Quantitative Finance Collector is a blog on Quantitative finance analysis, methods in mathematical finance focusing on derivative pricing, quantitative trading and quantitative risk management.
Oct
20
Heston stochastic volatility model is widely used in both industry and academia, due to the correlation between stock process and volatility process, plus a non-negative square root volatility, simulation of Heston volatility doesn't sound easy & stable. Professor Mark Joshi has a recent working paper Fast and Accurate Long Stepping Simulation of the Heston Stochastic Volatility Model, as its title suggests: this paper aims to provide a better simulation scheme.
looks brilliant indeed, interested readers shall download the C++ code and paper directly at http://www.markjoshi.com/downloads/index.htm
Quotation
In this paper, we present three new discretization schemes for the Heston stochastic volatility model - two schemes for simulating the variance process and one scheme for simulating the integrated variance process conditional on the initial and the end-point of the variance process. Instead of using a short time-stepping approach to simulate the variance process and its integral, these new schemes evolve the Heston process accurately over long steps without the need to sample the intervening values. Hence, prices of financial derivatives can be evaluated rapidly using our new approaches.
looks brilliant indeed, interested readers shall download the C++ code and paper directly at http://www.markjoshi.com/downloads/index.htm
Nov
24
Generates Heston stochastic volatility process at various frequencies,
% ds = mu dt + Vt^1/2 dW_1t
% dVt = b(a-Vt) dt + sig Vt^1/2 dW_2t
% Corr( dW_1t, dW_2t )=rho
% S0 is starting value of price proces
% NbD corresponds to numbers of days
http://www.hec.unil.ch/matlabcodes/option_pricing.html
% ds = mu dt + Vt^1/2 dW_1t
% dVt = b(a-Vt) dt + sig Vt^1/2 dW_2t
% Corr( dW_1t, dW_2t )=rho
% S0 is starting value of price proces
% NbD corresponds to numbers of days
http://www.hec.unil.ch/matlabcodes/option_pricing.html
Oct
20
Quotation
The Heston Model is one of the most widely used stochastic volatility (SV) models today. Its attractiveness lies in the powerful duality of its tractability and robustness relative to other SV models.
This project initially begun as one that addressed the calibration problem of this model. Attempting to solve such a problem was an impossible task due to the lack of exposure to such ‘advanced’ models.
I, therefore, decided to take a slight digression into the world of Heston and stochastic volatility. Enroute I realised that fundamental information that one would require to gain an intuitive understanding of such a model was very disjoint and hence incomplete. This project, therefore, evolved into something that could fill this gap.
A practical approach has been adopted since the focus of calibration is quite practical itself. All the relevant tools are provided to facilitate this calibration process, including MATLAB code. This code has been confined to the appendix to keep the main body clutter free and ‘quick-to-read’.
paper and code can be downloaded at http://math.nyu.edu/~atm262/fall06/compmethods/a1/nimalinmoodley.pdf
Oct
1
Calculate variance swap hedging portfolio under Heston vol model using MC simulation. The strategy is discussed in Gatheral p.136 and http://www.ederman.com/new/docs/gs-volatility_swaps.pdf.
The strategy works by exploiting the difference between percentage differences and log differences. A percentage difference is expressed as (S’ – S)/S or S’/S - 1. A log difference is log(S’) – log(S) or log(S’/S). Fore more detail refer to http://math.nyu.edu/~atm262/files/fall06/casestudies/a7/hestonvarswap.m and the above mentioned paper.
The strategy works by exploiting the difference between percentage differences and log differences. A percentage difference is expressed as (S’ – S)/S or S’/S - 1. A log difference is log(S’) – log(S) or log(S’/S). Fore more detail refer to http://math.nyu.edu/~atm262/files/fall06/casestudies/a7/hestonvarswap.m and the above mentioned paper.
Sep
16
Cliquet option, also called ratchet option, is an extended roll-down option, with strikes set at the barriers, which never knock out completely. It is a series of at the money options, with periodic settlement, resetting the strike value at the then current price level, at which time, the option locks in the difference between the old and new strike and pays that out as the profit. The profit can be accumulated until final maturity, or paid out at each reset date.
The Bates Model is a type of Jump-Difussion model that is able to improve calibration results for short term options. The Bates Model consists of Jumps processes built on top a Heston model.
http://www.javaquant.net/finalgo/BatesModel.html lists the C++ code to price Cliquet options using the Log-Jump variant of the Bates model with stochastic volatility.
The Bates Model is a type of Jump-Difussion model that is able to improve calibration results for short term options. The Bates Model consists of Jumps processes built on top a Heston model.
http://www.javaquant.net/finalgo/BatesModel.html lists the C++ code to price Cliquet options using the Log-Jump variant of the Bates model with stochastic volatility.
