Representation and approximation of martingale measures

In this paper we propose a class of financial market models which are based on telegraph processes with alternating tendencies and jumps. It is assumed that the jumps have random sizes and that they occur when the tendencies are switching. These models are typically incomplete, but the set of equivalent martingale measures can be described in detail. We provide additional suggestions which permit arbitrage-free option prices as well as hedging strategies to be obtained.

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Shifting martingale measures and the birth of a bubble as a submartingale

Finance and Stochastics ◽

10.1007/s00780-013-0221-8 ◽

2013 ◽

Vol 18 (2) ◽

pp. 297-326 ◽

Cited By ~ 29

Author(s):

Francesca Biagini ◽

Hans Föllmer ◽

Sorin Nedelcu

Keyword(s):

Martingale Measures

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GENERALIZED BN–S STOCHASTIC VOLATILITY MODEL FOR OPTION PRICING

International Journal of Theoretical and Applied Finance ◽

10.1142/s021902491650014x ◽

2016 ◽

Vol 19 (02) ◽

pp. 1650014 ◽

Cited By ~ 16

Author(s):

INDRANIL SENGUPTA

Keyword(s):

Option Pricing ◽

Stochastic Volatility ◽

Gaussian Distribution ◽

Implied Volatility ◽

Stochastic Volatility Model ◽

Inverse Gaussian ◽

Martingale Measures ◽

Structure Preserving ◽

Volatility Model ◽

Equivalent Martingale

In this paper, a class of generalized Barndorff-Nielsen and Shephard (BN–S) models is investigated from the viewpoint of derivative asset analysis. Incompleteness of this type of markets is studied in terms of equivalent martingale measures (EMM). Variance process is studied in details for the case of Inverse-Gaussian distribution. Various structure preserving subclasses of EMMs are derived. The model is then effectively used for pricing European style options and fitting implied volatility smiles.

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Arbitrage Pricing

Arbitrage Theory in Continuous Time ◽

10.1093/oso/9780198851615.003.0007 ◽

2019 ◽

pp. 95-118

Author(s):

Tomas Björk

Keyword(s):

Continuous Time ◽

Implied Volatility ◽

Futures Contracts ◽

Martingale Measures ◽

Bank Account ◽

No Arbitrage ◽

Black Scholes Model ◽

Black Scholes ◽

Financial Derivative ◽

Delta Hedging

The chapter starts with a detailed discussion of the bank account in discrete and continuous time. The Black–Scholes model is then introduced, and using the principle of no arbitrage we study the problem of pricing an arbitrary financial derivative within this model. Using the classical delta hedging approach we derive the Black–Scholes PDE for the pricing problem and using Feynman–Kač we also derive the corresponding risk neutral valuation formula and discuss the connection to martingale measures. Some concrete examples are studied in detail and the Black–Scholes formula is derived. We also discuss forward and futures contracts, and we derive the Black-76 futures option formula. We finally discuss the concepts and roles of historic and implied volatility.

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