scholarly journals Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Limin Tao ◽  
Liping Xu ◽  
Hani Jamal Sulaimani
Keyword(s):  
Nonlinear Differential Equations ◽  
Mathematical Finance ◽  
Investment Strategy ◽  
Financial Derivatives ◽  
Martingale Measure ◽  
Martingale Method ◽  
Hedging Strategy ◽  
Discrete State ◽  
Research Issues ◽  
Imperfect Market

Abstract The pricing and hedging of financial derivatives have become one of the hot research issues in mathematical finance today. In the case of non-risk neutrality, this article uses the martingale method and probability measurement method to study the pricing method and hedging strategy of financial derivatives. This paper also further studies the hedging strategy of financial derivatives in the incomplete market based on the BSM model and converts the solution of this problem into solving a vector on the Hilbert space to its closure. The problem of space projection is to use projection theory to decompose financial derivatives under a given martingale measure. In the imperfect market, the vertical projection theory is used to obtain the approximate pricing method and hedging strategy of financial derivatives in which the underlying asset follows the martingale process; the projection theory is further expanded, and the pricing problem of financial derivatives under the mixed-asset portfolio is obtained. Approximate pricing of financial derivatives; in the discrete state, the hedging investment strategy of financial derivatives H in the imperfect market is found through the method of variance approximation.

Arbitrage Theory in Continuous Time

2019 ◽  
Author(s):  
Tomas Björk
Keyword(s):  
Incomplete Markets ◽  
Continuous Time ◽  
Dynamic Equilibrium ◽  
Factor Model ◽  
Good Deal ◽  
Financial Derivatives ◽  
Equilibrium Theory ◽  
Martingale Measure ◽  
Fourth Edition ◽  
Optimal Stopping Theory

The fourth edition of this textbook on pricing and hedging of financial derivatives, now also including dynamic equilibrium theory, continues to combine sound mathematical principles with economic applications. Concentrating on the probabilistic theory of continuous time arbitrage pricing of financial derivatives, including stochastic optimal control theory and optimal stopping theory, the book is designed for graduate students in economics and mathematics, and combines the necessary mathematical background with a solid economic focus. It includes a solved example for every new technique presented, contains numerous exercises, and suggests further reading in each chapter. All concepts and ideas are discussed, not only from a mathematics point of view, but the mathematical theory is also always supplemented with lots of intuitive economic arguments. In the substantially extended fourth edition Tomas Björk has added completely new chapters on incomplete markets, treating such topics as the Esscher transform, the minimal martingale measure, f-divergences, optimal investment theory for incomplete markets, and good deal bounds. There is also an entirely new part of the book presenting dynamic equilibrium theory. This includes several chapters on unit net supply endowments models, and the Cox–Ingersoll–Ross equilibrium factor model (including the CIR equilibrium interest rate model). Providing two full treatments of arbitrage theory—the classical delta hedging approach and the modern martingale approach—the book is written in such a way that these approaches can be studied independently of each other, thus providing the less mathematically oriented reader with a self-contained introduction to arbitrage theory and equilibrium theory, while at the same time allowing the more advanced student to see the full theory in action.

scholarly journals Implicit incentives for fund managers with partial information

2021 ◽  
Author(s):  
Flavio Angelini ◽  
Katia Colaneri ◽  
Stefano Herzel ◽  
Marco Nicolosi
Keyword(s):  
Asset Allocation ◽  
Partial Information ◽  
Market Price ◽  
Investment Strategy ◽  
Expected Returns ◽  
Martingale Method ◽  
Fund Manager ◽  
Market Price Of Risk ◽  
Fund Managers ◽  
Price Of Risk

AbstractWe study the optimal asset allocation problem for a fund manager whose compensation depends on the performance of her portfolio with respect to a benchmark. The objective of the manager is to maximise the expected utility of her final wealth. The manager observes the prices but not the values of the market price of risk that drives the expected returns. Estimates of the market price of risk get more precise as more observations are available. We formulate the problem as an optimization under partial information. The particular structure of the incentives makes the objective function not concave. Therefore, we solve the problem by combining the martingale method and a concavification procedure and we obtain the optimal wealth and the investment strategy. A numerical example shows the effect of learning on the optimal strategy.

Hedging of Variable Annuities under Basis Risk

2020 ◽  
Vol 14 (2) ◽  
Author(s):  
Jan Bauer
Keyword(s):  
Financial Market ◽  
Negative Impact ◽  
Mathematical Finance ◽  
Insurance Company ◽  
Hedging Strategy ◽  
Variable Annuities ◽  
Basis Risk ◽  
Hedging Strategies ◽  
Cross Hedging ◽  
Geometric Brownian Motions

AbstractI study dynamic hedging for variable annuities under basis risk. Basis risk, which arises from the imperfect correlation between the underlying fund and the proxy asset used for hedging, has a highly negative impact on the hedging performance. In this paper, I model the financial market based on correlated geometric Brownian motions and analyze the risk management for a pool of stylized GMAB contracts. I investigate whether the choice of a suitable hedging strategy can help to reduce the risk for the insurance company. Comparing several cross-hedging strategies, I observe very similar hedging performances. Particularly, I find that well-established but complex strategies from mathematical finance do not outperform simple and naive approaches in the context studied. Diversification, however, could help to reduce the adverse impact of basis risk.

scholarly journals Mean-Variance Hedging Under Additional Market Information

2003 ◽  
Vol 06 (06) ◽  
pp. 613-636 ◽  
Author(s):  
F. Thierbach
Keyword(s):  
Martingale Measure ◽  
Option Prices ◽  
Market Information ◽  
Hedging Strategy ◽  
No Arbitrage ◽  
Finite Set ◽  
The Mean ◽  
Hedging Problem ◽  
Equivalent Martingale ◽  
Mean Variance

In this paper we analyze the mean-variance hedging approach in an incomplete market under the assumption of additional market information, which is represented by a given, finite set of observed prices of non-attainable contingent claims. Due to no-arbitrage arguments, our set of investment opportunities increases and the set of possible equivalent martingale measures shrinks. Therefore, we obtain a modified mean-variance hedging problem, which takes into account the observed additional market information. Solving this we obtain an explicit description of the optimal hedging strategy and an admissible, constrained variance-optimal signed martingale measure, that generates both the approximation price and the observed option prices.

QUADRATIC HEDGING FOR THE BATES MODEL

2007 ◽  
Vol 10 (05) ◽  
pp. 873-885 ◽  
Author(s):  
FRIEDRICH HUBALEK ◽  
CARLO SGARRA
Keyword(s):  
Option Pricing ◽  
Explicit Expression ◽  
Martingale Measure ◽  
Risk Minimization ◽  
Hedging Strategy ◽  
Preliminary Results ◽  
Quadratic Hedging ◽  
The Mean ◽  
Mean Variance ◽  
Martingale Case

In the present paper we give some preliminary results for option pricing and hedging in the framework of the Bates model based on quadratic risk minimization. We provide an explicit expression of the mean-variance hedging strategy in the martingale case and study the Minimal Martingale measure in the general case.

Change of Numeraire

2019 ◽  
pp. 202-218
Author(s):  
Tomas Björk
Keyword(s):  
General Theory ◽  
Option Pricing ◽  
General Formula ◽  
Financial Derivatives ◽  
Martingale Measure ◽  
Girsanov Transformation ◽  
Suitable Change ◽  
Pricing Formula ◽  
Exchange Option ◽  
Numeraire Portfolio

In this chapter we discuss how a suitable change of numeraire and the corresponding change of martingale measure, can simplify the computation of pricing formula for financial derivatives. We derive a general formula for the likelihood process related to an arbitrary numeraire, and we identify the corresponding Girsanov transformation. As an example, we compute the price of an exchange option. In particular we study the class of forward measures related to zero coupon bonds and we derive a general option pricing formula. As an application of the general theory we also study the so-called numeraire portfolio.

Using VIX to Dynamically Hedge Portfolio Risk

GIS Business ◽  
1970 ◽  
Vol 13 (3) ◽  
pp. 15-22
Author(s):  
Richard Cloutier
Keyword(s):  
Asset Allocation ◽  
Market Timing ◽  
Investment Strategy ◽  
Market Risk ◽  
Portfolio Risk ◽  
Dynamic Hedging ◽  
Hedging Strategy ◽  
Tactical Asset Allocation ◽  
Asset Class ◽  
Timing Strategies

Many investors accept buy and hold as their long-term investment strategy. However, during periods of heightened risk, staying disciplined can be problematic. Alternatively, market timing appeals to our emotions but is very difficult to employ successfully. Between these two extremes lies tactical asset allocation, where limited variances are allowed to take advantage of market conditions. Dynamic hedging is a form of tactical asset allocation. Instead of relying on future predictions of asset class returns, dynamic hedging strives to reduce portfolio risk when market risk is elevated. This paper presents a dynamic hedging strategy developed to accomplish this goal. It uses VIXs normal trading range to assess market risk. When VIX trades above its normal trading range and the upper Bollinger band, the dynamic hedging strategy is applied. The result is that portfolio risk is lowered when market risk is extreme. The application of this strategy provides better returns, lower volatility, and better downside protection than a strategic buy and hold allocation. It also avoids the deployment problems associated with market timing strategies.

A SHOT NOISE MODEL FOR FINANCIAL ASSETS

2008 ◽  
Vol 11 (01) ◽  
pp. 87-106 ◽  
Author(s):  
TIMO ALTMANN ◽  
THORSTEN SCHMIDT ◽  
WINFRIED STUTE
Keyword(s):  
Simulation Study ◽  
Shot Noise ◽  
Stock Prices ◽  
Continuous Time ◽  
Noise Model ◽  
Martingale Measure ◽  
Financial Assets ◽  
Hedging Strategy ◽  
Noise Effects ◽  
Abrupt Changes

In this article we propose and study a model for stock prices which allows for shot-noise effects. This means that abrupt changes caused by jumps may fade away as time goes by. This model is incomplete. We derive the minimal martingale measure in discrete and continuous time and discuss the associated hedging strategy. Finally, a simulation study is included to show that our model is able to produce smile effects.

A More General One Period Model

2019 ◽  
pp. 27-40
Author(s):  
Tomas Björk
Keyword(s):  
Fundamental Theorem ◽  
Financial Derivatives ◽  
Sample Space ◽  
Martingale Measure ◽  
Finite Sample ◽  
Equivalent Martingale Measure ◽  
No Arbitrage ◽  
Equivalent Martingale ◽  
Absence Of Arbitrage ◽  
Period Model

In this chapter we study a general one period model living on a finite sample space. The concepts of no arbitrage and completeness are introduced, as well as the concept of a martingale measure. We then prove the First Fundamental Theorem, stating that absence of arbitrage is equivalent to the existence of an equivalent martingale measure. We also prove the Second Fundamental Theorem which says that the market is complete if and only if the martingale measure is unique. Using this theory, we derive pricing and hedging formulas for financial derivatives.

PORTFOLIO ALLOCATION IN A LEVY-TYPE JUMP-DIFFUSION MODEL WITH NONLIFE INSURANCE RISK

2021 ◽  
Vol 24 (01) ◽  
pp. 2150005
Author(s):  
RAFAEL SERRANO
Keyword(s):  
Control Variable ◽  
Jump Diffusion ◽  
Investment Strategy ◽  
Martingale Method ◽  
Insurance Risk ◽  
Worst Case ◽  
Diffusion Dynamics ◽  
Insurance Portfolio ◽  
Income Ratio ◽  
Jump Diffusion Model

We propose a model that integrates investment, underwriting, and consumption/dividend policy decisions for a nonlife insurer by using a risk control variable related to the wealth-income ratio of the firm. This facilitates the efficient transfer of insurance risk to capital markets since it allows to select simultaneously investments and underwriting volume. The model is particularly valuable for business lines with significant exposure to extreme events and disaster risk, as it accounts for features usually depicted during negative economic shocks and catastrophic events, such as Levy-type jump-diffusion dynamics for the financial log-returns that are in turn correlated with insurance premiums and liabilities, as well as worst-case scenarios in which policyholders in the insurance portfolio report claims with the same severity simultaneously. Using the martingale method, we determine an optimal solvency threshold or wealth-income ratio, and investment strategy that maximizes the expected utility from dividend payouts that follows a (possibly stochastic) consumption clock. We illustrate the main results with numerical examples for log- and power-utility functions, and (bounded variation) tempered stable Levy jumps.

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