On Minimizing Risk in Incomplete Markets Option Pricing Models

1998 ◽  
Vol 01 (02) ◽  
pp. 227-233 ◽  
Author(s):  
Ola Hammarlid
Keyword(s):  
Incomplete Markets ◽  
Option Price ◽  
Market Model ◽  
Risk Minimization ◽  
Pricing Options ◽  
Fair Game ◽  
Local Risk ◽  
Game Condition ◽  
Stock Portfolio ◽  
Global And Local

I study the Bouchaud–Sornette, Schweizer and Schäl way of pricing options, presenting the methodology in accordance with Bouchaud–Sornette. The definitions of the wealth balance and risk from trading in options and stocks are presented. The problem of finding a risk minimizing strategy in an incomplete market model where a perfect hedge is not possible is analyzed. Using this strategy according to the approach of Bouchaud and Sornette the option is priced by a fair game condition. In this article I establish the equivalence between global and local risk minimization and prove an option price conjecture of Wolczyńska. I also investigate optimality for a stock portfolio with extra profit.

scholarly journals On Market Completions Approach to Option Pricing

2021 ◽  
Vol 9 (3) ◽  
pp. 77-93
Author(s):  
I. Vasilev ◽  
A. Melnikov
Keyword(s):  
Option Pricing ◽  
Incomplete Markets ◽  
Option Price ◽  
Optimal Investment ◽  
Market Model ◽  
Option Prices ◽  
End Points ◽  
Hedging Strategies ◽  
Risk Neutral ◽  
Interesting Approach

Option pricing is one of the most important problems of contemporary quantitative finance. It can be solved in complete markets with non-arbitrage option price being uniquely determined via averaging with respect to a unique risk-neutral measure. In incomplete markets, an adequate option pricing is achieved by determining an interval of non-arbitrage option prices as a region of negotiation between seller and buyer of the option. End points of this interval characterise the minimum and maximum average of discounted pay-off function over the set of equivalent risk-neutral measures. By estimating these end points, one constructs super hedging strategies providing a risk-management in such contracts. The current paper analyses an interesting approach to this pricing problem, which consists of introducing the necessary amount of auxiliary assets such that the market becomes complete with option price uniquely determined. One can estimate the interval of non-arbitrage prices by taking minimal and maximal price values from various numbers calculated with the help of different completions. It is a dual characterisation of option prices in incomplete markets, and it is described here in detail for the multivariate diffusion market model. Besides that, the paper discusses how this method can be exploited in optimal investment and partial hedging problems.

scholarly journals Hedging and pricing illiquid options with market impacts

2017 ◽  
Vol 04 (02n03) ◽  
pp. 1750030
Author(s):  
Taiga Saito
Keyword(s):  
Option Price ◽  
Critical Issue ◽  
Option Prices ◽  
Risk Minimization ◽  
Hedging Strategy ◽  
Time Model ◽  
Underlying Asset ◽  
Market Impacts ◽  
Local Risk ◽  
Liquidity Costs

In this paper, we consider hedging and pricing of illiquid options on an untradable underlying asset, where an alternative asset is used as a hedging instrument. Particularly, we consider the situation where the trade price of the hedging instrument is subject to market impacts caused by the hedger and the liquidity costs paid as a spread from the mid price. Pricing illiquid options, which often appears in trading of structured products, is a critical issue in practice because of its difficulties in hedging mainly due to untradability of the underlying asset as well as the liquidity costs and market impacts of the hedging instrument. First, by setting the problem under a discrete time model, where the optimal hedging strategy is defined by the local risk-minimization, we present algorithms to obtain the option price along with the hedging strategy by an asymptotic expansion. Moreover, we provide numerical examples. This model enables the estimation of the effect of both the market impacts and the liquidity costs on option prices, which is important in practice.

NUMERICAL ANALYSIS ON LOCAL RISK-MINIMIZATION FOR EXPONENTIAL LÉVY MODELS

2016 ◽  
Vol 19 (02) ◽  
pp. 1650008 ◽  
Author(s):  
TAKUJI ARAI ◽  
YUTO IMAI ◽  
RYOICHI SUZUKI
Keyword(s):  
Incomplete Markets ◽  
Contingent Claims ◽  
Risk Minimization ◽  
Gamma Model ◽  
Variance Gamma ◽  
Call Options ◽  
Estimated Parameters ◽  
Exponential Lévy Models ◽  
Local Risk ◽  
Variance Gamma Model

We illustrate how to compute local risk minimization (LRM) of call options for exponential Lévy models. Here, LRM is a popular hedging method through a quadratic criterion for contingent claims in incomplete markets. Arai & Suzuki (2015) have previously obtained a representation of LRM for call options; here we transform it into a form that allows use of the fast Fourier transform (FFT) method suggested by by Carr & Madan (1999). Considering Merton jump-diffusion models and variance gamma models as typical examples of exponential Lévy models, we provide the forms for the FFT explicitly; and compute the values of LRM numerically for given parameter sets. Furthermore, we illustrate numerical results for a variance gamma model with estimated parameters from the Nikkei 225 index.

scholarly journals Residue Sum Formula for Pricing Options under the Variance Gamma Model

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1143
Author(s):  
Pedro Febrer ◽  
João Guerra
Keyword(s):  
Complex Analysis ◽  
Asset Price ◽  
Option Price ◽  
Market Model ◽  
Call Option ◽  
Variance Gamma ◽  
Pricing Options ◽  
Variance Gamma Process ◽  
Multidimensional Complex Analysis ◽  
Multidimensional Complex

We present and prove a triple sum series formula for the European call option price in a market model where the underlying asset price is driven by a Variance Gamma process. In order to obtain this formula, we present some concepts and properties of multidimensional complex analysis, with particular emphasis on the multidimensional Jordan Lemma and the application of residue calculus to a Mellin–Barnes integral representation in C3, for the call option price. Moreover, we derive triple sum series formulas for some of the Greeks associated to the call option and we discuss the numerical accuracy and convergence of the main pricing formula.

scholarly journals HOLDER-EXTENDIBLE EUROPEAN OPTION: CORRECTIONS AND EXTENSIONS

2015 ◽  
Vol 56 (4) ◽  
pp. 359-372 ◽  
Author(s):  
PAVEL V. SHEVCHENKO
Keyword(s):  
Brownian Motion ◽  
Interest Rate ◽  
Real Life ◽  
Option Price ◽  
Geometric Brownian Motion ◽  
Fixed Amount ◽  
Underlying Asset ◽  
Closed Form Solutions ◽  
Pricing Options ◽  
Interest Rate Volatility

Financial contracts with options that allow the holder to extend the contract maturity by paying an additional fixed amount have found many applications in finance. Closed-form solutions for the price of these options have appeared in the literature for the case when the contract for the underlying asset follows a geometric Brownian motion with constant interest rate, volatility and nonnegative dividend yield. In this paper, option price is derived for the case of the underlying asset that follows a geometric Brownian motion with time-dependent drift and volatility, which is more important for real life applications. The option price formulae are derived for the case of a drift that includes nonnegative or negative dividend. The latter yields a solution type that is new to the literature. A negative dividend corresponds to a negative foreign interest rate for foreign exchange options, or storage costs for commodity options. It may also appear in pricing options with transaction costs or real options, where the drift is larger than the interest rate.

scholarly journals Uncertainty in Twenty-First-Century CMIP5 Sea Level Projections

2015 ◽  
Vol 28 (2) ◽  
pp. 838-852 ◽  
Author(s):  
Christopher M. Little ◽  
Radley M. Horton ◽  
Robert E. Kopp ◽  
Michael Oppenheimer ◽  
Stan Yip
Keyword(s):  
Sea Level ◽  
Model Uncertainty ◽  
Internal Variability ◽  
Climate System ◽  
First Century ◽  
The Arctic ◽  
Management Approach ◽  
Local Risk ◽  
Twenty First Century ◽  
Global And Local

Abstract The representative concentration pathway (RCP) simulations included in phase 5 of the Coupled Model Intercomparison Project (CMIP5) quantify the response of the climate system to different natural and anthropogenic forcing scenarios. These simulations differ because of 1) forcing, 2) the representation of the climate system in atmosphere–ocean general circulation models (AOGCMs), and 3) the presence of unforced (internal) variability. Global and local sea level rise projections derived from these simulations, and the emergence of distinct responses to the four RCPs depend on the relative magnitude of these sources of uncertainty at different lead times. Here, the uncertainty in CMIP5 projections of sea level is partitioned at global and local scales, using a 164-member ensemble of twenty-first-century simulations. Local projections at New York City (NYSL) are highlighted. The partition between model uncertainty, scenario uncertainty, and internal variability in global mean sea level (GMSL) is qualitatively consistent with that of surface air temperature, with model uncertainty dominant for most of the twenty-first century. Locally, model uncertainty is dominant through 2100, with maxima in the North Atlantic and the Arctic Ocean. The model spread is driven largely by 4 of the 16 AOGCMs in the ensemble; these models exhibit outlying behavior in all RCPs and in both GMSL and NYSL. The magnitude of internal variability varies widely by location and across models, leading to differences of several decades in the local emergence of RCPs. The AOGCM spread, and its sensitivity to model exclusion and/or weighting, has important implications for sea level assessments, especially if a local risk management approach is utilized.

LOCAL RISK-MINIMIZATION UNDER MARKOV-MODULATED EXPONENTIAL LÉVY MODEL

2015 ◽  
Vol 18 (05) ◽  
pp. 1550033
Author(s):  
OLIVIER MENOUKEU-PAMEN ◽  
ROMUALD MOMEYA
Keyword(s):  
Regime Switching ◽  
Risk Minimization ◽  
Martingale Representation ◽  
Option Hedging ◽  
Hedging Strategies ◽  
Local Risk ◽  
Hedging Problem ◽  
Markov Modulated ◽  
Martingale Representation Theorem ◽  
Minimization Approach

In this paper, the option hedging problem for a Markov-modulated exponential Lévy model is examined. We use the local risk-minimization approach to study optimal hedging strategies for Europeans derivatives when the price of the underlying is given by a regime-switching Lévy model. We use a martingale representation theorem result to construct an explicit local risk minimizing strategy.

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