Transcript Analisi e Gestione del Rischio

Financial Products and Markets
Lecture 7
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.
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) > )
• The function q(.) is the  level quantile of
the distribution of losses (Xi).
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).
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) > )
= inf(x: Prob(x + Xi > 0) > ) =
= inf(x: Prob(x + Xi  0)  1 – )
VaR and distribution
• Call FX the distribution of Xi. Notice that
• FX(–VaR(Xi)) = Prob(Xi –VaR(Xi))
= Prob(– Xi >VaR(Xi))
= Prob(– Xi > F–X –1())
= Prob(F–X (– Xi ) > ) = 1 – 
• So, we may conclude
Prob(Xi  –VaR(Xi)) = 1 – 
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%)
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.
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.
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
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)
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.
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)
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!
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.
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 10–160, almost
zero…
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
V
1  2V
1
2
2




V 
S 

S



S



S
S
2 S 2
2
V
S S 1 S  S 

 


V
V S
2 V  S 
2
2
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.
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
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
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)
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 )).
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.
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.
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.
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.
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.
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
Smile, please!
Smiles in the equity markets
4
3,5
Implied Volatility
3
2,5
Mib30
SP500
FTSE
Nikkei
2
1,5
1
0,5
0
0,8
0,85
0,9
0,95
1
1,05
Moneyness
1,1
1,15
1,2
1,25
1,3
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.