From Implied to Local Volatility Surface

2012 ◽  
Author(s):  
Daniel Alexandre Bloch
Keyword(s):  
Local Volatility ◽  
Volatility Surface ◽  
Local Volatility Surface

scholarly journals Control-Theoretic Framework for a Quasi-Newton Local Volatility Surface Inversion

2009 ◽  
Author(s):  
Gabriel Turinici ◽  
George Maroulis ◽  
Theodore E. Simos
Keyword(s):  
Local Volatility ◽  
Surface Inversion ◽  
Volatility Surface ◽  
Local Volatility Surface ◽  
Quasi Newton

Robust and accurate construction of the local volatility surface using the Black–Scholes equation

2021 ◽  
Vol 150 ◽  
pp. 111116
Author(s):  
Sangkwon Kim ◽  
Junseok Kim
Keyword(s):  
Local Volatility ◽  
Black Scholes ◽  
Volatility Surface ◽  
Local Volatility Surface

scholarly journals A penalty-based method from reconstructing smooth local volatility surface from American options

2015 ◽  
Vol 11 (2) ◽  
pp. 631-644 ◽  
Author(s):  
Kai Zhang ◽  
◽  
Kok Lay Teo ◽  
Keyword(s):  
American Options ◽  
Local Volatility ◽  
Volatility Surface ◽  
Local Volatility Surface

An Innovation for the Local Volatility Surface

2020 ◽  
Author(s):  
Hyuncheul Lim
Keyword(s):  
Local Volatility ◽  
Volatility Surface ◽  
Local Volatility Surface

Stochastic Implied Trees: Arbitrage Pricing with Stochastic Term and Strike Structure of Volatility

1998 ◽  
Vol 01 (01) ◽  
pp. 61-110 ◽  
Author(s):  
Emanuel Derman ◽  
Iraj Kani
Keyword(s):  
Stochastic Volatility ◽  
Continuous Time ◽  
Transition Probabilities ◽  
Arbitrage Pricing ◽  
Index Options ◽  
Local Volatility ◽  
Arbitrage Opportunities ◽  
Stochastic Term ◽  
Volatility Surface ◽  
Local Volatility Surface

In this paper we present an arbitrage pricing framework for valuing and hedging contingent equity index claims in the presence of a stochastic term and strike structure of volatility. Our approach to stochastic volatility is similar to the Heath-Jarrow-Morton (HJM) approach to stochastic interest rates. Starting from an initial set of index options prices and their associated local volatility surface, we show how to construct a family of continuous time stochastic processes which define the arbitrage-free evolution of this local volatility surface through time. The no-arbitrage conditions are similar to, but more involved than, the HJM conditions for arbitrage-free stochastic movements of the interest rate curve. They guarantee that even under a general stochastic volatility evolution the initial options prices, or their equivalent Black–Scholes implied volatilities, remain fair. We introduce stochastic implied trees as discrete implementations of our family of continuous time models. The nodes of a stochastic implied tree remain fixed as time passes. During each discrete time step the index moves randomly from its initial node to some node at the next time level, while the local transition probabilities between the nodes also vary. The change in transition probabilities corresponds to a general (multifactor) stochastic variation of the local volatility surface. Starting from any node, the future movements of the index and the local volatilities must be restricted so that the transition probabilities to all future nodes are simultaneously martingales. This guarantees that initial options prices remain fair. On the tree, these martingale conditions are effected through appropriate choices of the drift parameters for the transition probabilities at every future node, in such a way that the subsequent evolution of the index and of the local volatility surface do not lead to riskless arbitrage opportunities among different option and forward contracts or their underlying index. You can use stochastic implied trees to value complex index options, or other derivative securities with payoffs that depend on index volatility, even when the volatility surface is both skewed and stochastic. The resulting security prices are consistent with the current market prices of all standard index options and forwards, and with the absence of future arbitrage opportunities in the framework. The calculated options values are independent of investor preferences and the market price of index or volatility risk. Stochastic implied trees can also be used to calculate hedge ratios for any contingent index security in terms of its underlying index and all standard options defined on that index.

A NEW REPRESENTATION OF THE LOCAL VOLATILITY SURFACE

2008 ◽  
Vol 11 (07) ◽  
pp. 691-703
Author(s):  
MARIANITO R. RODRIGO ◽  
ROGEMAR S. MAMON
Keyword(s):  
Explicit Solution ◽  
Implied Volatility ◽  
Option Prices ◽  
Strike Price ◽  
Local Volatility ◽  
Forward Equation ◽  
Volatility Surface ◽  
Volatility Function ◽  
Ansatz Approach ◽  
Local Volatility Surface

In this paper, we address the problem of recovering the local volatility surface from option prices consistent with observed market data. We revisit the implied volatility problem and derive an explicit formula for the implied volatility together with bounds for the call price and its derivative with respect to the strike price. The analysis of the implied volatility problem leads to the development of an ansatz approach, which is employed to obtain a semi-explicit solution of Dupire's forward equation. This solution, in turn, gives rise to a new expression for the volatility surface in terms of the price of a European call or put. We provide numerical simulations to demonstrate the robustness of our technique and its capability of accurately reproducing the volatility function.

The Local Volatility Surface: Unlocking the Information in Index Option Prices

1996 ◽  
Vol 52 (4) ◽  
pp. 25-36 ◽  
Author(s):  
Emanuel Derman ◽  
Iraj Kani ◽  
Joseph Z. Zou
Keyword(s):  
Option Prices ◽  
Local Volatility ◽  
Volatility Surface ◽  
Index Option ◽  
Local Volatility Surface
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