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2φ0 (2y
’ x)

Inverting this relationship between (x, y) and (H, C),

P [H ∈ ∆H, C ∈ ∆C] = ’2φ0 (2y ’ x)∆x∆y

con¬rming that the joint density of (H, C) is given by ’2φ0 (2y ’ x) for x < y.
In order to get the joint density of the High and the Close when the drift is
non-zero, we need only multiply by the ratio of the two normal density functions
of the close
fµ (x)
f0 (x)
SIMULATING BARRIER AND LOOKBACK OPTIONS 295




Figure 5.8: Con¬rmation of the joint density of (H, C)


and this gives the more general result in the table below.
The table below summarizes many of our distributional results for a Brown-
ian motion process with drift on the interval [0, 1],

dSt = µdt + σdWt , with S0 = O.



Statistic Density Conditions
’∞ < x < y,
X = C ’ O,
f (y, x) = ’2φ0 (2y ’ x) exp(µx ’ µ2 /2) and y > 0, σ = 1
Y =H ’O
given O
fY |X (y|x) = 2(2y ’ x)e’2y(y’x)
Y |X y > x, σ = 1
exp((σ 2 /2)
Z = Y (Y ’ X) given O, X
exp(σ2 /2)
(L ’ O)(L ’ C) given (O, C)
exp(σ2 /2)
(H ’ O)(H ’ C) given (O, C)
H’O
drift ν = 0, given O, 2H ’ O ’ C
U [0, 1]
2H’O’C
L’O
drift ν = 0, given O, 2L ’ O ’ C
U [0, 1]
2L’O’C
C’O
drift ν = 0, given H, O
U [’1, 1]
2H’O’C
TABLE 5.1: Some distributional results for High, Close and Low.
296 CHAPTER 5. SIMULATING THE VALUE OF OPTIONS

We now consider brie¬‚y the case of non-zero drift for a geometric Brownian
motion. Fortunately, all that needs to be changed in the results above is the
marginal distribution of ln(C) since all conditional distributions given the value
of C are the same as in the zero-drift case. Suppose an option has payo¬ on
maturity ψ(C) if an upper barrier at level Oeb , b > 0 is not breached. We have
already seen that to accommodate the ¬letering e¬ect of this knock-out barrier
we should determine, numerically or by simulation, the expected value

b(b + ln(O/C))
E[ψ(C)(1 ’ exp{’2 })]
σ2 T

the expectation conditional (as always) on the value of the open O. The e¬ect
of a knock-out lower barrier at Oe’a is essentially the same but with b replaced
by a, namely
a(a + ln(C/O))
E[ψ(C)(1 ’ exp{’2 })].
σ2T
In the next section we consider the e¬ect of two barriers, both an upper and a
lower barrier.



One Process, Two barriers.

We have discussed a simple device above for generating jointly the high and the
close or the low and the close of a process given the value of the open. The joint
distribution of H, L, C given the value of O or the distribution of C in the case
of upper and lower barriers is more problematic. Consider a single factor model
and two barriers- an upper and a lower barrier. Note that the high and the
low in any given interval is dependent, but if we simulate a path in relatively
short segments, by ¬rst generating n increments and then generating the highs
and lows within each increment, then there is an extremely low probability
that the high and low of the process will both lie in the same short increment.
For example for a Brownian motion with the time interval partitioned into 5
equal subintervals, the probability that the high and low both occur in the
SIMULATING BARRIER AND LOOKBACK OPTIONS 297

same increment is less than around 0.011 whatever the drift. If we increase the
number of subintervals to 10, this is around 0.0008. This indicates that provided
we are willing to simulate highs, lows and close in ten subintervals, pretending
that within subintervals the highs and lows are conditionally independent, the
error in our approximation is very small.

An alternative, more computationally intensive, is to di¬erentiate the in¬nite
series expression for the probability P (H · b, L ≥ a, C = u|O = 0] A ¬rst step
in this direction is the the following result, obtained from the re¬‚ection principle
with two barriers.



Theorem 47 For a Brownian motion process


dSt = µdt + dWt , S0 = 0


de¬ned on [0, 1] and for ’a < u < b,


P (L < ’a or H > b|C = u)

1X
[φ{2n(a + b) + u} + φ{2n(a + b) ’ 2a ’ u}
=
φ(u) n=1

+ φ{’2n(b + a) + u} + φ{2n(b + a) + 2a + u}]


where φ is the N (0, 1) probability density function.



Proof. The proof is a well-known application of the re¬‚ection principal.
It is su¬cient to prove the result in the case µ = 0 since the conditional
distribution of L, H given C does not depend on µ (A statistician would say
that C is a su¬cient statistic for the drift parameter). Denote the following
paths determined by their behaviour on 0 < t < 1. All paths are assumed to
end at C = u.
298 CHAPTER 5. SIMULATING THE VALUE OF OPTIONS


= H > b (path goes above b)
A+1
path goes above b and then falls below ’a
=
A+2
goes above b then falls below ’a then rises above b
=
A+3
etc.
L < ’a
=
A’1
path falls below ’a then rises above b
=
A’2
falls below ’a then rises above b then falls below ’a
=
A’3
etc.
For an arbitrary event A, denote by P (A|u) probability of the event conditional
on C = u. Then according to the re¬‚ection principal the probability that the
Brownian motion leaves the interval [’a, b] is given from an inclusion-exclusion
argument by

P (A+1 |u) ’ P (A+2 |u) + P (A+3 |u) ’ · · · (5.30)

+P (A’1 |u) ’ P (A’2 |u) + P (A’3 |u) · · ·

This can be veri¬ed by considering the paths in Figure 5.9. (It should be noted
here that, as in our application of the re¬‚ection principle in the one-barrier case,
the re¬‚ection principle allows us to show that the number of paths in two sets is
the same, and this really only translates to probability in the case of a discrete
sample space, for example a simple random walk that jumps up or down by a
¬xed amount in discrete time steps. This result for Brownian motion obtains if
we take a limit over a sequence of simple random walks approaching a Brownian
motion process.)
Note that

φ(2b ’ u)
P (A+1 |u) =
φ(u)
φ{2n(a + b) + u}
P (A+2n |u) =
φ(u)
φ{2n(a + b) ’ 2a ’ u}
P (A+(2n’1) |u) =
φ(u)
SIMULATING BARRIER AND LOOKBACK OPTIONS 299




Figure 5.9: The Re¬‚ection principle with Two Barriers


and
φ(’2a ’ u)
P (A’1 |u) =
φ(u)
φ{’2n(b + a) + u}
P (A’2n |u) =
φ(u)
φ{2n(b + a) + 2a + u}
P (A’(2n+1) |u) = .
φ(u)
The result then obtains from substitution in (5.30).

As a consequence of this result we can obtain an expression for P (a < L ·
H < b, u < C < v) (see also Billingsley, (1968), p. 79) for a Brownian motion
on [0, 1] with zero drift:

P (a, b, u, v) = P (a < L · H < b, u < C < v)

X
¦[v + 2k(b ’ a)] ’ ¦[u + 2k(b ’ a)]
=
k=’∞

X
’ ¦[2b ’ u + 2k(b ’ a)] ’ ¦[2b ’ v + 2k(b ’ a)]. (5.31)
k=’∞
300 CHAPTER 5. SIMULATING THE VALUE OF OPTIONS

where ¦ is the standard normal cumulative distribution function. From (5.31)
we derive the joint density of (L, H, C) by taking the limit P (a, b, u, u + δ)/δ as
δ ’ 0, and taking partial derivatives with respect to a and b:

X
k 2 φ00 [u + 2k(b ’ a)] ’ k(1 + k)φ00 [2b ’ u + 2k(b ’ a)]
f (a, b, u) = 4
k=’∞

X
k φ [u + 2k(b ’ a)] ’ k(1 + k)φ00 [2b ’ u + 2k(b ’ a)]
2 00
=4
k=1

+ k2 φ00 [u ’ 2k(b ’ a)] + k(1 ’ k)φ00 [2b ’ u ’ 2k(b ’ a)] (5.32)

for a < u < b.
From this it is easy to see that the conditional cumulative distribution func-
tion of L given C = u, H = b is given by on a · u · b (where ’2φ0 (2b ’ u)
is the joint p.d.f. of H, C) by
‚2
‚b‚v P (a, b, u, v)| v=u
(5.33)
F (a|b, u) = 1 +
2φ0 (2b ’ u)

X
’1
{’kφ0 [u + 2k(b ’ a)] + (1 + k)φ0 [2b ’ u + 2k(b ’ a)]
=0
φ (2b ’ u)
k=1

+ kφ0 [u ’ 2k(b ’ a)] + (1 ’ k)φ0 [2b ’ u ’ 2k(b ’ a)]}

This allows us to simulate both the high and the low, given the open and the
close by ¬rst simulating the high and the close using ’2φ0 (2b ’ u) as the joint
p.d.f. of (H, C) and then simulating the low by inverse transform from the
cumulative distribution function of the form (5.33).



Survivorship Bias

It is quite common for retrospective studies in ¬nance, medicine and to be
subject to what is often called “survivorship bias”. This is a bias due to the
fact that only those members of a population that remained in a given class
(for example the survivors) remain in the sampling frame for the duration of
the study. In general, if we ignore the “drop-outs” from the study, we do so
SURVIVORSHIP BIAS 301

at risk of introducing substantial bias in our conclusions, and this bias is the
survivorship bias.




Suppose for example we have hired a stable of portfolio managers for a large
pension plan. These managers have a responsibility for a given portfolio over
a period of time during which their performance is essentially under continuous
review and they are subject to one of several possible decisions. If returns below
a given threshhold, they are deemed unsatisfactory and ¬red or converted to
another line of work. Those with exemplary performance are promoted, usually
to an administrative position with little direct ¬nancial management. And those
between these two “absorbing” barriers are retained. After a period of time,
T, an amibitious graduate of an unnamed Ivey league school working out of
head o¬ce wishes to compare performance of those still employed managing
portfolios. How are should the performance measures re¬‚ect the ¬ltering of
those with unusually good or unusually bad performance? This is an example
of a process with upper and lower absorbing barriers, and it is quite likely
that the actual value of these barriers di¬ers from one employee to another, for
example the son-in-law of the CEO has a substantially di¬erent barriers than the
math graduate fresh out of UW. However, let us ignore this di¬erence, at least
for the present, and concentrate on a di¬erence that is much harder to ignore
in the real world, the di¬erence between the volatility parameters of portfolios,
possibly in di¬erent sectors of the market, controlled by di¬erent managers.
For example suppose two managers were responsible for funds that began and
ended the year at the same level and had approximately the same value for the
lower barrier as in Table 5.2. For each the value of the volatility parameter
σ was estimated using individual historical volatilities and correlations of the
component investments.
302 CHAPTER 5. SIMULATING THE VALUE OF OPTIONS


Portfolio Open price Close Price Lower Barrier Volatility

56 5
1 40 30 .5
8

56 1
2 40 30 .2
4
Table 5.2


Suppose these portfolios (or their managers) have been selected retrospec-
tively from a list of “survivors” which is such that the low of the portfolio value
never crossed a barrier at l = Oe’a (bankruptcy of fund or termination or
demotion of manager, for example) and the high never crossed an upper barrier
at h = Oeb . However, for the moment let us assume that the upper barrier is so
high that its in¬‚uence can be neglected, so that the only absorbtion with any
substantial probability is at the lower barrier. We interested in the estimate of
return from the two portfolios, and a preliminary estimate indicates a continu-
ously compounded rate of return from portfolio 1 of R1 = ln(56.625/40) = 35%
and from portfolio two of R2 = ln(56.25/40) = 34%. Is this di¬erence signi¬cant
and are these returns reasonably accurate in view of the survivorship bias?
We assume a geometric Brownian motion for both portfolios,

(5.34)
dSt = µSt dt + σSt dWt ,

and de¬ne O = S(0), C = S(T ),

H = max S(t), L = min S(t)
0tT 0tT


with parameters µ, σ possibly di¬erent.
In this case it is quite easy to determine the expected return or the value of
any performance measure dependent on C conditional on survival, since this is
essentially the same as a problem already discussed, the valuation of a barrier
option. According to (5.27), the probability that a given Brownian motion
process having open 0 and close c strikes a barrier placed at l < min(0, c) is

zl
}
exp{’2
σ2 T
SURVIVORSHIP BIAS 303

with
zl = l(l ’ c).

Converting this statement to the Geometric Brownian motion (5.34), the prob-
ability that a geometric Brownian motion process with open O and close c
breaches a lower barrier at l is

zl
P [L · l|O, C] = exp{’2 }
σ2T

with
zl = ln(O/l) ln(C/l) = a(a + ln(C/O)).

Of course the probability that a particular path with this pair of values (O, C)
is a “survivor” is 1 minus this or

zl
1 ’ exp{’2 }. (5.35)
σ2 T

When we observe the returns or the closing prices C of survivors only, the results
have been ¬ltered with probability (5.35). In other words if the probability
density function of C without any barriers at all is f (c) (in our case this is a
lognormal density with parameters that depend on µ and σ) then the density
function of C of the survivors in the presence of a lower barrier is proportional
to

ln(O/l) ln(c/l)
f (c)[1 ’ exp{’2 }]
σ2 T
l 2 ln(O/l) 2a
= f (c)(1 ’ ( )» ), with » = = 2 > 0.
σ2T
c σT

It is interesting to note the e¬ect of this adjustment on the moments of C for
various values of the parameters. For example consider the expected value of C
conditional on survival
R∞ l
cf (c)(1 ’ ( c )» )dc
l
E(C|L ≥ l] = R ∞ l
f (c)(1 ’ ( c )» )dc
l
E[CI(C ≥ l)] ’ l» E[C 1’» I(C ≥ l)]
(5.36)
=
P [C ≥ l] ’ l» E[C ’» I(C ≥ l)]
304 CHAPTER 5. SIMULATING THE VALUE OF OPTIONS

and this is easy to evaluate in the case of interest in which C has a lognormal
distribution. In fact the same kind of calculation is used in the development of
the Black-Scholes formula. In our case C = exp(Z) where Z is N (µT, σ 2 T )
and so for any p and l > 0, we have from (3.11), using the fact that E(C|O) =
O exp{µT + σ2 T /2}, (and assuming O is ¬xed),

1
E[C p I(C > l)] = Op exp{pµT + p2 σ 2 T /2}¦( √ (a + µT ) + σ T p)
σT

To keep things slightly less combersome, let us assume that we observe the
geometric Brownian motion for a period of T = 1. Then (5.36) results in
2 2 2
1 1
Oeµ+σ /2
¦( σ (a + µ) + σ) ’ Oe’a»+(1’»)µ+(1’») σ /2 ¦( σ (a + µ) + σ(1 ’ »))
1 1
¦( σ (a + µ)) ’ e’»a’»µ+»2 σ2 /2 ¦( σ (a + µ) ’ σ»)

Let there be no bones about it. At ¬rst blush this is still a truly ugly and
opaque formula. We can attempt to beautify it by re-expressing it in terms more
like those in the Black-Scholes formula, putting

1 1
(µ ’ a), and d2 (0) = (a + µ),
d2 (») =
σ σ
d1 (») = d2 (») + σ, d1 (0) = d2 (0) + σ.

These are analogous to the values of d1, d2 in the Black-Scholes formula in the
case » = 0. Then
2 2 2
eµ+σ /2
¦(d1 (0)) ’ e’»a+(1’»)µ+(1’») σ /2 ¦(d1 (»))
E[C|L ≥ l] = O (5.37)
.
¦(d2 (0)) ’ e’»a’»µ+»2 σ2 /2 ¦(d2 (»))

What is interesting is how this conditional expectation, the expected close for
the survivors, behaves as a function of the volatility parameter σ. Although this
is a rather complicated looking formula, we can get a simpler picture (Figure
5.10) using a graph with the drift parameter µ chosen so that E(C) = 56.25
is held ¬xed. We assume a = ’ ln(30/40) (consistent with Table 5.2)and vary
the value of σ over a reasonable range from σ = 0.1 (a very stable investment)
through σ = .8 (a highly volatile investment). In Figure 5.10 notice that for
small volatility, e.g. for σ · 0.2, the conditional expectation E[C|L ≥ 30]
SURVIVORSHIP BIAS 305




Figure 5.10: E[C|L ≥ 30] for various values of (µ, σ) chosen such that E(C) =
56.25.



remains close to its unconditional value E(C) but for σ ≥ 0.3 it increases almost
linearly in σ to around 100 for σ = 0.8. The intuitive reason for this dramatic
increase is quite simple. For large values of σ the process ¬‚uctuates more,
and only those paths with very large values of C have abeen able to avoid
the absorbing barrier at l = 30. Two comparable portfolios with unconditional
return about 40% will show radically di¬erent apparent returns in the presence
of an absorbing barrier. If σ = 20% then the survivor™s return will still average
around 40%, but if σ = 0.8, the survivor™s returns average close to 150%. The
practical implications are compelling. If there is any form of survivorship bias
(as there usually is), no measure of performance should be applied to the returns
from di¬erent investments, managers, or portfolios without an adjustment for
the risk or volatility.

In the light of this discussion we can return to the comparison of the two
portfolios in Table 5.2. Evidently there is little bias in the estimate of returns
for portfolio 2, since in this case the volatility is small σ = 0.2. However there
is very substantial bias associated with the estimate for portfolio 1, σ = 0.5.
In fact if we repeat the graph of Figure 5.10 assuming that the unconditional
306 CHAPTER 5. SIMULATING THE VALUE OF OPTIONS




Figure 5.11: The E¬ect of Surivorship bias for a Brownian Motion



return is around 8% we discover that E[C|L ≥ 30] is very close to 56 5 when
8

σ = 0.5 indicating that this is a more reasonable estimator of the performance
of portfolio 1.

For a Brownian motion process it is easy to demonstrate graphically the
nature of the surivorship bias. In Figure 5.11, the points under the graph of
the probability density which are shaded correspond to those which whose low
fell below the absorbing barrier at l = 30. The points in the unshaded region
correspond to the survivors. The expected value of the return conditional on
survival is the mean return (x-cooredinate of the center of mass) of those points
chosen uniformly under the density but above the lower curve, in the region
labelled “survivors”. Note that if the mean µ of the unconditional density
approaches the barrier (here at 30) , this region approaches a narrow band
along the top of the curve and to the right of 30. Similarly if the unconditional
standard deviation or volatility increases, the unshaded region stretches out to
the right in a narrow band and the conditional mean increases.
SURVIVORSHIP BIAS 307

We arrive at the following seemingly paradoxical conclusions which make it
imperative to adjust for survivorship bias The conditional mean, conditional on
survivorship, may increase as the volatility increases even if the unconditional
mean decreases.

Let us return to the problem with both an upper and lower barrier and
consider the distribution of returns conditional on the low never passing a barrier
Oe’a and the high never crossing a barrier at Oeb ( representing a fund buyout,
recruitment of manager by competitor or promotion of fund manager to Vice
President).It is common in process control to have an upper and lower barrier
and to intervene if either is crossed, so we might wish to study those processes
for which no intervention was required. Similarly, in a retrospective study we
may only be able to determine the trajectory of a particle which has not left
a given region and been lost to us. Again as an example, we use the following
data on two portfolio managers, both observed conditional on survival, for a
period of one year.




Portfolio Open price Close Price Lower Barrier Upper Barrier Volatility

56 5
1 40 30 100 .5
8

56 1
2 40 30 100 .2
4



If φ denotes the standard normal p.d.f., then the conditional probability
density function of ln(C/O) given that Oe’a < L < H < Oeb is proportional
u’µ
1
to where, as before
σ φ( σ )w(u)




2 2 2 2
w(u) = 1 ’ e’2b(b’u)/σ + e’2(a+b)(a+b’u)/σ ’ e’2a(a+u)/σ + e’2(a+b)(a+b+u)/σ ’ E(W ),
’ ln(L)
ln(H) b a
], and
W = I[f rac1( )> ] + I[f rac1( )>
a+b a+b a+b a+b
a = ’ln(30/40).
b = ln(100/40),
308 CHAPTER 5. SIMULATING THE VALUE OF OPTIONS

The expected return conditional on survival when the drift is µ is given by
Z b
u’µ
1
E(ln(C/O)|30 < L < H < 100) = uw(u)φ( )du.
σ σ
’a


where w(u) is the weight function above. Therefore a moment estimator of the
drift for the two portfolios is determined by setting this expected return equal
to the observed return, and solving for µi the equation
Z b
u ’ µi
1
uw(u)φ( )du = Ri , i = 1, 2.
σi σi
’a


The solution is, for portfolio 1, µ1 = 0 and for portfolio 2, µ2 = 0.3. Thus the
observed values of C are completely consistent with a drift of 30% per annum
for portfolio 2 and a zero drift for portfolio 1. The bias again very strongly
e¬ects the portfolio with the greater volatility and estimators of drift should
account for this substantial bias. Ignoring the survivorship bias has led in the
past to some highly misleading conclusions about persistence of skill among
mutual funds.



Problems

1. If the values of dj are equally spaced, i.e. if dj = j∆, j = ..., ’2, ’1, 0, 1, ...and
with S0 = 0, ST = C and M = max(S0 , ST ), show that

P [C > u and C’M is even]

E[H|C = u] = M + ∆ .
P [C = u]

2. Let W (t) be a standard Brownian motion on [0, 1] with W0 = 0. De¬ne
C = W (1) and H = max{W (t); 0 · t · 1}. Show that the joint probabil-
ity density function of (C, H) is given by


f (c, h) = 2φ(c)(2h ’ c)e’2h(h’c) , for h > max(0, c)


where φ(c) is the standard normal probability density function.
PROBLEMS 309

3. Use the results of Problem 2 to show that the joint probability density
function of the random variables

Y = exp{’(2H ’ C)2 /2}

and C is a uniform density on the region {(x, y); y < exp(x2 /2)}.

4. Let X(t) be a Brownian motion on [0, 1], i.e. Xt satis¬es

dXt = µdt + σdWt , and X0 = 0.

De¬ne C = X(1) and H = max{X(t); 0 · t · 1}. Find the joint proba-
bility density function of (C, H).
310 CHAPTER 5. SIMULATING THE VALUE OF OPTIONS

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