Pricing currency options under two-factor Markov-modulated stochastic volatility models

This article investigates the valuation of currency options when the dynamic of the spot Foreign Exchange (FX) rate is governed by a two-factor Markov-modulated stochastic volatility model, with the first stochastic volatility component driven by a lognormal diffusion process and the second independent stochastic volatility component driven by a continuous-time finite-state Markov chain model. The states of the Markov chain can be interpreted as the states of an economy. We employ the regime-switching Esscher transform to determine a martingale pricing measure for valuing currency options under the incomplete market setting. We consider the valuation of the European-style and American-style currency options. In the case of American options, we provide a decomposition result for the American option price into the sum of its European counterpart and the early exercise premium. Numerical results are included.     

Analysis of VIX Markets with a Time-Spread Portfolio

This paper explores the relationship between option markets for the S&P500 (SPX) and Chicago Board Options Exchange’s CBOE’s Volatility Index (VIX). Results are obtained by using the so-called time-spread portfolio to replicate a future contract on the squared VIX. The time-spread portfolio is interesting because it provides a model-free link between derivative prices for SPX and VIX. Time spreads can be computed from SPX put options with different maturities, which results in a term structure for squared volatility. This term structure can be compared to the VIX-squared term structure that is backed-out from VIX call options. The time-spread portfolio is also used to measure volatility-of-volatility (vol-of-vol) and the volatility leverage effect. There may emerge small differences in these measurements, depending on whether time spreads are computed with options on SPX or options on VIX. A study of 2012 daily options data shows that vol-of-vol estimates utilizing SPX data will reflect the volatility leverage effect, whereas estimates that exclusively utilize VIX options will predominantly reflect the premia in the VIX-future term structure.     

Implied Filtering Densities on the Hidden State of Stochastic Volatility

Abstract

We formulate and analyse an inverse problem using derivative prices to obtain an implied filtering density on volatility’s hidden state. Stochastic volatility is the unobserved state in a hidden Markov model (HMM) and can be tracked using Bayesian filtering. However, derivative data can be considered as conditional expectations that are already observed in the market, and which can be used as input to an inverse problem whose solution is an implied conditional density on volatility. Our analysis relies on a specification of the martingale change of measure, which we refer to as separability. This specification has a multiplicative component that behaves like a risk premium on volatility uncertainty in the market. When applied to SPX options data, the estimated model and implied densities produce variance-swap rates that are consistent with the VIX volatility index. The implied densities are relatively stable over time and pick up some of the monthly effects that occur due to the options’ expiration, indicating that the volatility-uncertainty premium could experience cyclic effects due to the maturity date of the options.     

Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility

We consider the pricing of long-dated insurance contracts under stochastic interest rates and stochastic volatility. In particular, we focus on the valuation of insurance options with long-term equity or foreign exchange exposures. Our modeling framework extends the stochastic volatility model of Schöbel and Zhu (1999) by including stochastic interest rates. Moreover, we allow all driving model factors to be instantaneously correlated with each other, i.e. we allow for a general correlation structure between the instantaneous interest rates, the volatilities and the underlying stock returns. As insurance products often incorporate long-term exposures, they are typically more sensitive to changes in the interest rates, volatility and currencies. Therefore, having the flexibility to correlate the underlying asset price with both the stochastic volatility and the stochastic interest rates, yields a realistic model which is of practical importance for the pricing and hedging of such long-term contracts. We show that European options, typically used for the calibration of the model to market prices, and forward starting options can be priced efficiently and in closed-form by means of Fourier inversion techniques. We extensively discuss the numerical implementation of these pricing formulas, allowing for a fast and accurate valuation of European and forward starting options. The model will be especially useful for the pricing and risk management of insurance contracts and other exotic derivatives involving long-term maturities.     

Option pricing with mean reversion and stochastic volatility

Many underlying assets of option contracts, such as currencies, commodities, energy, temperature and even some stocks, exhibit both mean reversion and stochastic volatility. This paper investigates the valuation of options when the underlying asset follows a mean-reverting lognormal process with stochastic volatility. A closed-form solution is derived for European options by means of Fourier transform. The proposed model allows the option pricing formula to capture both the term structure of futures prices and the market implied volatility smile within a unified framework. A bivariate trinomial lattice approach is introduced to value path-dependent options with the proposed model. Numerical examples using European options, American options and barrier options demonstrate the use of the model and the quality of the numerical scheme.     

Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility

We consider the pricing of long-dated insurance contracts under stochastic interest rates and stochastic volatility. In particular, we focus on the valuation of insurance options with long-term equity or foreign exchange exposures. Our modeling framework extends the stochastic volatility model of Schöbel and Zhu (1999) by including stochastic interest rates. Moreover, we allow all driving model factors to be instantaneously correlated with each other, i.e. we allow for a general correlation structure between the instantaneous interest rates, the volatilities and the underlying stock returns. As insurance products often incorporate long-term exposures, they are typically more sensitive to changes in the interest rates, volatility and currencies. Therefore, having the flexibility to correlate the underlying asset price with both the stochastic volatility and the stochastic interest rates, yields a realistic model which is of practical importance for the pricing and hedging of such long-term contracts. We show that European options, typically used for the calibration of the model to market prices, and forward starting options can be priced efficiently and in closed-form by means of Fourier inversion techniques. We extensively discuss the numerical implementation of these pricing formulas, allowing for a fast and accurate valuation of European and forward starting options. The model will be especially useful for the pricing and risk management of insurance contracts and other exotic derivatives involving long-term maturities.     

Development and calibration of a currency trading strategy using global optimization

We have developed a new financial indicator—called the Interest Rate Differentials Adjusted for Volatility (IRDAV) measure—to assist investors in currency markets. On a monthly basis, we rank currency pairs according to this measure and then select a basket of pairs with the highest IRDAV values. Under positive market conditions, an IRDAV based investment strategy (buying a currency with high interest rate and simultaneously selling a currency with low interest rate, after adjusting for volatility of the currency pairs in question) can generate significant returns. However, when the markets turn for the worse and crisis situations evolve, investors exit such money-making strategies suddenly, and—as a result—significant losses can occur. In an effort to minimize these potential losses, we also propose an aggregated Risk Metric that estimates the total risk by looking at various financial indicators across different markets. These risk indicators are used to get timely signals of evolving crises and to flip the strategy from long to short in a timely fashion, to prevent losses and make further gains even during crisis periods. Since our proprietary model is implemented in Excel as a highly nonlinear “black box” computational procedure, we use suitable global optimization methodology and software—the Lipschitz Global Optimizer solver suite linked to Excel—to maximize the performance of the currency basket, based on our selection of key decision variables. After the introduction of the new currency trading model and its implementation, we present numerical results based on actual market data. Our results clearly show the advantages of using global optimization based parameter settings, compared to the typically used “expert estimates” of the key model parameters.     

Operator trigonometry of multivariate finance

We inquire into an operator-trigonometric analysis of certain multi-asset financial pricing models. Our goal is to provide a new geometric point of view for the understanding and analysis of such financial instruments. Among those instruments which we examine are quantos for currency hedging, spread options for multi-asset pricing, portfolio rebalancing under stochastic interest rates, Black-Scholes volatility models, and risk measures.     

Calibrating the Black-Derman-Toy model: some theoretical results

The Black-Derman-Toy (BDT) model is a popular one-factor interest rate model that is widely used by practitioners. One of its advantages is that the model can be calibrated to both the current market term structure of interest rate and the current term structure of volatilities. The input term structure of volatility can be either the short term volatility or the yield volatility. Sandmann and Sondermann derived conditions for the calibration to be feasible when the conditional short rate volatility is used. In this paper conditions are investigated under which calibration to the yield volatility is feasible. Mathematical conditions for this to happen are derived. The restrictions in this case are more complicated than when the short rate volatilities are used since the calibration at each time step now involves the solution of two non-linear equations. The theoretical results are illustrated by showing numerically that in certain situations the calibration based on the yield volatility breaks down for apparently plausible inputs. In implementing the calibration from period n to period n + 1, the corresponding yield volatility has to lie within certain bounds. Under certain circumstances these bounds become very tight. For yield volatilities that violate these bounds, the computed short rates for the period (n, n + 1) either become negative or else explode and this feature corresponds to the economic intuition behind the breakdown.     

Rare Shock,Two-Factor Stochastic Volatility and Currency Option Pricing

Abstract

In this paper, we develop an option valuation model where the dynamics of the spot foreign exchange rate is governed by a two-factor Markov-modulated jump-diffusion process. The short-term fluctuation of stochastic volatility is driven by a Cox–Ingersoll–Ross (CIR) process and the long-term variation of stochastic volatility is driven by a continuous-time Markov chain which can be interpreted as economy states. Rare events are governed by a compound Poisson process with log-normal jump amplitude and stochastic jump intensity is modulated by a common continuous-time Markov chain. Since the market is incomplete under regime-switching assumptions, we determine a risk-neutral martingale measure via the Esscher transform and then give a pricing formula of currency options. Numerical results are presented for investigating the impact of the long-term volatility and the annual jump intensity on option prices.     

Markov-modulated jump-diffusions for currency option pricing

This paper introduces dynamic models for the spot foreign exchange rate with capturing both the rare events and the time-inhomogeneity in the fluctuating currency market. For the rare events, we use a compound Poisson process with log-normal jump amplitude to describe the jumps. As for the time-inhomogeneity in the market dynamics, we particularly stress the strong dependence of the domestic/foreign interest rates, the appreciation rate and the volatility of the foreign currency on the time-varying sovereign ratings in the currency market. The time-varying ratings are formulated by a continuous-time finite-state Markov chain. Based on such a spot foreign exchange rate dynamics, we then study the pricing of some currency options. Here we will adopt a so-called regime-switching Esscher transform to identify a risk-neutral martingale measure. By determining the regime-switching Esscher parameters we then get an integral expression on the prices of European-style currency options. Finally, numerical illustrations are given.     

基于共轭几何变换程序的国债市场利率期限结构波动建模

本文变革已有的利率期限结构模型估计依赖于定价误差平方和最小化原则,引入几何双重变换程序解决非线性约束的误差绝对距离最小化问题,丰富国债市场利率波动和定价研究的理论体系和研究方法;运用负指数平滑立方L₁样条优化模型,克服B样条函数对节点数目与定位的过度敏感和放宽对贴现函数的二阶导数平滑要求,协同拟合误差绝对距离与贴现函数波动率最小化,保留B样条函数刻画中长期利率波动趋势的优势,增强对短期利率波动结构突变的估计、定价和预测能力,缓解B样条和NSS模型在利率期限结构拟合存在的过度波动问题。     

A delayed stochastic volatility correction to the constant elasticity of variance model

The Black-Scholes model does not account non-Markovian property and volatility smile or skew although asset price might depend on the past movement of the asset price and real market data can find a non-flat structure of the implied volatility surface. So, in this paper, we formulate an underlying asset model by adding a delayed structure to the constant elasticity of variance (CEV) model that is one of renowned alternative models resolving the geometric issue. However, it is still one factor volatility model which usually does not capture full dynamics of the volatility showing discrepancy between its predicted price and market price for certain range of options. Based on this observation we combine a stochastic volatility factor with the delayed CEV structure and develop a delayed hybrid model of stochastic and local volatilities. Using both a martingale approach and a singular perturbation method, we demonstrate the delayed CEV correction effects on the European vanilla option price under this hybrid volatility model as a direct extension of our previous work [12].     

Exact volatility calibration based on a Dupire-type Call–Put duality for perpetual American options

This paper investigates the calibration of a model with a time-homogeneous local volatility function to the market prices of the perpetual American Call and Put options. The main step is the derivation of a Call–Put duality equality for perpetual American options similar to the equality which is equivalent to Dupire’s formula (Dupire in Risk 7(1):18–20, 1994) in the European case. It turns out that in addition to the simultaneous exchanges between the spot price and the strike and between the interest and dividend rates which already appear in the European case, one has to modify the local volatility function in the American case. To show this duality equality, we exhibit non-autonomous nonlinear ODEs satisfied by the perpetual Call and Put exercise boundaries as functions of the strike variable. We obtain uniqueness for these ODEs and deduce that the mapping associating the exercise boundary with the local volatility function is one-to-one onto. Thanks to this Dupire-type duality result, we design a theoretical calibration procedure of the local volatility function from the perpetual Call and Put prices for a fixed spot price x ₀. The knowledge of the Put (resp. Call) prices for all strikes enables to recover the local volatility function on the interval (0, x ₀) (resp. (x ₀, +∞)). We last prove that equality of the dual volatility functions only holds in the standard Black-Scholes model with constant volatility.      

The American Foreign Exchange Option in Time-Dependent One-Dimensional Diffusion Model for Exchange Rate

The classical Garman-Kohlhagen model for the currency exchange assumes that the domestic and foreign currency risk-free interest rates are constant and the exchange rate follows a log-normal diffusion process. In this paper we consider the general case, when exchange rate evolves according to arbitrary one-dimensional diffusion process with local volatility that is the function of time and the current exchange rate and where the domestic and foreign currency risk-free interest rates may be arbitrary continuous functions of time. First non-trivial problem we encounter in time-dependent case is the continuity in time argument of the value function of the American put option and the regularity properties of the optimal exercise boundary. We establish these properties based on systematic use of the monotonicity in volatility for the value functions of the American as well as European options with convex payoffs together with the Dynamic Programming Principle and we obtain certain type of comparison result for the value functions and corresponding exercise boundaries for the American puts with different strikes, maturities and volatilities. Starting from the latter fact that the optimal exercise boundary curve is left continuous with right-hand limits we give a mathematically rigorous and transparent derivation of the significant early exercise premium representation for the value function of the American foreign exchange put option as the sum of the European put option value function and the early exercise premium. The proof essentially relies on the particular property of the stochastic integral with respect to arbitrary continuous semimartingale over the predictable subsets of its zeros. We derive from the latter the nonlinear integral equation for the optimal exercise boundary which can be studied by numerical methods.     

Risk arbitrage in the Nikkei put warrant market of 1989–1990

This paper discusses the Nikkei put warrant market in Toronto and New York during 1989–1990. Three classes of long term American puts were traded which when evaluated in yen are ordinary, product and exchange asset puts, respectively. Type I do not involve exchange rates for yen investors. Type II, called quantos, fix in advance the exchange rate to be used on expiry in the home currency. Type III evaluate the strike and spot prices of the Nikkei Stock Average in the home currency rather than in yen. For typically observed parameters, type I are theoretically more valuable than type II which in turn are more valuable than type III. In late 1989 and early 1990 there were significant departures from fair values in various markets. This was a market with a set of complex financial instruments that even sophisticated investors needed time to learn about to price properly. Investors in Canada were willing to buy puts at far more than fair value based on historical volatility. In addition, US investors overpriced type II puts fixed in dollars rather than the type I's in yen. This led to cross border and US traded (on the same exchange) low risk hedges. The market's convergence to efficiency (that is, all puts priced within transaction cost bands) took about one month after the introduction of the US puts in early 1990 leading to significant profits for the hedgers.     

中国股市期权波动率微笑特征的实证分析

波动率微笑现象显示了期权隐含波动率和执行价格之间的关系.在理想的完全符合Black-Scholes期权定价模型假设的情况下,期权隐含波动率关于执行价格应该是一条水平线.然而,在实证分析中,对隐含波动率和执行价格进行拟合并绘制曲线,会产生一个倾斜或微笑形状的曲线,证明Black-Scholes期权定价模型存在一定的缺陷.本文从Black-Scholes期权定价模型和回归分析出发,尝试用不同的函数形式(对数函数、二次函数和三角函数)拟合波动率的解析表达式并绘制图形,最终以调整的可决系数最大为最优.首先拟合截面数据,对一固定的时间期限拟合出波动率关于执行价格的解析表达式以及波动率微笑曲线,然后将不同时间期限的波动率微笑曲线排列成时间序列,拟合面板数据,即波动率微笑曲面.然而由于面板数据的复杂性,该模型的拟合优度相对于截面数据有所降低,但是在考虑了期限与执行价格对隐含波动率的交互影响后,面板数据模型调整的可决系数显著增大,拟合优度得到提高.     

我国股指期权对现货市场的波动性影响研究

2015年2月9日,上证50ETF期权在上交所正式上市交易,我国资本市场进入期权时代.股指期权推出是否会使我国现货市场大幅波动直接影响着我国证券市场的发展和稳定,首次以我国上证50ETF期权为研究对象,使用GARCH模型和TARCH模型对我国股指期权推出后现货市场的波动性进行了研究.实证结果表明,我国股指期权的推出使现货市场的波动性增大,并且波动存在非对称性;在上证50ETF期权推出后利好消息对现货市场的波动性影响减小而利空消息对现货市场的波动性影响增大,即不对称性加大.最后,基于实证分析结果分析我国股指期权推出后导致现货市场波动增大的原因,并为我国现货市场的风险管理以及沪深300指数期权的正式推出提供参考意见.     

Multifactor Heston's stochastic volatility model for European option pricing

In this study, we extend the multiscale stochastic volatility model of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254] by incorporating a slow varying factor of volatility. The resulting model can be viewed as a multifactor extension of the Heston model with two additional factors driving the volatility levels. An asymptotic analysis consisting of singular and regular perturbation expansions is developed to obtain an approximation to European option prices. We also find explicit expressions for some essential functions that are available only in integral formulas in the work of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254]. This finding basically leads to considerable reduction in computational time for numerical calculation as well as calibration problems. An accuracy result of the asymptotic approximation is also provided. For numerical illustration, the multifactor Heston model is calibrated to index options on the market, and we find that the resulting implied volatility surfaces fit the market data better than those produced by the multiscale stochastic volatility model of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254], particularly for long‐maturity call options.