An Empirical Study on Implied Volatility Skew Using PCA

2016 ◽  
Vol 24 (3) ◽  
pp. 365-397
Author(s):  
Jin Woo Kim ◽  
Joon H. Rhee
Keyword(s):  
Financial Crisis ◽  
Implied Volatility ◽  
Structural Changes ◽  
Explanatory Power ◽  
Asset Price ◽  
Option Price ◽  
Principal Component ◽  
Underlying Asset ◽  
Put Options ◽  
Two Factors

This paper extracts the factors determining the implied volatility skew movements of KOSPI200 index options by applying PCA (Principal Component Analysis). In particular, we analyze the movement of skew depending on the changes of the underlying asset price. As a result, it turned out that two factors can explain 94.6%~99.8% of the whole movement of implied volatility. The factor1 could be interpreted as ‘parallel shift’, and factor2 as the movement of ‘tilt or slope’. We also find some significant structural changes in the movement of skew after the Financial Crisis. The explanatory power of factor1 becomes more important on the movement of skew in both call and put options after the financial crisis. On the other hand, the influences of the factor2 is less. In general, after financial crisis, the volatility skew has the strong tendency to move in parallel. This implies that the changes in the option price or implied volatility due to the some shocks becomes more independent of the strike prices.

scholarly journals Series representation of the pricing formula for the European option driven by space-time fractional diffusion

2018 ◽  
Vol 21 (4) ◽  
pp. 981-1004 ◽  
Author(s):  
Jean-Philippe Aguilar ◽  
Cyril Coste ◽  
Jan Korbel
Keyword(s):  
Asset Price ◽  
Series Representation ◽  
Option Price ◽  
Double Series ◽  
Fractional Diffusion ◽  
Space Time ◽  
Esscher Transform ◽  
European Option ◽  
Underlying Asset ◽  
European Call Option

Abstract In this paper, we show that the price of an European call option, whose underlying asset price is driven by the space-time fractional diffusion, can be expressed in terms of rapidly convergent double-series. This series formula is obtained from the Mellin-Barnes representation of the option price with help of residue summation in ℂ2. We also derive the series representation for the associated risk-neutral factors, obtained by Esscher transform of the space-time fractional Green functions.

scholarly journals PENENTUAN HARGA OPSI BELI TIPE ASIA DENGAN METODE MONTE CARLO-CONTROL VARIATE

2017 ◽  
Vol 6 (1) ◽  
pp. 29
Author(s):  
NI NYOMAN AYU ARTANADI ◽  
KOMANG DHARMAWAN ◽  
KETUT JAYANEGARA
Keyword(s):  
Monte Carlo ◽  
Standard Error ◽  
Asset Price ◽  
Option Price ◽  
Asian Option ◽  
Underlying Asset ◽  
Average Value ◽  
Control Variate ◽  
Exercise Price ◽  
Maturity Date

Option is a contract between the writer and the holder which entitles the holder to buy or sell an underlying asset at the maturity date for a specified price known as an exercise price. Asian option is a type of financial derivatives which the payoff taking the average value over the time series of the asset price. The aim of the study is to present the Monte Carlo-Control Variate as an extension of Standard Monte Carlo applied on the calculation of the Asian option price. Standard Monte Carlo simulations 10.000.000 generate standard error 0.06 and the option price convergent at Rp.160.00 while Monte Carlo-Control Variate simulations 100.000 generate standard error 0.01 and the option price convergent at Rp.152.00. This shows the Monte Carlo-Control Variate achieve faster option price toward convergent of the Monte Carlo Standar.

Implied Volatility of the KOSPI 200 Index Option Market

2015 ◽  
Vol 23 (4) ◽  
pp. 517-541
Author(s):  
Dam Cho
Keyword(s):  
Financial Crisis ◽  
Trading Volume ◽  
Implied Volatility ◽  
Spot Price ◽  
Strike Price ◽  
Market Pricing ◽  
Put Options ◽  
The Difference ◽  
Index Option ◽  
Option Market

This paper analyzes implied volatilities (IVs), which are computed from trading records of the KOSPI 200 index option market from January 2005 to December 2014, to examine major characteristics of the market pricing behavior. The data includes only daily closing prices of option transactions for which the daily trading volume is larger than 300 contracts. The IV is computed using the Black-Scholes option pricing model. The empirical findings are as follows; Firstly, daily averages of IVs have shown very similar behavior to historical volatilities computed from 60-day returns of the KOSPI 200 index. The correlation coefficient of IV of the ATM call options to historical volatility is 0.8679 and that of the ATM put options is 0.8479. Secondly, when moneyness, which is measured by the ratio of the strike price to the spot price, is very large or very small, IVs of call and put options decrease days to maturity gets longer. This is partial evidence of the jump risk inherent in the stochastic process of the spot price. Thirdly, the moneyness pattern showed heavily skewed shapes of volatility smiles, which was more apparent during the global financial crises period from 2007 to 2009. Behavioral reasons can explain the volatility smiles. When the moneyness is very small, the deep OTM puts are priced relatively higher due to investors’ crash phobia and the deep ITM calls are valued higher due to investors’ overconfidence and confirmation biases. When the moneyness is very large, the deep OTM calls are priced higher due to investors’ hike expectation and the deep ITM puts are valued higher due to overconfidence and confirmation biases. Fourthly, for almost all moneyness classes and for all sub-periods, the IVs of puts are larger than the IVs of calls. Also, the differences of IVs of deep OTM put ranges minus IVs of deep OTM calls, which is known to be a measure of crash phobia or hike expectation, shows consistent positive values for all sub-periods. The difference in the financial crisis period is much bigger than in other periods. This suggests that option traders had a stronger crash phobia in the financial crisis.

scholarly journals DUPIRE'S EQUATION FOR BUBBLES

2012 ◽  
Vol 15 (06) ◽  
pp. 1250041 ◽  
Author(s):  
ERIK EKSTRÖM ◽  
JOHAN TYSK
Keyword(s):  
Option Price ◽  
Local Martingale ◽  
Option Prices ◽  
Uniqueness Of Solutions ◽  
Underlying Asset ◽  
Put Options ◽  
Price Process ◽  
Usual Form ◽  
Volatility Models ◽  
Strict Local Martingale

We study Dupire's equation for local volatility models with bubbles, i.e. for models in which the discounted underlying asset follows a strict local martingale. If option prices are given by risk-neutral valuation, then the discounted option price process is a true martingale, and we show that the Dupire equation for call options contains extra terms compared to the usual equation. However, the Dupire equation for put options takes the usual form. Moreover, uniqueness of solutions to the Dupire equation is lost in general, and we show how to single out the option price among all possible solutions. The Dupire equation for models in which the discounted derivative price process is merely a local martingale is also studied.

MARKETS AS A COUNTERPARTY: AN INTRODUCTION TO CONIC FINANCE

2010 ◽  
Vol 13 (08) ◽  
pp. 1149-1177 ◽  
Author(s):  
DILIP B. MADAN ◽  
ALEXANDER CHERNY
Keyword(s):  
Asset Price ◽  
Cash Flows ◽  
Index Options ◽  
Underlying Asset ◽  
Put Options ◽  
The Third ◽  
Conic Finance ◽  
Black Scholes ◽  
Using Data ◽  
Daily Returns

Markets are modeled as a counterparty accepting at zero cost a set of cash flows that are closed under addition, scaling and contain the nonnegative cash flows. Formulas are then provided for bid and ask prices in terms of this marketed cone. Additionally closed forms are obtained when parametric concave distortions introduced in Cherny and Madan (2009) define the marketed claims. Finally explicit expressions price call and put options at bid and ask. Three applications illustrate. The first estimates the movement of the cone through the financial crisis using data on bid and ask prices for S&P 500 index options. It is observed that the cone contracted significantly in 2008 and slowly opened up thereafter. The second application documents the improvements possible in terms of reduced ask prices by hedging at a flat Black-Scholes volatility even when the underlying assumptions for replication are violated. The third application considers a number of structured products written on daily returns to an underlying asset price and illustrates the use of our closed form expressions for the ask price as an objective function in designing hedges.

Alternative results for option pricing and implied volatility in jump-diffusion models using Mellin transforms

2016 ◽  
Vol 28 (5) ◽  
pp. 789-826 ◽  
Author(s):  
T. RAY LI ◽  
MARIANITO R. RODRIGO
Keyword(s):  
Differential Equation ◽  
Option Pricing ◽  
Implied Volatility ◽  
Asset Price ◽  
Jump Diffusion ◽  
Underlying Asset ◽  
Mellin Transforms ◽  
Diffusion Dynamics ◽  
Double Exponential ◽  
Jump Diffusion Model

In this article, we use Mellin transforms to derive alternative results for option pricing and implied volatility estimation when the underlying asset price is governed by jump-diffusion dynamics. The current well known results are restrictive since the jump is assumed to follow a predetermined distribution (e.g., lognormal or double exponential). However, the results we present are general since we do not specify a particular jump-diffusion model within the derivations. In particular, we construct and derive an exact solution to the option pricing problem in a general jump-diffusion framework via Mellin transforms. This approach of Mellin transforms is further extended to derive a Dupire-like partial integro-differential equation, which ultimately yields an implied volatility estimator for assets subjected to instantaneous jumps in the price. Numerical simulations are provided to show the accuracy of the estimator.

Research of the Heston Stochastic Volatility Model Within the Framework of the Options Pricing

2016 ◽  
Vol 4 (4) ◽  
pp. 33-36
Author(s):  
Насонов ◽  
A. Nasonov ◽  
Баранов ◽  
V. Baranov
Keyword(s):  
Stochastic Volatility ◽  
Stock Exchange ◽  
Asset Price ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Options Pricing ◽  
Put Options ◽  
Volatility Model ◽  
Anglo American

In this study the issues of the Heston Stochastic Volatility Model application to options pricing were researched. The Heston Model calibration problem in a particular market and time was considered. The comparison of two methods to solve it was carried out. As a result of the calibration the calculation of the price function for put options with different strikes, contract terms and interest rate was made. European options quotes to purchase Anglo American shares, traded on the London Stock Exchange, were used as initial data. The comparison of the Heston Model with the Black–Scholes Model was carried out. The dependencies of the option price on the underlying asset price were built, the estimates of discrepancy between model prices and market prices were found within the framework of these models. The results were analyzed.

scholarly journals Pricing Vulnerable European Options under Lévy Process with Stochastic Volatility

2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
Chaoqun Ma ◽  
Shengjie Yue ◽  
Yishuai Ren
Keyword(s):  
Stochastic Volatility ◽  
Asset Price ◽  
Option Price ◽  
Closed Form Solution ◽  
European Option ◽  
Short Term ◽  
Underlying Asset ◽  
Asset Value ◽  
Mean Reverting Process

This paper considers the pricing issue of vulnerable European option when the dynamics of the underlying asset value and counterparty’s asset value follow two correlated exponential Lévy processes with stochastic volatility, and the stochastic volatility is divided into the long-term and short-term volatility. A mean-reverting process is introduced to describe the common long-term volatility risk in underlying asset price and counterparty’s asset value. The short-term fluctuation of stochastic volatility is governed by a mean-reverting process. Based on the proposed model, the joint moment generating function of underlying log-asset price and counterparty’s log-asset value is explicitly derived. We derive a closed-form solution for the vulnerable European option price by using the Fourier inversion formula for distribution functions. Finally, numerical simulations are provided to illustrate the effects of stochastic volatility, jump risk, and counterparty credit risk on the vulnerable option price.

COMPUTING BOUNDS ON RISK-NEUTRAL DISTRIBUTIONS FROM THE OBSERVED PRICES OF CALL OPTIONS

2010 ◽  
Vol 27 (02) ◽  
pp. 211-225
Author(s):  
MICHI NISHIHARA ◽  
MUTSUNORI YAGIURA ◽  
TOSHIHIDE IBARAKI
Keyword(s):  
Lower Bounds ◽  
Asset Price ◽  
Option Price ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Upper And Lower Bounds ◽  
Underlying Asset ◽  
No Arbitrage ◽  
Call Options ◽  
Risk Neutral

This paper derives, in closed forms, upper and lower bounds on risk-neutral cumulative distribution functions of the underlying asset price from the observed prices of European call options, based only on the no-arbitrage assumption. The computed bounds from the option price data show that the gap between the upper and lower bounds is large near the underlying asset price but gets smaller away from the underlying asset price. Since the bounds can be easily computed and visualized, they could be practically used by investors in various ways.

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