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A Strategy for Reserve Accumulation via US Dollar Sales Options. Banco de México’s Case
Disclaimer: This note is not intended to substitute its original version in Spanish for any legal purpose. It is intended solely for guidance and didactic use.
1
CONTENTS I. ABSTRACT …….….……………………………………………………………….3
II. INTRODUCTION ….……………………………………………………………….3
III. US DOLLAR SALES OPTIONS………………………………………………….4
III.1. MAIN FEATURES ……………………………………………………...4
III.2. SIMULATION RESULTS………………………………………………5
IV. ANALYTICAL APPROACH FOR OPTION PRICES……………....................8
IV.1. PROB (NO RESTRICTION)…………………………………………...9
IV.2. W (STRIKE ON DAY T)……………………………………………….13
V. OPTION PRICE SENSITIVITY…………………………………………………..16
V.1. OPTION PRICE ESTIMATION…...……………………………………16
VI. CONCLUSIONS…………………………………………………………………..23
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I. Abstract
This research paper describes a strategy for Banco de México to accumulate reserves
through an option mechanism. The strategy seeks to reduce the impact of currency
purchases in the foreign exchange market. The document provides an analytical
approach for estimating the theoretical price of these options and analyzes the
parameters that determine their value.
Manuel Galán Medina1
Javier Duclaud González de Castilla
Alonso García Tamés
1 The authors Manuel Galán, Javier Duclaud, and Alonso García are officers from Banco de México and
hold the positions of Investment and Domestic Exchange Manager, Domestic Exchange Deputy Manager, and Director General of Central Bank Operations, respectively. The authors would like to thank the exhaustive review and comments of José Ramón Rodríguez and Ricardo Medina. The views and conclusions presented in this paper are exclusively those of the authors and do not necessarily reflect those of Banco de México.
3
II. Introduction
In the "Monetary Policy Statement for 1996," Banco de México anticipated the possibility
of acquiring foreign currency in the foreign exchange market, emphasizing the need to
not pressure the exchange rate and not send signals that could be interpreted incorrectly
by financial agents. While Banco de México wanted to pursue greater accumulation of
international reserves, it also warned that such an accumulation should be achieved
through a scheme that would promote dollar purchases when the market was offering
dollars and prevent purchases when the market was demanding dollars, so that the
mechanism would therefore have a minimal impact on the floating exchange rate
regime.
This document describes the central bank‟s strategy to accumulate reserves through
options to sell dollars. The paper is organized as follows. After the introduction, the
second section describes the characteristics of the option and presents the results of a
simulation of the dollar-buying mechanism if Banco de México had sold options from
January 1995 to July 1996. The third section presents an analytical approach to the
option pricing mechanism and the likelihood of exercise of the options. This latter section
serves to estimate an expected accumulation of reserves during a given period using
this strategy. The fourth section shows an analysis of price-sensitivity for changes in the
parameters used to price options. The last section presents the conclusions of this
paper.
4
III. US Dollar Sales Options
III.1. Key Features
On August 1, 1996, Banco de México issued a circular addressed to lending
institutions in Mexico2 informing them that the central bank would auction dollar
option contracts every month, subject to payment in pesos. Under this
mechanism, institutions could purchase the right to sell to the central bank a
predetermined amount of dollars in exchange for pesos. The characteristics of
the option are outlined below.
With the sale of the option, Banco de México is obliged to buy dollars in
exchange for pesos from the option holder on any business day that the holder
chooses during the time the contract is valid. However, unlike a traditional
European or American option, the strike exchange rate level is not fixed. In the
case of the option‟s exercise, the operation is carried out at the exchange rate
calculated the previous business day by Banco de México on the basis of the
survey that the bank performs every day among lending institutions in Mexico,
and which is known in the market as the "fix"3. Therefore, as discussed in detail
in Section III of this document, the option to sell dollars can be thought of as a
portfolio of options with 1-day maturities, provided that the options have not
been exercised previously.
One of the central bank‟s risks for accumulating reserves this way would be a
scenario where the peso followed a depreciating trend. If the peso appreciated
from one day to the next (i.e., if it saw an "overshooting" correction), it would be
2 Circular 71/96.
3 Exchange rate to settle liabilities denominated in foreign currency and payable in Mexico, as published
by Banco de México in the Official Federal Gazette on the bank day following the determination if this exchange rate.
5
optimal for option holders to exercise all their rights to sell dollars and
immediately thereafter recover their positions by buying the dollars back in the
foreign exchange market. If that were to occur, Banco de Mexico would
accumulate reserves through purchases in the market at a time of excess
demand for dollars, thus potentially magnifying the depreciating pressures on
the peso.
The way to mitigate this risk was to condition the exercise of the option on
having the exchange rate set below a predetermined level. Thus, an additional
feature of the option is that it can only be exercised when the exchange rate
strike level is not higher than the average exchange rate (“fix”) determined by
Banco de México during the 20 business days prior to the date the option is
intended to be exercised.
III.2. Simulation Results
In order to estimate the amount of possible reserve accumulation, an analysis
from January 1995 to July 1996 simulating the case for implementation of the
strategy was carried out. Figure 1 shows the "fix" exchange rate, the minimum
US dollar sales level in the interbank market every day, and the 20-day moving
average, which limits the option exercise. For this period, in 14 out of the 19
months analyzed, there was at least one day in which, without taking into
account the fee paid for the option, the exercise of the option would have yielded
profits. Therefore, if Banco de México had sold these options in the amount of
USD200 million per month, the total purchases of foreign currency by Banco de
México would have been USD2.8 billion4. The maximum profit for option holders
would have been 28 cents per dollar, and the month with the highest number of
positive differences would have been April 1995, with 16 differences. These
results are shown in Figure 1.
4 The analysis does not take into account the effect that the purchase of foreign currency by Banco de
México might have had on the exchange rate. Therefore, the results are only an approximation.
6
Figure 1 Simulation Results of US Dollar Sales Options
Table 1 Simulation Results of US Dollar Sales Options
Month Maximum Profit* Minimum Profit* Average Profit* Days with profit*
Jan-95 0 0 0 0
Feb-95 0.155 0.005 0.0283 7
Mar-95 0.08 0.08 0.0036 1
Apr-95 0.2875 0.0333 0.104 16
May-95 0.2117 0.0008 0.0368 12
Jun-95 0 0 0 0
Jul-95 0.0871 0.0025 0.0249 11
Aug-95 0.0042 0.0042 0.0002 1
Sep-95 0 0 0 0
Oct-95 0 0 0 0
Nov-95 0.2317 0.2317 0.0116 1
Dec-95 0.1609 0.0083 0.014 5
Jan-96 0.1196 0.0021 0.0286 14
Feb-96 0.0312 0.0045 0.0018 2
Mar-96 0.0411 0.0007 0.0047 6
Apr-96 0.058 0.002 0.019 13
May-96 0.0384 0.0071 0.0067 8
Resultados de la Simulación de Venta de Opciones.
4,0000
4,5000
5,0000
5,5000
6,0000
6,5000
7,0000
7,5000
8,0000
8,5000
2-e
ne
-95
16
-en
e-9
5
30
-en
e-9
5
13
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b-9
5
27
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b-9
5
13
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r-9
5
28
-ma
r-9
5
11
-ab
r-9
5
27
-ab
r-9
5
15
-ma
y-9
5
29
-ma
y-9
5
12
-ju
n-9
5
26
-ju
n-9
5
10
-ju
l-9
5
24
-ju
l-9
5
7-a
go
-95
21
-ag
o-9
5
5-s
ep
-95
19
-se
p-9
5
3-o
ct-9
5
17
-oct-
95
31
-oct-
95
15
-no
v-9
5
30
-no
v-9
5
15
-dic
-95
2-e
ne
-96
16
-en
e-9
6
30
-en
e-9
6
14
-fe
b-9
6
28
-fe
b-9
6
13
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r-9
6
28
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r-9
6
15
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r-9
6
29
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6
14
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6
28
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y-9
6
11
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6
25
-ju
n-9
6
9-j
ul-
96
23
-ju
l-9
6
PE
SO
S P
OR
DO
LA
R
0,0000
0,0500
0,1000
0,1500
0,2000
0,2500
0,3000
UT
ILID
AD
PE
SO
S P
OR
DO
LA
R
Tipo de Cambio Fix Tipo de Cambio de Venta Mínimo Promedio Móvil de 20 Días. Utilidad„Fix‟ exchange rate Minimum US dollar sales level 20-day moving average Profit
Simulation Results of US Dollar Sales Options
PE
SO
S
PE
R
DO
LLA
R
PE
SO
S
PE
R
DO
LLA
R P
RO
FIT
7
Jun-96 0 0 0 0
Jul-96 0.0415 0.0079 0.004 4
Average 0.0815 0.0217 0.0152 6
*Does not include the fee paid for the option
It is interesting to note that in September, October, and November 1995, a
period during which the peso had clearly been depreciating, making it
undesirable for Banco de México to buy dollars in the market, on only one
day would it have been possible for holders to exercise the options. This is
because during this period, the exchange rate generally remained at levels
higher than the 20-day moving average (see Figure 2).
Figure 2 Difference between the ‘Fix’ exchange rate and its 20-day moving
average
Once the main characteristics of the option were assessed, it is possible to
proceed to describe an analytical approach to value their price.
-100
-50
0
50
100
150
200
3-J
an
-95
23-J
an
-95
12-F
eb-9
5
4-M
ar-
95
24-M
ar-
95
13-A
pr-
95
3-M
ay-9
5
23-M
ay-9
5
12-J
un
-95
2-J
ul-
95
22-J
ul-
95
11-A
ug
-95
31-A
ug
-95
20-S
ep
-95
10-O
ct-
95
30-O
ct-
95
19-N
ov-9
5
9-D
ec-9
5
29-D
ec-9
5
18-J
an
-96
7-F
eb-9
6
27-F
eb-9
6
18-M
ar-
96
7-A
pr-
96
27-A
pr-
96
17-M
ay-9
6
6-J
un
-96
26-J
un
-96
16-J
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96
CE
NTS
OF
PE
SO
S
Difference between the 'Fix' exchange rate and its 20-day moving average
8
IV. Analytical Approximation of the Option Price
This section provides an analytical framework for approximating the value of the USD
put option, and that is useful to estimate the amount of foreign currency that Banco de
Mexico would purchase through this mechanism. Its major components are presented
below.
The USD sale options are similar to a portfolio of European-style “put at the money,”
with a one-day maturity and with a strike exchange rate that is determined the previous
day through Banco de México‟s survey. However, once the option is exercised, the
remaining options in the portfolio are lost. Furthermore, as mentioned in the previous
section, the currency option can only be exercised when the strike exchange rate for the
peso vs. the dollar is equal to or lower than the simple average of the “n” exchange rates
surveyed by Banco de México prior to the exercise date.
This is useful to break down the value of the option into the present value of a portfolio
of put options weighted by two factors:
a) The probability of meeting the restriction of average “n” days, and
b) The probability of investors exercising the option on a particular day.
The price of an option can be approximated by:
Where the previous equation adds the product of the following factors for each option in
the portfolio: a discount factor , the value of a put at-the-money type option, the
probability that on day t the restriction for exercising the option is met, and function W,
which represents the exercise strategy.
From this, the focus is on finding an analytical expression for the third and fourth terms
of the previous equation, since the first two terms correspond to a discount factor and to
the Black-Scholes formula modified by Garman M. and Kohlhagen S. to value currency
options.
9
What follows is an estimation of the analytical expression that calculates the probability
of the exchange rate not being higher than the observed average of the last “n”
observations, which will be identified as Prob(No restriction).
IV.1. Prob(No restriction)
Let:
et: The exchange rate (fix) determined by Banco de México on day t.
: The natural logarithm of et.
Yt: The moving average of the “n” previous observations to .
Assume that the exchange rate follows a stochastic process represented by a
“random walk with a drift” described by the following equation.
(a)
The parameter represents the expected one-day depreciation (percentage),
that for a neutral risk-averse investor would be given by the differences between
the nominal interest rates in pesos r, and in US dollars r*, both risk-free5. The
variables denote random errors, uncorrelated, with zero mean and variance
. Additionally, normality for such errors is assumed, that is:
The standard deviation corresponds to what financial market analysts
frequently identify as exchange rate “volatility.”6
The stochastic process previously described in equation (a) starts from a known
value . For simplicity, the vales that are known in will be denoted with an
asterisk. From equation (a) the exchange rate for day would be given by:
5 Taking into account that the expected devaluation is for one day only, the parameter would be given by
, where both rates are continuously composed. 6 corresponds to the one-day exchange rate volatility.
10
And for day :
Hence, the solution for the difference equation described in (a) can be written
as:
(b)
The other important variable for valuing the currency option is the average of the
exchange rate , of the “n” previous observations. In order to write a usable
expression of the mentioned average, the value of the variable is assumed to
be known in the (n-1) previous days to , (i.e., the values of
are known), and is defined as:
The value of this expression is known because it is composed of predetermined
values. For day , the average can be expressed as:
Where the first observation is subtracted from the known average ,
and the new observation is added. The previous expression allows generalizing
the equation for the average as follows:
(c) , for
By substituting equation (b) in (c):
11
(d)
In addition, it is know that:
This expression can be used for rewriting the fourth term of the right side of
equation (d). Further, the last term can also be rewritten as:
(e)
After applying the corresponding substitutions in (d) the following equation is
obtained:
(f)
Once expression (f) is known, it is possible to proceed to estimate the probability
that the level of the exchange rate at a certain date , is equal to or less than
the average of the “n” previous observations,7 that is:
By substituting expressions (b) and (f) in the previous equation the following is
obtained:
That is:
7 Note that the previously mentioned restriction for the exercise of the option works over the exchange rate
levels and not over logarithms. Hence it is an approximation made in the present model.
12
(g)
−12 −1
Where the random variable Zt has the following characteristics:
;
;
(h)
Note that independence and normality in errors was used in order to
determine the distribution and variance of the random variable Zt. With the result
obtained in (h), it is possible to proceed to estimate the probability described in
the right-hand side of equation (g) as follows:
(i)
Where:
And:
N( ) denotes the cumulative distributive function of the normal standard.
Therefore, expression (i) provides an estimation of the probability that on day t in
the future, the exchange rate will be at the same level or below the previous “n”
days‟ average.
13
IV.2. W(Strike on day t)
In this section a function that represents the exercise strategy for the currency
option is suggested. The function‟s purpose is to model the exercise of the
option, eliminating the subsequent options from the portfolio once the holder has
decided to exercise the option.
An assumption must therefore be made concerning the strategy to be followed
by an option holder. It is important to note that there are incentives to exercise
the option as soon as possible, because as time passes, earnings will have to
be greater to compensate for the financial cost incurred by holding and not
exercising the option. This implies that once the restriction mentioned in the
previous section is satisfied, it is likely that the option will be exercised as soon
as the profit is greater than the price paid for the option. Thus, function W in t =
1 is defined as the probability of exercising the option on the first day t = 1, that
is:
(j)
Where Oc represents prime paid for the option. The formula that determines the
price of an option will also depend on its own price. Thus, a recursive procedure
is required to determine the price.
By substituting equation (b) in (j), and rewriting the latter equation, the following
is obtained:
Or:
(k)
Where:
14
Equation (k) provides an analytical way to evaluate the probability that a holder
will exercise the option on the first day.
Assume that the option is not exercised on the first day, but on the second. The
probability of occurrence for this event would be given by:
The second term of the right-hand side of the equation incorporates the
information that the option was not exercised on the first day. Using a similar
procedure as the one used in equation (k), the following result is obtained:
(l)
Generalizing the previous result, the probability of exercising the currency option
on day t would be given by:
(m)
Where:
Equation (m) generalizes the way to estimate the probability of exercising the
currency option on day t. By using this expression it is possible to finally write a
formula that approximates the value of the currency option. Nevertheless, it is
important to point out that this valuation is associated with the exercise strategy
mentioned above,
15
(n)
Where the valuation of the Put(At the moneyt)BS was made using the Black-
Scholes model modified by Garman M. and Kohlhagen S. for valuation options
over foreign exchange operations. In addition, it can be seen in expression (n)
that all put type options are considered to be worth practically the same, since
all are “at the money”, they all have a one-day maturity, and they also have the
same exchange rate volatility.
V. Sensitivity of the Price of the Option
This section analyzes the sensitivity of the price of the option and the probability of
exercise to changes in the parameters used for its valuation. In particular, the effect of
modifying the volatility and the expected depreciation of the currency is studied. As it will
be shown, the direction and magnitude of these effects will depend heavily on the
difference between the spot exchange rate and the average which limits the possibility of
exercising the options, and on the history of the average.
V.1. Price estimation
In the previous section an expression that approximates the value of the option
was found and it was defined as:
Where:
Discount factor
Black-Scholes Probability of no restriction
Probability of strike on t
16
And N( ) denotes the cumulative distributive function of the normal standard. As
can be seen in the previous expressions, on the day of the purchase of the
option, all the variables that contribute to the determination of its price are
known, except for the exchange rate volatility and the expected depreciation.
Figure 2 shows the value of the option at changes in the value of these
parameters. For this exercise, the assumption that the number of observations
included in the average was 20 was made, and that all the observations that
determine the exchange rate average have the same value of 7.5 pesos per
dollar.
Table 2 Value of the Option Pesos for every one thousand dollars
As the expected peso depreciation increases, the price of the option falls. This is
due to two factors:
(a) the probability that exercising the option will generate profits decreases;
and
(b) the probability that the strike exchange rate will sometimes be higher
than the average of the previous 20 observed exchange rates increases.
On the other hand, it is possible to see that the greater the exchange
rate volatility, the greater the value of the option. This always occurs
% 10 15 17 19 21 23 25 30
5 3.12 2.78 2.66 2.54 2.43 2.32 2.22 1.99
6 3.89 3.54 3.41 3.29 3.17 3.06 2.95 2.70
7 4.67 4.31 4.18 4.01 3.93 3.81 3.70 3.42
8 5.45 5.09 4.95 4.82 4.69 4.56 4.45 4.17
9 6.24 5.86 5.73 5.59 5.46 5.33 5.21 4.92
10 7.02 6.64 6.50 6.36 6.23 6.10 5.93 5.68
15 10.95 10.56 10.41 10.26 10.12 9.98 9.85 9.53
20 14.89 14.49 14.33 14.18 14.04 13.89 13.76 13.42
Expected Depreciation
Annualized
exchange
rate
volatility
17
with options, because while the maximum loss for the buyer is limited,
the expected payoff increases along with the degree of volatility8.
In order to calculate the probability of exercising the option, the sum of the third
and fourth iterations of expression (n) is defined as the probability of exercising
the option during the whole term of its validity.
Table 3 shows the probability of exercising the option under the same
assumptions as in the previous example. When the expected depreciation of the
peso increases, the probability of exercising the option is reduced. Also,
increases in volatility increase the probability of option exercising (although this
effect is only significant when the expected depreciation is high).
Table 3 Probability of Exercising the Option
Applying the probabilities obtained in Table 3, the expected accumulation of
foreign reserves if Banco de Mexico were to apply this mechanism for a certain
period of time can be estimated. For example, in a one-month period, the
expected purchase of foreign exchange currency will be calculated by
8 Because these are very short-term options, the relevant exchange rate volatility is that observed during
the trading day.
% 10 15 17 19 21 23 25 30
5 0.44 0.42 0.41 0.40 0.39 0.38 0.37 0.35
6 0.45 0.43 0.42 0.42 0.41 0.40 0.39 0.37
7 0.46 0.44 0.44 0.43 0.42 0.41 0.41 0.39
8 0.47 0.45 0.44 0.44 0.43 0.43 0.42 0.41
9 0.47 0.45 0.45 0.44 0.44 0.43 0.43 0.42
10 0.47 0.46 0.45 0.45 0.44 0.44 0.43 0.42
15 0.48 0.47 0.47 0.47 0.46 0.46 0.46 0.45
20 0.49 0.48 0.48 0.47 0.47 0.47 0.47 0.46
Expected Depreciation
Annualized
exchange
rate
volatility
18
multiplying the probability of exercise by the amount of option sales for that
month.
Tables 4 and 5 show the same results but using the values for the parameters
that existed in the market as of August 7, 1996, the date on which Banco de
Mexico annpunced the first dollar put option auction. As in the previous exercise,
the price of the options decreases when the expected peso depreciation is
greater, and increases with greater levels of exchange rate volatility.
Nevertheless, the size of these changes expressed in percentages is smaller.
This is due to the fact that in this new example, there is a difference of almost 9
cents between the average that limits the exercising of the options and the spot
exchange rate. Hence, the probability of exercise is much greater and the value
of the option responds to a lesser degree to changes in the values of the
parameters.
Table 4 Value of the Option Pesos for every one thousand dollars
Table 5 Probability of Exercising the Option
% 10 15 17 19 21 23 25 30
5 6.85 6.43 6.27 6.12 5.97 5.82 5.68 5.35
6 8.30 7.87 7.71 7.55 7.39 7.24 7.09 6.75
7 9.71 9.27 9.10 8.93 8.77 8.62 8.47 8.10
8 11.06 10.61 10.44 10.27 10.11 9.95 9.79 9.42
9 12.37 11.91 11.73 11.56 11.39 11.23 11.07 10.69
10 13.62 13.15 12.97 12.80 12.63 12.46 12.30 11.91
15 19.20 18.71 18.53 18.35 18.17 17.99 17.82 17.41
20 24.03 23.54 23.35 23.17 22.99 22.81 22.64 22.22
Expected Depreciation
Annualized
exchange
rate
volatility
19
Contrary to what was observed in the previous exercise, in the latter, increases
in exchange rate volatility reduce the probability of the option‟s exercise, a result
that could be counterintuitive. What explains this is the particular information
contained in the 20-day moving average for that date. As new exchange rates
are included in the average and previous values are deleted, the average
exchange rate decreases9. Therefore, as exchange rate volatility increases, the
20-day average exchange rate is lower than the spot exchange rate much more
frequently.
Figure 3 shows this effect. Assume that the 20-day moving average of the
exchange rate on the day the option is sold is above the spot exchange rate.
Assume also that the exchange rate is formed by observed levels that decrease
in time. Figure 3 assumes that the moving average follows this behavior and
sketches the probability distribution of the exchange rate on day 5 for 2 different
volatilities. The area defined as “b” corresponds to volatility and contains the
observed levels for the peso that would end up above the date‟s 20-day moving
average. If the volatility increases to , the area that contains such observed
levels, defined as “a”, increases. Thus, the conclusion for this particular case
can be that, if the exchange rate volatility increases, the probability of exercising
the option decreases.
Figure 3
9 This happens when the average exchange rate is formed by observations that decrease their value from
S-19 to S0.
% 10 15 17 19 21 23 25 30
5 0.98 0.98 0.98 0.97 0.97 0.97 0.97 0.97
6 0.97 0.97 0.96 0.96 0.96 0.96 0.96 0.95
7 0.96 0.95 0.95 0.95 0.95 0.95 0.94 0.94
8 0.95 0.94 0.94 0.94 0.93 0.93 0.93 0.93
9 0.93 0.93 0.92 0.92 0.92 0.92 0.92 0.91
10 0.92 0.91 0.91 0.91 0.91 0.90 0.90 0.90
15 0.84 0.84 0.84 0.83 0.83 0.83 0.83 0.82
20 0.78 0.78 0.78 0.78 0.77 0.77 0.77 0.77
Expected Depreciation
Annualized
exchange
rate
volatility
20
Tipo
De
Cambio
Promedio
"Spot"
Dias
Promedio
'
'
Tendenciaa
1 2 43 5
b
a
Figure 4 shows the same effect described in the previous paragraph for three
different scenarios: one in which the level of the exchange rate decreases,
another for which it is constant, and a third in which the level of the exchange
rate increases. For all scenarios, it is possible to assume that the original spot
exchange rate is the same. It is shown in the same graph how the probability of
exercise changes as the annualized exchange rate volatility varies.
Figure 4
Average
Exchange rate
Trend
Average
Days
21
PROBABILIDADES DE EJERCICIO
0
0,
1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
2% 4% 6% 8% 10
%
12
%
14
%
16
%
18
%
20% 22% 24% 26% 28% 30%
VOLATILIDAD ANUALIZADA DEL TIPO DE CAMBIO
P
R
O
B
A
B
I
L
I
D
A
D
Caso-1
Caso-2
Caso-3
Figure 5 shows how the price of the option changes as the difference between
the average exchange rate and the spot exchange rate changes10. As shown, as
the difference increases, the value of the option increases, which is an intuitive
result because the probability of exercising the option is greater. However, as
the difference grows ever larger, the result is the opposite. This is due to the fact
that when a certain difference is reached, and the previous probability of
exercising has already attained 100%, what determines the value of the option is
the (base) level of the exchange rate used.
10
For this example, it is assumed that the number of observations in the average is 20, and that all observations that determine the average have the same value of 7.5 pesos per dollar, except the last one.
Probabilities of Exercise
Probability
Annualized exchange rate volatility
22
Figure 5
From the analysis made in this section, the conclusion is that the depreciation
and the expected volatility of the exchange rate determine the theoretical price
of the option and the probability of exercising it. However, the size and the
direction of these changes depend on the information contained in the 20-day
moving average.
Sensibilidad del Precio de la Opción ante variaciones en la diferencia entre el
promedio y el tipo de cambio al contado.
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
0.9
5
0.8
6
0.7
6
0.6
7
0.5
7
0.4
8
0.3
8
0.2
9
0.1
9
0.1
0
0.0
0
-0.0
9
-0.1
9
-0.2
8
-0.3
8
-0.4
7
-0.5
7
Diferencia entre el promedio y el tipo de cambio al contado
Pre
cio
(pe
sos p
or
cada
mil
dóla
res)
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
Pro
bab
ilid
ad d
e E
jerc
icio
Precio Probabilidad
Sensitivity of the option price to changes in the difference between
the average and spot exchange rates
Price (
pesos f
or
every
thousand d
olla
rs)
Pro
babili
ty o
f E
xerc
ise
Difference between the average and spot exchange rates
Price Probability
23
VI. Sensitivity of the Price of the Option
This article presents a mechanism for reserve accumulation and its valuation for a
central bank through options. This mechanism has several advantages. Among them, it
offers an intervention strategy for those central banks whose direct participation in the
exchange market alters or affects significantly the behavior of market participants, in
particular, exacerbating exchange rate volatility.
The mechanism proposed in this paper helps to reduce the impact purchases of foreign
exchange currency made by the central bank may have in the exchange market. This is
achieved by selling or auctioning a small amount of put options, whose price of exercise
varies with time according to the exchange rate known as the “fix”. The exercise of the
options is limited to satisfying one restriction: that the strike exchange rate is not greater
than the moving average of the previous 20 observed exchange rate levels. Another
advantage of the mechanism is the fact that the mentioned purchase of foreign currency
by the central bank occurs in a passive way, because the central bank is the seller of the
option.