scholarly journals The averaging principle for non-autonomous slow-fast stochastic differential equations and an application to a local stochastic volatility model

2021 ◽  
Vol 302 ◽  
pp. 406-443
Author(s):  
Filippo de Feo
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Averaging Principle ◽  
Volatility Model

scholarly journals Effect of Variance Swap in Hedging Volatility Risk

Risks ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 70
Author(s):  
Yang Shen
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Backward Stochastic Differential Equations ◽  
Stochastic Volatility Model ◽  
Linear Quadratic ◽  
Bank Account ◽  
Volatility Risk ◽  
Variance Swap ◽  
Volatility Model ◽  
Mean Variance

This paper studies the effect of variance swap in hedging volatility risk under the mean-variance criterion. We consider two mean-variance portfolio selection problems under Heston’s stochastic volatility model. In the first problem, the financial market is complete and contains three primitive assets: a bank account, a stock and a variance swap, where the variance swap can be used to hedge against the volatility risk. In the second problem, only the bank account and the stock can be traded in the market, which is incomplete since the idiosyncratic volatility risk is unhedgeable. Under an exponential integrability assumption, we use a linear-quadratic control approach in conjunction with backward stochastic differential equations to solve the two problems. Efficient portfolio strategies and efficient frontiers are derived in closed-form and represented in terms of the unique solutions to backward stochastic differential equations. Numerical examples are provided to compare the solutions to the two problems. It is found that adding the variance swap in the portfolio can remarkably reduce the portfolio risk.

scholarly journals Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation

2020 ◽  
pp. 1-27
Author(s):  
Shao-Qin Zhang ◽  
Chenggui Yuan
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Stochastic Volatility Model ◽  
Type Model ◽  
Interest Rate Models ◽  
Negative Power ◽  
Volatility Model ◽  
Locally Lipschitz

In this paper, we study a class of one-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $ H \gt \frac{{1}\over{2}}$ . The drift term of the equation is locally Lipschitz and unbounded in the neighbourhood of the origin. We show the existence, uniqueness and positivity of the solutions. The estimates of moments, including the negative power moments, are given. We also develop the implicit Euler scheme, proved that the scheme is positivity preserving and strong convergent, and obtain rate of convergence. Furthermore, by using Lamperti transformation, we show that our results can be applied to stochastic interest rate models such as mean-reverting stochastic volatility model and strongly nonlinear Aït-Sahalia type model.

A non-Gaussian stochastic volatility model

1998 ◽  
Vol 2 (2) ◽  
pp. 33-47 ◽  
Author(s):  
Yuichi Nagahara ◽  
Genshiro Kitagawa
Keyword(s):  
Stochastic Volatility ◽  
Stochastic Volatility Model ◽  
Volatility Model ◽  
Non Gaussian
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