scholarly journals Implied Volatility in Strict Local Martingale Models

2018 ◽  
Vol 9 (1) ◽  
pp. 171-189 ◽  
Author(s):  
Antoine Jacquier ◽  
Martin Keller-Ressel
Keyword(s):  
Implied Volatility ◽  
Local Martingale ◽  
Strict Local Martingale

scholarly journals Uniform convergence of conditional distributions for absorbed one-dimensional diffusions

2018 ◽  
Vol 50 (01) ◽  
pp. 178-203 ◽  
Author(s):  
Nicolas Champagnat ◽  
Denis Villemonais
Keyword(s):  
Sufficient Conditions ◽  
Initial Distribution ◽  
Initial Position ◽  
Local Martingale ◽  
One Dimensional ◽  
Positive Time ◽  
Strict Local Martingale ◽  
Necessary And Sufficient ◽  
Stationary Behavior ◽  
Uniformly Bounded

Abstract In this paper we study the quasi-stationary behavior of absorbed one-dimensional diffusions. We obtain necessary and sufficient conditions for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. An important tool is provided by one-dimensional strict local martingale diffusions coming down from infinity. We prove, under mild assumptions, that their expectation at any positive time is uniformly bounded with respect to the initial position. We provide several examples and extensions, including the sticky Brownian motion and some one-dimensional processes with jumps.

scholarly journals STRICT LOCAL MARTINGALES VIA FILTRATION ENLARGEMENT

2019 ◽  
Vol 23 (01) ◽  
pp. 2050001
Author(s):  
ADITI DANDAPANI ◽  
PHILIP PROTTER
Keyword(s):  
Stochastic Volatility ◽  
Sufficient Conditions ◽  
Local Martingale ◽  
Change Of Measure ◽  
Stochastic Volatility Models ◽  
Strict Local Martingales ◽  
Volatility Models ◽  
Strict Local Martingale

A strict local martingale is a local martingale that is not a martingale. We investigate how such a process might arise from a true martingale as a result of an enlargement of the filtration and a change of measure. We study and implement a particular type of enlargement, initial expansion of filtration, for stochastic volatility models with and without jumps and provide sufficient conditions in each of these cases such that initial expansion can create a strict local martingale. We provide examples of initial enlargement that effect this change.

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.

scholarly journals Foreign Exchange Expectation Errors and Filtration Enlargements

Stats ◽  
2019 ◽  
Vol 2 (2) ◽  
pp. 212-227
Author(s):  
Pedro Chaim ◽  
Márcio Laurini
Keyword(s):  
Exchange Rate ◽  
Local Martingale ◽  
Future Market ◽  
Forward Rates ◽  
Forecast Bias ◽  
Market Participants ◽  
Martingale Process ◽  
New Information ◽  
Information Set ◽  
Strict Local Martingale

Extrapolations of future market forward rates are a better predictor of the 30-days ahead BRL-USD exchange rate than forecasts from the Central Bank Focus survey of Brazilian market participants. This is puzzling because market participants observe forward rates as they submit predictions, and thus these agents perform biased forecasts even though they have access to a set of unbiased forecasts, consistent with a martingale process for the exchange rate. We argue that this rational conundrum can be explained by a mechanism through which new information enlarges the information set (a filtration), changing the underlying measure and inducing a drift into the martingale process, turning the process into a strict local martingale and generating a forecast bias. Empirical results suggest that Focus survey forecasts indeed display characteristics of a strict local martingale, while spot exchange rates and forward rates are consistent with a martingale process.

Strict local martingale deflators and valuing American call-type options

2011 ◽  
Vol 16 (2) ◽  
pp. 275-291 ◽  
Author(s):  
Erhan Bayraktar ◽  
Constantinos Kardaras ◽  
Hao Xing
Keyword(s):  
Local Martingale ◽  
Call Type ◽  
Strict Local Martingale

scholarly journals Asset price bubbles: Invariance theorems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Robert Jarrow ◽  
Philip Protter ◽  
Jaime San Martin
Keyword(s):  
Asset Price ◽  
Martingale Measure ◽  
Local Martingale ◽  
Statistical Probability ◽  
Asset Price Bubbles ◽  
Price Bubbles ◽  
New Family ◽  
Price Bubble ◽  
Markov Diffusion ◽  
Strict Local Martingale

<p style='text-indent:20px;'>This paper provides invariance theorems that facilitate testing for the existence of an asset price bubble in a market where the price evolves as a Markov diffusion process. The test involves only the properties of the price process' quadratic variation under the statistical probability. It does not require an estimate of either the equivalent local martingale measure or the asset's drift. To augment its use, a new family of stochastic volatility price processes is also provided where the processes' strict local martingale behavior can be characterized.</p>

PERFECT HEDGING OF INDEX DERIVATIVES UNDER A MINIMAL MARKET MODEL

2002 ◽  
Vol 05 (07) ◽  
pp. 757-774 ◽  
Author(s):  
DAVID HEATH ◽  
ECKHARD PLATEN
Keyword(s):  
Term Structure ◽  
Financial Market ◽  
Market Model ◽  
Martingale Measure ◽  
Local Martingale ◽  
Minimal Set ◽  
Benchmark Portfolio ◽  
Strict Local Martingale ◽  
Minimal Market Model ◽  
Standard Risk

The paper presents a financial market model that generates stochastic volatility using a minimal set of factors. These factors, formed by transformations of square root processes, model the dynamics of different denominations of a benchmark portfolio. Benchmarked prices are assumed to be local martingales. Numerical results for the pricing and hedging of basic derivatives on indices are described for the minimal market model. This includes cases where the standard risk neutral pricing methodology fails because of the presence of a strict local martingale measure. However, payoffs can be perfectly hedged using self-financing strategies and a form of arbitrage exists. This is illustrated by hedge simulations. The different term structure of implied volatilities is documented for calls and puts on an index.

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