Financial Products and Markets

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Financial Products and Markets

• Lecture 7

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Risk measurement

• The key problem for the construction of a risk
  measurement system is then the joint distribution
  of the percentage changes of value r1, r2,rn.
• The simplest hypothesis is a multivariate normal
  distribution. The RiskMetrics approach is
  consistent with a model of locally normal
  distribution, consistent with a GARCH model.

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Value-at-Risk

• Define Xi riciP(t,ti) the profit and loss on
  bucket i. The loss is then given by Xi. A risk
  measure is a function ?(Xi).
• Value-at-Risk
• VaR(Xi) q?(Xi) inf(x Prob(Xi? x) gt ?)
• The function q?(.) is the ? level quantile of the
  distribution of losses ?(Xi).

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VaR as margin

• Value-at-Risk is the corresponding concept of
  margin in the futures market.
• In futures markets, positions are
  marked-to-market every day, and for each position
  a margin (a cash deposit) is posted by both the
  buyer and the seller, to ensure enough capital is
  available to absorb the losses within a trading
  day.
• Likewise, a VaR is the amount of capital
  allocated to a given risk to absorb losses within
  a holding period horizon (unwinding period).

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VaR as capital

• It is easy to see that VaR can also be seen as
  the amount of capital that must be allocated to a
  risk position to limit the probability of loss to
  a given confidence level.
• VaR(Xi) q?(Xi) inf(x Prob(Xi? x) gt ?)
  inf(x Prob(x Xi gt 0) gt ?)
• inf(x Prob(x Xi ? 0) ? 1 ?)

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VaR and distribution

• Call FX the distribution of Xi. Notice that
• FX(VaR(Xi)) Prob(Xi ?VaR(Xi))
• Prob( Xi gtVaR(Xi)) Prob( Xi gt FX
  1(?))
• Prob(FX ( Xi ) gt ?) 1 ?
• So, we may conclude
• Prob(Xi ? VaR(Xi)) 1 ?

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VaR in a parametric approach

• pi marking-to-market of cash flow i
• ri, percentage daily change of i-th factor
• Xi, profits and losses piri
• Example ri has normal distribution with mean ?i
  and volatility ?i, Take ? 99
• Prob(ri ? ?i ?i 2.33) 1
• If ?i 0, Prob(Xi ri pi ? ?i pi 2.33) 1
• VaRi ?i pi 2.33 Maximum probable loss (1)

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VaR methodologies

• Parametric assume profit and losses to be
  (locally) normally distributed.
• Monte Carlo assumes the probability distribution
  to be known, but the pay-off is not linear (i.e
  options)
• Historical simulation no assumption about profit
  and losses distribution.

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VaR methodologies

• Parametric approach assume a distribution
  conditionally normal (EWMA model ) and is based
  on volatility and correlation parameters
• Monte Carlo simulation risk factors scenarios
  are simulated from a given distributon, the
  position is revaluated and the empirical
  distribution of losses is computed
• Historical simulation risk factors scenarios are
  simulated from market history, the position is
  revaluated, and the empirical distribution of
  losses is computed.

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Value-at-Risk criticisms

• The issue of coherent risk measures (aximoatic
  approach to risk measures)
• Alternative techniques (or complementary)
  expected shorfall, stress testing.
• Liquidity risk

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Coherent risk measures

• In 1999 Artzner, Delbaen-Eber-Heath addressed the
  following problems
• Which features must a risk measure have to be
  considered well defined?
• Risk measure axioms
•   Positive homogeneity ?(?X) ??(X)
•  Translation invariance ?(X ?) ?(X) ?
•  Subadditivity ?(X1 X2) ? ?(X1) ?(X2)

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Flaws of VaR

• Value-at-Risk is the quantile corresponding to a
  probability level.
• Critiques
• VaR does not give any information on the shape of
  the distribution of losses in the tail
• VaR of two businesses can be super-additive
  (merging two businesses, the VaR of the
  aggregated business may increase
• In general, the problem of finding the optimal
  portfolio with VaR constraint is extremely
  complex.

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Expected shortfall

• Expected shortfall is the expected loss beyond
  the VaR level. Notice however that, like VaR, the
  measure is referred to the distribution of
  losses.
• Expected shortfall is replacing VaR in many
  applications, and it is also substituting VaR in
  regulation (Base III).
• Consider a position X, the extected shortfall is
  defined as
• ES E(X X? VaR)

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Elicitability

• A new concept is elicitability, that means that
  there exists a function such that one can measure
  whether a measure is better then another.
• In other words, a measure is elicitable if it
  results from the optimization of a function. For
  example, minimizing a quadratic function yields
  the mean, while minimizing the absolute distance
  yields the median.
• Surprise VaR is elicitable, while ES is not.
• A new class of measures, both coherent and
  eligible? Expectiles!

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Economic capital and regulation

• Since the 80s the regulation has focussed on the
  concept of economic capital, defined as the
  distance between expected value of an investment
  and its VaR.
• In Basel II and Basel III the banks are required
  to post capital in order to face unexpected
  losses. The capital is measured by VaR
• In Solvency II and Basel IV VaR will be
  substituted by expected shortfall.

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Non normality of returns

• The assumption of normality of of returns is
  typically not borne out by the data. The reason
  is evidence of
• Asimmetry
• Leptokurtosis
• Other casual evidence on non-normality
• People make a living on that, so it must exist
• If nornal distribution of retruns were normal the
  crash of 1987 would have a probability of 10160,
  almost zero

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Why not normal? Options

• Assume to have a derivative sensitive to a single
  risk factor identified by the underlying asset S.
  
• Using a Taylor series expansion up to the second
  order

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Why non-normal? Leverage

• One possible reason for non normality,
  particularly for equity and corporate bonds, is
  leverage.
• Take equity, of a firm whose asset value is V and
  debt is B. Limited liability implies that at
  maturity
• Equity max(V(T) B, 0)
• Notice that if at some time t the call option
  (equity) is at the money, the return is not
  normal.

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Why not normal? Volatility

• Saying that a distribution is not normal amounts
  to saying that volatility is constant.
• Non normality may mean that variance either
• Does not exist
• It is a stochastic variable

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Dynamic volatility

• The most usual approach to non normality amounts
  to assuming that the volatility changes in time.
  The famous example is represented by GARCH models
• ht ? ? shock2t-1 ? ht -1

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Arch/Garch extensions

• In standard Arch/Garch models it is assumed that
  conditional distribution is normal, i.e. H(.) is
  the normal distribution
• In more advanced applications one may assume that
  H be nott normally distributed either. For
  example, it is assumed that it be Student-t or
  GED (generalised error distribution).
  Alternatively, one can assume non parametric
  conditonal distribution (semi-parametric Garch)

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Volatility asymmetry

• A flow of GARCH model is that the response of the
  return to an exogenous shock is the same no
  matter what the sign of the shock.
• Possible solutions consist in
• distinguishing the sign in the dynamic equation
  of volatility. Threshold-GARCH (TGARCH)
• ht ? ? shock2t-1 ? D shock2t-1 ? ht -1
• D 1 if shock is positive and zero otherwise.
• modelling the log of volatility (EGARCH)
• log(ht ) ? g (shockt-1 / ht -1 ) ? log( ht
  -1 )
• with g(x) ?x ?( ?x ?- E(?x ?)).

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High frequency data

• For some markets high frequency data is available
  (transaction data or tick-by-tick).
• Pros possibility to analyze the price dynamics
  on very small time intervals
• Cons data may be noisy because of microstructure
  of financial markets.
• Realised variance using intra-day statistics
  to represent variance, instead of the daily
  variation.

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Subordinated stochastic processes

• Consider the sequence of log-variation of prices
  in a given price interval. The cumulated return
• R r1 r2 ri rN
• is a variable that depends on the stoochastic
  processes
• a) log-returns ri.
• b) the number of transactions N.
• R is a subordinated stochastic process and N is
  the subordinator. Clark (1973) shows that R is a
  fat-tail process. Volatility increases when the
  number of transactions increases, and it is then
  correlated with volumes.

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Stochastic clock

• The fact that the number of transactions induces
  non normality of returns suggest the possibility
  to use a variable that, changing the pace of
  time, could restore normality.
• This variable is called stochastic clock. The
  technique of time change is nowadays one of the
  most used tools in mathematical finance.

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Implied volatility

• The volatility that in the Black and Scholes
  formula gives the option price observed in the
  market is called implied volatility.
• Notice that the Black and Scholes model is based
  on the assumption that volatility is constant.

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The Black and Scholes model

• Volatility is constant, which is equivalent to
  saying that returns are normally distributed
• The replicating portfolios are rebalanced without
  cost in continuous time, and derivatives can be
  exactly replicated (complete market)
• Derivatives are not subject to counterpart risk.

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Beyond Black Scholes

• Black Scholes implies the same volatility for
  every derivative contract.
• From the 1987 crash, this regularity is not
  supported by the data
• The implied volatility varies across the strikes
  (smile effect)
• The implied volatility varies across different
  maturities (volatility term structure)
• The underlying is not log-normally distributed

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Smile, please!
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Trading strategies with options

• Trade the skew betting on a reduction of the
  skewness flattening of the smile
• Trade of the fourth moment betting on a decrease
  of out and in the money options and increase of
  the at-the-money options.
• Volatility surface change of volatility across
  strike prices and maturities.