adapted to H in certain sense. Finally, in the second case (c ∈ [0,∞)) we obtain a duality result for the robust utility maximization on an enlarged space.
{"title":"Utility Maximization When Shorting American Options","authors":"Zhou Zhou","doi":"10.2139/ssrn.3464257","DOIUrl":"https://doi.org/10.2139/ssrn.3464257","url":null,"abstract":"Abstract. An investor initially shorts a divisible American option f and dynamically trades stock S to maximize her expected utility. The investor faces the uncertainty of the exercise time of f, yet by observing the exercise time she would adjust her dynamic trading strategy accordingly. We thus investigate the robust utility maximization problem V (x) = sup(H,c) infη E[U(x+H·S−c(η(f)−p))], where H is the dynamic trading strategy for S, c represents the amount of f the investor initially shorts, η is the liquidation strategy for f, and p is the initial price of f. We mainly consider two cases: In the first case the investor shorts a fixed amount of f, i.e., w.l.o.g., c = 1 and p = 0; in the second case she statically trades f, i.e., c can be any nonnegative number. \u0000 \u0000We first show that in both cases V (x) = sup(H,c) infτ E[U(x+H ·S−c(fτ −p))] = infρ sup(H,c) E[U(x + H · S − c(fρ − p))], where τ is a pure stopping time, ρ is a randomized stopping time, and H satisfies certain non-anticipation condition. Then in the first case (i.e., c = 1), we show that when U is exponential, V (x) = infτ supH E[U(x+H·S−fτ)]; for general utility this equality may fail, yet can be recovered if we in addition let τ be adapted to H in certain sense. Finally, in the second case (c ∈ [0,∞)) we obtain a duality result for the robust utility maximization on an enlarged space.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114604609","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}
In this short note, we describe a simple yet accurate way to set up a rate curve defined by daily forward rates that are computed as a spread over the daily forward rates of a reference rate curve. One current use case of interest is to build an Ester curve from an Eonia curve using the (constant) Ester-Eonia spread defined by the ECB (-8.5 bp). We derive error bounds and test the method with real market data in ORE resp. QuantLib.
{"title":"Daily Spread Curves and Ester","authors":"P. Caspers","doi":"10.2139/ssrn.3500090","DOIUrl":"https://doi.org/10.2139/ssrn.3500090","url":null,"abstract":"In this short note, we describe a simple yet accurate way to set up a rate curve defined by daily forward rates that are computed as a spread over the daily forward rates of a reference rate curve. One current use case of interest is to build an Ester curve from an Eonia curve using the (constant) Ester-Eonia spread defined by the ECB (-8.5 bp). We derive error bounds and test the method with real market data in ORE resp. QuantLib.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130102485","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 show how specific features of the microstructure information from VPIN and DPIN can volatile the futures market and can link with the price discover and investor sentiment. We develop an investor (institutional, noise, and both) sentiment index for the Shanghai Stock Exchange 50 (SSE 50) Index Futures, and analyze relations among the index futures return, the investor sentiment, VPIN and DPIN, illiquidity, and volatility. We first specify the informed investor sentiment index for traders who invest based on market information and uninformed investor sentiment index for irrational noise traders who provide market liquidity. Empirically, the VPIN and the investor sentiment can predict the SSE 50 futures returns in a low frequency environment, and there is a significantly negative correlation between the informed transaction and the next level of liquidity in a high frequency environment. We also show that the futures market is relatively stable under moderate investor sentiment, and the trading volume can correspond to both investor sentiment and liquidity levels.
{"title":"Investor Sentiment and Microstructure Information in Index Futures Markets","authors":"Weiping Li, Liu Wen Wen","doi":"10.2139/ssrn.3459011","DOIUrl":"https://doi.org/10.2139/ssrn.3459011","url":null,"abstract":"We show how specific features of the microstructure information from VPIN and DPIN can volatile the futures market and can link with the price discover and investor sentiment. We develop an investor (institutional, noise, and both) sentiment index for the Shanghai Stock Exchange 50 (SSE 50) Index Futures, and analyze relations among the index futures return, the investor sentiment, VPIN and DPIN, illiquidity, and volatility. We first specify the informed investor sentiment index for traders who invest based on market information and uninformed investor sentiment index for irrational noise traders who provide market liquidity. Empirically, the VPIN and the investor sentiment can predict the SSE 50 futures returns in a low frequency environment, and there is a significantly negative correlation between the informed transaction and the next level of liquidity in a high frequency environment. We also show that the futures market is relatively stable under moderate investor sentiment, and the trading volume can correspond to both investor sentiment and liquidity levels.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"154 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124043028","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}
Under the recent negative interest rate situation, the Bachelier model has been attracting attention and adopted for evaluating the price of interest rate options. In this paper, we will derive an option pricing formula based on the Bachelier model and compare it with the prior researches. We will derive it by eight methods and clarify the property of the Bachelier model. Then we will confirm the validity of the Normal model that is actually used in the valuation of interest rate options under negative interest rate, while comparing it with the Bachelier model for stocks. We start from the natural setting of modeling the undiscounted stock price by the Ornstein=Uhlenbeck process, and derive the Bachelier formula in consideration of discount. On the other hand, since the major prior researches start from modeling the discounted stock price by the Brownian motion, their models of the undiscounted stock price has an unnatural setting that the price of the numeraire asset is included. Furthermore, It has been confirmed that their formulas are not consistent among them. During the derivation process, we have obtained various results concerning the Bachelier model. In particular, in the case of the Bachelier model, it has been confirmed that the utility function of a representative agent is the CARA utility function unlike the Black-Scholes model. The assumption of the exponential type utility function is quite natural setting. In addition, we have derived other expressions of the Bachelier's formula (the formula decomposed into the intrinsic value and the time value and the formula using a characteristic function). As for the Normal model used for pricing interest rate options, we have derived an original pricing formula (Modified Normal model) in which the unnatural points of the Normal model of the forward LIBOR and forward swap rate have been partially corrected.
{"title":"On the Option Pricing Formula Based on the Bachelier Model","authors":"Satoshi Terakado","doi":"10.2139/ssrn.3428994","DOIUrl":"https://doi.org/10.2139/ssrn.3428994","url":null,"abstract":"Under the recent negative interest rate situation, the Bachelier model has been attracting attention and adopted for evaluating the price of interest rate options. In this paper, we will derive an option pricing formula based on the Bachelier model and compare it with the prior researches. We will derive it by eight methods and clarify the property of the Bachelier model. \u0000 \u0000Then we will confirm the validity of the Normal model that is actually used in the valuation of interest rate options under negative interest rate, while comparing it with the Bachelier model for stocks. We start from the natural setting of modeling the undiscounted stock price by the Ornstein=Uhlenbeck process, and derive the Bachelier formula in consideration of discount. \u0000 \u0000On the other hand, since the major prior researches start from modeling the discounted stock price by the Brownian motion, their models of the undiscounted stock price has an unnatural setting that the price of the numeraire asset is included. Furthermore, It has been confirmed that their formulas are not consistent among them. During the derivation process, we have obtained various results concerning the Bachelier model. In particular, in the case of the Bachelier model, it has been confirmed that the utility function of a representative agent is the CARA utility function unlike the Black-Scholes model. The assumption of the exponential type utility function is quite natural setting. In addition, we have derived other expressions of the Bachelier's formula (the formula decomposed into the intrinsic value and the time value and the formula using a characteristic function). As for the Normal model used for pricing interest rate options, we have derived an original pricing formula (Modified Normal model) in which the unnatural points of the Normal model of the forward LIBOR and forward swap rate have been partially corrected.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114615627","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 present some formulations of the Cox-Ross-Rubinstein and Black-Scholes formulas for European options obtained through a suitable change of measure, which corresponds to a change of numèraire for the underlying price process. Among other consequences, a closed formula for the price of an European call option at each node of the multi-period binomial tree is achieved, too. Some of the results contained herein, though comparable with analogous ones appearing elsewhere in the financial literature, provide however a supplementary widening and deepening in view of useful applications in the more challenging framework of incomplete markets. This last issue, having the present paper as a preparatory material, will be treated extensively in a forthcoming paper.
{"title":"Option Pricing Formulas Under a Change of Numèraire","authors":"Antonio Attalienti, Michele Bufalo","doi":"10.2139/ssrn.3451116","DOIUrl":"https://doi.org/10.2139/ssrn.3451116","url":null,"abstract":"We present some formulations of the Cox-Ross-Rubinstein and Black-Scholes formulas for European options obtained through a suitable change of measure, which corresponds to a change of numèraire for the underlying price process. Among other consequences, a closed formula for the price of an European call option at each node of the multi-period binomial tree is achieved, too. Some of the results contained herein, though comparable with analogous ones appearing elsewhere in the financial literature, provide however a supplementary widening and deepening in view of useful applications in the more challenging framework of incomplete markets. This last issue, having the present paper as a preparatory material, will be treated extensively in a forthcoming paper.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127243399","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}
I examine pricing of credit securities after a credit event for a sample of rms on which CDS are traded. Secondary market prices of bonds along with those discovered at Credit Event Auctions are estimates of terminal or ultimate recovery on these securities. I use hand-collected data on ultimate recovery to jointly test for bias in prices at the auction and in secondary markets. I find that ultimate recovery is mispriced. Credit Event Auctions are biased in a manner consistent with theory and generate prices that, on average, underestimate ultimate recovery resulting in higher payouts to credit protection buyers. Moreover, bond prices in secondary markets are more informed about ultimate recovery before the auction than after it suggesting that existence of open CDS positions enriches the information environment for these bonds.
{"title":"Pricing Recovery - Evidence from Markets, CDS Auctions and Ultimate Recovery","authors":"Sunil Teluja","doi":"10.2139/ssrn.3239797","DOIUrl":"https://doi.org/10.2139/ssrn.3239797","url":null,"abstract":"I examine pricing of credit securities after a credit event for a sample of rms on which CDS are traded. Secondary market prices of bonds along with those discovered at Credit Event Auctions are estimates of terminal or ultimate recovery on these securities. I use hand-collected data on ultimate recovery to jointly test for bias in prices at the auction and in secondary markets. I find that ultimate recovery is mispriced. Credit Event Auctions are biased in a manner consistent with theory and generate prices that, on average, underestimate ultimate recovery resulting in higher payouts to credit protection buyers. Moreover, bond prices in secondary markets are more informed about ultimate recovery before the auction than after it suggesting that existence of open CDS positions enriches the information environment for these bonds.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126943360","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 paper analyzes optimal hedge ratios for foreign exchange (FX) rate risk of companies. Our contribution to the literature is twofold: (i) We present a theoretical two-period regret model that allows us to analyze the determinants of the optimal hedge ratio given the outcome of past hedging decisions and future expectations. The model implies that the optimal hedge ratio depends on the past hedge ratio, the past exchange rate return, the expected exchange rate return and the skewness of its distribution, its covariance to the foreign market return, as well as the company's risk and regret aversion. (ii) We test the related model-derived hypotheses on a broad sample of US non-financial companies over the period 1995 to 2015 and find strong evidence for the model's predictions. By adding a dynamic regret approach to the hedging and FX literature we shed further light on the rationale behind selective hedging.
{"title":"Foreign Exchange Rate Exposure of Companies under Dynamic Regret","authors":"Oliver Entrop, Fabian U. Fuchs","doi":"10.2139/ssrn.3443487","DOIUrl":"https://doi.org/10.2139/ssrn.3443487","url":null,"abstract":"This paper analyzes optimal hedge ratios for foreign exchange (FX) rate risk of companies. Our contribution to the literature is twofold: (i) We present a theoretical two-period regret model that allows us to analyze the determinants of the optimal hedge ratio given the outcome of past hedging decisions and future expectations. The model implies that the optimal hedge ratio depends on the past hedge ratio, the past exchange rate return, the expected exchange rate return and the skewness of its distribution, its covariance to the foreign market return, as well as the company's risk and regret aversion. (ii) We test the related model-derived hypotheses on a broad sample of US non-financial companies over the period 1995 to 2015 and find strong evidence for the model's predictions. By adding a dynamic regret approach to the hedging and FX literature we shed further light on the rationale behind selective hedging.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131933339","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 estimate investor disagreement from synthetic long and short stock trades in the equity options market. We show that high disagreement predicts low stock returns after positive earnings surprises and high stock returns after negative earnings surprises. The negative effect is stronger for high-beta stocks and stocks that are more difficult to sell short. In the cross-section of all stocks and the subset of the 500 largest companies, high disagreement robustly predicts low monthly and weekly stock returns.
{"title":"Disagreement in the Equity Options Market and Stock Returns","authors":"Benjamin Golez, Ruslan Goyenko","doi":"10.2139/ssrn.3443241","DOIUrl":"https://doi.org/10.2139/ssrn.3443241","url":null,"abstract":"\u0000 We estimate investor disagreement from synthetic long and short stock trades in the equity options market. We show that high disagreement predicts low stock returns after positive earnings surprises and high stock returns after negative earnings surprises. The negative effect is stronger for high-beta stocks and stocks that are more difficult to sell short. In the cross-section of all stocks and the subset of the 500 largest companies, high disagreement robustly predicts low monthly and weekly stock returns.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124941339","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}
Papers treating variance swap replication often mention that the replicating portfolio consists of a static position in an appropriately weighted continuous strip of options, and a dynamic position in the underlying asset that can be regarded as the delta-hedge of the strip of options. Most papers, however, do not explicate the impact of delta-hedging the options, and in particular do not mention what volatility to use when delta-hedging the options. Although no new results are derived, in this educational note we clarify the aforementioned two points.
{"title":"Delta-Hedging and Variance Swap Replication","authors":"Frido Rolloos","doi":"10.2139/ssrn.3442808","DOIUrl":"https://doi.org/10.2139/ssrn.3442808","url":null,"abstract":"Papers treating variance swap replication often mention that the replicating portfolio consists of a static position in an appropriately weighted continuous strip of options, and a dynamic position in the underlying asset that can be regarded as the delta-hedge of the strip of options. Most papers, however, do not explicate the impact of delta-hedging the options, and in particular do not mention what volatility to use when delta-hedging the options. Although no new results are derived, in this educational note we clarify the aforementioned two points.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126558298","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}
While the standard to calculate model-free option-implied skewness (MFIS) relies on out-of-the-money (OTM) options, we examine the empirical implications of using in-the-money (ITM) options. First, we show that discarding ITM-options based on liquidity arguments appears unreasonable for individual stock options. Second, we show that the information content of ITM-options provides new economic insights. The positive short-term return predictability of OTM-based MFIS significantly reverses if ITM-options are used instead. This return pattern allows to better attribute the return predictability of MFIS to superior information of investors embedded in option prices rather than skewness preferences. Based on these findings, we introduce a new measure of sophisticated option trading called Delta-MFIS.
{"title":"The Information Content of ITM-Options for Risk-Neutral Skewness and Informed Trading","authors":"Hannes Mohrschladt, Judith C. Schneider","doi":"10.2139/ssrn.3439906","DOIUrl":"https://doi.org/10.2139/ssrn.3439906","url":null,"abstract":"While the standard to calculate model-free option-implied skewness (MFIS) relies on out-of-the-money (OTM) options, we examine the empirical implications of using in-the-money (ITM) options. First, we show that discarding ITM-options based on liquidity arguments appears unreasonable for individual stock options. Second, we show that the information content of ITM-options provides new economic insights. The positive short-term return predictability of OTM-based MFIS significantly reverses if ITM-options are used instead. This return pattern allows to better attribute the return predictability of MFIS to superior information of investors embedded in option prices rather than skewness preferences. Based on these findings, we introduce a new measure of sophisticated option trading called Delta-MFIS.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126854370","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}
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