On the Cost of Delayed Currency Fixing Announcements

Cent re for Pr ac tic al Qua ntitati ve Fin anc e
No. 3
On the Cost of Delayed Currency Fixing
Announcements
Christoph Becker
Uwe Wystup
June 2005
Author:
Christoph Becker
MathFinance AG
Germany
http://www.mathfinance.de
Publisher:
Prof. Dr. Uwe Wystup
HfB – Business School of
Finance & Management
Frankfurt/Main
Germany
HfB - Business School of Finance & Management
Phone: +49 (0) 69 154 008-0 § Fax: +49 (0) 69 154 008-728
Sonnemannstr. 9-11 § D-60314 Frankfurt/M. § Germany
2
Becker, C. and Wystup, U.
Abstract. In Foreign Exchange Markets vanilla and barrier options are traded
frequently. The market standard is a cutoff time of 10:00 a.m. in New York for
the strike of vanillas and a knock-out event based on a continuously observed
barrier in the inter bank market. However, many clients, particularly from Italy,
prefer the cutoff and knock-out event to be based on the fixing published by the
European Central Bank on the Reuters Page ECB37. These barrier options are
called discretely monitored barrier options. While these options can be priced
in several models by various techniques, the ECB source of the fixing causes
two problems. First of all, it is not tradable, and secondly it is published with
a delay of about 10 - 20 minutes. We examine here the effect of these problems
on the hedge of those options and consequently suggest a cost based on the
additional uncertainty encountered.
Keywords: Exotic Options, Currency Fixings
JEL classification: C15, G12
Acknowledgments. We would like to thank Michael Braun, Ansua DuttaWystup, Thomas Heidorn, Wolfgang Schmidt and Dagmar Wicht for their help
and comments.
Contents
1 Introduction
1.1 The Currency Fixing of the European Central Bank . . . . . . .
1.2 Model and Payoff . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
4
4
2 Error Estimation
2.1 European Style up-and-out Call . . . . . . . . . . . . . . . . . . .
2.2 Discretely Monitored up-and-out Call . . . . . . . . . . . . . . .
2.3 Calculating the Delta-Hedge Quantity . . . . . . . . . . . . . . .
5
5
6
7
3 Analysis of EUR-USD
7
4 Conclusion
10
5 Appendix
11
5.1 Analysis of USD-CHF . . . . . . . . . . . . . . . . . . . . . . . . 11
5.2 Analysis of USD-JPY . . . . . . . . . . . . . . . . . . . . . . . . 13
5.3 Analysis of EUR-GBP . . . . . . . . . . . . . . . . . . . . . . . . 15
On the Cost of Delayed Currency Fixing Announcements
1
3
Introduction
1.1
The Currency Fixing of the European Central Bank
The European Central Bank (ECB) sets currency fixings every working day in
Frankfurt at 2:15 p.m. Frankfurt time. The actual procedure of this fixing is
done by observing the spot rates in the inter bank market, in which the ECB also
participates. Traders of the ECB in various locations get together to decide on
how to set the fixing. The quantity quoted is not a bid price or an offer price the
ECB or anybody else necessarily trades at, but is rather used for statistical and
official means, for instance tax computation or economic research. An example
of the ECB37 REUTERS screen is presented in Figure 1.
Figure 1: Reuters screen ECB37 of 21 February 2005 showing the fixings of all currencies against EUR
Corporate treasures often prefer an independent source for currency exchange
rates that provides a reference rate for their underlying under consideration.
This way they are not bound to their own bank that might move the quoted
cut-off rate in favor of their own position. The key features to stress are the
following.
1. The ECB fixing is not tradable.
2. The ECB fixing is published with a delay of 10-20 minutes.
4
Becker, C. and Wystup, U.
In this paper we analyze the impact on the value for the short position of
a discretely monitored reverse knock-out, as the problems mentioned above
impose additional uncertainty when it comes to determining a proper hedge.
Most of the hedging error is expected in the case of jumps in the payoff of the
option, which is why we restrict ourselves to the liquidly traded up-and-out
call option. The currency-pairs under consideration are EUR-USD, USD-JPY,
USD-CHF and EUR-GBP.
1.2
Model and Payoff
To model the exchange rate we choose a geometric Brownian motion,
dSt = St [(rd − rf ) dt + σ dWt ],
(1)
under the risk-neutral measure. As usual, rd denotes the domestic interest rate,
rf the foreign interest rate, σ the volatility. These parameters are assumed to
be constant in this paper. For contract parameters maturity in years T , strike
K and knock-out barrier B, fixing schedule 0 = t0 < t1 < t2 . . . , tn = T , the
payoffs for the vanilla and for a discretely monitored up-and-out barrier option
under consideration are
V (FT , T ) = (φ(FT − K))+ ,
V (F, T ) = (φ(FT − K))+ II{max(Ft0 ,...,Ftn )<B} ,
(2)
(3)
where Ft denotes the fixing of the underlying exchange rate at time t, II the
indicator function and φ a put-call indicator taking the value +1 for a call and
−1 for a put. Of course, Ft is usually close to St , the spot at time t, but it may
differ as well. We start with payoffs
V (ST , T ) = (φ(ST − K))+ ,
V (S, T ) = (φ(ST − K))+ II{max(St0 ,...,Stn )<B} ,
(4)
(5)
whose values are explicitly known in the Black-Scholes model. In this model,
the values are called theoretical value (TV).
1.3
Analysis Procedure
1. We simulate the spot process with a Monte Carlo simulation using an
Euler-discretization. Furthermore, we use a Mersenne Twister pseudo
random number generator by Takuji Nishimura and Makoto Matsumoto [5]
and a library to compute the inverse of the normal cumulative distribution
function written by Barry W. Brown et al. [2].
2. We model the ECB-fixing Ft by
Ft = St + ϕ,
ϕ ∈ N (µ, σ),
(6)
where µ and σ are estimated from historic data. Note that Ft denotes the
ECB-fixing at time t, which is nonetheless only announced 10 - 20 minutes
later. We denote this time delay by ∆ECB (T ). This means that we model
the error, i.e. the difference of fixing and traded spot, as a normally
distributed random variable. The estimated values for the mean and the
On the Cost of Delayed Currency Fixing Announcements
5
standard-deviation of the quantity Spot - ECB Fixing from historic time
series are listed in Table 1. For the cross rates, where EUR is not one of
the currencies, we take the respective ratios of fixings against EUR, e.g.
for the USD/JPY fixing we divide the EUR/JPY fixing by the EUR/USD
fixing, which is also common market practise in trade confirmations.
Currency pair
Expected value
Standard deviation
Time horizon
EUR / USD
-3.125E-6
0.0001264
23.6 - 08.8.04
USD / YEN
-4.883E-3
0.0134583
22.6 - 26.8.04
USD / CHF
-1.424E-5
0.0001677
11.5 - 26.8.04
EUR / GBP
-1.33E-5
0.00009017
04.5 - 26.8.04
Table 1: Estimated values for mean and standard-deviations of the quantity
Spot - ECB-fixing from historic time series. The time series were provided by
Commerzbank AG
3. We evaluate the payoffs for barrier options for each path and run the
simulations with the appropriate delta hedge quantities to hold. Then we
compute for each path the error encountered due the fixing being different
from the spot, and then average over all paths.
4. We do this for various currency pairs, parameter scenarios, varying the
rates, volatilities, maturities, barriers and strikes. We expect a significant
impact particularly for reverse knock-out barrier options due to the jump
of the payoff and hence the large delta hedge quantity.
2
Error Estimation
Note that since we expect the resulting errors to be fairly small, we introduce a
bid/offer-spread δ for the spot, which is of the size of 2 basis points in the inter
bank market. We consider the following options in detail.
2.1
European Style up-and-out Call
To determine the possible hedging error we propose the following to be appropriate. Note that the error is measured for a nominal of 1 unit of the underlying.
We consider three cases.
1. Let ST ≤ K. In this case, the seller who is short the option decides not
to hedge as the option is probably out of the money, i.e. delta = 0. If
the option turns out to be in the money, i.e. FT > K, the holder of the
short position faces a P&L of K − (S(T + ∆ECB (T )) + δ) (units of the
base currency).
2. Let ST > K and ST < B. Hence, one assumes that the option is in the
money and delta is 1. If now FT ≤ K or FT ≥ B, there is a P&L of
S(T + ∆ECB (T )) − (S(T ) + δ).
6
Becker, C. and Wystup, U.
3. Let ST ≥ B and FT < B. Here we have a P&L of K −(S(T +∆ECB (T ))+
δ). Note that other than in the first case, this P&L is of order K − B due
to the jump in the payoff.
2.2
Discretely Monitored up-and-out Call
We consider a time to maturity of one year with 250 knock-out-events, i.e., the
possible knock out occurs every working day at 2:15 p.m. Frankfurt time, when
the ECB fixes the reference rate of the underlying currency pair. We propose
the following error determination to be appropriate. First of all, we adopt the
procedure above for the maturity time. In addition, we consider every knockout-event and examine the following cases.
1. Let St < B and Ft ≥ B. At time t the trader holds ∆(St ) shares of stock
in the delta hedge. He does not unwind the hedge at time t, as the spot is
below the barrier. Only after the fixing announcement, it turns out that
the hedge needs to be unwound, so he does this with delay and encounters
a P&L of
∆(St ) · (St+∆ECB (T ) − St ),
(7)
where ∆(St ) denotes the theoretical delta of the option under consideration, if the spot is at St . To see this, it is important to note, that the
theoretical delta is negative if the underlying is near the barrier B. In this
way, the seller of the option has been short the underlying at time t and
must buy it in t + ∆ECB (T ) minutes to close out the hedge. Therefore, he
makes profit if the underlying is cheaper in t+∆ECB (T ), which is reflected
in our formula. We shall elaborate later how to compute the theoretical
delta, but we would like to point out that whenever we need a spot price
at time t to calculate such a delta or to compute the value of a hedge, we
refer to S as the tradable instrument instead of the contractually specified
underlying F in order to account for the ECB fixing being non-tradable.
2. Let St ≥ B and Ft < B. Here the seller of the option closed out the
hedge at time t, though she shouldn’t have done so, and in t + ∆ECB (T )
she needs to build a new hedge. Note again that the theoretical delta is
negative. This means that at time t the seller bought the underlying with
the according theoretical delta-quantity, and in t + ∆ECB (T ) she goes
short the underlying with the appropriate new delta-quantity. The profit
and loss (P&L) is calculated via
P&L = ∆(St ) · (St + δ) − ∆(St+∆ECB (T ) ) · St+∆ECB (T )
(8)
The other cases do not lead to errors due to an unexpected fixing announcement.
Of course, delta hedging an option in the Black-Scholes model can lead to errors,
because of hedge adjustments at discrete times and and because of model risk
in general, see, e.g. [1].
On the Cost of Delayed Currency Fixing Announcements
2.3
7
Calculating the Delta-Hedge Quantity
The valuation of continuously monitored barrier options has been treated, e.g.,
in [3]. In order to compute the theoretical delta for the discretely monitored upand-out call, for which no closed-form solution is known, in acceptable time and
precision, we refer to an approximation proposed by Per Hörfelt in [4], which
works in the following way. Assume the value of the spot is observed at times
iT /n, i = 0, . . . , n, and the payoff of the discretely monitored up-and-out call is
given by Equation (5). We define the value and abbreviations
rd − rf ± σ 2 /2 √
T,
σ
ln(K/S0 )
∆
√
c =
,
σ T
ln(B/S0 )
∆
√
d =
,
σ T
p
∆
β = −ζ(1/2)/ (2π) ≈ 0.5826,
θ±
∆
=
(9)
(10)
(11)
(12)
where ζ denotes the Riemann zeta function. We define the function
∆
F+ (a, b; θ) = N (a − θ) − e2bθ N (a − 2b − θ)
(13)
and obtain for the value of the discretely monitored up-and-out call
√
√
V (S0 , 0) ≈ S0 e−rf T F+ (d, d + β/ n; θ+ ) − F+ (c, d + β/ n; θ+ ) (14)
√
√
−Ke−rd T F+ (d, d + β/ n; θ− ) − F+ (c, d + β/ n; θ− ) .
Using this approximation for the value, we take a finite difference approach for
the computation of the theoretical delta,
∆ = VS (S, t) ≈
3
V (S + , t) − V (S − , t)
.
2
(15)
Analysis of EUR-USD
Considering the simulations for a maturity T of one year, huge hedging errors
can obviously only occur near the barrier. The influence of the strike is comparatively small, as we discussed in the error determination procedure above.
In this way we chose the values listed in Table 2 to remain constant and only
to vary the barrier.
Using Monte Carlo simulations with one million paths we show the average of
the profit and loss with 99.9% confidence bands and how the probability of a
mishedge depends on the position of the barrier in Figure 2. It appears that
the additional costs for the short position are negligible. We also learn that the
mishedge is larger for a barrier in a typical traded distance from the spot, i.e.
not too close and not too far.
In Figure 3 we plot the barrier against the ratio Hedging Error / TV of the upand-out call and the ECB-fixing as underlying. This relationship is an important
message for the risk-averse trader. For a one-year reverse knock-out call we see
an average relative hedge error below 5% of TV if the barrier is at least 4 big
8
Becker, C. and Wystup, U.
Spot
1.2100
Strike
1.1800
Trading days
250
Domestic interest rate
2.17% (USD)
Foreign interest rate
2.27% (EUR)
Volatility
10.4%
Time to maturity
1 year
Notional
1,000,000 EUR
Table 2: EUR-USD testing parameters
Hedge error with 99.9% - confidence interval
Probability of mishedging
14.00%
1.22 1.24 1.26 1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.42 1.44 1.46
0
12.00%
-2
10.00%
-4
probability
-6
-8
error
-10
-12
8.00%
6.00%
4.00%
-14
2.00%
-16
-18
error in USD
confidence band
-20
0.00%
1.22 1.24 1.26 1.28
1.3
1.32 1.34 1.36 1.38
barrier
1.4
1.42 1.44 1.46
barrier
Figure 2: Additional average hedge costs and probability of a mishedge for the short
position of a discretely monitored up-and-out call in EUR-USD
Rel. hedge error with 99.9% bands
Rel. hedge error with 99.9% bands
0.00%
1.
46
1.
44
1.
4
1.
42
1.
38
1.
36
1.
34
1.
3
1.
32
1.
28
1.
26
-20.00%
1.25 1.27 1.29 1.31 1.33 1.35 1.37 1.39 1.41 1.43 1.45
1.
24
1.
22
0.00%
-0.50%
-1.00%
-1.50%
probability
probability
-40.00%
-60.00%
-80.00%
-2.00%
-2.50%
-3.00%
-3.50%
-100.00%
error
confidence band
-120.00%
-4.00%
error
confidence band
-4.50%
barrier
barrier
Figure 3: Hedging Error /TV for a discretely monitored up-and-out call in EUR-USD
figures away from the spot. Traders usually ask for a risk premium of 10 basis
points.
On the Cost of Delayed Currency Fixing Announcements
9
Finally, we would like to point out that the average loss is not the full story
as an average is very sensitive to outliers. Therefore, we present in Figure 4
the distribution of the maximal profits and losses, both in absolute as well as
in relative numbers. The actual numbers are presented in Table 3. We have
found other parameter scenarios to behave similarly. The crucial quantity is
the intrinsic value at the knock-out barrier. The higher this value, the more
dangerous the trade. In particular we do not exhibit the results for the vanilla
as there is hardly anything to see. Varying rates and volatilities do not yield
any noticeably different results.
1.
24
1.
26
1.
28
1.
3
1.
42
1.
44
1.
46
Relative P & L
1.
32
1.
34
1.
36
1.
38
1.
4
Extremal P & L for the short position
50000
5000%
0
0%
-50000
-5000%
-10000%
P & L in USD
-100000
P&L
-150000
-200000
-250000
-300000
-15000%
-20000%
-25000%
-30000%
max downside error
max. upside error
-35000%
1.22 1.25 1.28 1.31 1.34 1.37
barrier
1.4
max downside error
max upside error
-40000%
1.43 1.46
-45000%
barrier
Figure 4: Absolute and relative maximum profit and loss distribution for a discretely
monitored up-and-out call in EUR-USD. The upside error is the unexpected gain a
trader will face. The downside error is his unexpected loss. On average the loss seems
small, but the maximum loss can be extremely high. The effect is particularly dramatic
for knock-out calls with a large intrinsic value at the barrier as shown in the left hand
side. The right hand side shows the maximum gain and loss relative to the TV. Of
course, the further the barrier is away from the spot, the smaller the chance of hedging
error occurring.
barrier 1.2500
upside error
<1k$
<2k$
<3k$
<39k$
<40k$
<41k$
<42k$
<43k$
951744
20
1
0
0
0
0
0
downside error
48008
54
2
5
59
85
21
1
barrier 1.3000
<1k$
< 2k$
< 3k$
<89k$
< 90k$
<91k$
<92k$
974340
20
1
0
0
0
0
25475
43
0
2
40
59
20
upside error
downside error
barrier 1.4100
upside error
downside error
<1k$
<2k$
<3k$
<199k$
<200k$
<201k$
<202k$
<203k$
994854
78
0
0
0
0
0
0
4825
194
3
1
19
17
8
1
Table 3: EUR-USD distribution of absolute errors in USD. The figures are the
number of occurrences out of 1 million. For instance, for the barrier at 1.2500,
there are 54 occurrences out of 1 million, where the trader faces a loss between
1000 and 2000 $.
As the analysis of the other currency pairs is of similar nature, we list it in the
appendix and continue with the conclusion.
10
4
Becker, C. and Wystup, U.
Conclusion
We have seen that even though a trader can be in a time interval where he does
not know what delta hedge he should hold for an option due to the delay of
the fixing announcement, the loss encountered is with probability 99.9% within
less than 5% of the TV for usual choices of barriers and strikes and the liquid
currency pairs, in which complex barrier options such as a discretely monitored
up-and-out call are traded. However, the maximum loss quantity in case of a
hedging error can be rather substantial. So in order to take this into account, it
appears generally sufficient to charge a maximum of 10% of the TV to cover the
potential loss with probability 99.9%. This work shows that the extra premium
of 10 basis points per unit of the notional of the underlying, which traders argue
is needed when the underlying is the ECB-fixing instead of the spot, is justified
and well in line with our results. However, charging 10 basis points extra may
be easy to implement, but is not really precise as we have seen, since the error
depends heavily on the distance of the barrier from the spot.
Of course the level of complexity of the model can be elaborated further arbitrarily, but using a geometric Brownian motion and a Monte Carlo simulation
appears sufficient. The relative errors are small enough not to pursue any further
investigation concerning this problem.
On the Cost of Delayed Currency Fixing Announcements
5
5.1
11
Appendix
Analysis of USD-CHF
We used the market and contract data listed in Table 4. We summarize the
results in Table 5 and Figure 5. The results are similar to the analysis of EURUSD.
Spot
1.2800
Strike
1.2000
Trading days
250
Domestic interest rate
0.91% (CHF)
Foreign interest rate
2.17% (USD)
Volatility
11.25%
Time to maturity
1 year
Notional
1,000,000 USD
Table 4: USD-CHF testing parameters
barrier 1.3300
upside error
downside error
barrier 1.3900
upside error
downside error
barrier 1.5000
upside error
downside error
< 1k
< 2k
< 3k
< 49k
< 50k
< 51k
< 52k
< 54k
950902
29
1
0
0
0
0
0
48846
65
2
9
44
72
28
2
< 113k
< 1k
< 2k
< 3k
< 109k
< 110k
< 111k
< 112k
975939
26
3
0
0
0
0
0
23847
72
5
4
23
52
27
2
< 222k
< 1k
< 2k
< 3k
< 4k
< 5k
< 219k
< 220k
993544
749
90
2
0
0
0
0
4523
880
158
12
3
2
12
25
Table 5: USD-CHF distribution of absolute errors in CHF. The figures are the
number of occurrences out of 1 million
12
Becker, C. and Wystup, U.
Hedge error with 99,9% confidence intervall
Probability to mishedge
16.00%
1.
49
1.
45
1.
47
1.
43
1.
41
1.
39
1.
37
1.
35
1.
33
1.
31
1.
29
0
14.00%
12.00%
probability
-10
10.00%
8.00%
6.00%
-15
4.00%
2.00%
-20
0.00%
hedge error
barrier
barrier
relative hedge error
relative hedge error
1.
49
1.
43
1.
41
1.
39
1.
37
1.
35
-2.00%
-40.00%
1.
33
1.
31
1.
49
1.
47
1.
45
1.
43
1.
41
1.
39
1.
37
1.
35
1.
33
-20.00%
0.00%
1.
31
1.
29
0.00%
1.
47
-25
1.
29
1.
3
1.
31
1.
32
1.
33
1.
34
1.
35
1.
36
1.
37
1.
38
1.
39
1.
4
1.
41
1.
42
1.
43
1.
44
1.
45
1.
46
1.
47
1.
48
1.
49
1.
5
confidence intervall
1.
45
hedge error in CHF
-5
-4.00%
error/TV
-80.00%
-100.00%
-6.00%
-8.00%
-10.00%
-120.00%
rel error
confidence intervall
-12.00%
rel error
confidence intervall
-14.00%
barrier
barrier
Extremal P & L for the short position
1.
45
1.
43
1.
41
1.
37
0%
rel. P & L in CHF
1.
49
1.
47
1.
45
1.
43
1.
41
1.
39
1.
37
1.
35
1.
33
1.
31
1.
29
P&L
-100000
-150000
-200000
-250000
1.
35
1.
31
10000%
0
-50000
1.
33
Extremal relative P & L
50000
1.
49
-160.00%
1.
47
-140.00%
1.
39
error/TV
-60.00%
-10000%
-20000%
-30000%
-40000%
-50000%
upside error
downside error
max rel upside error
max rel downside err
-60000%
barrier
barrier
Figure 5: Analysis of the discretely monitored up-and-out call in USD-CHF
On the Cost of Delayed Currency Fixing Announcements
5.2
13
Analysis of USD-JPY
We used the market and contract data listed in Table 6. Again the analysis
looks very much like the EUR-USD case. We summarize the results in Table 7
and Figure 6.
Spot
110.00
Strike
100.00
Trading days
250
Domestic interest rate
0.03% (JPY)
Foreign interest rate
2.17% (USD)
Volatility
9.15%
Time to maturity
1 year
Notional
1,000,000 USD
Table 6: USD-JPY testing parameters
barrier 133.00
< 10k
<20k
< 30k
< 40k
< 50k
< 60k
< 70k
949563
957
29
34
18
20
19
downside error
47623
1260
32
25
23
23
21
barrier 133.00
< 80k
< 155k
< 175k
more
upside error
21
39
3
0
downside error
13
34
0
234
upside error
Table 7: USD-JPY distribution of absolute errors in JPY. The figures are the
number of occurrences out of 1 million
14
Becker, C. and Wystup, U.
Probability of mishedging
hedging error with 99,9% confidence band
111 113 115 117 119 121 123 125 127 129 131 133 135
18.00%
0
16.00%
-200
14.00%
probability
-400
error in JPY
-600
-800
-1000
12.00%
10.00%
8.00%
6.00%
-1200
4.00%
-1400
2.00%
-1600
relative error with 99,9% confidence band
5
3
1
5
3
13
1
13
9
13
7
12
0%
13
9
13
7
13
5
12
3
12
1
12
9
12
7
12
5
11
11
11
3
113 115 117 119 121 123 125 127 129 131 133 135
11
1
5
relative error with 99,9% confidence band
0.00%
11
12
3
barrier
barrier
-20.00%
-1%
-2%
-40.00%
-3%
-60.00%
error
error
12
1
12
9
12
7
11
5
11
11
1
11
11
-2000
3
0.00%
hedge error
confidence intervall
-1800
-80.00%
-4%
-5%
-6%
-100.00%
rel. error
confidence band
-120.00%
-7%
rel. error
confidence band
-8%
barrier
barrier
Extremal P & L for the short position
Extremal relative error
11
1
11
3
11
5
11
7
11
9
12
1
12
3
12
5
12
7
12
9
13
1
13
3
13
5
5000%
5000000
0%
11
3
11
5
11
7
11
9
12
1
12
3
12
5
12
7
12
9
13
1
13
3
13
5
0
relative error
P & L in JPY
-5000000
-10000000
-15000000
-5000%
-10000%
-15000%
-20000000
-25000000
-30000000
-20000%
upside error
downside error
max positive rel error
max negative rel error
-25000%
barrier
barrier
Figure 6: Analysis of the discretely monitored up-and-out call in USD-JPY
On the Cost of Delayed Currency Fixing Announcements
5.3
15
Analysis of EUR-GBP
To compare we take now EUR-GBP as a currency pair without the USD. We
used the market and contract data listed in Table 8. Again, we observe the usual
picture, i.e., a similar hedging error curvature as in EUR-USD. We summarize
the results in Table 9 and Figure 7.
Spot
0.6700
Strike
0.6700
Trading days
250
Domestic interest rate
5.17% (GBP)
Foreign interest rate
2.27% (EUR)
Volatility
7.65%
Time to maturity
1 year
Notional
1,000,000 EUR
Table 8: EUR-GBP testing parameters
barrier 0.7100
< 1k
< 2k
< 4k0
< 41k
< 42k
950005
0
0
0
0
downside error
49711
3
59
220
2
barrier 0.7400
< 1k
< 2k
< 7k0
< 71k
< 72k
0
upside error
upside error
978884
139
0
0
downside error
20493
273
48
162
1
barrier 0.7800
< 1k
< 2k
< 3k
< 4k
< 5k
994032
678
122
11
2
4001
860
183
27
2
< 109k
< 11k0
< 111k
< 112k
< 114k
upside error
downside error
barrier 0.7800
upside error
0
0
0
0
0
downside error
1
15
63
2
1
Table 9: EUR-GBP distribution of absolute errors in GBP. The figures are the
number of occurrences out of 1 million
16
Becker, C. and Wystup, U.
probability of mishedging
Hedge error with 99,9% confidence band
0.
92
0.
88
0.
9
0.
86
0.
84
0.
82
0.
8
0.
78
0.
76
0.
74
0.
72
0.
7
0.
68
20.00%
18.00%
5
16.00%
14.00%
probability
error in GBP
0
-5
-10
12.00%
10.00%
8.00%
6.00%
4.00%
-15
2.00%
barrier
barrier
0.
92
0.
9
0.
88
0.
86
0.
84
0.
82
0.
8
0.
78
0.00%
0.
9
0.
92
0.
88
0.
86
0.
84
0.
82
0.
8
0.
78
0.
76
0.
74
0.
72
0.
7
0.
68
0.
76
0.
7
0.20%
0.00%
-0.20%
-0.40%
-10.00%
-0.60%
error
error
0.
74
rel hedge error with 99,9% confidence band
0.
72
rel hedge error with 99,9% confidence band
5.00%
-5.00%
0.
9
0.
92
0.
8
0.
82
0.
84
0.
86
0.
88
0.
78
0.
76
0.
74
0.
68
-25
probability
0.00%
hedge error
confidence band
0.
7
0.
72
-20
-15.00%
-0.80%
-1.00%
-20.00%
-1.20%
rel hedge error
confidence band
-1.40%
rel hedge error
confidence band
-1.60%
-1.80%
barrier
barrier
-500%
P & L in GBP
0%
-50000
-100000
-150000
-250000
0.
85
0.
83
0.
81
0.
77
0.
75
500%
0
-200000
0.
73
0.
71
0.
92
0.
82
0.
84
0.
86
0.
88
0.
9
0.
72
0.
74
0.
76
0.
78
0.
8
0.
68
0.
7
P&L
50000
0.
79
Relative P & L
Extremal P & L for the short position
0.
89
0.
91
-30.00%
0.
87
-25.00%
-1000%
-1500%
-2000%
upside error
downside error
max upside error
max downside error
-2500%
barrier
barrier
Figure 7: Analysis of the discretely monitored up-and-out call in EUR-GBP
On the Cost of Delayed Currency Fixing Announcements
17
References
[1] Anagnou-Basioudis, I. and Hodges, S. (2004) Derivatives Hedging and
Volatility Errors. Warwick University Working Paper.
[2] Brown, B., Lovato, J. and Russell, K. (2004). CDFLIB
- C++ - library, http://www.csit.fsu.edu/~burkardt/cpp_src/
dcdflib/dcdflib.html
[3] Fusai G. and Recchioni, C. (2003). Numerical Valuation of Discrete Barrier Options Warwick University Working Paper.
[4] Hörfelt, P. (2003). Extension of the corrected barrier approximation by
Broadie, Glasserman, and Kou. Finance and Stochastics, 7, 231-243.
[5] Matsumoto, M. (2004). Homepage of Makoto Matsumoto on the server
of the university of Hiroshima: http://www.math.sci.hiroshima-u.ac.
jp/~m-mat/eindex.html
HFB – WORKING PAPER S ERIES
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The Relevance of real-time data in estimating reaction functions for the euro area
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52.
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46.
45.
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41.
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Modeling Default Dependence with Threshold Models
2003
Balthasar, Daniel / Cremers, Heinz / Schmidt, Michael
Portfoliooptimierung mit Hedge Fonds unter besonderer Berücksichtigung der Risikokomponente
2002
Heidorn, Thomas / Kantwill, Jens
Eine empirische Analyse der Spreadunterschiede von Festsatzanleihen zu Floatern im Euroraum
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2001
35.
Heidorn, Thomas / Weier, Sven
Einführung in die fundamentale Aktienanalyse
2001
34.
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International Accounting Standards (IAS)
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33.
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Strategic Positioning of E-Commerce Business Models in the Portfolio of Corporate Banking
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Behavioral Finance - Eine Alternative zur vorherrschenden Kapitalmarkttheorie?
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Event Risk Covenants
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Biswas, Rita / Löchel, Horst
Recent Trends in U.S. and German Banking: Convergence or Divergence?
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28.
Löchel, Horst / Eberle, Günter Georg
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Economic Value Added zur Prognose der Performance europäischer Aktien
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Entscheidungsorientierte Mindestmargenkalkulation
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CatBonds
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1999
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The EMU and the Theory of Optimum Currency Areas
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1998
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Die Umstellung auf die Stückaktie für deutsche Aktiengesellschaften
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Stand und Perspektiven der Informationsverarbeitung in Banken
1998
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04.
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Die Bank als Betreiberin einer elektronischen Shopping-Mall
1997
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Jahresbericht 1996
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02.
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Interpolation of Discount Factors
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