H.F¨o llmer,A.Schied:Stochastic Finance–An Introduction in Discrete Time.de Gruyter Studies in Mathematics27,Walter de Gruyter,Berlin,New York,2002,IX+422 pages,Hardcover.ISBN3-11-017119-8.
This book can be regarded as one of the standard text books offinancial mathematics in discrete time.Mathematically,it is more demanding than other introductions,like Pliska’s“Introduction to Mathematical Finance”,while on the other hand it avoids continuous-time markets and thus also Itˆo-calculus altogether so that the prerequisites are still relatively low.
The book is structured into two large parts:The one-period model in Part I and dynamic hedging in the more advanced multi-period model in Part II.While the one-period model is very simple,it still displays almost all of the fundamental concepts of the no-arbitrage theory.The multi-period model in Part II on the other hand allows for a restructuring of the portfolio and thus allows one to adjust to newly-available information at time t, modelled by afiltration F t.In order to replicate a given payoff,this is even required,and the theory of dynamic hedging deals with the question how to restructure the portfolio at each intermediate time.
Part I(“Mathematicalfinance in one period”)starts with a general introduction to the no-arbitrage theory in one he market model that is bad on the assumption that there should be no possib
ility of a riskless gain.Martingale measures are introduced and the connection between the existence of martingale measures and the abnce of arbitrage is shown,as well as their u for pricing derivative curities. Also,the completeness of a market is discusd and its equivalence to the uniqueness of the risk-neutral measure is shown in the one-period model.
In Chapter2follows the introduction of subjective risk evaluation,using utility func-tions and comparing different portfolios using their expected utilities.In contrast to other introductions to expected utility,this chapter also focus on uniform preferences, meaning that the particular choice of a utility function is not relevant for an investment strategy to be chon over another.placket
Chapter3deals with the question of portfolio optimization for a given initial wealth. First,uniqueness of the maximizer is considered and relative entropy methods for special utility functions are prented.The rest of the chapter deals with the classical portfolio optimization methods and with the theory of microeconomic market equilibrium(Arrow-Debreu equilibrium).
Thefinal chapter of thefirst part,Chapter4,gives an introduction to risk measures in the one-period model,introducing convex and coherent measures of risk.A large part of this chapter is dedicated to the most common(but unfortunately not coherent)risk measure,Value at Risk,and its generalizations to form coherent risk measures.
Part II(“Dynamic hedging”)develops a dynamic version of the market model and the corresponding portfolio theory bad on the no-arbitrage condition.
Chapter5generalizes the one-period market model of Chapter1to the classical discrete-time multi-period model.The structure of this fundamental chapter is straightforward: After a definition of the basic properties like lf-financing trading strategies and its con-quences,the connection between arbitrage opportunities and the lack of an equivalent martingale measure is shown.Using martingale measures,simple European-type con-tingent claims(only exercisable at thefinal time T)can now be introduced and priced
using expectation under an equivalent martingale measure.After a quick discussion of market completeness,the Cox-Ross-Rubinstein binomial model is prented as the sim-plest multi-period market model.Using the proper rescaling and taking limits in the binomial model,the chapter ends with the convergence of the binomial model to the Black-Scholes model in continuous time.
开学主题As the multi-period model allows a richer structure of contingent claims than European-type claims,Chapter6discuss American-type contingent claims,which the holder can also exerci at certain times before T.Hedging strategies for the ller using Doob’s decomposition theorem for the s
王字旁加深的右边upermartingale price process are discusd as well as stopping strategies for the ller of American-type claims.Using the Snell envelope and stopping times,the optimal exerci time is investigated and arbitrage-free prices are determined using the methods developed in the previous chapters.相思之情的诗句
最懂你的人Thefinal four chapters of the book discuss the problem of hedging in incomplete markets, where the martingale measure–and thus also the arbitrage-free price–is not unique. In incomplete markets,a claim can not necessarily be replicated exactly,so Chapter7 deals with superhedging,which means to replicate a claim that generates at least the required amount in every possible market state.The methods ud are again super-martingales in connection with Snell envelopes and the Doob decomposition.
Chapter8prents efficient hedging techniques,as superhedging strategies are always on the safe side and thus require unnecessarily high prices.Using various measures of risk, efficient hedging results in hedging strategies that not necessarily generate the required amount in every possible market state,but still with a high probability.This is discusd using the quantile method,which corresponds to the Value at Risk,and the method of minimal shortfall risk.
When the trading of asts and thus the restructuring of the portfolio is not always possible as it was
assumed in the frictionless market of thefirst eight chapters,veral convex trading constraints are introduced into the market model.Their effects on the abnce of arbitrage are discusd in Chapter9,as well as superhedging in markets with friction.diyt恤
Thefinal Chapter10takes an alternative approach to hedging in incomplete markets by minimizing the quadratic hedging error.Another approach prented is the variance-optimal hedging method.
批判思维
计算机实习总结In summary,the book is a very good introduction to mathematicalfinance in the discrete-time tting.The structure of the book follows the well-established pattern of typical financial mathematics cours and books.It is well suited for an advanced introductury lecture onfinancial mathematics,although the lack of exercis is a significant drawback in that regard.Thefinal chapters deal with more advanced topics that might be treated in a subquent lecture.
Reinhold Kainhofer,Vienna
