5,742 research outputs found
Modeling electricity spot prices - Combining mean-reversion, spikes and stochastic volatility
Starting with the liberalization of electricity trading, this market grew rapidly over the last decade. However, while spot and future markets are rather liquid nowadays, option trading is still limited. One of the potential reasons for this is that the spot price process of electricity is still puzzling researchers and practitioners. In this paper, we propose an approach to model spot prices that combines mean-reversion, spikes and stochastic volatility. Thereby we use different mean-reversion rates for 'normal' and 'extreme' (spike) periods. Another feature of the model is its ability to capture correlation structures of electricity price spikes. Furthermore, all model parameters can easily be estimated with help of historical data. Consequently, we argue that this model does not only extend academic literature on electricity spot price modeling, but is also suitable for practical purposes, e.g. as underlying price model for option pricing. --Electricity,Energy markets,Lévy processes,Mean-reversion,Spikes,Stochastic volatility,GARCH
Currency option pricing with Wishart process
AbstractIt has been well-documented that foreign exchange rates exhibit both mean reversion and stochastic volatility. In addition to these, recent empirical evidence shows a stochastic skew of implied volatility surface from currency option data, which means that the slope of implied volatility curve of a given maturity is stochastically time varying. This paper develops a currency option pricing model which accommodates for this phenomena. The proposed model postulates that the log-currency value follows a mean reverting process with stochastic volatility driven by Wishart process under risk-neutral measure. Pricing formula for European currency option is derived in terms of Fourier Transform. Benchmarking against the Monte Carlo simulation, our numerical examples reveal that the pricing formula is accurate and remarkably efficient. The proposed model is also generalized to include jumps. The ability of the our model on capturing stochastic skew is illustrated through a numerical example
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Analysis of model implied volatility for jump diffusion models: Empirical evidence from the Nordpool market
In this paper we examine the importance of mean reversion and spikes in the stochastic behaviour of the underlying asset when pricing options on power. We propose a model that is flexible in its formulation and captures the stylized features of power prices in a parsimonious way. The main feature of the model is that it incorporates two different speeds of mean reversion to capture the differences in price behaviour between normal and spiky periods. We derive semi-closed form solutions for European option prices using transform analysis and then examine the properties of the implied volatilities that the model generates. We find that the presence of jumps generates prominent volatility skews which depend on the sign of the mean jump size. We also show that mean reversion reduces the volatility smile as time to maturity increases. In addition, mean reversion induces volatility skews particularly for ITM options, even in the absence of jumps. Finally, jump size volatility and jump intensity mainly affect the kurtosis and thus the curvature of the smile with the former having a more important role in making the volatility smile more pronounced and thus increasing the kurtosis of the underlying price distribution
Foreign Exchange Option Valuation under Stochastic Volatility
>Magister Scientiae - MScThe case of pricing options under constant volatility has been common practise for decades. Yet market data proves that the volatility is a stochastic phenomenon, this is evident in longer duration instruments in which the volatility of underlying asset is dynamic and unpredictable. The methods of valuing options under stochastic volatility that have been extensively published focus mainly on stock markets and on options written on a single reference asset. This work probes the effect of valuing European call option written on a basket of currencies, under constant volatility and under stochastic volatility models. We apply a family of the stochastic models to investigate the relative performance of option prices. For the valuation of option under constant volatility, we derive a closed form analytic solution which relaxes some of the assumptions in the Black-Scholes model. The problem of two-dimensional random diffusion of exchange rates and volatilities is treated with present value scheme, mean reversion and non-mean reversion stochastic volatility models. A multi-factor Gaussian distribution function is applied on lognormal asset dynamics sampled from a normal distribution which we generate by the Box-Muller method and make inter dependent by Cholesky factor matrix decomposition. Furthermore, a Monte Carlo simulation method is adopted to approximate a general form of numeric solution The historic data considered dates from 31 December 1997 to 30 June 2008. The basket
contains ZAR as base currency, USD, GBP, EUR and JPY are foreign currencies
Option Pricing under Fast-varying and Rough Stochastic Volatility
Recent empirical studies suggest that the volatilities associated with
financial time series exhibit short-range correlations. This entails that the
volatility process is very rough and its autocorrelation exhibits sharp decay
at the origin. Another classic stylistic feature often assumed for the
volatility is that it is mean reverting. In this paper it is shown that the
price impact of a rapidly mean reverting rough volatility model coincides with
that associated with fast mean reverting Markov stochastic volatility models.
This reconciles the empirical observation of rough volatility paths with the
good fit of the implied volatility surface to models of fast mean reverting
Markov volatilities. Moreover, the result conforms with recent numerical
results regarding rough stochastic volatility models. It extends the scope of
models for which the asymptotic results of fast mean reverting Markov
volatilities are valid. The paper concludes with a general discussion of
fractional volatility asymptotics and their interrelation. The regimes
discussed there include fast and slow volatility factors with strong or small
volatility fluctuations and with the limits not commuting in general. The
notion of a characteristic term structure exponent is introduced, this exponent
governs the implied volatility term structure in the various asymptotic
regimes.Comment: arXiv admin note: text overlap with arXiv:1604.0010
Sequential Monte Carlo pricing of American-style options under stochastic volatility models
We introduce a new method to price American-style options on underlying
investments governed by stochastic volatility (SV) models. The method does not
require the volatility process to be observed. Instead, it exploits the fact
that the optimal decision functions in the corresponding dynamic programming
problem can be expressed as functions of conditional distributions of
volatility, given observed data. By constructing statistics summarizing
information about these conditional distributions, one can obtain high quality
approximate solutions. Although the required conditional distributions are in
general intractable, they can be arbitrarily precisely approximated using
sequential Monte Carlo schemes. The drawback, as with many Monte Carlo schemes,
is potentially heavy computational demand. We present two variants of the
algorithm, one closely related to the well-known least-squares Monte Carlo
algorithm of Longstaff and Schwartz [The Review of Financial Studies 14 (2001)
113-147], and the other solving the same problem using a "brute force" gridding
approach. We estimate an illustrative SV model using Markov chain Monte Carlo
(MCMC) methods for three equities. We also demonstrate the use of our algorithm
by estimating the posterior distribution of the market price of volatility risk
for each of the three equities.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS286 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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