A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility with Rough Market Price of Volatility Risk

2017 ◽  
Author(s):  
Hideharu Funahashi ◽  
Masaaki Kijima
Keyword(s):  
Time Scale ◽  
Implied Volatility ◽  
Market Price ◽  
Volatility Risk

A Simple Model for Option Pricing with Jumping Stochastic Volatility

1998 ◽  
Vol 01 (04) ◽  
pp. 487-505 ◽  
Author(s):  
Stefano Herzel
Keyword(s):  
Implied Volatility ◽  
Market Price ◽  
Volatility Risk ◽  
Simple Modification ◽  
Closed Form Solutions ◽  
Black Scholes Model ◽  
A Value ◽  
Black Scholes ◽  
Delta Hedging ◽  
Actual Price

This paper proposes a simple modification of the Black–Scholes model by assuming that the volatility of the stock may jump at a random time τ from a value σa to a value σb. It shows that, if the market price of volatility risk is unknown, but constant, all contingent claims can be valued from the actual price C0, of some arbitrarily chosen "basis" option. Closed form solutions for the prices of European options as well as explicit formulas for vega and delta hedging are given. All such solutions only depend on σa, σb and C0. The prices generated by the model produce a "smile"-shaped curve of the implied volatility.

scholarly journals MEAN-REVERTING STOCHASTIC VOLATILITY

2000 ◽  
Vol 03 (01) ◽  
pp. 101-142 ◽  
Author(s):  
JEAN-PIERRE FOUQUE ◽  
GEORGE PAPANICOLAOU ◽  
K. RONNIE SIRCAR
Keyword(s):  
Stochastic Volatility ◽  
Stock Price ◽  
Implied Volatility ◽  
Asset Price ◽  
Market Price ◽  
Price Volatility ◽  
Estimation Procedure ◽  
Derivative Pricing ◽  
Volatility Risk ◽  
Index Value

We present derivative pricing and estimation tools for a class of stochastic volatility models that exploit the observed "bursty" or persistent nature of stock price volatility. An empirical analysis of high-frequency S&P 500 index data confirms that volatility reverts slowly to its mean in comparison to the tick-by-tick fluctuations of the index value, but it is fast mean-reverting when looked at over the time scale of a derivative contract (many months). This motivates an asymptotic analysis of the partial differential equation satisfied by derivative prices, utilizing the distinction between these time scales. The analysis yields pricing and implied volatility formulas, and the latter is used to "fit the smile" from European index option prices. The theory identifies the important group parameters that are needed for the derivative pricing and hedging problem for European-style securities, namely the average volatility and the slope and intercept of the implied volatility line, plotted as a function of the log-moneyness-to-maturity-ratio. The results considerably simplify the estimation procedure, and the data produces estimates of the three important parameters which are found to be stable within periods where the underlying volatility is close to being stationary. These segments of stationarity are identified using a wavelet-based tool. The remaining parameters, including the growth rate of the underlying, the correlation between asset price and volatility shocks, the rate of mean-reversion of the volatility and the market price of volatility risk can be roughly estimated, but are not needed for the asymptotic pricing formulas for European derivatives. The extension to American and path-dependent contingent claims is the subject of future work.

scholarly journals A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility

2017 ◽  
Vol 1 (1) ◽  
pp. 14 ◽  
Author(s):  
Hideharu Funahashi ◽  
Masaaki Kijima
Keyword(s):  
Time Scale ◽  
Implied Volatility

Information Content of Adjusted Implied Volatility in the KOSPI 200 Index Options Market

2009 ◽  
Vol 17 (4) ◽  
pp. 75-103
Author(s):  
Byung Jin Kang ◽  
Sohyun Kang ◽  
Sun-Joong Yoon
Keyword(s):  
Implied Volatility ◽  
Risk Premia ◽  
Risk Averse ◽  
Options Market ◽  
Jump Risk ◽  
Index Options ◽  
Volatility Risk ◽  
Model Free ◽  
Volatility Model ◽  
Forecasting Errors

This study examines the forecasting ability of the adjusted implied volatility (AIV), which is suggested by Kang, Kim and Yoon (2009), using the horserace competition with historical volatility, model-free implied volatility, and BS implied volatility in the KOSPI 200 index options market. The adjusted implied volatility is applicable when investors are not risk averse or when underlying returns do not follow a normal distribution. This implies that AIV is consistent with the presence of risk premia for other risk such as volatility risk and jump risk. Using KOSPI 200 index options, it is shown that the AIV outperforms other volatility estimates in terms of the unbiasedness for future realized volatilities as well as the forecasting errors.

REPLICATION OF NON-TRIVIAL DIRECTIONAL MOTION IN MULTI-SCALES OBSERVED BY THE RUNS TEST

2012 ◽  
Vol 16 ◽  
pp. 136-148 ◽  
Author(s):  
YOSHIHIRO YURA ◽  
TAKAAKI OHNISHI ◽  
KENTA YAMADA ◽  
HIDEKI TAKAYASU ◽  
MISAKO TAKAYASU
Keyword(s):  
Financial Market ◽  
Time Scale ◽  
Market Price ◽  
Short Time Scale ◽  
Price Model ◽  
Tuning Parameters ◽  
Multi Scale ◽  
Directional Motion ◽  
Short Time ◽  
Multi Scales

Non-trivial autocorrelation in up-down statistics in financial market price fluctuation is revealed by a multi-scale runs test(Wald-Wolfowitz test). We apply two models, a stochastic price model and dealer model to understand this property. In both approaches we successfully reproduce the non-stationary directional price motions consistent with the runs test by tuning parameters in the models. We find that two types of dealers exist in the markets, a short-time-scale trend-follower and an extended-time-scale contrarian who are active in different time periods.

scholarly journals Pricing Poseidon: Extreme Weather Uncertainty and Firm Return Dynamics

2021 ◽  
pp. 01-57
Author(s):  
Mathias S. Kruttli ◽  
◽  
Brigitte Roth Tran ◽  
Sumudu W. Watugala ◽  
◽  
...  
Keyword(s):  
Stock Returns ◽  
Implied Volatility ◽  
Extreme Weather ◽  
Hurricane Sandy ◽  
Extreme Weather Events ◽  
Volatility Risk ◽  
Ex Post ◽  
Significant Incidence ◽  
Weather Uncertainty ◽  
Expected Stock Returns

We present a framework to identify market responses to uncertainty faced by firms regarding both the potential incidence of extreme weather events and subsequent economic impact. Stock options of firms with establishments in forecast and realized hurricane landfall regions exhibit large increases in implied volatility, reflecting significant incidence uncertainty and long-lasting impact uncertainty. Comparing ex ante expected volatility to ex post realized volatility by analyzing volatility risk premia changes shows that investors significantly underestimate extreme weather uncertainty. After Hurricane Sandy, this underreaction diminishes and, consistent with Merton (1987), these increases in idiosyncratic volatility are associated with positive expected stock returns.

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