• Know the various measures for measures for calculating
  portfolio VAR including diversified VAR and undiversified VAR.
• Be familiar with the computation of incremental VAR and
  component VAR. Incremental VAR is the change in VAR given an
  addition to the portfolio. Component VAR is used to calculate the
  change in portfolio VAR from deleting a position.
• Know that using VAR as a risk budgeting technique is becoming
  more popular than using historical measures of risk. Investors are
  relying more on VAR because of increased globalization, complexity,
  and dynamics of the investment industry.
• Memorize the formulas for expected return and the variance of a
  two-asset portfolio. A question asking you to calculate the
  variance could give you the correlation or covariance between two
  assets. Know the relationship between the correlation and
  covariance.
• Learn the concepts of beta, the security market line, the CAPM,
  and how they are related to the expected rate of return.
• Memorize the formula for the CAPM.
• Be familiar with the forms of market efficiency.
• Know how to calculate and interpret the Treynor measure, Sharpe
  measure, Jensen’s alpha, tracking error, the information ratio, and
  the Sortino ratio. Memorizing some relatively simple formulas is in
  order here.
• Know the difference between returns-based and portfolio-based
  style analysis.
• Be able to calculate the three components of active systematic
  returns.
• Be able to compare and contrast mutual funds and hedge funds.
  Hedge funds have been growing at a higher rate recently; however,
  this growth will likely slow with changes in regulation.
• Know the two methods used to replicate hedge fund returns(i.e.,
  how hedge fund portfolios are "cloned"). The two methods are
  fix-weight and rolling-window clones.
• Reading "Individual Hedge Fund Strategies" is a great way to
  learn about the various styles of hedge funds in the marketplace.
  If asked about a specific type of hedge fund, be able to describe
  general performance and risk characteristics about that style.
• Selecting a hedge fund manager involves strategy/sector
  selection, individual manager selection, manager due diligence, and
  ongoing risk management. Be familiar with the main considerations
  in each of these areas.
• Know how style drift for a hedge fund differs from style drift
  for traditional long-only investments and be aware of the potential
  red flags for style drift.
• Know the various forms of risk for a fund of hedge funds. A
  fund of hedge funds is exposed to risks at both the investment and
  portfolio levels.
• The issue of hedge fund transparency is a hot topic in the
  marketplace.
• Know when transparency is and is not appropriate for a fund,
  and know the four main attributes of transparency.
• Know that the best way to present hedge fund returns is to
  compare the overall performance of the fund with the performance of
  the underlying risk premium returns. Also, be familiar with
  attributes of a good hedge fund index and the problems associated
  with hedge fund indices.
• Be cognizant of the recommendations set forth by the
  President’s Working Group on Financial Markets(PWG). PWG has
  provided recommendations for investors, fiduciaries, supervisors,
  and key creditors and counterparties regarding private pool of
  capital.

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• The Bank of International Settlement(BIS) defines operational
  risk as the risk of losses due to inadequate or failed pocesses,
  persons, and systems that are unable to protect a firm from outside
  events. This definition focuses on the impact of operational
  losses.
• Be familiar with the methodologies for measuring operational
  risk and calculating capital charges, The approaches include basic
  indicator, standadized, and advanced measurement.
• There are two categories used to describe operational losses.
  Low frequency, high severity(LFHS) risks are the greatest area of
  concern for operational risk managers because there is little data
  available to study such risks, and their cost to the firm
  could be catastrophic. Expect a test question in some form
  regarding this concept.
• Distinguish between top-down and bottom-up operational risk
  models and know the strengths and weaknesses of each approach. You
  should also know examples of operational risk models for each
  approach.
• Know the characteristics of catastrophe options and catastrophe
  bonds.
• Be familiar with the general structure of loss distribution
  approach(LDA) models and understand the application of frequency
  and severity distributions when modeling losses.
• Have a general understanding of the types of distributions that
  are used to model operational risk severity with LDA models.
• Know the two risks of implementing technological
  innovations.
• Know the difference between economies of scale and economies of
  scope.
• Daylight overdraft risk is a significant risk for the banking
  system. Know this concept.
• Model risk is the risk associated with using financial models
  to simulate complex relationships. It may arise from incorrect
  model application, implementation risk, calibration errors,
  programming errors, and data problems. Managers can protect against
  model risk by employing reality checks of the model.
• Know that under an EMH framework, the best way to manager model
  risk is to find a better model. Under a non-EMH framework, the
  focus is on how current pricing methodologies will change in the
  future.
• Read and be familiar with the various case studies presented,
  including Metallgesellschaft, Sumitomo, Barings, and Long-Term
  Capital Management. Lessons from history are important, and as a
  potential certified Financial Risk Manager, GARP will test your
  knowledge of the factors that caused the enormous losses in these
  cases, hoping that your knowledge will prevent history from
  repeating itself. Be able to cite examples of risk controls that
  could have been in place to stop these disasters from
  occurring.
• Be familiar with the various ways corporations can manage risk
  through the use of an Enterprise Risk Management(ERM) process. ERM
  can be used to better carry out a company’s strategic plan, gain a
  competitive advantage, and create shareholder value.
• The formula for the RAROC of a loan is one of the few equations
  in this section that you need to memorize. You should also know the
  formula for adjusted RAROC(ARAROC), which overcomes the deficiency
  in RAROC of assuming the probability of default is constant.
• Know that liquidity risk is comprised of funding risk and
  market liquidity risk. Alternatives to measuring liquidity risk
  include: liquidity gap and liquidity risk
  elasticity.
• Be aware that VAR can be adjusted for liquidity risk(LVAR) by
  incorporating the impact of the bid-ask spread.
• Review the recommendations and guiding principles of the
  Counterparty Risk Management Policy Group II report. Understand
  that it was drafted with the goal of promoting global financial
  stability. Do not memorize all the recommendations, but rather
  focus on the big picture concepts for each category and recognize
  that much of the material addressed is related to other areas of
  the FRM curriculum.
• Be familiar with the four primary justifications for the
  existence of banking regulation. They include:
  protect bank depositor from a loss in bankruptcy, provide stability
  for transactions, avoid domino effects on the banking system, and
  maintain stability in the economy.
• The new Basel Capital Accord is very important. Questions about
  Basel are the closest to a "sure thing" there is for the FRM exam.
  Read the Basel material more than once to make sure you understand
  the big picture of the Accord’s purpose, as well as the many
  details in this material that may show up as test questions.
• Be able to define what is included in tier 1, tier 2, and tier
  3 capital. Ways you could see this material tested on the exam
  include giving you a balance sheet and asking you to calculate one
  of the forms of capital, or asking you which asset mix would best
  satisfy Basel’s rules regarding the capital used to satisfy capital
  requirements.
• Be able to relate to the expectation of the amount of overall
  capital that should result, either through the standardized
  approach or the internal ratings-based(IRB) approach for the
  risk-weighted capital calculations.
• Know which factors the bank calculates internally for the
  fundation and advanced IRB approaches.
• Know the difference between the simple and comprehensive
  approaches of addressing credit risk mitigation practices.
• Securitization exposures under Basel II are addressed
  differently than other exposures; be able to explain how they are
  addressed.
• Know the three methods for addressing operational risk under
  Basel II. Be careful not to confuse the terminology of the
  operational risk approaches with the terms used for credit
  risk.
• Be familiar with the four principles of the new Accord’s Second
  Pillar. Read about the three pillars of the new Basel II
  Accord.
• Know that current regulations regarding financial conglomerates
  require updating. Current regulation views each subsidiary of a
  financial conglomerate independently. To better handle the unique
  circumstances of financial conglomerates, a 3+1 Pillar framework
  was established.

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08年FRM考试Credit Risk Mgt.考点解析

• Know how to compute the cumulative default probability over a
  multiyear period given the marginal default probability for each
  year. Be able to do this computation in both directions. The
  computation could be turned around if you were given a 1-year
  cumulative default  probability and asked to solve for the
  quarterly marginal default probability.
• Note that if the rates on the corporate bond and Treasury bond
  are equivalent, the implied probability of repayment is 100%.
  Therefore, since corporate bond rates exceed Treasury bond rates,
  realistic implied probabilities of repayment will be less than
  100%.
• When calculating the probability of default, be sure to use
  debt securities of the same maturity. Mismatching maturities will
  give incorrect implied probabilities. Also, if p equals
  the implied probability of repayment, then 1-p equals the
  implied probability of default. Know how to make adjustments on a
  quarterly, semmiannual, or annual basis.
• Memorize the fomula for the contractually promised gross return
  on a loan. Remember that this is a promised return, which is
  potentially different from the actual or realized return.
• Sovereign risk is the risk that a foreign government may
  default on a loan. There are five key variables that measure the
  probability of a foreign government rescheduling its debt payments;
  focus on the interpretation of these variables. Review the example
  for a multiyear restructuring agreement(MYRA).
• Memorize the formulas for expected loss and unexpected loss and
  be able to apply these concepts to a portfolio setting.
• Understanding the relationship between commitments and
  outstandings is vital toward correctly computing adjusted exposure
  which is utilized in the calculation of expected and unexpected
  loss.
• Know the difference between current, potential, and peak
  exposure for a derivative contract. Current and potential exposures
  are both essential for properly assessing credit risk.
• Know the difference between right-way and wrong-way exposures
  and be able to cite examples of each.
• The various forms of credit risk mitigants are important. Make
  sure you are familiar with terminology such as credit triggers,
  netting agreements, and liquidity puts.
• A credit valuation adjustment(CVA) adjusts payments to reflect
  changes in credit risk changes relative to the counterparties in
  derivative transactions.
• Understand the difference between the mean loss rate and the
  risk-neutral mean loss rate. The risk-neutral mean loss is an
  artificially higher loss rate that makes an investor indifferent
  between buying a risky security and a risk-free security with the
  same expected payoff. The risk-neutral mean loss rate is a key
  input to many credit risk pricing applications.
• Know the difference between managing the risks of a derivatives
  portfolio and managing the risks associated with traditional bank
  lending. Derivatives contracts have features that allow the fim to
  offset risks between counterparties or with a single
  counterparty(netting). Understand the various strategies used to
  offset these risks.
• Be familiar with purposes of external and internal ratings and
  know the process that is conducted to arrive at a particular
  rating.
• Learn the differences in default rate computations with various
  credit scoring models.
• Know that seniority and collateralization are two important
  factors that determine the recovery rates of bonds.
• Understand how the Merton model assesses the value of a firm’s
  stocks and bonds. Know that the Merton model has some unrealistic
  assumptions, and that the KMV model uses the same basic approach,
  but relaxes some of the assumptions.
• The value of a firm’s equity can be viewed as a long call on
  the firm’s assets with a strike price equal to the value of a
  firm’s debt, while the value of a firm’s debt can be viewed as a
  risk-free bond and a short put on the firm’s assets.
• Do not get lost with the ugly equations in the Merton
  model/firm value material. Focus on the concepts and the
  relationships between Merton model inputs and the value of debt and
  equity . The value of subordinate debt under high and low firm
  values is also important.
• Vulnerable options are options with default risk. Know how
  correlation impacts vulnerable options.
• Be familiar with the various credit risk portfolio models.
  Focus your attention on CreditMetrics, CreditPortfolio View, and
  CreditRisk+.
• Know the definitions of the numerous portfolio risk indicators.
  They include: expected loss, unexpected loss, VAR,
  economic capital, and expected shortfall.
• Be able to differentiate among credit derivative
  instruments(credit default swap, total return swaps, and structured
  instruments such as credit linked notes). All of those instruments
  provide the transfer of credit risk. Recognize that with a first to
  default swap, payment is only based on the first loan in a basket
  defaulting. First to default swaps and total return swaps seem to
  be a popular source of GARP questions.
• Be familiar with the characteristics of synthetic structures
  such as CLOs and CDOs(collateralized debt obligations) and review
  the variants of these two instruments, such as cash CDOs.
• Strategic capital allocation is the process of determining
  customized return objectives for different business units of the
  bank. Be familiar with the six methods of allocating economic
  capital, which include: stand-alone, scaling,
  internal betas, margnical capital, Arbitrage Pricing Theory, and
  fair value.
• Be familiar with the players in the securitization marketplace
  and explain their motivation for securitizing assets.
• It is important that FRM candidates know how securitization may
  affect the financial condition  of the originator.
• Understand the frictions that exist in the subprime mortgage
  market. Expect a question or two on subprime mortgages given the
  relevance to the current U.S. housing market troubles.

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• Debt
  service ratio（债务/出口）DSR+
• Import ratio （进口系数）IR+
• Investment ratio （投资系数）INVR_OR+
• Variance of export revenue （出口波动）VAREX+
• Domestic money supply growth （国内货币增长）MG+

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Contractually promised gross loan return

• Base lending rate plus
  margin: likely the bank’s cost of
  capital plus a profit margin. Note this pricing already
  includes expected losses (EL) on the loan. The promised
  return is greater than the expected return due to default risk; the
  expected losses due to default risk are priced into a component of
  the margin.
• Origination fee
• Compensation balance (a.k.a., offsetting
  balance): held by bank
• Reserve requirement: This is a
  determined by regulators. Reserve requirements are one of central
  banks’ tools for influencing the demand for liquidity.
  Varies by country and, within country, often by complex rules
  pertaining to type/size of deposits. Could be 0%, 2%, 5%, 10%.

• 8% base lending rate (BR) + 2% margin (m) = 10% loan rate
• Origination fee (f): 0.125% (one-eight of one
  point)
• Compensation balance (b): 10%
• Reserve requirement (R): 5%

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Details

Varieties of VaR

The definition of VaR is nonconstructive,
it specifies a
property VaR must have, but not how to compute VaR. Moreover,
there is wide scope for interpretation in the
definition.^[8]
This has led to two broad types of VaR, one used primarily in
risk
management and the other primarily for risk measurement. The
distinction is not sharp, however, and hybrid versions are
typically used in financial control,

financial reporting and computing regulatory
capital.
^[9]

To a risk manager, VaR is a system, not a number. The system
is run periodically (usually daily) and the published number is
compared to the computed price movement in opening positions over
the time horizon. There is never any subsequent adjustment to the
published VaR, and there is no distinction between VaR breaks
caused by input errors (including
Information Technology breakdowns, fraud and
rogue
trading), computation errors (including failure to produce a
VaR on time) and market movements.^[10]

A
frequentist claim is made, that the long-term frequency of VaR
breaks will equal the specified probability, within the limits of
sampling error, and that the VaR breaks will be
independent in time and
independent of the level of VaR. This claim is validated by a
backtest,
a comparison of published VaRs to actual price movements. In this
interpretation, many different systems could produce VaRs with
equally good backtests,
but wide disagreements on daily VaR values.^[1]

For risk measurement a number is needed, not a system. A

Bayesian probability claim is made, that given the information
and beliefs at the time, the
subjective probability of a VaR break was the specified level.
VaR is adjusted after the fact to correct errors in inputs and
computation, but not to incorporate information unavailable at the
time of computation.^[7]
In this context, “backtest”
has a different meaning. Rather than comparing published VaRs to
actual market movements over the period of time the system has been
in operation, VaR is retroactively computed on scrubbed data over
as long a period as data are available and deemed relevant. The
same position data and pricing models are used for computing the
VaR as determining the price movements.^[2]

Although some of the sources listed here treat only one kind
of VaR as legitimate, most of the recent ones seem to agree that
risk management VaR is superior for making short-term and tactical
decisions today, while risk measurement VaR should be used for
understanding the past, and making medium term and strategic
decisions for the future. When VaR is used for financial
control or
financial reporting it should incorporate elements of both. For
example, if a trading
desk is held to a VaR limit, that is both a risk-management
rule for deciding what risks to allow today, and an input into the
risk measurement computation of the desk’s
risk-adjusted return
at the end of the reporting period.^[4]

VAR in Governance

An interesting takeoff on VaR is its application in Governance
for endowments, trusts, and pension plans. Essentially trustees
adopt portfolio Values-at-Risk metrics for the entire pooled
account and the diversified parts individually managed. Instead of
probability estimates they simply define maximum levels of
acceptable loss for each. Doing so provides an easy metric for
oversight and adds accountability as managers are then directed to
manage, but with the additional constraint to avoid losses within a
defined risk parameter. VAR utilized in this manner adds relevance
as well as an easy to monitor risk measurement control far more
intuitive than Standard Deviation of Return. Use of VAR in this
context, as well as a worthwhile critque on board governance
practices as it relates to investment management oversight in
general can be found in ‘Best Practices in
Governance".^[11]

Risk measure and risk metric

The term “VaR” is used both for a risk
measure and a risk metric. This sometimes leads to confusion.
Sources earlier than 1995 usually emphasize the risk measure, later
sources are more likely to emphasize the metric.

The VaR risk measure defines risk as
mark-to-market loss on a fixed portfolio over a fixed time
horizon, assuming normal markets. There are many alternative risk
measures in finance. Instead of mark-to-market, which uses market
prices to define loss, loss is often defined as change in fundamental
value. For example, if an institution holds a loan
that declines in market price because interest
rates go up, but has no change in cash flows or credit quality,
some systems do not recognize a loss. Or we could try to
incorporate the economic
cost of things not measured in daily
financial statements, such as loss of market confidence or
employee morale, impairment of brand names or
lawsuits.^[4]

Rather than assuming a fixed portfolio over a fixed time
horizon, some risk measures incorporate the effect of expected
trading (such as a stop
loss order) and consider the expected holding period of
positions. Finally, some risk measures adjust for the possible
effects of abnormal markets, rather than excluding them from the
computation.^[4]

The VaR risk metric summarizes the
distribution of possible losses by a quantile, a point with a
specified probability of greater losses. Common alternative metrics
are standard
deviation, mean absolute
deviation, expected
shortfall and downside
risk.^[1]

VaR risk management

Supporters of VaR-based risk management claim the first and
possibly greatest benefit of VaR is the improvement in systems
and modeling it forces on an institution. In 1997, Philippe Jorion
wrote:^[12]

[T]he greatest benefit of VAR lies in the imposition of a
structured methodology for critically thinking about risk.
Institutions that go through the process of computing their VAR are
forced to confront their exposure to financial risks and to set up
a proper risk management function. Thus the process of getting to
VAR may be as important as the number itself.

Publishing a daily number, on-time and with specified
statistical
properties holds every part of a trading organization to a high
objective standard. Robust backup systems and default assumptions
must be implemented. Positions that are reported, modeled or priced
incorrectly stand out, as do data feeds that are inaccurate or late
and systems that are too-frequently down. Anything that affects
profit and loss that is left out of other reports will show up
either in inflated VaR or excessive VaR breaks. “A risk-taking
institution that does not compute VaR might escape disaster,
but an institution that cannot compute VaR will not.”

^[13]

The second claimed benefit of VaR is that it separates risk
into two regimes.
Inside the VaR limit, conventional statistical
methods are reliable. Relatively short-term and specific data can
be used for analysis. Probability estimates are meaningful, because
there are enough data to test them. In a sense, there is no true
risk because you have a sum of many
independent observations with a left bound on the outcome. A
casino doesn’t worry about whether red or black will come up on the
next roulette spin. Risk managers encourage productive risk-taking
in this regime, because there is little true cost. People tend to
worry too much about these risks, because they happen frequently,
and not enough about what might happen on the worst
days.^[14]

Outside the VaR limit, all bets are off. Risk should be
analyzed with stress
testing based on long-term and broad market data.^[15]
Probability statements are no longer meaningful.^[16]
Knowing the distribution of losses beyond the VaR point is both
impossible and useless. The risk manager should concentrate instead
on making sure good plans are in place to limit the loss if
possible, and to survive the loss if not.^[1]

One specific system uses three regimes.^[17]

1. Out to three times VaR are normal occurrences. You expect
   periodic VaR breaks. The loss distribution typically has fat
   tails, and you might get more than one break in a short period
   of time. Moreover, markets may be abnormal and trading may
   exacerbate losses, and you may take losses not measured in daily
   
   marks such as lawsuits, loss of employee morale and market
   confidence and impairment of brand names. So an institution that
   can’t deal with three times VaR losses as routine events probably
   won’t survive long enough to put a VaR system in place.
2. Three to ten times VaR is the range for stress
   testing. Institutions should be confident they have examined
   all the foreseeable events that will cause losses in this range,
   and are prepared to survive them. These events are too rare to
   estimate probabilities reliably, so risk/return calculations are
   useless.
3. Foreseeable events should not cause losses beyond ten times
   VaR. If they do they should be hedged
   or insured, or the business plan should be changed to avoid them,
   or VaR should be increased. It’s hard to run a business if
   foreseeable losses are orders of magnitude larger than very large
   everyday losses. It’s hard to plan for these events, because they
   are out of scale with daily experience. Of course there will be
   unforeseeable losses more than ten times VaR, but it’s pointless to
   anticipate them, you can’t know much about them and it results in
   needless worrying. Better to hope that the discipline of preparing
   for all foreseeable three-to-ten times VaR losses will improve
   chances for surviving the unforeseen and larger losses that
   inevitably occur.

"A risk manager has two jobs: make people take
more risk the 99% of the time it is safe to do so, and survive the
other 1% of the time. VaR is the border."^[13]

VaR risk measurement

The VaR risk measure is a popular way to aggregate risk across
an institution. Individual business units have risk measures such
as duration
for a fixed
incomeportfolio
or beta
for an equity
business. These cannot be combined in a meaningful
way.^[1]
It is also difficult to aggregate results available at different
times, such as positions marked in different time
zones, or a high frequency trading desk with a business holding
relatively illiquid
positions. But since every business contributes to profit and loss
in an additive
fashion, and many financial
businesses mark-to-market daily, it is natural to define firm-wide
risk using the distribution of possible losses at a fixed point in
the future.^[4]

In risk measurement, VaR is usually reported alongside other
risk metrics such as standard
deviation, expected
shortfall and “greeks”
(partial
derivatives of portfolio value with respect to market factors).
VaR is a
distribution-free metric, that is it does not depend on
assumptions about the probability distribution of future gains and
losses.^[13]
The probability level is chosen deep enough in the left tail of the
loss distribution to be relevant for risk decisions, but not so
deep as to be difficult to estimate with accuracy.^[18]

Risk measurement VaR is sometimes called
parametric VaR. This usage can be confusing, however, because
it can be estimated either parametrically (for examples, variance-covariance
VaR or delta-gamma
VaR) or nonparametrically (for examples, historical simulation
VaR or resampled
VaR). The inverse usage makes more logical sense, because risk
management VaR is fundamentally nonparametric, but it is seldom
referred to as nonparametric VaR.^[4][6]

History of VaR

The problem of risk measurement is an old one in statistics,
economics
and finance.
Financial risk management has been a concern of regulators and
financial executives for a long time as well. Retrospective
analysis has found some VaR-like concepts in this history. But VaR
did not emerge as a distinct concept until the late 1980s. The
triggering event was the stock market crash
of 1987. This was the first major financial crisis in which a
lot of academically-trained
quants were in high enough positions to worry about firm-wide
survival.^[1]

The crash was so unlikely given standard statistical
models, that it called the entire basis of
quant finance into question. A reconsideration of history led
some quants to decide there were recurring crises, about one or two
per decade, that overwhelmed the statistical assumptions embedded
in models used for trading,

investment management and
derivative pricing. These affected many markets at once,
including ones that were usually not correlated,
and seldom had discernible economic cause or warning (although
after-the-fact explanations were plentiful).^[16]
Much later, they were named "Black
Swans" by
Nassim Taleb and the concept extended far beyond finance.^[19]

If these events were included in
quantitative analysis they dominated results and led to
strategies that did not work day to day. If these events were
excluded, the profits made in between "Black Swans" could be much
smaller than the losses suffered in the crisis. Institutions could
fail as a result.^[13][16][19]

VaR was developed as a systematic way to segregate extreme
events, which are studied qualitatively over long-term history and
broad market events, from everyday price movements, which are
studied quantitatively using short-term data in specific markets.
It was hoped that "Black Swans" would be preceded by increases in
estimated VaR or increased frequency of VaR breaks, in at least
some markets. The extent to which has proven to be true is
controversial.^[16]

Abnormal markets and trading were excluded from the VaR
estimate in order to make it observable.^[14]
It is not always possible to define loss if, for example, markets
are closed as after
9/11, or severely illiquid, as happened several times in
2008.^[13]
Losses can also be hard to define if the risk-bearing institution
fails or breaks up.^[14]
A measure that depends on traders taking certain actions, and
avoiding other actions, can lead to self
reference.^[1]

This is risk management VaR. It was well-established in

quantative trading groups at several financial institutions,
notably Bankers
Trust, before 1990, although neither the name nor the
definition had been standardized. There was no effort to aggregate
VaRs across trading desks.^[16]

The financial events of the early 1990s found many firms in
trouble because the same underlying bet had been made at many
places in the firm, in non-obvious ways. Since many trading desks
already computed risk management VaR, and it was the only common
risk measure that could be both defined for all businesses and
aggregated without strong assumptions, it was the natural choice
for reporting firmwide risk. J. P.
Morgan CEO Dennis
Weatherstone famously called for a “4:15
report” that combined all firm risk on one
page, available within 15 minutes of the market close.^[8]

Risk measurement VaR was developed for this purpose.
Development was most extensive at J. P.
Morgan, which published the methodology and gave free access to
estimates of the necessary underlying parameters in 1994. This was
the first time VaR had been exposed beyond a relatively small group
of quants. Two years later, the methodology was spun off into an
independent for-profit business now part of RiskMetrics Group.^[8]

In 1997, the
U.S. Securities and Exchange Commission ruled that public
corporations must disclose quantitative information about their

derivatives activity. Major banks and
dealers chose to implement the rule by including VaR information in
the notes to their
financial statements.^[1]

Worldwide adoption of the Basel
II Accord, beginning in 1999 and nearing completion today, gave
further impetus to the use of VaR. VaR is the preferred
measure of market
risk, and concepts similar to VaR are used in other parts of
the accord.^[1]

Mathematics

"Given some confidence level
the VaR of the portfolio at the confidence level α is
given by the smallest number l such that the
probability that the loss L exceeds
l is not larger than (1 −
α)"^[3]

The left equality is a definition of VaR. The right equality
assumes an underlying probability distribution, which makes it true
only for parametric VaR. Risk managers typically assume that some
fraction of the bad events will have undefined losses, either
because markets are closed or illiquid, or because the entity
bearing the loss breaks apart or loses the ability to compute
accounts. Therefore, they do not accept results based on the
assumption of a well-defined probability distribution.^[6]Nassim
Taleb has labeled this assumption, "charlatanism."^[20]
On the other hand, many academics prefer to assume a well-defined
distribution, albeit usually one with fat
tails.^[1]
This point has probably caused more contention among VaR theorists
than any other.^[8]

Criticism

VaR has been controversial since it moved from trading desks
into the public eye in 1994. A famous 1997 debate between
Nassim Taleb and Philippe Jorion set out some of the major
points of contention. Taleb claimed
VaR:^[21]

1. Ignored 2,500 years of experience in favor of untested models
   built by non-traders
2. Was charlatanism because it claimed to estimate the risks of
   rare events, which is impossible
3. Gave false confidence
4. Would be exploited by traders

More recently
David Einhorn and Aaron
Brown debated VaR in Global Association of Risk Professionals
Review^[13][22]
Einhorn compared VaR to “an airbag that works all the time, except
when you have a car accident.” He further charged that
VaR:

1. Led to excessive risk-taking and leverage at financial
   institutions
2. Focused on the manageable risks near the center of the
   distribution and ignored the tails
3. Created an incentive to take “excessive but remote risks”
4. Was “potentially catastrophic when its use creates a false
   sense of security among senior executives and watchdogs.”

New
York Times reporter Joe
Nocera wrote an extensive piece Risk Mismanagement^[23]
on January 4, 2009 discussing the role VaR played in the
Financial crisis of 2007-2008. After interviewing risk managers
(including several of the ones cited above) the article suggests
that VaR was very useful to risk experts, but nevertheless
exacerbated the crisis by giving false security to bank executives
and regulators. A powerful tool for professional risk managers, VaR
is portrayed as both easy to misunderstand, and dangerous when
misunderstood.

A common complaint among academics is that VaR is not
subadditive.^[4]
That means the VaR of a combined portfolio can be larger than the
sum of the VaRs of its components. To a practicing risk manager
this makes sense. For example, the average bank branch in the
United States is robbed about once every ten years. A single-branch
bank has about 0.004% chance of being robbed on a specific day, so
the risk of robbery would not figure into one-day 1% VaR. It would
not even be within an order of magnitude of that, so it is in the
range where the institution should not worry about it, it should
insure against it and take advice from insurers on precautions. The
whole point of insurance is to aggregate risks that are beyond
individual VaR limits, and bring them into a large enough portfolio
to get statistical predictability. It does not pay for a one-branch
bank to have a security expert on staff.

As institutions get more branches, the risk of a robbery on a
specific day rises to within an order of magnitude of VaR. At that
point it makes sense for the institution to run internal stress
tests and analyze the risk itself. It will spend less on insurance
and more on in-house expertise. For a very large banking
institution, robberies are a routine daily occurrence. Losses are
part of the daily VaR calculation, and tracked statistically rather
than case-by-case. A sizable in-house security department is in
charge of prevention and control, the general risk manager just
tracks the loss like any other cost of doing business.

As portfolios or institutions get larger, specific risks
change from low-probability/low-predictability/high-impact to
statistically predictable losses of low individual impact. That
means they move from the range of far outside VaR, to be insured,
to near outside VaR, to be analyzed case-by-case, to inside VaR, to
be treated statistically.^[13]

Even VaR supporters generally agree there are common abuses of
VaR:^[6][8]

1. Referring to VaR as a "worst-case" or "maximum tolerable" loss.
   In fact, you expect two or three losses per year that exceed
   one-day 1% VaR.
2. Making VaR control or VaR reduction the central concern of risk
   management. It is far more important to worry about what happens
   when losses exceed VaR.
3. Assuming plausible losses will be less than some multiple,
   often three, of VaR. The entire point of VaR is that losses can be
   extremely large, and sometimes impossible to define, once you get
   beyond the VaR point. To a risk manager, VaR is the level of losses
   at which you stop trying to guess what will happen next, and start
   preparing for anything.
4. Reporting a VaR that has not passed a backtest.
   Regardless of how VaR is computed, it should have produced the
   correct number of breaks (within sampling
   error) in the past. A common specific violation of this is to
   report a VaR based on the unverified assumption that everything
   follows a
   multivariate normal distribution.

写下VAR几个字母的时候，心里很是激动，这是这一章节的重点，也是整个risk
manaement的重点。甚至是考试的核心，并不仅仅是计算，包括这些概念和以后的应用。

写题，看书。

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• The payoff at maturity depends not just on the value of the
  underlying index at maturity, but at its value at several times
  during the contract’s life (it could be an Asian
  option depending on some average, a lookback
  option depending on the maximum or minimum, a barrier
  option which ceases to exist if a certain level is reached or
  not reached by the underlying, a digital
  option,
  peroni options,
  range options, etc.)
• It could depend on more than one index (as in a basket
  options,
  Himalaya options,
  Peroni options, or other
  mountain range options,
  outperformance options, etc.)
• There could be callability and putability rights.
• It could involve foreign exchange rates in various ways, such
  as a quanto or
  composite option.

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Volatility smiles and implied volatility

Implied volatility and historical
volatility

Term structure of volatility

Implied volatility surface

Evolution: Sticky

• "sticky strike" (or "sticky-by-strike", or
  "stick-to-strike"): if spot changes, the implied
  volatility of an option with a given absolute strike does
  not change.
• "sticky moneyness"
  (aka, "sticky delta"; see moneyness
  for why these are equivalent terms): if spot
  changes, the implied volatility of an option with a given
  moneyness does not change.

Modeling volatility

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Charm

Charm or delta decay, measures the instantaneous
rate of change of delta over the passage of time. Charm has also
been called DdeltaDtime. Charm can be an important greek to
measure/monitor when delta-hedging a position over a weekend. Charm
is a second order derivative of the option value, once to price and
once to time. It is also then the (minus) derivative of theta with
respect to the underlying’s price.

delta对时间的。反应delta的变化情况。

Practical use

The mathematical result of the formula for charm (see below)
is expressed in delta/year. It is often useful to divide this by
the number of days per year to arrive at the delta decay per day.
This use is fairly accurate when the number of days remaining until
option expiration is large. When an option nears expiration, charm
itself may change quickly, rendering full day estimates of delta
decay inaccurate.

Color

Color, gamma decay or DgammaDtime
measures the rate of change of gamma over the passage of time.
Color is a third-order derivative of the option value, twice to
underlying asset price and once to time. Color can be an important
sensitivity to monitor when maintaining a gamma-hedged portfolio as
it can help the trader to anticipate the effectiveness of the hedge
as time passes.

Practical use

The mathematical result of the formula for color (see below)
is expressed in gamma/year. It is often useful to divide this by
the number of days per year to arrive at the change in gamma per
day. This use is fairly accurate when the number of days remaining
until option expiration is large. When an option nears expiration,
color itself may change quickly, rendering full day estimates of
gamma change inaccurate.

Delta

Delta^[8]Δ
measures the rate of change of option value with respect to changes
in the underlying asset’s price. Delta is the first
derivative of the value, V, of a portfolio
of derivative securities on a single underlying instrument,
S, with respect to the underlying instrument’s
price.

Practical use

Even though delta will be a number between 0.0 and 1.0 for a
call and 0.0 and -1.0 for a put, these numbers are commonly
presented as a percentage of the total number of shares represented
by the option contract(s). This is convenient because the option
will (instantaneously) behave like the number of shares indicated
by the delta. For example, if an American call option on XYZ has a
delta of 0.25, it will gain or lose value just like 25% of 100
shares or 25 shares of XYZ as the price changes for small price
movements.

Delta is always positive for long calls and short puts and
negative for long puts and short calls. The total delta of a
complex portfolio of positions on the same underlying asset can be
calculated by simply taking the sum of the deltas for each
individual position. Since the delta of underlying asset is always
1.0, the trader could delta-hedge
his entire position in the underlying by buying or shorting the
number of shares indicated by the total delta. For example, if a
portfolio of options in XYZ (expressed as shares of the underlying)
is +275, the trader would be able to delta-hedge the portfolio by
selling short 275 shares of the underlying. This portfolio will
then retain its total value regardless of which direction the price
of XYZ moves. (Albeit for only small movements of the underlying, a
short amount of time and not-withstanding changes in other market
conditions such as volatility and the rate of return for a
risk-free investment).

As a proxy for probability

Some option traders also use the absolute value of delta as
the probability that the option will expire in-the-money
(if the market moves under Brownian
motion). For example, if an out-of-the-money
call option has a delta of 0.15, the trader might estimate that the
option has appropriately a 15% chance of expiring in-the-money.
Similarly, if a put contract has a delta of -0.25, the trader might
expect the option to have a 25% probability of expiring
in-the-money. At-the-money
puts and calls have a delta of approximately^[9]
0.5 and -0.5 respectively, or each will have a 50% chance of
expiring in-the-money.

Relationship between call and put delta

Given a call and put option for the same underlying, strike
price and time to maturity, the sum of the absolute values of the
delta of each option will be 1.00

If the value of delta for an option is known, one can compute
the value of the option of the same strike price, underlying and
maturity but opposite right by subtracting 1 from the known value.
For example, if the delta of a call is .42 then one can compute the
delta of the corresponding put at the same strike price by 0.42 – 1
= -0.58.

DvegaDtime

DvegaDtime^[10],
measures the rate of change in the vega with respect to the passage
of time. DvegaDtime is the second derivative of the value function;
once to volatility and once to time.

Practical use

It is common practice to divide the mathematical result of
DvegaDtime by 100 times the number of days per year to reduce the
value to the percentage change in vega per one day.

Gamma

Gamma^[11]Γ
measures the rate of change in the delta with respect to changes in
the underlying price. Gamma is the second derivative
of the value function with respect to the underlying price. Gamma
is important because it corrects for the convexity
of value.

When a trader seeks to establish an effective delta-hedge for
a portfolio, the trader may also seek to neutralize the portfolio’s
gamma, as this will ensure that the hedge will be effective over a
wider range of underlying price movements.

Lambda

Lambdaλ,
omegaΩ
or elasticity^[12]
is the percentage
change in option value per percentage
change in the underlying price, a measure of leverage.

[edit]Rho

Rho^[13]ρ
measures sensitivity to the applicable interest rate. Rho is the
derivative of the option value with respect to the risk free rate.
Except under extreme circumstances, the value of an option is least
sensitive to changes in the risk-free-interest rates. For this
reason, rho is the least used of the primary Greeks.

Practical use

Rho is typically expressed as the amount of money, per share,
that the value of the option will gain or lose as the rate of
return of a risk-free investment rises or falls by 1.0%.

Speed

Speed^[14]
measures the rate of change in Gamma with respect to changes in the
underlying price. This is also sometimes referred to as the
gamma of the gamma^[15]
or DgammaDspot^[16].
Speed is the third derivative of the value function with
respect to the underlying spot price. Speed can be important to
monitor when delta-hedging
or gamma-hedging a portfolio.

Theta

Theta^[17]Θ,
or "time decay", measures sensitivity of the value of an option to
the passage of time (see Option
time value).

Practical use

The mathematical result of the formula for theta (see below)
is expressed in value/year. By convention, it is useful to divide
the result by the number of days per year to arrive at the amount
of money, per share of the underlying that the option loses in one
day. Theta is always negative for long calls and puts and positive
for short (or written) calls and puts. The total theta for a
portfolio of options can be determined by simply taking the sum of
the thetas for each individual position.

The value of an option is made up of two
parts: the
intrinsic value and the time value. The intrinsic value is the
amount of money you would gain if you exercised the option
immediately, so a call with strike $50 on a stock with price $60
would have intrinsic value of $10, whereas the corresponding put
would have zero intrinsic value. The time value is the worth of
having the option of waiting longer when deciding to exercise. Even
a deeply out
of the money put will be worth something as there is some
chance the stock price will fall below the strike. However, as time
approaches maturity, there is less chance of this happening, so the
time value of an option is decreasing with time. Thus if you are
long an option you are short theta: your portfolio
will lose value with the passage of time (unless there is enough
volatility to offset this).

Ultima

Ultima^[18]
measures the sensitivity of the option vomit with respect to change
in volatility. Ultima has also been referred to as
DvomitDvol^[19].
Ultima is a third order derivative of the option value to
volatility.

Vanna

Vanna^[20],
also referred to as DvegaDspot and
DdeltaDvol^[21],
is a second order derivative of the option value, once to the
underlying spot price and once to volatility. It is mathematically
equivalent to DdeltaDvol^[22],
the sensitivity of the option delta with respect to change in
volatility; or alternately, the partial of vega with respect to the
underlying instrument’s price. Vanna can be a useful sensitivity to
monitor when maintaining a delta- or vega-hedged portfolio as vanna
will help the trader to anticipate changes to the effectiveness of
a delta-hedge as volatility changes or the effectiveness of a
vega-hedge against change in the underlying spot price.

Vega

Vega^[23],
is not actually a Greek letter (The Greek letter nu,

is used instead). Vega measures sensitivity to
volatility. Vega is the derivative of the option value with
respect to the
volatility of the underlying. The term kappa,
κ, is sometimes used (by academics) instead of
vega, as is tau, τ, though this is
rare.

Practical use

Vega is typically expressed as the amount of money, per
underlying share the option’s value will gain or lose as volatility
rises or falls by 1%.

Vega can be an important Greek to monitor for an option
trader, especially in volatile markets since some the value of
option strategies can be particularly sensitive to changes in
volatility. The value of an option
straddle, for example, is extremely dependent on changes to
volatility.

vomit

vomit^[24],
Volga^[25],
Vega Convexity^[26]
or Vega gamma measures second order sensitivity to
volatility. vomit is the second derivative of the option value
with respect to the volatility, or, stated another way, vomit
measures the rate of change to vega as volatility changes.

Zomma

Zomma^[27]
measures the rate of change of gamma with respect to changes in
volatility. Zomma has also been referred to as
DgammaDvol^[28].
Zomma is the third derivative of the option value, twice to
underlying asset price and once to volatility. Zomma can be a
useful sensitivity to monitor when maintaining a gamma-hedged
portfolio as zomma will help the trader to anticipate changes to
the effectiveness of the hedge as volatility changes.

Black-Scholes

The Greeks under the Black-Scholes
model are calculated as follows, where φ (phi) is
the standard
normalprobability
density function and Φ is the standard
normalcumulative
distribution function. Note that the gamma and vega formulas
are the same for calls
and puts.

For a given: Stock Price , Strike Price , Risk-Free Rate , Annual Dividend Yield , Time to Maturity, , and Volatility …

	Calls	Puts
value
delta
vega
theta
rho
gamma
vanna
charm
speed
zomma
color
DvegaDtime
vomit
dual delta
dual gamma

where

其实我们要知道只是那几个一阶导就可以了，而且我们平时用的比较多的也是那写个一阶导的，还是和以前一样，概念是重要的，习题时必须的。结果是没有必要记的。

对于不同的distribution，得到的表达式自然不一样。

注意的是delta的表达式比较特别，比起其他的比较简单，这就是为什么我们一直用delts的原因了。

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Black–Scholes model

• It is possible to borrow and lend cash at a known constant
  
  risk-free interest rate.
• The price follows a
  geometric Brownian motion with constant drift and
  volatility.
• There are no transaction
  costs.
• The stock does not pay a dividend (see
  below for extensions to handle dividend payments).
• All securities are perfectly divisible (i.e. it is
  possible to buy any fraction of a share).
• There are no restrictions on short
  selling.
• There is no arbitrage opportunity

Notation

S, the price of the stock (please note as

below).

V(S,t), the price of a
derivative as a function of time and stock price.

C(S,t) the price of a
European call and P(S,t) the
price of a European put option.

K, the strike of the option.

r, the annualized
risk-free interest rate,
continuously compounded.

μ, the drift
rate of S, annualized.

σ, the volatility of the stock; this is the square
root of the quadratic
variation of the stock’s log price process.

t a time in years; we generally use now =
0, expiry = T.

Π, the value of a portfolio.

R, the accumulated profit or loss following
a delta-hedging
trading strategy.

N(x) denotes the standard
normalcumulative
distribution function, .

N‘(x) denotes the standard normal

probability density function,.

Black–Scholes PDE

Simulated Geometric Brownian Motions with Parameters from
Market Data

Now consider a trading strategy under which one holds a single
option and continuously trades in the stock in order to hold
shares. At time t, the value of these holdings
will be

Black–Scholes formula

Black-Scholes European Call Option Pricing Surface

• N(•) is the standard
  normal or
  cumulative distribution function
• T – t is the time to maturity
• S is the spot
  price of the underlying asset
• K is the strike
  price
• r is the risk
  free rate (annual rate, expressed in terms of
  continuous compounding)
• σ is the
  volatility in the log-returns of the underlying

Interpretation

Derivation

The last condition gives the value of the option at the time
that the option matures. The solution of the PDE gives the value of
the option at any earlier time, . In order to solve the PDE we transform the equation
into a diffusion
equation which may be solved using standard methods. To this
end we introduce the change-of-variable transformation

BS模型的理解：

bs模型，一个让人又爱又恨的model，开始入门的时候，觉得这个模型简直就是天才的杰作，虽然说看着公式复杂，其实计算还是很简单的，那么一折腾就能得到想要的结果，最重要的是这个结果还不算太坏。于是很是兴奋的拿着公式满世界挥舞，却发现这也不行，那也不行，一时间完全失去了对这些个模型的信心。

后来慢慢看见的模型多了，开始慢慢理解bs了。
我曾经和bobo说过，虽然我们用着bs，其实bs的含义是很简单的，就是一个有价值的部分的实现加权加上一个没有价值部分的不实现加权。这样理解的话，那些歌N(D1)和N（d2）就变成了纯粹的概率问题，我们在用正态的时候这个值可能容易算出来，可是要是把这个过程看成其他的分布就难了。假设这个过程是一个power-law的过程，假设这个power-law的a和效用函数是有关系的，这个a和市场也是有关系的，怎么样得到一个简单的表达和一个完整的推导？假设这个过程是一个levy
process，比如说是VG，是CGMY，怎么样求那个default的概率，这又是一大堆问题了，有空我真想把power-law的那个想法做一下，不过现在看起来要等这个考试结束了。

anyway，看习题，写习题，这一章没有难点，难点我也基本搞定了。过。

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Binomial options pricing model

Use of the model

Methodology

1. price tree generation
2. calculation of option value at each final node
3. progressive calculation of option value at each earlier node;
   the value at the first node is the value of the option.

STEP 1: Create the binomial price
tree

At each step, it is assumed that the
underlying instrument will move up or down by a specific factor
(u or d) per step of the tree (where, by definition,
and ). So, if S is the current price, then in the
next period the price will either be or .

N_u: Number of up
ticks

N_d: Number of down
ticks

STEP 2: Find Option value at each final
node

Max
[ (S – K), 0 ], for a call
option

Max [ (K – S), 0 ], for a put
option:

Where: K is the Strike
price and S is the spot price of the underlying asset

STEP 3: Find Option value at earlier nodes

The following formula to compute the expectation
value is applied at each node:

Binomial Value = [ p × Option up + (1-p) × Option down] × exp
(- r × Δt), or

where

is the option’s value for the node at time ,

is chosen such that the related
Binomial distribution simulates the
geometric Brownian motion of the underlying stock with
parameters r and σ,

q is the dividend
yield of the underlying corresponding to the life of the
option.

(Note that the alternative valuation approach, arbitrage-free
pricing, yields identical results; see "delta-hedging".)

• For a European
  option, there is no option of early exercise, and the binomial
  value applies at all nodes.

• For an American
  option, since the option may either be held or exercised prior
  to expiry, the value at each node is: Max (Binomial
  Value, Exercise Value).

• For a Bermudan
  option, the value at nodes where early exercise is allowed
  is: Max (Binomial Value, Exercise Value); at nodes
  where early exercise is not allowed, only the binomial value
  applies.

Discrete dividends

• At each time step, i, calculate , for all k< i where
  PV(D_k) is the present value
  of the k-th dividend. Subtract this value from the value of
  the security price S at each node (i, j).

Relationship with Black-Scholes

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Contract specifications

• whether the option holder has the right to buy (a call
  option) or the right to sell (a put
  option)
• the quantity and class of the underlying
  asset(s) (e.g. 100 shares of XYZ Co. B stock)
• the strike
  price, also known as the exercise price, which is the price at
  which the underlying transaction will occur upon exercise
• the
  expiration date, or expiry, which is the last date the option
  can be exercised
• the
  settlement terms, for instance whether the writer must deliver
  the actual asset on exercise, or may simply tender the equivalent
  cash amount
• the terms by which the option is quoted in the market to
  convert the quoted price into the actual premium–the
  total amount paid by the holder to the writer of the option.

Types of options

• Exchange traded options (also called "listed options")
  are a class of
  exchange traded derivatives. Exchange traded options have
  standardized contracts, and are settled through a clearing
  house with fulfillment guaranteed by the credit of the
  exchange. Since the contracts are standardized, accurate pricing
  models are often available. Exchange traded options
  include:^[4][5]
  • stock options,
  • 
    commodity options,
  • bond
    options and other
    interest rate options
  • 
    stock market index options or, simply, index options and
  • 
    options on futures contracts

• 
  Over-the-counter options (OTC options, also called "dealer
  options") are traded between two private parties, and are not
  listed on an exchange. The terms of an OTC option are unrestricted
  and may be individually tailored to meet any business need. In
  general, at least one of the counterparties to an OTC option is a
  well-capitalized institution. Option types commonly traded over the
  counter include:

1. interest rate options
2. currency cross rate options, and
3. options on swaps or
   swaptions.

• 
  Employee stock options are issued by a company to its
  employees as compensation.

Option styles

• European option – an option that may only be exercised
  on
  expiration.
• American option – an option that may be exercised on any
  trading day on or before expiration.
• Bermudan option – an option that may be exercised only
  on specified dates on or before expiration.
• Barrier option – any option with the general
  characteristic that the underlying security’s price must pass a
  certain lever or "barrier" before it can be exercised
• Exotic option – any of a broad category of options that
  may include complex financial structures.^[6]
• Vanilla option – by definition, any option that is not
  exotic.

Options Pricing

Intrinsic Value (Calls):

Intrinsic Value (Puts):

Time
Value:

Six Major Factors Influencing Options Premium

• 
  A change in price of the underlying security
• 
  Strike price
• 
  Time until expiration
• 
  Volatility of the underlying security
• 
  Dividends/Risk-free interest rate
  Dividends and risk-free interest rate have a
  lesser effect.

Call option

Put option

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远期与期货市场的构成

远期与期货的定价

1、远期合约在到期时的价格

原则1：远期合约在到期日的价格应该等于其现货价格。即：

F（T，T）=ST

（1）为什么要对远期合约到期前的价值进行评价？

n
远期合约的交易属于表外业务，在没有进行最后交割之前是不进入资金平衡表的，即合约既不是资产，也不是负债。但在合约的有效期内，由于市场条件的变化，合约的价值既可能增加，也可能减少。如果在合约到期前的某时点，投资者需要对一企业整个资产与负债的价值进行评价，就有必要对远期合约的价值进行评价。如果远期合约价值为正，企业的资产价值就增加，如果远期合约的价值为负，企业的负债值就增加。

n       期货合约也同样。

（2）远期合约在到期前的价值

现设某投资者持有一份在0时点上购入的远期合约，在时点t又卖出一到期日为T的相同标的资产的新合约。我们将原有的在0时点上购入的到期日为T的合约称之为旧合约，t时点上购入的称之为新合约。

原则2：旧多头远期合约在t时点的价值就是新合约与旧合约价格差的现值。即：Vt(0,T)=
[ F(t,T)- F(0,T)](1+r)-(T-t)

（三）期货合约的价值

1、期货合约到期时的价格

原则1：期货合约到期时的价格等于现货价格。即：fT(0,T)=ST

与远期合约到期时的价格形成原理一样，如果两期货价格不等于现货价格，就产生套利利润，市场会迅速进行套利而使两者一致。

2、在任意交易日，但在进行盯市之前期货合约的价值

原则2：当天开仓的期货合约，其价值是自合约开仓之时起至当日交易结束时（但在最后结算前）的价格变化。如是头一天开仓的合约，其价值就是自头一天盯市之时起至当日最后交易时止（但在盯市前）的价格变化。即：

Vt(T)=ft(T)-ft-1(T)（盯市前）

（1）合约价值有正有负。

（2）空头合约的价值与多头合约价值的符号相反。

3、进行盯市后那一时刻期货合约的价值

原则3：进行盯市后的那一时刻，期货合约价值为0。即：Vt(T)=0

感觉他讲这个讲的很不清楚。没有折现，没有引入interest，用一个符号代替了折现的价格，这种表达并不适合定价。找了下wiki：

The forward price (or sometimes forward
rate) is the agreed upon price of an asset in a
forward
contract. Using the rational
pricing assumption, we can express the forward price in terms
of the spot
price and any dividends etc., so that there is no possibility
for arbitrage.

Forward Price Formula

The forward price is given by:

where

F is the forward price to be paid at time T

e^x is the
exponential function (used for calculating compounding
interests)

r is the
risk-free interest rate

q is the cost-of-carry

S₀ is the spot
price of the asset (i.e. what it would sell for at time 0)

D_i is a dividend
which is guaranteed to be paid at time t_i
where 0 <
t_i< T.

Proof of the forward price formula

The two questions here are what price the short position (the
seller of the asset) should offer to maximize his gain, and what
price the long position (the buyer of the asset) should accept to
maximize his gain?

At the very least we know that both do not want to lose any
money in the deal.

The short position knows as much as the long position
knows: the short/long positions are both aware of
any schemes that they could partake on to gain a profit given some
forward price.

So of course they will have to settle on a fair price or else
the transaction cannot occur.

An economic articulation would be:

(fair price + future value of asset’s dividends) – spot price
of asset = cost of capital

Forward price = Spot Price + cost of carry

The future value of that asset’s dividends (this could also be
coupons from bonds, monthly rent from a house, fruit from a crop,
etc.) is calculated using the risk-free force of interest. This is
because we are in a risk-free situation (the whole point of the
forward contract is to get rid of risk or to at least reduce it) so
why would the owner of the asset take any chances? He would
reinvest at the risk-free rate (i.e. U.S. T-bills which are
considered risk-free). The spot price of the asset is simply the
market value at the instant in time when the forward contract is
entered into. So OUT – IN = NET GAIN and his net gain can only come
from the opportunity cost of keeping the asset for that time period
(he could have sold it and invested the money at the risk-free
rate).

let:

K = fair price

C = cost of capital

S = spot price of asset

F = future value of asset’s dividend

I = present value of F (discounted using r
)

r = risk-free interest rate compounded continuously

T = length of time from when the contract was entered
into

Solving for fair price and substituting mathematics we
get:

where:

(since where j is the effective rate of interest per
time period of T )

where c_i is the i ^th
dividend paid at time t ⁱ.

Doing some reduction we end up with:

Futures Pricing

Arbitrage arguments

Arbitrage arguments ("Rational
pricing") apply when the deliverable asset exists in plentiful
supply, or may be freely created. Here, the forward price
represents the expected future value of the underlying discounted
at the
risk free rate—as any deviation from the theoretical price will
afford investors a riskless profit opportunity and should be
arbitraged away; see
rational pricing of futures.

Thus, for a simple, non-dividend paying asset, the value of
the future/forward, F(t), will be found by compounding the
present value S(t) at time t to maturity T by
the rate of risk-free return r.

or, with
continuous compounding

This relationship may be modified for storage costs,
dividends, dividend yields, and convenience yields.

In a perfect market the relationship between futures and spot
prices depends only on the above variables; in practice there are
various market imperfections (transaction costs, differential
borrowing and lending rates, restrictions on short selling) that
prevent complete arbitrage. Thus, the futures price in fact varies
within arbitrage boundaries around the theoretical price.

Forward versus Futures prices

There is a difference between forward and futures prices when
interest rates are stochastic.
This difference disappears when interest rates are
deterministic.

In the language of
stochastic processes, the forward price is a
martingale under the forward
measure, whereas the futures price is a martingale under the
risk
neutral measure. The forward measure and the risk neutral
measure are the same when interest rates are deterministic.

在interest ratio
相同的情况下，他们的价格应该是一样，不过如果利率是随机变化的，他们的价格是不一样的，一个定价是用forward
measure的熵，一个定价是基于risk neutral
measure，这两个算法不一样，最后得到的价格可能也就不一样了，不过由于熵的算法不是那么容易，所以应该不会考这个的计算，还是把重点放在他们的概念上。

See Musiela and Rutkowski’s book on Martingale Methods in
Financial Markets for a continuous time proof of this result. See
van der Hoek and Elliott’s book on Binomial Models in Finance for
the discrete time version of this result.

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