Portfolio optimization and a factor model in a stochastic volatility market

This paper uses an extensive set of market data of forward swap rates and swaptions covering 3 July 2002 to 21 May 2009 to identify a two-dimensional stochastic volatility process for the level of rates. The process is identified step by step by increasing the requirement of the model and introducing appropriate adjustments. The first part of the paper investigates the smile dynamics of forward swap rates at their setting dates. Comparing the SABR (with different ) and Heston stochastic volatility models informs us what different specifications of the driving SDEs have to offer in terms of reflecting the dynamics of the smile across dates. The outcome of the analysis is that a normal SABR model () satisfactorily passes all tests and seems to provide a good match to the market. In contrast, we find that the Heston model does not. The next step is to seek a model of the forward swap rates (in their own swaption measure) based on only two Brownian motions that enables a specification with common parameters. It turns out that this can be done by extending the SABR model with a time-dependent volatility function and a mean-reverting volatility process. The performance of the extended (SABR with mean-reversion) model is analysed over several historical dates and is shown to be a stable and flexible choice that allows for good calibration across expiries and strikes. Finally, a time-homogeneous candidate stochastic volatility process that can be used as a driver for all swap rates is identified. This candidate process may in future work be used as a building block for a separable stochastic volatility LIBOR market model or a stochastic volatility Markov-functional model.

In this article, we are interested in an investment problem with stochastic volatilities and Portfolio constraints on proportions. We model the risky assets by jump diffusion processes and we consider a power utility function. The objective is to maximize the expected utility from terminal wealth. The value function is known to be a viscosity solution of an integro-differential Hamilton-Jacobi-Bellman (HJB) equation. Thanks to an exponential transformation, we reduce the nonlinearity of the HJB equation to a semi-linear equation. We prove existence of a smooth solution to the latter equation and we state a verification theorem which relates this solution to the value function. We derive the backward stochastic differential equation (BSDE) associated with the semi-linear equation. We propose a numerical scheme for the resolution of the BSDE based on iterative regressions on function bases and Monte-Carlo method. We illustrate an example which shows the importance of this reduction for the optimal portfolio numerical study. Therefore, we compute the optimal investment strategy by solving the associated BSDE.

In this paper, we propose a fractional stochastic volatility jump-diffusion model which extends the Bates(1996) model. Where we model the volatility as a fractional process. Extensive empirical studies show that the distributions of the logarithmic returns of financial asset usually exhibit properties of self-similarity and long-range dependence and since the fractional Brownian motion has these two important properties, it has the ability to capture the behavior of underlying asset price. Further incorporating jumps into the stochastic volatility framework gives further freedom to financial mathematicians to fit both the short and long end of the implied volatility surface. We propose a stochastic model which contains both fractional and jump process. Then we price options using Monte Carlo simulations along with a variance reduction technique(antithetic variates). We use market data from the S&P 500 index and we compare our results with the Heston and Bates model using error measures. The results show our model greatly outperforms previous models in terms of estimation accuracy.

This paper is concerned with the optimal portfolio problem for a company that can invest in two risky assets, where a novel Lévy-process-driven model is constructed to describe the dynamics of the wealth process by using a constant elasticity of variance model and a jump-diffusion process. A delicately designed value function is proposed under the mean–variance criterion to reflect the optimal portfolio for the stochastic volatility model. By using the verification theorem, the desired optimal portfolio strategy is proposed by the solution to certain Hamilton–Jacobi–Bellman equations. Furthermore, the corresponding expressions are achieved by using the stochastic analysis theory. Finally, a numerical simulation example is provided to verify the effectiveness of the proposed optimal portfolio strategy.

RECENTLY IN THIS JOURNAL, Clark Holloway employed a novel and interesting approach to testing for superior performance [1]. However, there appear to be errors in his methodology and in inferences he draws from his evidence. Holloway holds that a Buy-and-Hold policy based on Value Line (VL) recommendations yields abnormal returns even when transaction costs are considered. He calculates that annual compound rate of return of VL Rank-I stocks was 11.3 for 1965-78 period. And indeed this is result reported by VL when dividends are not included. Holloway then applies a 2 one-way commission up to May 1975 and 1.25 thereafter. He reports that the annual abnormal return after commissions was found to be 8.6 percent [italics added, p. 711]. This abnormal return is presumably amount compound rate of return of Rank-I stocks exceeds compound rate of (AllVL stocks), prior to an adjustment for risk. It is obviously too large to be correct, and except for his repeating same figure in his conclusion [p. 718], one would assume it is a typographical error. For if we adjust Rank-I returns with his own commission adjustment equation [p. 711], a terminal value (TV) per dollar originally invested of 2.697 emerges. This amounts to a postcommission annual rate of return of 7.34 percent. The market TV for same period before commissions (also without dividends) is 1.728. As for commissions, there are only two commission occasions with respect to this market portfolio for entire 1965-78 period: at entry and at exit fourteen years later. The postcommission return of market is 3.75 annually. The margin by which Rank-I stocks exceeds market, therefore, is difference between 7.34 and 3.75 percent, namely 3.6 prior to risk adjustment, not 8.6 percent. Since 3.6 extra return just derived did not include dividends, differential would be narrowed even more if yield on market is greater than on Rank-I stocks. We can get an estimate of differential yields using Holloway's Table I [p. 712]. In Table I he compares VL reported returns with his own calculations using those VL stocks included in CRSP file. According to Table IV [p. 717], number of Rank-I stocks included therein ranged from 86 to 90 (out of 100) over four-year period he chooses as representative of whole 1965-78 period. The number of stocks which compose market of AllVL stocks on File ranged from 1476 to 1482 (out of about 1700). Holloway's calculations of return include dividends; VL's do not. Holloway uses table to show how close his results were to VL's if consideration is given to dividends.

Fractional Brownian Motions, Fractional Noises and Applications

Real world systems have been created using differential equations, this has made it possible to predict future trends and behavior.Specifically stochastic differential equations have been fundamental in describing and understanding random phenome na. So far the Black-Scholes-Merton partial differential equation used in deriving the famous Black -Scholes-Merton model has been one of the greatest breakthroughs in finance as far as prediction of asset prices in the stock market is concerned. In this model we use the Logistic Brownian motion as opposed to the usual Brownian motion and we also consider volatility to be stochastic. In this stud y we have incorporated the stochastic nature of volatility and derived a Logistic Black -Scholes-Merton partial differential equation with stochastic volatility. This has been done by analyzing the Logistic Brownian motion and the Brownian motion, using the Ito pr ocess, Ito's lemma, stochastic volatility model and reviewing the derivation of the Black -Scholes-Merton partial differential equation. The formulated Differential equation may enhance reliable decision making based on more rational prediction of asset prices.

Portfolio Optimization in a Defaultable LLvy Driven Market Model

Portfolio Benchmarking under Drawdown Constraint and Stochastic Sharpe Ratio

Estimation of cost of equity and expected return in financial decision-making are gaining momentum in recent days. This study analyses FAMA French Three Factor model and Capital asset pricing model in Indian stock market. To maximize share price, the financial manager must learn to assess two key determinants viz., risk and return. Each financial decision presents certain risk and return characteristics, and the unique combination of these characteristics has an impact on share price. The risk and return of single asset was discussed using CAPM and FAMA French Three factor model. The objectives of the study included, measuring and analysing the performance of the stock using FAMA French three factor model and capital asset pricing model. This study was done for the Indian Stock market by choosing the first leading stock market Bombay Stock Exchange as the sample. The index selected for the study was S&P BSE 200, and only 120 companies were selected as the sample for conducting the study. CAPM being single factor model gave only 7.5percent significant result to the single asset. This showed that market return alone cannot determine the risk and return of the company stock. The predictability of the variables of FAMA French Three Factor model namely market return, size (SMB) and value (HML) factors are also tested in this study. The result showed that about 58.3 percent of company stocks showed significant results towards SMB and 52.5 percent of company stocks are showing significant results towards HML. Hence, the findings were generally supportive of the FAMA French model applied to Indian equities.

This paper derives optimal investment strategies for the 4/2 stochastic volatility model proposed in [Grasselli, M., The 4/2 stochastic volatility model: a unified approach for the Heston and the 3/2 model. Math. Finance, 2017, 27(4), 1013–1034] and the embedded 3/2 model [Heston, S.L., A simple new formula for options with stochastic volatility. 1997]. We maximize the expected utility of terminal wealth for a constant relative risk aversion (CRRA) investor, solving the corresponding Hamilton–Jacobi–Bellman (HJB) equations in closed form for both complete and incomplete markets. Conditions for the verification theorems are provided. Interestingly, the optimal investment strategy displays a very intuitive dependence on current volatility levels, a trend which has not been previously reported in the literature of stochastic volatility models. A full empirical analysis comparing four popular embedded models—i.e. the Merton (geometric Brownian motion), Heston (1/2), 3/2 and 4/2 models—is conducted using S&P 500 and VIX data. We find that the 1/2 model carries the larger weight in explaining the 4/2 behaviour, and optimal investments in the 1/2 and 4/2 models are similar, while investments in the 3/2 model are the most conservative in high-variance settings (20% of Merton's solution). Despite the similarities between the 1/2 and 4/2 models, wealth-equivalent losses due to deviations from the 4/2 model are largest for the1/2 and GBM models (40% over 10 years). Meanwhile, the wealth losses due to market incompleteness are harsher for the 1/2 model than for the 4/2 and 3/2 models (60% versus 40% and 30% respectively), highlighting the benefits of choosing the 4/2 or the 3/2 over the 1/2 model.

This paper studies the question of filtering and maximizing terminal wealth from expected utility in a partially information stochastic volatility models. The special features is that the only information available to the investor is the one generated by the asset prices, and the unobservable processes will be modeled by a stochastic differential Using the change of measure techniques, the partial observation context can be transformed into a full information context such that coefficients depend only on past history of observed prices (filters processes). Adapting the stochastic non-linear filtering, we show that under some assumptions on the model coefficients, the estimation of the filters depend on a priori models for the trend and the stochastic volatility. Moreover, these filters satisfy a stochastic partial differential equations named Kushner-Stratonovich equations. Using the martingale duality approach in this partially observed incomplete model, we can characterize the value function and the optimal portfolio. The main result here is that the dual value function associated to the martingale approach can be expressed, via the dynamic programming approach, in terms of the solution to a semilinear partial differential equation. We illustrate our results with some examples of stochastic volatility models popular in the financial literature.

Mean–variance asset–liability management with asset correlation risk and insurance liabilities

The calendar spread options (CSOs) on agricultural commodities, most notably corn, soybeans and wheat, allow market participants to hedge the roll-over risk of futures contracts. Despite the interest from agricultural businesses, there is lack of both theoretical and empirical research on pricing and hedging performances of CSOs. We propose to price and hedge CSOs under geometric Brownian motion (GBM) and stochastic volatility (SV) models. We estimate the model parameters by using implied state-generalized method of moments (IS-GMM) and evaluate the in-sample and out- of-sample pricing and hedging performances. We find that the average pricing errors of the SV model are 0.79% for corn, 0.75% for soybeans and 1.2% for wheat; the pricing and hedging performance of the SV model are mostly superior to the benchmark GBM model, both in and out of sample, with only one exception where the out-of-sample hedging error for the GBM model for market makers is slightly better than the SV model.

The purpose of this paper is to consider the optimal proportional reinsurance and investment strategies for an insurance company. The insurer’s surplus process is approximated by a Brownian motion with drift. The insurance company can purchase proportional reinsurance and invest the surplus in a financial market which includes one risk-free asset and one risky asset whose price is modeled by a CEV model. The primary problem is changed to the dual problem by implying Legendre transform. When the objective of the insurance company is to maximize the expected logarithmic utility from terminal wealth, the closed-form expressions for the optimal reinsurance-investment policy which is different to the Merton case to the primal optimal problem are obtained and numerical simulations are provided to demonstrate our results. Moreover, we find an interesting result that risk exposure is non-monotonic in the cost of reinsurance.