VAR for a Portfolio of Options

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VAR for a Portfolio of Options Commodities as
an asset class going forward
By NGUYEN Thi My Hoang NGUYEN Tuong
Tam CHAMNINAWAKUN Phawina
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Contents
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• VAR of portfolio of options

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Options basic strategies

• Buying calls
• Buying calls to participate in upward price
  movements.
• Buying calls as part of an investment plan
• Buying calls to lock in a stock purchase price
• Buying calls to hedge short stock sales
• Buying puts
• Buying puts to participate in downward price
  movements.
• Buying puts to protect a long stock position
• Buying puts to protect unrealized profit in long
  stock
• Selling calls
• Covered call writing
• Uncovered call writing
• Selling puts
• Covered put writing
• Uncovered put writing

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• It is paradoxical that instruments such as
  derivatives originally conceived as a hedging
  instrument could be blamed for the generation of
  bigger risks, and more difficult to control
• An empirical surveys proposed to send a series
  of portfolios of different kinds including
  non-linear to all Australian banks to have the
  daily VaR of each portfolio calculated with their
  own methods. Of all the banks which answered
  (only twenty-two), just two banks were able to
  calculate the VaR for all the portfolios

Source Maria Coronado, 2000
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Schematic of How VAR measures work

• A mapping procedure describes exposure
• An inference procedure describes uncertainty by
  characterizing the joint distribution for key
  factors
• A transformation procedure combines exposure and
  uncertainty to describe the distribution of the
  portfolio's current market value , which it then
  summarizes with the value of some VaR metric

Source www.riskglossary.com
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Mapping and inference procedures

• The purpose of a mapping procedure is to
  characterize a portfolio's exposures. It does so
  by expressing the portfolio's value as a function
  of applicable market variables (key factors),
  such as stock prices, exchange rates, commodity
  prices or interest rates.
• In case of options portfolio, we would be using
  option prices as key factors, which means that
  the inference procedure would need to
  characterize their joint distribution.
• Designing an inference procedure to do
  characterize their joint distribution would be
  difficult. Because of
• the limited downside risk of options,
• their prices have skewed price distributions,
• their standard deviations would be highly
  dependent upon whether the options were
  in-the-money or out-of-the-money.

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Mapping procedures

• A simpler solution is to not employ options
  prices as key factors, but to use more
  fundamental risk factors such as the options'
  underlier prices and implied volatilities .
• For many VaR measures, output of the mapping
  procedure is a primary mapping, but not for all.
  Use of primary mappings can pose certain
  problems.
• The most common of these is that primary mappings
  can be mathematically complicated. This is
  especially true for large portfolios of
  instruments such as mortgage-backed securities or
  exotic derivatives.
• Applying a transformation procedure to a
  complicated portfolio mapping can be
  computationally expensive. For this reason, many
  mapping procedures replace primary mappings with
  simpler approximations. Those approximations are
  called portfolio remappings.

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Remapping

• portfolio remapping, which approximate a
  portfolio's value with some other random
  variable.
• A function remapping approximates a portfolios
  value by replacing its primary mapping functions
  with an approximate mapping function
• A variables remapping approximates a portfolios
  value by replacing its key factors with
  alternative and simpler key factors
• A dual remapping approximates a portfolios value
  by replacing both mapping functions and key
  factor vector
• Many function remapping approximate a portfolio
  mapping function with a linear or quadratic
  polynomial to facilitate use of a linear
  transformation or quadratic transformation.

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Transformation procedures

• Four basic forms of transformations are used
• linear transformations simple and run in real
  time, applied only if a portfolio mapping
  function is a linear polynomial.
• quadratic transformations slightly more
  complicated, but also run in real time (or
  near-real time). They apply only if a portfolio
  mapping function is a quadratic polynomial and
  key factors distribution is joint-normal (using
  Central Limit Theorem)
• Monte Carlo transformations, and
• historical transformations

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VAR measures

• Traditionally, VaR measures have been categorized
  according to the transformation procedures they
  employ. There are
• linear VaR measures (other names include
  parametric, variance-covariance, closed form, or
  delta normal VaR measures)
• quadratic VaR measures (also called delta-gamma
  VaR measures)
• Monte Carlo VaR measures, and
• historical VaR measures.

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Source Maria Coronado, 2000
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Normality assumption in the case of options
Source www.riskglossary.com
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Normality assumption in the case of options
portfolio

• The normality of the underlying asset does not
  mean that the option should follow a normal
  distribution, since the price of the options does
  not change in a linear fashion with reference to
  the underlying ones.
• However, with sufficiently large portfolios of
  independent options, the normality assumption can
  be used by applying Central Limit Theorem
• To determine when the options portfolio returns
  can be described by a normal distribution, we
  have to identify two aspects
• The number of options in the portfolio (i.e. its
  size).
• Analyze if even though the options are not
  completely independent but with a small degree of
  serial correlation

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Analytical methods

• Computing delta, gamma, vega risk of a single
  instruments is straightforward.
• But they often become ad hoc when applied to
  portfolios
• The famous RikMetricsTM of J. P. Morgan is an
  analytic variance-covariance matrix method

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Pros Cons

• Advantages
• It facilitates VaR calculations , making this
  method easily understandable for all the
  connected people in the risk management. Its
  implementation could be effortless.
• Rapidity in such calculations, a very important
  feature when working in real time.
• Drawbacks
• The VaR estimation through the variance-covariance
  matrix approach gives VaR overestimates for
  small confidence levels and VaR underestimates
  for big levels of probability.
• The linearity assumption makes this method only
  applicable, theoretically, to linear portfolios
• Even by increasing the portfolio value
  approximation to a quadratic one (deltagamma
  methods), success would not be guaranteed since
  an accurate VaR estimate of the non-linear
  portfolios is not feasible

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Delta-normal VAR

• The delta-normal approach originally introduced
  by J.P. Morgan's RiskMetrics software is based on
  two important assumptions (J.P. Morgan 1996)
• Linearity The change in the value of the
  portfolio over a given interval of time is linear
  in the returns of N lt 8 risk factors
• Normality The returns of the risk factors1
  follow a multivariate normal distribution.

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Historical simulation method

• The historical simulation approach is an easy
  method both to understand and to explain. It is
  also quite easy to implement. It is a
  non-parametric method, which does not depend on
  any assumption about the probability distribution
  of the underlying asset. Therefore, it allows
  capturing fat tails (and other non-normal
  characteristics), while eliminating the need for
  estimating and working with volatilities and
  correlations. It also avoids greatly the
  modelisation risk.
• It can be applied to all kinds of instruments,
  both linear and non-linear.
• Advantages This method allows capturing fat
  tails this is not the case of the
  variance-covariance matrix approach.
• Drawbacks This method depends completely on the
  specific historical data set used, and ignores
  any event not represented in such database.

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Monte-Carlo simulation method

• Both parametric (partial MC simulation or
  delta-gamma simulation) and non-parametric (full
  MC simulation)
• the non-parametric Monte Carlo simulation method,
  as it does not rely on any assumption about the
  probability distribution of the underlying asset,
  greatly avoids the modelisation risk and allows
  capturing fat tails (and other non-normal
  properties). At the same time, it excludes the
  necessity of estimating and working with
  volatilities and correlations, keeping out
  historical simulation method drawbacks as opposed
  to the variance-covariance matrix one.
• the parametric Monte Carlo simulation method
  presents a theoretical superiority as opposed to
  the variance-covariance matrix method. Although
  the former does call for the specification of a
  particular stochastic process for risk factors
  (dealing, therefore, with modelisation risk), it
  can be applied to non-linear positions and, it
  does not require the normality assumption.

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Monte-Carlo simulation method

• In contrast to the historical simulation method,
  its advantage lies in the random character of
  future prices paths, whereas prices generated by
  historical simulation represent only one of the
  possible paths that may happen.
• Its drawback is time consuming and more
  difficult, and requiring a bigger computational
  effort, indeed are more exact in case of complex
  portfolios

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Conclusion

• Most of the problems originate from
• VAR depends on the combined distribution of the
  instruments, which can become very complex when
  the number of different instruments is big.
• The non-linearity of the instruments
• The computational effort deriving from the number
  size and complexity of valuation formulas
  involved
• Covariance matrix of underlying assets

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• Commodities as a future

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What is commodities?

• Commodities are things of value, of uniform
  quality, that were produced in large quantities
  by many different producers the items from each
  different producer are considered equivalent.
• It is the contract and this underlying standard
  that define the commodity, not any quality
  inherent in the product.
• Commodities exchanges include
• Chicago Board of Trade
• Euronext.liffe
• London Metal Exchange
• New York Mercantile Exchange
• Multi Commodity Exchange
• Dalian Commodity Exchange

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Why commodities impact in the worlds capital
market?

• Three main reasons to invest in commodities
  products
• Rising global demand from emerging economies and
  low worldwide supplies
• Shifts in world diets
• Greater demand for alternative energy.

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Why we need to forecast commodities price?

• Since commodities and stocks have a negative
  correlation, which means that when stocks go
  down, commodities tend to move up and vice versa.
• If a portfolio is diversified with stocks and
  commodities, it will likely experience less
  volatility than a portfolio that is comprised of
  only stocks. That's because as one asset class
  underperforms, the other asset class can
  outperform to offset the volatility.
• Therefore, commodities price is important for
  investors to forecast return and risk from their
  portfolios.

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Commodities products

• 3 Main product categories
• Agricultural Products
• Metal and Minerals
• Oil and Gas

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Composite of each group
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Historical price of commodities products
Agricultural Products
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Historical price of commodities products
Metal and Minerals
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Historical price of commodities products
Oil and Gas
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Model for price forecasting
The mathematical models of the persistence, or
autocorrelation, in a time series which calls .
Auto regression Integrated Moving Average Model
(ARIMA )
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Box and Jenkins ARIMA model
ARIMA is a method for determining two
things 1. How much of the past should be used
to predict the next observation (length of
weights)2. The values of the weights.For
example y(t) 1/3 y(t-3) 1/3 y(t-2)
1/3 y(t-1) y(t) 1/6 y(t-3) 4/6
y(t-2) 1/6 y(t-1)
Source http//www.valuebasedmanagement.net
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ARIMA(p,d,q)

• A nonseasonal ARIMA model is classified as an
  "ARIMA(p,d,q)" model, where
• p is the number of autoregressive terms,
• d is the number of nonseasonal differences, and
• q is the number of lagged forecast errors in the
  prediction equation
• ( In this case, we choose d 0)

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Rules for identifying ARIMA model
Identifying the order of differencing and the
constant Rule 1 If the series has positive
autocorrelations out to a high number of lags,
then it probably needs a higher order of
differencing. Rule 2 If the lag-1
autocorrelation is zero or negative, or the
autocorrelations are all small and pattern less,
then the series does not need a higher order of
differencing. If the lag-1 autocorrelation is
-0.5 or more negative, the series may be over
differenced.  BEWARE OF OVERDIFFERENCING!! Rule
3 The optimal order of differencing is often the
order of differencing at which the standard
deviation is lowest.
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Rules for identifying ARIMA model
Rule 4 A model with no orders of differencing
assumes that the original series is stationary
(among other things, mean-reverting). A model
with one order of differencing assumes that the
original series has a constant average trend
(e.g. a random walk or SES-type model, with or
without growth). A model with two orders of total
differencing assumes that the original series has
a time-varying trend (e.g. a random trend or
LES-type model). Rule 5 A model with no
orders of differencing normally includes a
constant term (which represents the mean of the
series). A model with two orders of total
differencing normally does not include a constant
term. In a model with one order of total
differencing, a constant term should be included
if the series has a non-zero average trend.
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Rules for identifying ARIMA model
Identifying the order of autoregression
and moving average Rule 6 If the partial
autocorrelation function (PACF) of the
differenced series displays a sharp cutoff and/or
the lag-1 autocorrelation is positive--i.e., if
the series appears slightly "underdifferenced"--th
en consider adding one or more AR terms to the
model. The lag beyond which the PACF cuts off is
the indicated number of AR terms. Rule 7 If
the autocorrelation function (ACF) of the
differenced series displays a sharp cutoff and/or
the lag-1 autocorrelation is negative--i.e., if
the series appears slightly "overdifferenced"--the
n consider adding an MA term to the model. The
lag beyond which the ACF cuts off is the
indicated number of MA terms.
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Rules for identifying ARIMA model
Rule 8 It is possible for an AR term and an MA
term to cancel each other's effects, so if a
mixed AR-MA model seems to fit the data, also try
a model with one fewer AR term and one fewer MA
term--particularly if the parameter estimates in
the original model require more than 10
iterations to converge. Rule 9 If there is a
unit root in the AR part of the model i.e., if
the sum of the AR coefficients is almost exactly
1 you should reduce the number of AR terms by one
and increase the order of differencing by one.
Rule 10 If there is a unit root in the MA part
of the model--i.e., if the sum of the MA
coefficients is almost exactly 1--you should
reduce the number of MA terms by one and reduce
the order of differencing by one. Rule 11 If
the long-term forecasts appear erratic or
unstable, there may be a unit root in the AR or
MA coefficients.
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Gold Price Forecasting
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Gold Price Forecasting
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Gold Price Forecasting
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Gold Price Forecasting
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Gold Price Forecasting
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Forecasting results
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Forecasting results
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Forecasting results
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Forecasting results
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Agricultural price forecasting
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Metals and Minerals price forecasting
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Oil and Gas price forecasting
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All commodities
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All commodities
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All commodities
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All commodities price forecasting
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Thank You !