scholarly journals EXACT PRICING AND LARGE-TIME ASYMPTOTICS FOR THE MODIFIED SABR MODEL AND THE BROWNIAN EXPONENTIAL FUNCTIONAL

2011 ◽  
Vol 14 (04) ◽  
pp. 559-578 ◽  
Author(s):  
MARTIN FORDE
Keyword(s):  
Stock Price ◽  
Implied Volatility ◽  
Large Time ◽  
Transition Density ◽  
Closed Form Expression ◽  
Modified Model ◽  
Exponential Functional ◽  
Price Distribution ◽  
Zero Correlation ◽  
Sabr Model

We derive a closed-form expression for the stock price density under the modified SABR model [see section 2.4 in Islah (2009)] with zero correlation, for β = 1 and β < 1, using the known density for the Brownian exponential functional for μ = 0 given in Matsumoto and Yor (2005), and then reversing the order of integration using Fubini's theorem. We then derive a large-time asymptotic expansion for the Brownian exponential functional for μ = 0, and we use this to characterize the large-time behaviour of the stock price distribution for the modified SABR model; the asymptotic stock price "density" is just the transition density p(t, S0, S) for the CEV process, integrated over the large-time asymptotic "density" [Formula: see text] associated with the Brownian exponential functional (re-scaled), as we might expect. We also compute the large-time asymptotic behaviour for the price of a call option, and we show precisely how the implied volatility tends to zero as the maturity tends to infinity, for β = 1 and β < 1. These results are shown to be consistent with the general large-time asymptotic estimate for implied variance given in Tehranchi (2009). The modified SABR model is significantly more tractable than the standard SABR model. Moreover, the integrated variance for the modified model is infinite a.s. as t → ∞, in contrast to the standard SABR model, so in this sense the modified model is also more realistic.

FORWARD AND FUTURE IMPLIED VOLATILITY

2011 ◽  
Vol 14 (03) ◽  
pp. 407-432 ◽  
Author(s):  
PAUL GLASSERMAN ◽  
QI WU
Keyword(s):  
Implied Volatility ◽  
Transition Density ◽  
Option Prices ◽  
Piecewise Constant ◽  
Model Based ◽  
Model Free ◽  
Out Of Sample ◽  
Sabr Model ◽  
Using Data ◽  
Better Than

We address the problem of defining and calculating forward volatility implied by option prices when the underlying asset is driven by a stochastic volatility process. We examine alternative notions of forward implied volatility and the information required to extract these measures from the prices of European options at fixed maturities. We then specialize to the SABR model and show how the asymptotic expansion of the bivariate transition density in Wu (forthcoming) allows calibration of the SABR model with piecewise constant parameters and calculation of forward volatility. We then investigate empirically whether current option prices at multiple maturities contain useful information in predicting future option prices and future implied volatility. We undertake this investigation using data on options on the euro-dollar, sterling-dollar, and dollar-yen exchange rates. We find that prices across maturities do indeed have predictive value. Moreover, we find that model-based forward volatility extracts this predicative information better than a standard "model-free" measure of forward volatility and better than spot implied volatility. The enhancement to out-of-sample forecasting accuracy gained from model-based forward volatility is greatest at longer forecasting horizons.

THE LARGE-MATURITY SMILE FOR THE SABR AND CEV-HESTON MODELS

2013 ◽  
Vol 16 (08) ◽  
pp. 1350047 ◽  
Author(s):  
MARTIN FORDE ◽  
ANDREY POGUDIN
Keyword(s):  
Implied Volatility ◽  
Large Time ◽  
Heston Model ◽  
Large Time Asymptotics ◽  
Cev Model ◽  
Constant Elasticity ◽  
Constant Elasticity Of Variance ◽  
Time Estimates ◽  
Time Asymptotics ◽  
Sabr Model

Large-time asymptotics are established for the SABR model with β = 1, ρ ≤ 0 and β < 1, ρ = 0. We also compute large-time asymptotics for the constant elasticity of variance (CEV) model in the large-time, fixed-strike regime and a new large-time, large-strike regime, and for the uncorrelated CEV-Heston model. Finally, we translate these results into a large-time estimates for implied volatility using the recent work of Gao and Lee (2011) and Tehranchi (2009).

AN IMPLIED VOLATILITY MODEL DETERMINED BY CREDIT DEFAULT SWAPS

2012 ◽  
Vol 15 (07) ◽  
pp. 1250049
Author(s):  
PASCAL HEIDER
Keyword(s):  
Capital Structure ◽  
Stock Price ◽  
Implied Volatility ◽  
Contingent Claims ◽  
Stock Volatility ◽  
Numerical Examples ◽  
Volatility Model ◽  
Credit Default ◽  
Spread Dynamics ◽  
A Company

In this paper we propose a diffusion model relating the stock price dynamics to the CDS spread dynamics of a company by assuming a linear relationship between instantaneous stock volatility and CDS spread. To value contingent claims under this model we apply a finite elements discretization to the associated pricing partial differential equation. A robust calibration strategy is presented and numerical examples are studied to validate the model assumptions. Besides option pricing, we discuss further applications which are e.g. the identification of market situations allowing volatility and capital structure arbitrage.

scholarly journals ANALYSIS OF MARKET VOLATILITY VIA A DYNAMICALLY PURIFIED OPTION PRICE PROCESS

2014 ◽  
Vol 09 (03) ◽  
pp. 1450006 ◽  
Author(s):  
CHUONG LUONG ◽  
NIKOLAI DOKUCHAEV
Keyword(s):  
Stock Price ◽  
Implied Volatility ◽  
Financial Time Series ◽  
Option Price ◽  
Option Prices ◽  
Price Process ◽  
Volatility Index ◽  
Quadratic Interpolation ◽  
Dynamic Estimation ◽  
The Impact

The paper studies methods of dynamic estimation of volatility for financial time series. We suggest to estimate the volatility as the implied volatility inferred from some artificial "dynamically purified" price process that in theory allows to eliminate the impact of the stock price movements. The complete elimination would be possible if the option prices were available for continuous sets of strike prices and expiration times. In practice, we have to use only finite sets of available prices. We discuss the construction of this process from the available option prices using different methods. In order to overcome the incompleteness of the available option prices, we suggests several interpolation approaches, including the first order Taylor series extrapolation and quadratic interpolation. We examine the potential of the implied volatility derived from this proposed process for forecasting of the future volatility, in comparison with the traditional implied volatility process such as the volatility index VIX.

Informed Trading in the Stock Market and Option-Price Discovery

2020 ◽  
pp. 1-40 ◽  
Author(s):  
Pierre Collin-Dufresne ◽  
Vyacheslav Fos ◽  
Dmitry Muravyev
Keyword(s):  
Stock Market ◽  
Stock Price ◽  
Implied Volatility ◽  
Common Knowledge ◽  
Option Price ◽  
Price Volatility ◽  
Informed Trading ◽  
Flow Dynamics ◽  
Option Prices ◽  
Market Makers

Abstract When activist shareholders file Schedule 13D filings, the average stock-price volatility drops by approximately 10%. Prior to filing days, volatility information is reflected in option prices. Using a comprehensive sample of trades by Schedule 13D filers that reveals on what days and in what markets they trade, we show that on days when activists accumulate shares, option-implied volatility decreases, implied volatility skew increases, and implied volatility time slope increases. The evidence is consistent with a theoretical model where it is common knowledge that informed trading occurs only in the stock market and market makers update option prices based on stock-price and order-flow dynamics.

Mass at Zero and Small-Strike Implied Volatility Expansion in the SABR Model

2015 ◽  
Author(s):  
Archil Gulisashvili ◽  
Blanka Horvath ◽  
Antoine Jacquier
Keyword(s):  
Implied Volatility ◽  
Sabr Model
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