A closed-form exact solution for pricing fixed-income variance swaps with affine-jump model

Fixed-income variance swaps became popular for investors to trade and hedge the fluctuation of interest rates after the recent global financial crisis over the past few decades, however, their valuations and risk management have not been studied sufficiently. This paper presents an analytic approach for pricing some discretely sampled fixed-income variance swaps under an affine-jump model with stochastic mean, stochastic volatility, and jumps. We employ a generalized characteristic function to derive the closed-form pricing formulas of these swaps, including two kinds of zero-coupon bond variance swap, Libor variance swap, and bond yield variance swap, to be precise. We also perform some numerical studies based on these models, which suggest that the fair strike values of these variance swaps are within a reasonable range regardless of estimation risk with data dependence and near-zero short rate regime. Our numerics show that the influences of varying sampling frequency and time-to-maturity on the values of these swaps are significant, and highlight the risks of specifying short rate model. Furthermore, the sensitivity analysis on the key parameters finds that the risks of stochastic volatility and jumps play prominent roles in pricing these variance swaps under the near-zero short rate regime.     

Identifying contagion

Identifying contagion effects during periods of financial crisis is known to be complicated by the changing volatility of asset returns during periods of stress. To untangle this we propose a GARCH (generalized autoregressive conditional heteroskedasticity) common features approach, where systemic risk emerges from a common factor source (or indeed multiple factor sources) with contagion evident through possible changes in the factor loadings relating to the common factor(s). Within a portfolio mimicking factor framework this can be identified using moment conditions. We use this framework to identify contagion in three illustrations involving both single and multiple factor specifications: to the Asian currency markets in 1997–1998, to US sectoral equity indices in 2007–2009 and to the CDS (credit default swap) market during the European sovereign debt crisis of 2010–2013. The results reveal the extent to which contagion effects may be masked by not accounting for the sources of changed volatility apparent in simple measures such as correlation.     

Pricing currency options with scheduled and unscheduled announcement effects on volatility

It is well documented that exchange rate volatility is time-varying and that it can be affected by scheduled events such as money supply announcements and unscheduled ones such as spot market interventions and interest rate changes. This study provides a European event model (E model) for currency call options that explicitly addresses the volatility effects of these two classes of events. Managers who are concerned with hedging in an environment of changing volatility may find the E model useful. The E and modified Black-Scholes (MBS) models have similar average errors in predicting option price changes across event windows and do better than a naive no-change prediction. The E model tends to reduce the underpricing of convex, short-term out-of-the-money options and the mispricing of most classes of convex options.     

Embedded options and interest rate risk for insurance companies,banks and other financial institutions

We explore how embedded options in assets and liabilities of financial institutions impact interest rate risk, which is measured by equity value change with interest rate movements. We find that both asset and liability durations decline when embedded options are present where liability duration declines more substantially. This leads to a duration-mismatch and a negative change in equity value when interest rates rise, but a positive change for interest rate declines. In a more sophisticated model, an option adjusted duration-matching strategy eliminates interest rate risk caused by duration-mismatch, but the convexity-mismatch remains large due to large negative convexity of callable assets and large positive convexity of putable liabilities. The interest rate risk introduced by convexity mismatch is quite large in comparison to that of duration mismatch. The pattern of this impact is complex and strongest for roughly intermediate maturities. We propose and show that a simple convexity hedging strategy with putable assets and callable liabilities (or caps and floors combined with floating rate assets and liabilities) reduces the interest rate risk substantially.     

Premia for correlated default risk

Using data on corporate default experience in the U.S. and market rates of CDX index and tranche swaps of various maturities, we estimate reduced-form models of correlated default timing in the CDX High Yield and Investment Grade portfolios under actual and risk-neutral probabilities. The striking contrast between the estimated processes followed by the actual and risk-neutral arrival intensities of defaults, and between the parameters governing the actual and risk-neutral dynamics of the risk-neutral intensities, indicates the presence of substantial default risk premia in CDX swap market rates. The effects of risk premia on swap rates covary strongly across maturities, and depend on general stock market volatility and several measures of credit spreads. Large moves in the effects of these premia on swap rates have natural interpretations in terms of economic and financial market developments during the sample period, April 2004 to October 2007. Our results suggest that a large portion of the movements in CDX swap market rates observed during the sample period may be caused by changing attitudes toward correlated default risk rather than changes in the economic factors affecting the actual risk of clustered defaults, which ultimately governs swap payoffs.     

Do interest rate options contain information about excess returns?

There is strong empirical evidence that long-term interest rates contain a time-varying risk premium. Options may contain valuable information about this risk premium because their prices are sensitive to the underlying interest rates. We use the joint time series of swap rates and interest rate option prices to estimate dynamic term structure models. The risk premiums that we estimate using option prices are better able to predict excess returns for long-term swaps over short-term swaps. Moreover, in contrast to the previous literature, the most successful models for predicting excess returns have risk factors with stochastic volatility. We also show that the stochastic volatility models we estimate using option prices match the failure of the expectations hypothesis.     

Pricing Range Accrual Interest Rate Swap employing LIBOR market models with jump risks

This research derives the LIBOR market model with jump risks, assuming that interest rates follow a continuous time path and tend to jump in response to sudden economic shocks. We then use the LIBOR model with jump risk to price a Range Accrual Interest Rate Swap (RAIRS). Given that the multiple jump processes are independent, we employ numerical analysis to further demonstrate the influence of jump size, jump volatility, and jump frequency on the pricing of RAIRS. Our results show a negative relation between jump size, jump frequency, and the swap rate of RAIRS, but a positive relation between jump volatility and the swap rate of RAIRS.     

Implications of Executive Hedge Markets for Firm Value Maximization

This paper analyzes the incentive implications of executive hedge markets. The manager can promise the return from his shares to third parties in exchange for a fixed payment—swap contracts—and/or he can trade a customized security correlated with his firm-specific risk. The customized security improves incentives by diversifying the manager's firm-specific risk. However, unless they are exclusive, swap contracts lead to a complete unraveling of incentives. When security customization is sufficiently high, the manager only trades the customized security—but not any nonexclusive swap contracts, and incentives improve. Access to highly customized hedge securities and/or exclusive swap contracts increases the manager's pay-performance sensitivity .     

Conditional value-at-risk forecasts of an optimal foreign currency portfolio

This study provides daily conditional value-at-risk (C-VaR) forecasts for a foreign currency portfolio comprising the USD/EUR, USD/JPY, and USD/BRL currencies. To do so, we estimate multivariate stochastic volatility models with time-varying conditional correlations using a Bayesian Markov chain Monte Carlo algorithm. Then, given the model-specific currency return density forecasts, we make the optimal portfolio choice by minimizing the C-VaR through numerical optimization. According to out-of-sample experiment, including emerging markets into the currency basket is essential for downside risk management, and considering model uncertainty as well as the parameter uncertainty can improve the portfolio performance.     

Region-wide connectedness of Asian equity and currency markets

This study systemically analyzes the dynamics of interdependence between the Asian equity and currency markets. The novelty of our study is that unlike other studies that explore either co-movements among equity markets or co-movements among currency markets, we pay particular attention to the interdependence between the two in terms of both return and volatility connectedness. We find that the contribution of crossspillovers between the Asian equities and currencies is substantial for the region-wide connectedness of both the returns and volatilities. We also find that the short-term spillovers are far more important for the return spillovers, while the long-term spillovers are far more important for the volatility spillovers, presumably reflecting the long-lasting effects of volatility shocks. All the results consistently underline the pivotal role of cross-interdependence between equity and currency markets, both as channels for integrating Asian financial markets and as sources of financial contagion across these markets. Our findings will provide useful guidance for portfolio risk management to adopt better hedging strategies for foreign exchange risks involved in the international investment of Asian equities.     

Option pricing with discrete time jump processes

In this paper we propose new option pricing models based on class of models with jumps contained in the Lévy-type based models (NIG-Lévy, Schoutens, 2003, Merton-jump, Merton, 1976 and Duan based model, Duan et al., 2007). By combining these different classes of models with several volatility dynamics of the GARCH type, we aim at taking into account the dynamics of financial returns in a realistic way. The associated risk neutral dynamics of the time series models is obtained through two different specifications for the pricing kernel: we provide a characterization of the change in the probability measure using the Esscher transform and the Minimal Entropy Martingale Measure. We finally assess empirically the performance of this modelling approach, using a dataset of European options based on the S&P 500 and on the CAC 40 indices. Our results show that models involving jumps and a time varying volatility provide realistic pricing and hedging results for options with different kinds of time to maturities and moneyness. These results are supportive of the idea that a realistic time series model can provide realistic option prices making the approach developed here interesting to price options when option markets are illiquid or when such markets simply do not exist.     

Volatility estimation from observed option prices

It is well established that the standard Black-Scholes model does a very poor job in matching the prices of vanilla European options. The implied volatility varies by both time to maturity and by the moneyness of the option. One approach to this problem is to use the market option prices to back out a local volatility function that reproduces the market prices. Since option price observations are only available for a limited set of maturities and strike prices, most algorithms require a smoothing technique to implement this approach. In this paper we modify the implementation of Andersen and Brotherton-Ratcliffe to provide another way of dealing with this issue. Numerical examples indicate that our approach is reasonably successful in reproducing the input prices.     

Volatility forecasting in European government bond markets

In this paper we examine the predictive power of the heterogeneous autoregressive (HAR) model for the return volatility of major European government bond markets. The results from HAR-type volatility forecasting models show that past short- and medium-term volatility are significant predictors of the term structure of the intraday volatility of European bonds with maturities ranging from 1 year up to 30 years. When we decompose bond market volatility into its continuous and discontinuous (jump) component, we find that the jump component is a significant predictor. Moreover, we show that feedback from past short-term volatility to forecasts of future volatility is stronger in the days that precede monetary policy announcements.     

Modeling common volatility characteristics and dynamic risk premia in European equity markets

The paper applies a Factor-GARCH model to evaluate the impact of the market portfolio, as a single common dynamic risk factor, on conditional volatility and risk premia for the returns on size-based equity portfolios of three major European markets; France, Germany and the United Kingdom. The results show that for the size-based portfolios the factor loading for the dynamic market factor is significant and positive but the association between the risk premia and the conditional market volatility is weak. However, the dynamic market factor is shown to explain common characteristics in the conditional variance such as asymmetry and persistence. This finding is consistent across markets and portfolio sizes.     

The valuation of nature-linked bonds with exchange rate risk

This paper develops an arbitrage approach to pricing insurance bonds that bear currency risk. Bondholders are shown to have a short position on path-dependent digital options written on risk-tracking indices. It implements the technique of forward-neutral change of numeraire and comes down to computing first-passage-time distributions of drifted Brownian motions. We derive closed-form formulas or perform simulations depending on whether interest rates are deterministic or stochastic. Along the way, we evaluate outside-barrier currency call options. Then this research studies the effects of both nature risk and exchange rate uncertainty and shows that the former is more significant than the latter.     

Valuation of callable accreting interest rate swaps: Least squares Monte-Carlo method under Hull-White interest rate model

Using the Hull-White interest rate model, this paper proposes a valuation method of callable accreting interest rate swap (CAIRS) and how it can be used for managing the risk of zero callable bonds (ZCBs). Firstly, CAIRS can be decomposed into accreting payer interest rate swaps and Bermudan options. Considering the financial valuation of both components, the former can be valued directly while the latter has no close-form due to its early exercise characteristics. Using the Least Squares Monte-Carlo method (LSM) proposed by Longstaff and Schwartz (2001), we find that the two options embedded in ZCB and CAIRS have the same exercise strategy since the terms of the swaps will include the bonds in practice. However, the cash flow of risk management in swaps and bonds can differ when considering the time value. Hence, CAIRS is not the best financial instrument for managing risks of ZCB under the current design.     

Evidence of information transmission across currency futures markets using frequency domain tests

In this article, I examine the returns and volatility spillovers in the currency futures market incorporating the recently developed frequency domain tests. Such analysis allows differentiating between permanent (long-run) and transitory (short-run) linkages among the currency futures markets by investigating the causality dynamics at low and high frequencies respectively. I detect significant informational linkages between USD, EUR, GBP and JPY futures contracts in the Indian currency futures market. Evidence of innovations from USD futures market to other markets is the most significant for returns spillover and for volatility spillover, EUR is found to be the most significant compared to other currency futures contracts. The results would have implications for the market participants and policymakers.     

Modelling the conditional volatility of commodity index futures as a regime switching process

Commodity index futures offer a versatile tool for gaining different forms of exposure to commodity markets. Volatility is a critical input in many of these applications. This paper examines issues in modelling the conditional variance of futures returns based on the Goldman Sachs Commodity Index (GSCI). Given that commodity markets tend to be ‘choppy’ (Webb, 1987 ), a general econometric model is proposed that allows for abrupt changes or regime shifts in volatility, transition probabilities which vary explicitly with observable fundamentals such as the basis, GARCH dynamics, seasonal variations and conditional leptokurtosis. The model is applied to daily futures returns on the GSCI over 1992–1997. The results show clear evidence of regime shifts in conditional mean and volatility. Once regime shifts are accounted for, GARCH effects are minimal. Consistent with the theory of storage, returns are more likely to switch to the high‐variance state when the basis is negative than when the basis is positive. The regime switching model also performs well in forecasting the daily volatility compared to standard GARCH models without regime switches. The model should be of interest to sophisticated traders who base their trading strategies on short‐term volatility movements, managed commodity funds interested in hedging an underlying diversified portfolio of commodities and investors of options and other derivatives tied to GSCI futures contracts. Copyright © 2001 John Wiley & Sons, Ltd.     

国际金融市场波动溢出效应与动态相关性

在DCC GARCH、DCC EGARCH、DCC TGARCH方法下,采用中、美、日、德、英等国家1993年1月至2013年12月的金融数据,实证得出如下结论：样本国市场利率和股指波动率呈现尖峰、肥尾、有偏的特征,更为符合t分布。样本国市场利率波动表现出显著的溢出效应、杠杆效应和联动效应。样本国股指波动率对中国股指波动率的溢出效应趋于增强,特别在美国金融危机后。样本国利率波动对中国股指波动率具有一定的溢出效应和杠杆效应,但影响程度非常低。治理世界性金融风险,各国当局应加强政策协调性,合理进行风险分担。     

Volatility of exchange rate futures and high-low price spreads

This study explores the time-series behavior and the predictability of daily percentage changes in the Japanese Yen futures contracts. The relationship between currency futures volatility and high-low price spreads in the Japanese Yen futures contracts is examined. In addition, this study explores the issue of first- and second-order dependencies in the Japanese Yen futures contract prices changes, address the issue of asymmetric volatility, and examine the extent to which the information contained in the high-low price spreads can be used to predict future Japanese Yen currency futures contract price changes. The analysis is carried out using the EGARCH model. The volatility of the Japanese Yen currency futures price changes is adequately modeled by an EGARCH process and is predictable using information contained in the high-low price spread variables constructed in this study. This study also finds a positive and significant relationship between the spread variable and the conditional mean of price changes, suggesting that current information contained in the spread variable can be used to predict future Japanese Yen currency futures contract price changes. The hypothesis that volatility is an asymmetric function of past innovations is confirmed.