<<

. 8
( 16)



>>

Let us now show the calculation of the value of the option based on the ¬nal formula.
The auxiliary probability is given by:

1.07477 · 0.5047
q= = 0.5406
1.003274

Table 5.3 Binomial tree for underlying equity

0 1 2 3 4 5 6 7

100 107.477 115.513 124.150 133.432 143.409 154.132 165.656
93.043 100.000 107.477 115.513 124.150 133.432 143.409
86.570 93.043 100.000 107.477 115.513 124.150
80.548 86.570 93.043 100.000 107.477
74.944 80.548 86.570 93.043
69.731 74.944 80.548
64.880 69.731
60.366



Table 5.4 Binomial tree for option

0 1 2 3 4 5 6 7

4.657 7.401 11.462 17.196 24.809 34.126 44.491 55.656
1.891 3.312 5.696 9.555 15.482 23.791 33.409
0.456 0.906 1.801 3.580 7.118 14.150
0 0 0 0 0
0 0 0 0
0 0 0
0 0
0
168 Asset and Risk Management

ln 110 ’ ln(100 · d 7 )
= 4.1609, we ¬nd that J = 5. This will allow us to
In addition, as
ln u ’ ln d
calculate:

7 7 7
Pr[B(7; p) ≥ 5] = p 5 (1 ’ p)2 + p 6 (1 ’ p) + p7
5 6 7
= p 5 (21 ’ 35p + 15p 2 )

and therefore: Pr[B(7; q) ≥ 5] = 0.2343 and Pr[B(7; q ) ≥ 5] = 0.2984.
The price of the call therefore equals: C0 = 100 · 0.2984 ’ 110 · (1 + RF )’7 · 0.2343 =
4.657.
Meanwhile, the premium for the put with the same characteristics is:

P0 = 100 · (1 ’ 0.2984) + 100 · (1 + RF )’7 · (1 ’ 0.2343) = 12.168

Note that it is logical for the price of the put to be higher than that of the call, as the
option is currently ˜out of the money™.

5.3.1.4 Taking account of dividends
We have assumed until now that the underlying equity does not pay a dividend. Let us
now examine a case in which dividends are paid.
If only one dividend is paid during the i th period (interval [i ’ 1; i]), and the rate of the
dividend is termed δ (ratio of the dividend amount to the value of the security), the value
of the security will be reduced to the rate δ when the dividend is paid and the binomial
tree diagram for the underlying equity must therefore be modi¬ed as follows:

• up to the time (i ’ 1), no change: the values carried by the nodes in the tree diagram
for the period j ¤ i ’ 1 will be S0 uk d j ’k (k = 0, . . . , j );
• from the time i onwards (let us say for j ≥ i), the values become14 S0 (1 ’ δ)uk d j ’k
(k = 0, . . . , j );
• the tree diagram for the option is constructed in the classic backward style from
that point;
• if several dividends are paid at various times during the option contract, the procedure
described above must be applied whenever a payment is made.

5.3.2 Black and Scholes model for equity options
We now develop the well-known continuous time model compiled by Black and Scholes.15
In this model the option, concluded at moment 0 and maturing at moment T , can be
evaluated at any moment t ∈ [0; T ], and as usual, we note „ = T ’ t.
We further assume that the risk-free rate of interest does not change during this period,
that it is valid for any maturity date (¬‚at and constant yield curve) and that it is the same

This means that when the tree diagram is constructed purely numerically, taking account of the factor (1 ’ δ) will only
14

be effective for the passage from the time i ’ 1 to the time i.
15
Black F. and Scholes M., The pricing of options and corporate liabilities, Journal of Political Economy, Vol. 81, 1973,
pp. 637“59.
Options 169

for an investment as for a loan. The annual rate of interest, termed RF up until now, is
replaced in this continuous model by the corresponding instant rate r = ln(1 + RF ), so
that a unitary total invested during a period of t years becomes (1 + RF )t = ert .
Remember (see Section 3.4.2) that the evolution of the underlying equity value is
governed by the stochastic differential equation

dSt
= ER · dt + σR · dwt
St

We will initially establish16 the Black and Scholes formula for a call option the value
of which is considered to be a function of the value St of the underlying equity and of
time t, the other parameters being considered to be constant: Ct = C(St , t).
By applying Itˆ ™s formula to the function C(St , t), we obtain:
o

σR 2
2
dC(St , t) = Ct + ER St CS + SC · dt + σR St CS · dwt
2 t SS

Let us now put together a portfolio that at moment t consists of:

• the purchase of X underlying equities with a value of St ;
• the sale of one call on then underlying equity, with value C(St , t).

The value Vt of this portfolio is given by Vt = X · St ’ C(St , t). This, by differentiation,
gives:

σR 2
2
dVt = X · [ER St · dt + σR St · dwt ] ’ Ct + ER St CS + SC · dt + σR St CS · dwt
2 t SS
σR 2
2
= X · ER St ’ Ct + ER St CS + SC · dt + [X · σR St ’ σR St CS ] · dwt
2 t SS


We then choose X so that the portfolio no longer has any random components (the
coef¬cient of dwt in the preceding relation must be zero). The hypothesis of absence of
arbitrage opportunity shows that if possible, the return on the portfolio should be given
by the risk-free rate r:
dVt
= r · dt + 0 · dwt
Vt

We therefore arrive at:
±
 X · ER St ’ C + ER St C + σR S 2 C
 2

 t
2 t SS
 S

=r
X · St ’ C(St , t)


 X · σR St ’ σR St CS


 =0
X · St ’ C(St , t)

16
We will only develop the ¬nancial part of the logic, as the end of the demonstration is purely analytical. Readers interested
´
in details of calculations can consult the original literature or Devolder P., Finance Stochastique, Editions de l™ULB, 1993.
170 Asset and Risk Management

or, in the same way:
±
σR 2
2

X · (ER ’ r)St ’ Ct + ER St CS + S C ’ rC(St , t) = 0
2 t SS

X ’ CS = 0

The second equation provides the value of X, which cancels out the random component
of the portfolio: X = C S . By making a substitution in the ¬rst equation, we ¬nd:

σR 2
2
(ER ’ r)St · CS ’ Ct + ER St CS + S C ’ rC(St , t) = 0
2 t SS

In other words:
σR 2
2
Ct + rSt CS + S C ’ rC(St , t) = 0
2 t SS

In this equation, the instant mean return ER has disappeared.17
We are looking at a partial derivative equation (in which none of the elements are now
random) of the second order for the unknown function C(St , t). It allows a single solution
if two limit conditions are imposed:

C(0, t) = 0
C(ST , T ) = (ST ’ K)+

Through a change in variables, this equation can be turned into an equation well known
to physicists: the heat equation.18 It is in fact easy, although demanding, to see that if the
new unknown function u(x, s) = C(St , t)ert is introduced where the change of variables
± « 
occurs:


 ¬2 ·


 St = K · exp ¬ σR (x ’ s) ·

¬ 2·


  σR 
 2 r’
2


 sσR2
t = T ’



 22
σR

 2 r’

2

which inverts to:
±
 x = 2 r ’ σR St σ2
2

+ r’ R
 · ln „
 K
σR
2 2 2
2

 s = 2 r ’ σR
2
 „

σR
2 2

In the same way as the independence of the result obtained by the binomial model with respect to the probability ±
17

governing the evolution of the subjacent share price was noted.
18
See for example: Krasnov M., Kisilev A., Makarenko G. and Chikin E., Math´ matique sup´ rieures pour ing´ nieurs et
e e e
polytechniciens, De Boeck, 1993. Also: Sokolnikov I. S. and Redheffer R. M., Mathematics of Physics and Modern Engineering,
McGraw-Hill, 1966.
Options 171

The equation obtained turns into: uxx = us .
With the conditions limit:
±
 lim u(x, s) = 0
 x’’∞
 ± ® «  


 σR
2

 

¬ x2
  · 

 exp ¬ · ’ 1 if x ≥ 0
 u(x, 0) = v(x) = K · °   »
 σR2
 
 r’

 
 
 2


if x < 0
0
this heat equation has the solution:
+∞
1
v(y)e’(x’y) /4s
2
u(x, s) = √ dy
2 πs ’∞

By making the calculations with the speci¬c expression of v(y), and then making the
inverse change of variables, we obtain the Black and Scholes formula for the call option

C(St , t) = St (d1 ) ’ Ke’r„ (d2 ),

where we have:
St σR2
ln + r ± „
K
d1 2
= √
d2 σR „
and the function represents the standard normal distribution function:
t
1
e’x /2
2
(t) = √ dx

’∞

The price Pt of a put option can be evaluated on the basis of the price of the call option,
thanks to the relation of ˜call“put™ parity: Ct + K · e’rT = Pt + St .
In fact:

P (St , t) = C(St , t) + Ke’r„ ’ St
= St (d1 ) ’ Ke’r„ (d2 ) + Ke’r„ ’ St
(d1 )] + Ke’r„ [1 ’
= ’St [1 ’ (d2 )]

and therefore:
P (St , t) = ’St (’d1 ) + Ke’r„ (’d2 )

because:
1’ (t) = (’t)

Example
Consider an option with the same characteristics as in Section 5.3.1: S0 = 100, K = 110,
t = 0, T = 7 months, σR = 0.25 on an annual basis and RF = 4 % per year.
172 Asset and Risk Management

We are working with the year as the time basis, so that: „ = 7/12, r = ln 1.04 = 0.03922.

0.252
100 7
+ 0.03922 ± ·
ln
d1 ’0.2839
110 2 12
= =
d2 ’0.4748
7
0.25 ·
12

Hence (d1 ) = 0.3823 and (d2 ) = 0.3175. This allows the price of the call to be cal-
culated:

7
(d1 ) ’ 110 · e’0.03922· 12 ·
C = C(S0 , 0) = 100 · (d2 ) = 4.695

As the put premium has the same characteristics, it totals:

7
(d1 )] + 110 · e’0.03922· 12 · [1 ’
P = P (S0 , 0) = ’100 · [1 ’ (d2 )] = 12.207

The similarity of these ¬gures to the values obtained using the binomial model (4.657
and 12.168 respectively) will be noted.


5.3.2.2 Sensitivity parameters

When the price of an option is calculated using the Black and Scholes formula, the
sensitivity parameters or ˜Greeks™ take on a practical form.
Let us examine ¬rst the case of a call option delta. If the reduced normal density is
termed φ
1
φ(x) = (x) = √ e’x /2
2




we arrive, by derivation, at:

1
St φ(d1 ) ’ Ke’r„ φ(d2 )
(C) = CS = (d1 ) + √
St σR „

It is easy to see that the quantity between the square brackets is zero and that therefore
(C) = (d1 ), and that by following a very similar logic, we will arrive at a put of:
(P ) = (d1 ) ’ 1.
The above formula provides a very simple means of determining the number of equities
that should be held by a call issuer to hedge his risk (the delta hedging). This is a common
use of the Black and Scholes relation: the price of an option is determined by the law
of supply and demand and its ˜inversion™ provides the implied volatility. The latter is
therefore used in the relation (C) = (d1 ), which is then known as the hedging formula.
Options 173

The other sensitivity parameters (gamma, theta, vega and rho) are obtained in a simi-
lar way:
φ(d1 )
(C) = (P ) = √
St σR „
±
 (C) = ’ St σR√ 1 ) ’ rKe’r„ (d2 )
φ(d

 2„

 (P ) = ’ St σR√ 1 ) + rKe’r„ (’d )
φ(d
 2
2„
V (C) = V (P ) = „ St φ(d1 )
ρ(C) = „ Ke’r„ (d2 )
ρ(C) = ’„ Ke’r„ (’d2 )

In ¬nishing, let us mention a relationship that links the delta, gamma and theta parameters.
The partial derivative equation obtained during the demonstration of the Black and Scholes
formula for a call is also valid for a put (the price therefore being referred to as p without
being speci¬ed):
σR 2
2
pt + rSt pS + S p ’ rp(St , t) = 0
2 t SS

This, using the sensitivity parameters, will give:

σR 2
2
+ rSt + S =r ·p
2t


5.3.2.3 Taking account of dividends
If a continuous rate dividend19 δ is paid between t and T and the underlying equity
is worth St (resp. ST ) at the moment t (resp. T ), it can be said that had it not paid a
dividend, it would have passed from value St to value eδ„ ST . It can also be said that the
same equity without dividend would pass from value e’δ„ St at moment t to value ST at
moment T . In order to take account of the dividend, therefore, it will suf¬ce within the
Black and Scholes formula to replace St by e’δ„ St , thus giving:

C(St , t) = St e’δ„ (d1 ) ’ Ke’r„ (d2 )
P (St , t) = ’St e’δ„ (’d1 ) + Ke’r„ (’d2 ).

where, we have:
St σR2
ln + r ’ δ ± „
K
d1 2
= √
d2 σR „

19
An discounting/capitalisation factor of the exponential type is used here and throughout this paragraph.
174 Asset and Risk Management

5.3.3 Other models of valuation
5.3.3.1 Options on bonds
It is not enough to apply the methods shown above (binomial tree diagram or Black and
Scholes formula) to options on bonds. In fact:

• Account must be taken of coupons regularly paid.
• The constancy of the underlying equity volatility (a valid hypothesis for equities) does
not apply in the case of bonds as their values on maturity converge towards the repay-
ment value R.

The binomial model can be adapted to suit this situation, but is not an obvious generali-
sation of the method set out above.20
Adapting the Black and Scholes model consists of replacing the geometric Brownian
motion that represents the changes in the value of the equity with a stochastic process that
governs the changes in interest rates, such as those used as the basic for the Vasicek and
Cox, Ingersoll and Ross models (see Section 4.5). Unfortunately, the partial derivatives
equation deduced therefrom does not generally allow an analytical solution and numeric
solutions therefore have to be used.21

5.3.3.2 Exchange options
For an exchange option, two risk-free rates have to be taken into consideration: one
relative to the domestic currency and one relative to the foreign currency.
For the discrete model, these two rates are referred to respectively as RF and RF ) .
(D) (F

By altering the logic of Section 5.3.1 using this generalisation, it is possible to determine
the price of an exchange option using the binomial tree diagram technique. It will be
seen that the principle set out above remains valid with a slight alteration of the close
formulae: C0 is the discounted expected value of the call on maturity (for a period):
(D)
C0 = (1 + RF )’1 q · C(u) + (1 ’ q) · C(d)

with the neutral risk probability:

1 + (RF ’ RF ) ) ’ d
(D) (F
u ’ 1 + (RF ’ RF ) )
(D) (F

q= 1’q =
u’d u’d
For the continuous model, the interest rates in the domestic and foreign currencies are
referred to respectively as r (D) and r (F ) . Following a logic similar to that accepted for
options on dividend-paying equities, we see that the Black and Scholes formula is still
(F )
valid provided the underlying equity price St is replaced by St e’r „ , which gives the
formulae: (F ) (D)
C(St , t) = St e’r „ (d1 ) ’ Ke’r „ (d2 )
(F ) (D)
P (St , t) = ’St e’r „
(’d1 ) + Ke’r „
(’d2 )
20
Read for example Copeland T. E. and Weston J. F., Financial Theory and Corporate Policy, Addison-Wesley, 1988.
21
See for example Cortadon G., The pricing of options on default-free bonds, Journal of Financial and Quantitative
Analysis, Vol. 17, 1982, pp. 75“100.
Options 175

where, we have:
St σ2
+ (r (D) ’ r (F ) ) ± R „
ln
K
d1 2
= √
d2 σR „

This is known as the Garman“Kohlhagen formula.22

5.4 STRATEGIES ON OPTIONS23
5.4.1 Simple strategies
5.4.1.1 Pure speculation
As we saw in Section 5.1, the asymmetrical payoff structure particular to options allows
investors who hold them in isolation to pro¬t from the fall in the underlying equity
price while limiting the loss (on the reduced premium) that occurs when a contrary
variation occurs.
The issue of a call/put option, on the other hand, is a much more speculative oper-
ation. The pro¬t will be limited to the premium if the underlying equity price remains
lower/higher than the exercise price, while considerable losses may arise if the price of
the underlying equity rises/falls more sharply. This type of operation should therefore
only be envisaged if the issuer is completely con¬dent that the price of the underlying
equity will fall/rise.

5.4.1.2 Simultaneous holding of put option and underlying equity
As the purchase of a put option allows one to pro¬t from a fall in the underlying equity
price, it seems natural to link this fall to the holding of the underlying equity, in order to
limit the loss in¬‚icted by the fall in the price of the equity held alone.

5.4.1.3 Issue of a call option with simultaneous holding of underlying equity
We have also seen (Example 1 in Section 5.1.2) that it is worthwhile issuing a call option
while holding the underlying equity at the same time. In fact, when the underlying equity
price falls (or rises slightly), the loss incurred thereon is partly compensated by encashment of
the premium, whereas when the price rises steeply, the pro¬t that would have been realised
on the underlying equity is limited to the price of the option increased by the difference
between the exercise price and the underlying equity price at the beginning of the contract.

5.4.2 More complex strategies
Combining options allows the creation of payoff distributions that do not exist for classic
assets such as equities or bonds. These strategies are usually used by investors trying to
turn very speci¬c forecasts to pro¬t. We will look brie¬‚y at the following:

• straddles;
• strangles;
22
Garman M. and Kohlhagen S., Foreign currency option values, Journal of International Money and Finance, No. 2,
1983, pp. 231“7.
23
Our writings are based on Reilly F. K. and Brown K. C., Investment Analysis and Portfolio Management, South-
Western, 2000.
176 Asset and Risk Management

• spreads;
• range forwards.

5.4.2.1 Straddles
A straddle consists of simultaneously purchasing (resp. selling) a call option and a put
option with identical underlying equity, exercise price and maturity date. The concomitant
call (resp. put) corresponds to a long (resp. short) straddle.
Clearly it is a question of playing volatility, as in essence, it is contradictory to play
the rise and the fall in the underlying equity price at the same time.
We saw in Section 5.2.3 (The Greeks: vega) that the premium of an option increases
along with volatility. As a result, the short straddle (resp. long straddle) is the action of
an investor who believes that the underlying equity price will vary more (resp. less) than
historically regardless of direction of variation.
It is particularly worth mentioning that with the short straddle, it is possible to make
money with a zero variation in underlying equity price.
Finally, note that the straddle (Figure 5.12) is a particular type of option known as the
chooser option.24

5.4.2.2 Strangles
The strangle is a straddle except for the exercise price, which is not identical for the call
option and the put option, the options being ˜out of the money™. As a result:

• The premium is lower.
• The expected variation must be greater than that associated with the straddle.

Certainly, this type of strategy presents a less aggressive risk-return pro¬le in comparison
with the straddle. A comparison is shown in Figure 5.13.

5.4.2.3 Spreads
Option spreads consist of the concomitant purchases of two contracts that are identical
but for just one of their characteristics:


Profit
Long
straddle

p
K
S
’p

Short
straddle


Figure 5.12 Long straddle and short straddle

24
Reilly F. K. and Brown K. C., suggest reading Rubinstein M., Options for the Undecided, in From Black“Scholes to
black holes, Risk Magazine, 1992.
Options 177
Profit

Long
straddle Long
strangle



S
Put exercise Call exercise
price price


Figure 5.13 Long strangle compared with long straddle


Profit

Call 1
exercise
price


S
Call 2
exercise
price


Figure 5.14 Bull money spread


• The money spread consists of the simultaneous sale of an out of the money call option
and the purchase of the same option in the money. The term bull money spread (resp.
bear money spread ) is used to describe a money spread combination that gains when
the underlying equity price rises (resp. falls) (see Figure 5.14). The term butter¬‚y money
spread is used to de¬ne a combination of bear and bull money spreads with hedging
(limitation) for potential losses (and, obviously, reduced opportunities for pro¬t).
• The calendar spread consists of the simultaneous sale and purchase of call or put
options with identical exercise prices but different maturity dates.

Spreads are used when a contract appears to have an aberrant value in comparison with
another contract.

5.4.2.4 Range forwards
For memory, range forwards consist of a combination of two optional positions. This
combination is used for hedging, mainly for options on exchange rates.
Part III
General Theory of VaR




Introduction
6 Theory of VaR
7 VaR estimation techniques
8 Setting up a VaR methodology
180 Asset and Risk Management

Introduction
As we saw in Part II, the sheer variety of products available on the markets, linear and
otherwise, together with derivatives and underlying products, implies a priori a multi-
faceted understanding of risk, which by nature is dif¬cult to harmonise.
Ideally, therefore, we should identify a single risk indicator that estimates the loss
likely to be suffered by the investor with the level of probability of that loss arising. This
indicator is VaR.
There are three classic techniques for estimating VaR:

1. The estimated variance“covariance matrix method.
2. The Monte Carlo simulation method.
3. The historical simulation method.

An in-depth analysis of each of these methods will show their strong and weak points
from both a theoretical and a practical viewpoint.
We will now show, in detail, how VaR can be calculated using the historical simulation
method. This method is the subject of the following chapter as well as a ¬le on the
accompanying CD-ROM entitled ˜Ch 8™, which contains the Excel spreadsheets relating
to these calculations.
6
Theory of VaR

6.1 THE CONCEPT OF ˜RISK PER SHARE™
6.1.1 Standard measurement of risk linked to ¬nancial products
The various methods for measuring risks associated with an equity or portfolio of equities
have been studied in Chapter 3. Two types of measurement can be de¬ned: the intrinsic
method and the relative method.
The intrinsic method is the variance (or similarly, the standard deviation) in the return
of the equity. In the case of a portfolio, we have to deal not only with variances but also
with correlations (or covariances) two by two. They are evaluated practically by their
ergodic estimator, that is, on the basis of historical observations (see Section 3.1).
The relative method takes account of the risk associated with the equity or portfolio of
equities on the basis of how it depends upon market behaviour. The market is represented
by a stock-exchange index (which may be a sector index). This dependence is measured
using the beta for the equity or portfolio and gives rise to the CAPM type of valuation
model (see Section 3.3).
The risk measurement methods for the other two products studied (bonds and options)
fall into this second group.
Among the risks associated with a bond or portfolio of bonds, those that are linked
to interest-rate ¬‚uctuations can be expressed as models. In this way (see Section 4.1) we
see the behaviour of the two components of the risk posed by selling the bond during its
lifetime and reinvesting the coupons, according to the time that elapses between the issue
of the security and its repayment. If we wish to summarise this behaviour in a simple
index, we have to consider the duration of the bond; as we are looking in this context
at a ¬rst-level approximation, a second measurement, that of convexity (see Section 4.2)
will de¬ne the duration more precisely.
Finally, the value of an option depends on a number of variables: underlying equity
price, exercise price, maturity, volatility, risk-free rate.1 The most important driver is of
course the underlying equity price, and for this reason two parameters, one of the ¬rst
order (delta) and another of the second order (gamma), are associated with it. The way
in which the option price depends on the other variables gives rise to other sensitivity
parameters. These indicators are known as ˜the Greeks™ (see Section 5.2).


6.1.2 Problems with these approaches to risk
The ways of measuring the risks associated with these products or a portfolio of them,
whatever they may make to the management of these assets, bring features with them
that do not allow for immediate generalisation.

1
Possibly in two currencies if an exchange option is involved.
182 Asset and Risk Management

1. The representation of the risk associated with an equity through the variance in its
returns (or through its square root, the standard deviation), or of the risk associated
with an option through its volatility, takes account of both good and bad risks. A
signi¬cant variance corresponds to the possibility of seeing returns vastly different
from the expected return, i.e. very small values (small pro¬ts and even losses) as well
as very large values (signi¬cant pro¬ts).
This method does not present many inconveniences in portfolio theory (see
Section 3.2), in which equities or portfolios with signi¬cant variances are volatile
elements, little appreciated by investors who prefer ˜certainty™ of return with low risk
of loss and low likelihood of signi¬cant pro¬t. It is no less true to say that in the context
of risk management, it is the downside risk that needs to be taken into consideration.
Another parameter must therefore be used to measure this risk.
2. The approach to the risks associated with equities in Markowitz™s theory limits the
description of a distribution to two parameters: a measure of return and a measure
of deviation. It is evident that an in¬nite number of probability laws correspond to
any one expected return“variance pairing. We are, in fact, looking at skewed distribu-
tions: Figure 6.1 shows two distributions that have the same expectation and the same
variance, but differ considerably in their skewness.
In the same way, distributions with the same expectation, variance and skewness
coef¬cient γ1 may show different levels of kurtosis, as shown in Figure 6.2.
The distributions with higher peaks towards the middle and with fatter tails than
a normal distribution2 (and therefore less signi¬cant for intermediate values) are
described as leptokurtic and characterised by a positive kurtosis coef¬cient γ2 (for the


f(x)




x

Figure 6.1 Skewness of distributions





√3
3

√6

6
“√3 “ √6
  
√6 √3

2 2

Figure 6.2 Kurtosis of distributions

2
The de¬nition of this law is given in Point (3) below.
Theory of VaR 183

distributions in Figure 6.2, this coef¬cient totals ’0.6 for the triangular and ’1.2 for
the rectangular).
Remember that this variance of expected returns approach is sometimes justi¬ed
through utility theory. In fact, when the utility function is quadratic, the expected
utility of the return on the portfolio is expressed solely from the single-pair expectation
variance (see Section 3.2.7).
3. In order to justify the mean-variance approach, the equity portfolio theory deliberately
postulates that the return follows a normal probability law, which is characterised
speci¬cally by the two parameters in question; if µ and σ respectively indicate the
mean and the standard deviation for a normal random variable, this variable will have
the density of:
1 x’µ 2
1
f (x) = √ exp ’
σ
2
2πσ

This is a symmetrical distribution, very important in probability theory and found
everywhere in statistics because of the central limit theorem. The graph for this density
is shown in Figure 6.3.
A series of studies shows that normality of return of equities is a hypothesis that
can be accepted, at least in an initial approximation, provided the period over which
the return is calculated is not too short. It is admitted that weekly and monthly returns
do not diverge too far from a normal law, but daily returns tend to diverge and follow
a leptokurtic distribution instead.3
If one wishes to take account of the skewness and the leptokurticity of the distribution
of returns, one solution is to replace the normal distribution with a distribution that
depends on more parameters, such as the Pearson distribution system,4 and to estimate
the parameters so that µ, σ 2 , γ1 and γ2 correspond to the observations. Nevertheless,
the choice of distribution involved remains wholly arbitrary.
Finally, for returns on securities other than equities, and for other elements involved
in risk management, the normality hypothesis is clearly lacking and we do not therefore
need to construct a more general risk measurement index.
4. Another problem, by no means insigni¬cant, is that concepts such as duration and
convexity of bonds, variances of returns on equities, or the delta, gamma, rho or theta


f (x)




x

Figure 6.3 Normal distribution

3
We will deal again with the effects of kurtosis on risk evaluation in Section 6.2.2.
4
Johnson N. L. and Kotz S., Continuous Univariate Distributions, John Wiley and Sons, Ltd, 1970.
184 Asset and Risk Management

option parameters do not, despite their usefulness, actually ˜say™ very much as risk
measurement indices. In fact, they do not state the kind of loss that one is likely
to suffer, or the probability of it occurring. At the very most, the loss“probability
pairing will be calculated on the basis of variance in a case of normal distribution (see
Section 6.2.2).
5. In Section 6.1.1 we set out a number of classical risk analysis models associated with
three types of ¬nancial products: bonds, equities and options. These are speci¬c models
adapted to speci¬c products. In order to take account of less ˜classical™ assets (such as
certain sophisticated derivatives), we will have to construct as many adapted models as
are necessary and take account in those models of exchange-rate risks, which cannot
be avoided on international markets.
Building this kind of structure is a mammoth task, and the complexity lies not only
in building the various blocks that make up the structure but also in assembling these
blocks into a coherent whole. A new technique, which combines the various aspects
of market risk analysis into a uni¬ed whole, therefore needs to be elaborated.

6.1.3 Generalising the concept of ˜risk™
The market risk is the risk with which the investor is confronted because of his lack
of knowledge of future changes in basic market variables such as security rates, interest
rates, exchange rates etc. These variables, also known as risk factors, determine the price
of securities, conditional assets, portfolios etc.
If the price of an asset is expressed as p and the risk factors that explain the price
as X1 , X2 , . . . , Xn , we have the wholly general relation p = f (X1 , X2 , . . . , Xn ) + µ, in
which the residue µ corresponds to the difference between reality (the effective price p)
and the valuation model (the function f ).
If the price valuation model is a linear model (as for equities), the risk factors com-
bine, through the central limit theorem, to give a distribution of the variable p that is
normal (at least in an rough approximation) and is therefore de¬ned only by the two
expectation“variance parameters.
On the other hand, for some types of security such as options, the valuation model
ceases to be linear. The above logic is no longer applicable and its conclusions cease to
be valid.
We would point out that alongside the risk factors that we have just mentioned, the
following can be added as factors in market risk:

• the imperfect nature of valuation models;
• imperfect knowledge of the rules and limits particular to the institution;
• the impossibility of anticipating regulatory and legislative changes.

Note
As well as market risk, investors are confronted with other types of risk that correspond
to the occurrence of exceptional events such as wars, oil crises etc. This group of risks
cannot of course be estimated using techniques designed for market risk. The techniques
shown in this Part III do not therefore deal with these ˜event-related™ risks. This should
not, however, prevent the wise risk manager from analysing his positions using value at
risk theory, or from using ˜catastrophe scenarios™, in an effort to understand this type of
exceptional risk.
Theory of VaR 185

6.2 VaR FOR A SINGLE ASSET
6.2.1 Value at Risk
In view of what has been set out in the previous paragraph, an index that allows estimation
of the market risks facing an investor should:

• be independent of any distributional hypothesis;
• concern only downside risk, namely the risk of loss;
• measure the loss in question in a certain way;
• be valid for all types of assets and therefore either involve the various valuation models
or be independent of these models.

Let us therefore consider an asset the price5 of which is expressed as pt at moment t.
The variation observed for the asset in the period [s; t] is expressed as ps,t and is
therefore de¬ned as ps,t = pt ’ ps . Note that if ps,t is positive, we have a pro¬t; a
negative value, conversely, indicates a loss.
The only hypothesis formulated is that the value of the asset evolves in a stationary
manner; the random variable ps,t has a probability law that only depends on the interval
in which it is calculated through the duration (t ’ s) of that interval. The interval [s; t]
is thus replaced by the interval [0; t ’ s] and the variable p will now only have the
duration of the interval as its index. We therefore have the following de¬nitive de¬nition:
p t = pt ’ p0 .
The ˜value at risk™ of the asset in question for the duration t and the probability level
q is de¬ned as an amount termed VaR, so that the variation pt observed for the asset
during the interval [0; t] will only be less than VaR with a probability of (1 ’ q):

Pr[ pt ¤ VaR] = 1 ’ q

Or similarly:
Pr[ pt > VaR] = q

By expressing as F p and f p respectively the distribution function and density function
of the random variable pt , we arrive at the de¬nition of VaR in Figures 6.4 and 6.5.


F∆p(x)
1




1“q

VaR x

Figure 6.4 De¬nition of VaR based on distribution function

5
In this chapter, the theory is presented on the basis of the value, the price of assets, portfolios etc. The same developments
can be made on the basis of returns on these elements. The following two chapters will show how this second approach is the
one that is adopted in practice.
186 Asset and Risk Management

f∆p(x)




1“q

x
VaR

Figure 6.5 De¬nition of VaR based on density function


It is evident that two parameters are involved in de¬ning the concept of VaR: duration
t and probability q. In practice, it is decided to ¬x t once for everything (one day or one
week, for example), and VaR will be calculated as a function of q and expressed VaR q
if there is a risk of confusion. It is in fact possible to calculate VaR for several different
values of q.

Example
If VaR at 98 % equals ’500 000, this means that there are 98 possibilities out of 100
of the maximum loss for the asset in question never exceeding 500 000 for the period
in question.

Note 1
As we will see in Chapter 7, some methods of estimating VaR are based on a distribution
of value variation that does not have a density. For these random variables, as for the
discrete values, the de¬nition that we have just given is lacking in precision. Thus, when
1 ’ q corresponds to a jump in the distribution function, no suitable value for the loss
can be given and the de¬nition will be adapted as shown in Figure 6.6.

In the same way, when q corresponds to a plateau in the distribution function, an
in¬nite number of values will be suitable; the least favourable of these values, that is the
smallest, is chosen as a safety measure, as can be seen in Figure 6.7.
In order to take account of this note, the very strict de¬nition of VaR will take the
following form:
VaR q = min {V : Pr[ pt ¤ V ] ≥ 1 ’ q}



F∆p(x)




1“q

VaR x

Figure 6.6 Case involving jump
Theory of VaR 187

F∆p(x)




1“q

VaR x

Figure 6.7 Case involving plateau


Table 6.1 Probability distribution of loss

p Pr

’5 0.05
’4 0.05
’3 0.05
’2 0.10
’1 0.15
0 0.10
1 0.20
2 0.15
3 0.10
4 0.05



Example
Table 6.1 shows the probability law for the variation in value.
For this distribution, we have VaR 0.90 = ’4 and VaR 0.95 = ’5.

Note 2
Clearly VaR is neither the loss that should be expected nor the maximum loss that is likely
to be incurred, but is instead a level of loss that will only be exceeded with a level of
probability ¬xed a priori. It is a parameter that is calculated on the basis of the probability
law for the variable (˜variation in value™) and therefore includes all the parameters for
that distribution. VaR is not therefore suitable for drawing up a classi¬cation of securities
because, as we have seen for equities, the comparison of various assets is based on the
simultaneous consideration of two parameters: the expected return (or loss) and a measure
of dispersion of the said return.

Note 3
On the other hand, it is essential to be fully aware when de¬ning VaR of the duration on the
basis of which this parameter is evaluated. The parameter, calculated for several different
portfolios or departments within an institution, is only comparable if the reference period
is the same. The same applies if VaR is being used as a comparison index for two or
more institutions.
188 Asset and Risk Management

Note 4
Sometimes a different de¬nition of VaR is found,6 one that takes account not of the varia-
tion in the value itself but the difference between that variation and the expected variation.
More speci¬cally, this value at risk (for the duration t and the probability level q) is de¬ned
as the amount (generally negative) termed VaR — , so that the variation observed during the
interval [0; t] will only be less than the average upward variation in |VaR — | with a prob-
ability of (1 ’ q). Thus, if the expected variation is expressed as E( pt ), the de¬nition
Pr[ pt ’ E( pt ) ¤ VaR — ] = 1 ’ q. Or, again: Pr[ pt > VaR — + E( pt )] = q.
It is evident that these two concepts are linked, as we evidently have
VaR = VaR — + E( pt ).

6.2.2 Case of a normal distribution
In the speci¬c case where the random variable pt follows a normal law with mean
E( pt ) and standard deviation σ ( pt ), the de¬nition can be changed to:

VaR q ’ E( pt )
pt ’ E( pt )
¤ =1’q
Pr
σ ( pt ) σ ( pt )

VaR q ’ E( pt )
This shows that the expression is the quantile of the standard normal
σ ( pt )
distribution, ordinarily expressed as z1’q . As z1’q = ’zq , this allows VaR to be written
in a very simple form VaR q = E( pt ) ’ zq · σ ( pt ) according to the expectation and
standard deviation for the loss. In the same way, the parameter VaR — is calculated simply,
for a normal distribution, VaR q — = ’zq · σ ( pt ).
The values of zq are found in the normal distribution tables.7 A few examples of these
values are given in Table 6.2.


Table 6.2 Normal distribution quantiles

q zq

0.500 0.0000
0.600 0.2533
0.700 0.5244
0.800 0.8416
0.850 1.0364
0.900 1.2816
0.950 1.6449
0.960 1.7507
0.970 1.8808
0.975 1.9600
0.980 2.0537
0.985 2.1701
0.990 2.3263
0.995 2.5758

6
Jorion P., Value At Risk, McGraw-Hill, 2001.
7
Pearson E. S. and Hartley H. O., Biometrika Tables for Statisticians, Biometrika Trust, 1976, p. 118.
Theory of VaR 189

Example
If a security gives an average pro¬t of 100 over the reference period with a standard
deviation of 80, we have E( pt ) = 100 and σ ( pt ) = 80, which allows us to write:

VaR 0.95 = 100 ’ (1.6449 — 80) = ’31.6
VaR 0.975 = 100 ’ (1.9600 — 80) = ’56.8
VaR 0.99 = 100 ’ (2.3263 — 80) = ’86.1

The loss incurred by this security will only therefore exceed 31.6 (56.8 and 86.1 respec-
tively) ¬ve times (2.5 times and once respectively) in 100 times.

Note
It has been indicated in Section 6.1.2 that the normality hypothesis was far from being
valid in all circumstances. In particular, it has been shown that the daily returns on equities
are better represented by a Pareto or Student distribution,8 that is, leptokurtic distributions.
Thus, for a Student distribution with ν degrees of freedom (where ν > 2), the variance is
2
σ2 = 1 +
ν ’2
and the kurtosis coef¬cient (for ν > 4) will then be:
6
γ2 =
ν’4
This last quantity is always positive, and this proves that the Student distribution is
leptokurtic in nature. With regard to the number of degrees of freedom ν, Table 6.3 shows
the coef¬cient γ2 and the quantiles zq for q = 0.95, q = 0.975 and q = 0.99 relative to
these Student distributions,9 reduced beforehand (the variable is divided by its standard
deviation) in order to make a useful comparison between these ¬gures and those obtained
on the basis of the reduced normal law.

Table 6.3 Student distribution quantiles

ν γ2 z0.95 z0.975 z0.99

5 6.00 2.601 3.319 4.344
10 1.00 2.026 2.491 3.090
15 0.55 1.883 2.289 2.795
20 0.38 1.818 2.199 2.665
25 0.29 1.781 2.148 2.591
30 0.23 1.757 2.114 2.543
40 0.17 1.728 2.074 2.486
60 0.11 1.700 2.034 2.431
120 0.05 1.672 1.997 2.378
normal 0 1.645 1.960 2.326

8
Blattberg R. and Gonedes N., A comparison of stable and student distributions as statistical models for stock prices,
Journal of Business, Vol. 47, 1974, pp. 244“80.
9
Pearson E. S. and Hartley H. O., Biometrika Tables for Statisticians, Biometrika Trust, 1976, p. 146.
190 Asset and Risk Management

This clearly shows that when the normal law is used in place of the Student laws, the
VaR parameter is underestimated unless the number of degrees of freedom is high.

Example

With the same data as above, that is, E( pt ) = 100 and σ ( pt ) = 80, and for 15 degrees
of freedom, we ¬nd the following evaluations of VaR, instead of 31.6, 64.3 and 86.1
respectively.

VaR 0.95 = 100 ’ (1.883 — 80) = ’50.6
VaR 0.975 = 100 ’ (2.289 — 80) = ’83.1
VaR 0.99 = 100 ’ (2.795 — 80) = ’123.6


6.3 VaR FOR A PORTFOLIO
6.3.1 General results
Consider a portfolio consisting of N assets in respective quantities10 n1 , . . . , nN . If the
price of the j th security is termed pj , the price pP of the portfolio will of course be
given by:
N
pP = n j pj
j =1


The price variation will obey the same relation:

N
pP = n j pj
j =1


Once the distribution of the various pj elements is known, it is not easy to determine
the distribution of the pP elements: the probability law of a sum of random variables
will only be easy to determine if these variables are independent, and this is clearly not
the case here. It is, however, possible to ¬nd the expectation and variance for pP on
the basis of expectation, variance and covariance in the various pj elements:

N
E( pP ) = nj E( pj )
j =1

N N
var( pP ) = ni nj cov( pi , pj )
i=1 j =1


It can be shown that when prices are replaced by returns, the numbers nj of assets in the portfolio must be replaced by
10

proportions Xj (positive numbers the sum of which is 1), representing the respective stock-exchange capitalisation levels of
the various securities (see Chapter 3).
Theory of VaR 191

where we have, when the two indices are equal:

cov( pi , pi ) = var( pi )

The relation that gives var ( pP ) is the one that justi¬es the principle of diversi¬cation
in portfolio management: the imperfect correlations (<1) between the securities allows
the portfolio risk to be diminished (see Section 3.2).
Under the hypothesis of normality, the VaR of the portfolio can thus be calculated on
the basis of these two elements using the formula:

VaR q = E( pP ) ’ zq · σ ( pP )

A major problem with using these relations is that they require knowledge not only of
the univariate parameters E( pi ) and var( pi ) for each security, but also of the bivariate
parameters cov( pi , pj ) for each pair of securities. If the portfolio contains N = 100
different securities, for example, the ¬rst parameters will be 2N = 200 in number, while
the second parameters will be much more numerous (N (N ’ 1)/2 = 4950) and it will
not be easy to determine whether they have all been obtained or whether they are accu-
rate. Thankfully, some models allow these relations to be simpli¬ed considerably under
certain hypotheses.
Thus, for a portfolio of equities, it is possible to use Sharpe™s simple index model (see
Section 3.2.4) and consider that the relative variations in price11 of the various equities
are ¬rst-degree functions of the relative variation in a single market index:

pj I
= aj + bj + µj j = 1, . . . , N
pj I

Remember that this model, the parameters of which are estimated using linear regression,
relies on the following hypotheses: the variables µ1 , . . . , µN and ( I /I ) are not correlated
and E(µj ) = 0 for j = 1, . . . , N .
By using the notations:

I I
E = EI = σI2 var(µj ) = σµ2j
var
I I

pj elements:
we have, for the expectation and variance of the

pj
E = aj + bj EI
pj
pj
= bj σI2 + σµ2j
2
var
pj

The covariances are calculated in the same way:

pj
pi
, = bi bj σI2
cov
pi pj
11
That is, the returns.
192 Asset and Risk Management

pP ,
By making a substitution in the formulae that give the expectation and variance for
we ¬nd successively:
N
E( pP ) = nj (aj pj + bj pj EI )
j =1
«  « 
N N
= nj aj pj  +  nj bj pj  EI
j =1 j =1

= aP + bP EI
and:
N N N
var( pP ) = n2 (bj pj σI2 + pj σµ2j ) + ni nj bi bj pi pj σI2
22 2
j
j =1 i=1 j =1
j =i
« 
N N N
= ni nj bi bj pi pj  σI2 + n2 pj σµ2j
2
j
i=1 j =1 j =1

N
= bP σI2 + n2 pj σµ2j
2 2
j
j =1

The VaR relative to this portfolio is therefore given by:
N
VaR q = aP + bP EI ’ zq bP σI2 + n2 pj σµ2j
2 2
j
j =1


Example
Let us take the example of Section 3.2.4 again and consider a portfolio consisting of three
securities in the following numbers: n1 = 3, n2 = 6, n3 = 1.
The relative price variations of these securities are expressed on the basis of the prices
in a stock-exchange index for the following regressive relations:
R1 = 0.014 + 0.60RI (σµ21 = 0.0060)
R2 = ’0.020 + 1.08RI (σµ22 = 0.0040)
R3 = 0.200 + 1.32RI (σµ23 = 0.0012)
The index is characterised by EI = 0.0031 and σI = 0.0468. If the current prices of these
securities are p1 = 120, p2 = 15 and p3 = 640, we have:
aP = 3 — 0.014 — 120 ’ 6 — 0.020 — 15 + 1 — 0.200 — 640 = 131.24
bP = 3 — 0.60 — 120 + 6 — 1.08 — 15 + 1 — 1.32 — 640 = 1158.00
N
n2 pj σµ2j = 32 — 1202 — 0.0060 + 62 — 152 — 0.0040 + 12 — 6402 — 0.0012 = 1283.52
2
j
j =1
Theory of VaR 193

and therefore:
VaR 0.99 = 131.24 + 1158.00 — 0.04 ’ 2.3263 1158.002 — 0.0045 + 1283.52
= ’21.4421
Note
Once more, the results obtained in this paragraph assume that we are in the context of the
hypothesis of normality and that the distributions are therefore characterised entirely by
the expectation“variance pairing. If one requires an estimation of the VaR that is entirely
independent from this hypothesis, the problem will need to be approached in a different
way, that is, without going through an evaluation of the various assets separately.

6.3.2 Components of the VaR of a portfolio
In this and the following paragraph, we will be working under the hypothesis of normality
and with the version of VaR that measures the risk in relation to the average variation in
value:12
VaR — = VaR ’ E( p) = ’zq · σ ( p)

The argument developed in the previous paragraph expressed the portfolio on the basis
of nj numbers (j = 1, . . ., N ) of securities of each type in the portfolio. Here, we
will use the presentation used in Chapter 3 and introduce the proportions of the various
securities expressed in terms of stock-exchange capitalisation: Xj (j = 1, . . . , N ). We can
therefore write:
N
pP pj
= Xj
pP pj
j =1

with:
n j pj
Xj = N
n k pk
k=1

By using the return notation as in Chapter 3, the variable being studied for the VaR
is therefore:
N
pP = pP · Xj Rj
j =1

The average and the variances for this are given by:
N
E( pP ) = pP · Xj Ej = pP · EP
j =1

N N
var( pP ) = pP · Xi Xj σij = pP · σP
2 2 2

i=1 j =1

We will also be omitting the index q relative to probability, so as not to make the notation too laborious.
12
194 Asset and Risk Management

where we have, as usual:

Ej = E(Rj ) σj2 = var(Rj ) σij = cov(Ri , Rj )

This will allow the VaR for the portfolio to be written as:

N N

VaR P = ’pP · zq · σP = ’pP · zq · Xi Xj σij
i=1 j =1

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