The Pricing Of Options On Assets With Stochastic Volatilities, John Hull, Alan White, 1987

by Dale F. Gray, Robert C. Merton, Zvi Bodie 1.IntroductionThis paper proposes a new approach to improve the way central bank can analyze and manage the financial risks of a national economy. It is based on the modern theory and practice of contingent claims analysis (CCA). Vulnerability of a national economy to volatility in the global markets for credit, currencies, commodities, and other assets has become a central concern of policymakers. When applied to the credit risk, the CCA is commonly called the Merton Model. The basic analytical tool is the risk-adjusted balance sheet, which shows the sensitivity of the enterprise’s assets and liabilities to external “shocks”. At the national level, the sectors of an economy are viewed as interconnected portfolios of assets, liabilities, and guarantees----some explicit and some implicit. In addition to their traditional focus on inflation and output, central banks are increasingly focusing on the resilience of the national financial system.2.Contingent Claim AnalysisA contingent claim is any financial asset whose future payoff dependes on the value of another asset. There are three principles: a) the values of liabilities are derived from assets b) liabilities have different priority c) assets follow a stochastic process. As total assets decline, the value of risky debt declines and credit spreads on risky debt rise.Balance sheet risk is the key to understanding credit risk and crisis probabilities. Default happens when assets cannot service debt payments. For more info about Merton Model, see Box 1 Merton Model Equations for Pricing Contingent Claims & Figure 1.Financial fragility is intimately related to probability of default. Shocks to flows, prices, or liquidity frequently end up being converted into credit risk in a crisis. In addition, flow-of-funds and accounting balance sheets cannot provide measures of risk exposures which are forward-looking estimates of losses.3.Contingent Claim Balance sheets for SectorsWe view an economy as a set of interrelated balance sheets with four types of aggregate sectors----corporate, financial, household, and sovereign. In fact, passage 2 introduce a balance sheet with three sectors----corporate, financial, public sector (government and monetary authorities). Treating the corporate sector as one large firm and the financial sector as one large institution a very simplified way of looking at the balance sheet but we will initially start out with this stylized framework to illustrate risk characteristics of the sector for the purposes of this analysis.More info in Figure 2.4.Economy-wide Macro Contingent Claim BalanceSheets and Risk ExposuresBuilding upon the theory of contingent claims laid out above, the macrofinance valuation identities user put-call parity relationships, which state that the asset value A of each sector is equal to the value of its equity plus the value of its risky debt. Risky debt is equal to the default-free value of debt minus the value of the implicit put option ( the expected losses associated with the debt).For the corporate sector, the simplest formula with 4 factors.by Dale F. Gray, Robert C. Merton, Zvi Bodie For the financial sector, the support from government is added into its function with 4 factors.For the sovereign, passage 2 has more info. See in passage 2 Figure 2. Assets include: foreign currency reserves and contingent foreign currency reserves; net fiscal asset (present value of taxes and revenues, including seignorage, less present value of government expenditures); and other public assets. Liabilities include: local-currency debt; foreign-currency debt; financial guarantees; base money. Public sector is divided into two parts----government & monetary authority. A case can be made that foreign currency debt is senior to local currency debt.For the household sector, the formula is different. See more in Passage 2.Problem: How to understand “The household sector asset is equal to the household net worth plus c which is consumption modeled as a “dividend” payment out of the household asset up to time T”?5.Interrelationship of Macro Financial Contingent ClaimBalance Sheets, Risk Exposures and TraditionalMacroeconomic Flows.Flow of funds can be seen as a special deterministic case of the CCA balance sheet equations when volatility is set to zero and annual changes are calculated. The risk transmission between sectors is lost. Risk managers would find it difficult to analyze the risk exposure of their firm or financial institution by relying solely on the income an cashflow statements, and not taking into account balance sheet or information on their institution’s derivative or option positions. Country risk analysis that relies only on macroeconomic flow-based approach is deficient in a similar way.6.Measuring Implied Asset Value and Volatilities UsingMarket PricesFirms and Financial InstitutionsThe simplest method solves two equations for two unknowns, asset value and asset volatility.SovereignA simplified balance sheet (subtracting the “senior” guarantee to too-big-to-fail entities from both sides the balance sheet as shown). See more in passage 2.Household Sector Balance SheetIn the household sector, we can user macroeconomic data and info from household surveys to construct measures of the portfolio of household assets directly, for the most part, and try to estimate the volatility of household assets directly. Household balance sheet assets include financial assets and estimated labor income. For the household “subsidiary” balance sheet, direct estimation of the real estate prices, volatilities and debt obligations is likely to be the most practical approach.by Dale F. Gray, Robert C. Merton, Zvi Bodie 7.Some Important Extensions and Refinements of CCAModelsRecent research has studied the relationship between the volatility skew implied by equity options and CDS spreads. They establish a relationship between implied volatility of two equity options.Merton Model has been extended to include stochastic interest rates.8.Measuring Risk ExposuresRisk exposures in risky debt, probabilities of default, distance-to-distress, spreads on debt and the sensitivity of the implicit options to the change in the underlying asset and other measures.9.Risk Transmission Between SectorsThe risk-transmission patterns can be dampened or may be magnified depending on the capital structure and linkages.corporate to banking; banking to government; government to banking/financial system; Pension to government; sovereign to debt holders; markets to household and then to consumption; potential highly non-linear risk transmission when assets of one sector are linked (through implicit put options) to the assets of another sector.10.Balance Sheet Risk Framework for Stress Testing,Scenario, and Simulation AnalysisExample of financial stability stress-testing with CCA model, factor model and Macro Model.Example of Banking Stability Stress-testing with links to corporate and household sub-sectorsExample of Stress-Testing and Accessing Capital Adequacy Using CCA Models of Financial Institutions11.Integrating Financial Risk Models and Indicatorswith Macro ModelsCredit risk, and the market risk of the claims held in agent’s financial portfolios, are generally absent even in the majority of state-of-the-art macroeconomic models.In order to understand the interaction of balance sheet risk and the macroeconomy a promising area of future research is the integration of the financial risk analytic models and indicators with traditional macroeconomic models. Macroeconomic models are primarily stock-flow models in discrete time and are usually geared to try to forecast the mean of macroeconomic variables. Financial risk analytics is critical in risk analysis. CCA is a framework with volatile assets relative to a distress barrier using option pricing concepts to calculate credit risk indicators. Whatever level of aggregation is chosen, the time pattern of the following could be calculated and used with macroeconomic models:Time series of CCA balance sheet components and sensitivity measures.Time series of CCA derived credit risk indicators (distance-to-distress, estimated default probability, CCA credit spreads)by Dale F. Gray, Robert C. Merton, Zvi Bodie Time series of market indicators such as observed CDS and bond spreads or market risk appetite indicators (such as VIX)12.Financial Risk Analytic Indicators and MonetaryPolicy ModelsFinancial stability models and monetary stability models, by their nature, are very different framework. There is keen interest in relating these two types of analysis, but no consensus on how it can be done.The primary tool for macroeconomic management is the interest rate set by the central bank. A simple for module monetary policy model of this type consists of an equation for the GDP output gap, an equation for inflation, an equation for exchange rate and real interest rates, and a Taylor rule for setting the domestic policy rate. The domestic policy rate is a short-term interest rate set by the central bank, such as the Federal Funds rate in the United States.Since the economy and interest rates affects financial sector credit risk, and the financial sector affects the economy, an important issue is whether credit risk indicators should be included in monetary policy models and, if so, how. The next step could be to add a fifth equation relating the CRIs to GDP and interest rates. Using past data, it might be interesting to include the CRI in the policy rate reaction function to examine whetherfinancial stability appears to have been taken into account when setting interest rates in the past.13.conclusions。

Theoretically, day is the standard deviation of the continuously compounded return in one day In practice we assume that it is the standard deviation of the percentage change in one day
Options, Futures, and Other Derivatives, 8th Edition,
Copyright © John C. Hull 2012
4
Historical Simulation to Calculate the One-Day VaR
Create a database of the daily movements in all market variables.
…..
……
5,109.67 4,113.33
5,197.00 4,226.81
Nikkei 225 14,023.44 14,300.91 14,467.09 14,413.32
…… 12,159.59 12,006.53
Options, Futures, and Other Derivatives, 8th Edition,
…
…….
…….
…….
……..
499 10,831.43 5,079.84 4,125.61 12,115.90
……. 9,857.465
…….. 142.535
500 11,222.53 5,285.82 4,343.42 11,855.40 10,126.439 −126.439
Example of Calculation:

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
CP
S0 X
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– +
–+
– +
T
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r D
–++
–+ +
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Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull
7.4
American vs European Options
An American option is worth at least as much as the corresponding European option
7.11
Early Exercise
• Usually there is some chance that an American option will be exercised early
• An exception is an American call on a non-dividend paying stock
• Consider the following 2 portfolios: – Portfolio A: European call on a stock + PV of the strike price in cash – Portfolio B: European put on the stock + the stock
insurance against stock price falling below strike price

JohnCochraneAssetPricing：约翰Cochrane资产定价COPYRIGHT NOTICE:John Cochrane: Asset Pricingis published by Princeton University Press and copyrighted, ? 2000, by Princeton University Press. All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher, except for reading and browsing via the World Wide Web. Users are not permitted to mount this file on any network servers.For COURSE PACK and other PERMISSIONS, refer to entry on previous page. For more information, send e-mail to permissions@/doc/9eccf262d7bbfd0a795 63c1ec5da50e2524dd1fa.htmlPrefaceAsset pricing theory tries to understand the prices or values of claims to uncertain payments.A low price implies a high rate of return,so one can also think of the theory as explaining why some assets pay higher average returns than others.To value an asset,we have to account for the delay and for the risk of its payments.The effects of time are not too dif?cult to work out. However,corrections for risk are much more important determinants of many assets’valu es.For example,over the last50years U.S.stocks have given a real return of about9%on average.Of this,only about1%is due to interest rates;the remaining8%is a premium earned for holding risk.Uncertainty,or corrections for risk make asset pricing interesting and challenging.Asset pricing theory shares the positive versus normative tension present in the rest of economics.Does it describe the way the world does work,or the way the world should work?We observe the prices or returns of many assets.We can use the theory positively,to try to understand why prices or returns are what they are.If the world does not obey a model’s predictions,we can decide that the model needs improvement.However, we can also decide that the world is wrong,that some assets are“mis-pr iced”and present trading opportunities for the shrewd investor.This latter use of asset pricing theory accounts for much of its popularity and practical application.Also,and perhaps most importantly,the prices of many assets or claims to uncertain cash?ows are not observed,such as potential public or private investment projects,new?nancial securities, buyout prospects,and complex derivatives.We can apply the theory to establish what the prices of these claims should be as well;the answers are important guides to public and private decisions.Asset pricing theory all stems from one simple concept,presented in the?rst page of the?rst chapter of this book:price equals expected discounted payoff.The rest is elaboration,special cases,and a closetxiiixiv Preface full of tricks that make the central equation useful for one or another application.There are two polar approaches to this elaboration.I call them abso-lute pricing and relative pricing.In absolute pricing,we price each asset by reference to its exposure to fundamental sources of macroeconomic risk. The consumption-based and general equilibrium models are the purest examples of thisapproach.The absolute approach is most common in academic settings,in which we use asset pricing theory positively to give an economic explanation for why prices are what they are,or in order to predict how prices might change if policy or economic structure changed.In relative pricing,we ask a less ambitious question.We ask what we can learn about an asset’s value given the p rices of some other assets.We do not ask where the prices of the other assets came from,and we use as lit-tle information about fundamental risk factors as possible.Black–Scholes option pricing is the classic example of this approach.While limited in scope,this approach offers precision in many applications.Asset pricing problems are solved by judiciously choosing how much absolute and how much relative pricing one will do,depending on the assets in question and the purpose of the calculation.Almost no prob-lems are solved by the pure extremes.For example,the CAPM and its successor factor models are paradigms of the absolute approach.Yet in applications,they price assets“relative”to the market or other risk fac-tors,without answering what determines the market or factor risk premia and betas.The latter are treated as free parameters.On the other end of the spectrum,even the most practical?nancial engineering questions usually involve assumptions beyond pure lack of arbitrage,assumptions about equilibrium“mar ket prices of risk.”The central and un?nished task of absolute asset pricing is to under-stand and measure the sources of aggregate or macroeconomic risk that drive asset prices.Of course,this is also the central question of macroeco-nomics,and this is a particularly exciting time for researchers who want to answer thesefundamental questions in macroeconomics and?nance.A lot of empirical work has documented tantalizing stylized facts and links between macroeconomics and?nance.For example,expected returns vary across time and across assets in ways that are linked to macroeco-nomic variables,or variables that also forecast macroeconomic events;a wide class of models suggests that a“recession”or“?nancial distress”fac-tor lies behind many asset prices.Yet theory lags behind;we do not yet have a well-described model that explains these interesting correlations.In turn,I think that what we are learning about?nance must feed back on macroeconomics.To take a simple example,we have learned that the risk premium on stocks—the expected stock return less inter-est rates—is much larger than the interest rate,and varies a good dealPreface xv more than interest rates.This means that attempts to line investment up with interest rates are pretty hopeless—most variation in the cost of capi-tal comes from the varying risk premium.Similarly,we have learned that some measure of risk aversion must be quite high,or people would all borrow like crazy to buy stocks.Most macroeconomics pursues small devi-ations about perfect-foresight equilibria,but the large equity premium means that volatility is a?rst-order effect,not a second-order effect.Stan-dard macroeconomic models predict that people really do not care much about business cycles(Lucas[1987]).Asset prices reveal that they do—that they forego substantial return premia to avoid assets that fall in reces-sions.This fact ought to tell us something about recessions!This book advocates a discount factor/generalized method of moments view of asset pricing theory and associated empiricalprocedures.I summarize asset pricing by two equations:p t=E(m t+1x t+1)m t+1=f(data,parameters)where p t=asset price,x t+1=asset payoff,m t+1=stochastic discount factor.The major advantages of the discount factor/moment condition approach are its simplicity and universality.Where once there were three apparently different theories for stocks,bonds,and options,now we see each as special cases of the same theory.The common language also allows us to use insights from each?eld of application in other?elds.This approach allows us to conveniently separate the step of specify-ing economic assumptions of the model(second equation)from the step of deciding which kind of empirical representation to pursue or under-stand.For a given model—choice of f(·)—we will see how the?rst equa-tion can lead to predictions stated in terms of returns,price-dividend ratios,expected return-beta representations,moment conditions,continu-ous versus discrete-time implications,and so forth.The ability to translate between such representations is also very helpful in digesting the results of empirical work,which uses a number of apparently distinct but funda-mentally connected representations.Thinking in terms of discount factors often turns out to be much simpler than thinking in terms of portfolios.For example,it is easier to insist that there is a positive discount factor than to check that every possible portfolio that dominates every other portfolio has a larger price, and the long arguments over the APT stated in terms of portfolios are easy to digest when stated interms of discount factors.The discount factor approach is also associated with a state-space geometry in place of the usual mean-variance geometry,and this book emphasizes the state-space intuition behind many classic results.xvi Preface For these reasons,the discount factor language and the associated state-space geometry are common in academic research and high-tech practice.They are not yet common in textbooks,and that is the niche that this book tries to?ll.I also diverge from the usual order of presentation.Most books are structured following the history of thought:portfolio theory,mean-variance frontiers,spanning theorems,CAPM,ICAPM,APT,option pricing,and?nally consumption-based model.Contingent claims are an esoteric extension of option pricing theory.I go the other way around: contingent claims and the consumption-based model are the basic and simplest models around;the others are specializations.Just because they were discovered in the opposite order is no reason to present them that way.I also try to unify the treatment of empirical methods.A wide variety of methods are popular,including time-series and cross-sectional regres-sions,and methods based on generalized method of moments(GMM)and maximum likelihood.However,in the end all of these apparently differ-ent approaches do the same thing:they pick free parameters of the model to make it?t best,which usually means to minimize pricing errors;and they evaluate the model by examining how big those pricing errors are.As with the theory,I do not attempt an encyclopediccompilation of empirical procedures.The literature on econometric methods contains lots of methods and special cases(likelihood ratio analogues of common Wald tests;cases with and without risk-free assets and when factors do and do not span the mean-variance frontier,etc.)that are seldom used in practice.I try to focus on the basic ideas and on methods that are actually used in practice.The accent in this book is on understanding statements of theory,and working with that theory to applications,rather than rigorous or general proofs.Also,I skip very lightly over many parts of asset pricing theory that have faded from current applications,although they occupied large amounts of the attention in the past.Some examples are portfolio sep-aration theorems,properties of various distributions,or asymptotic APT. While portfolio theory is still interesting and useful,it is no longer a cor-nerstone of pricing.Rather than use portfolio theory to?nd a demand curve for assets,which intersected with a supply curve gives prices,we now go to prices directly.One can then?nd optimal portfolios,but it is a side issue for the asset pricing question.My presentation is consciously informal.I like to see an idea in its simplest form and learn to use it before going back and understanding all the foundations of the ideas.I have organized the book for similarly minded readers.If you are hungry for more formal de?nitions and back-ground,keep going,they usually show up later on.Preface xvii Again,my organizing principle is that everything can be traced back to specializations of the basic pricing equation p=E(mx).Therefore, after reading the?rst chapter,one can pretty much skip around and read topics in as much depth or order asone likes.Each major subject always starts back at the same pricing equation.The target audience for this book is economics and?nance Ph.D.stu-dents,advanced MBA students,or professionals with similar background.I hope the book will also be useful to fellow researchers and?nance pro-fessionals,by clarifying,relating,and simplifying the set of tools we have all learned in a hodgepodge manner.I presume some exposure to under-graduate economics and statistics.A reader should have seen a utility func-tion,a random variable,a standard error,and a time series,should have some basic linear algebra and calculus,and should have solved a maxi-mum problem by setting derivatives to zero.The hurdles in asset pricing are really conceptual rather than mathematical.。