lorexpansion

σ2λk +2
τ
(12.18)(12.19)
Strike pricesand premiums
Melick &Thomas21
Output
Linear combinationof 2 log-normaldistributions
RND
ATMVolatilityRiskReversalStrangle
OTC optionvolatilities(observed)
Garman &Kohlhagen(market convention)
Figure 12.6
Estimation procedure of RND functions from option spreadsSource: Adapted from McCauley and Melick (1996).
21Using a mixture of two log-normal distributions with no jumps or with up to one jump, following Merton(1976), Bates (1991) or Malz (1996, 1997).
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The whole estimation procedure, including both RND estimation techniques, can thenbe brieﬂy presented using Figure 12.6, based on that presented in McCauley and Melick(1996).
12.4 MEASURES OF CORRELATION AND OPTION PRICESThe implied volatilities of options on cross exchange rates provide enough information tocompute implied correlations between the exchange rates involved. Assuming no arbitrageopportunities, the exchange rate between currencies X and Y at time t, denoted by S1,t ,may be written as:S1,t = S2,t S3,t(12.20)where S2,t and S3,t are the exchange rates, respectively, between X and a third currencyZ and between Z and Y .Let si,t = ln(Si,t ), with i = 1, 2, 3. Thus:s1,t = s2,t + s3,t
(12.21)
Denoting the daily exchange rate variation si,t − si,t−1 by vi,t , we have:v1,t = v2,t + v3,t
(12.22)
Let σi,t,T (with i = 1, 2, 3) be the standard deviation of daily returns over a period oftime from t to t + T and let Cov(v2,t,T , v3,t,T ) be the covariance between v2,t and v3,tover the same period. The variance of v1,t from t to t + T is given by:222σ1,t,T= σ2,t,T+ σ3,t,T+ 2 Cov(v2,t,T , v3,t,T )
(12.23)
As Cov(v2,t,T , v3,t,T ) = ρt,T σ2,t,T σ3,t,T , where ρt,T is the correlation coefﬁcient betweenv2,t and v3,t , solving equation (12.23) in order to ρt,T we get:ρt,T =
222σ1,t,T− σ2,t,T− σ3,t,T
2σ2,t,T σ3,t,T
(12.24)
In equation (12.24), the correlation coefﬁcient between the daily returns of the exchangerates of two currencies vis-à-vis a third currency may be obtained from the variance ofthe daily returns of the exchange rates between the three currencies.22 Accordingly, whent is the current time, it is possible to estimate at t the correlation between future dailyreturns of two exchange rates, using forecasts of the variances of the daily returns of theexchange rates between the three currencies:23ρ̂t,T =
222σ̂1,t,T− σ̂2,t,T− σ̂3,t,T
2σ̂2,t,T σ̂3,t,T
(12.25)
22Notice that it is irrelevant how exchange rates are expressed, since the variance of the growth rate of avariable is equal to the variance of the growth rate of its inverse.23The variances used are σ 2 (T − t).
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There are several ways to estimate the correlation coefﬁcient in equation (12.25), fromthe information available at time t. The simplest way is to compute the historical correlation over a window of t − T days:T
[(v2,t−T +j − v 2,t−T ,T )(v3,t−T +j − v 3,t−T ,T )]
j =0
ρ̂t,T =  T
(v2,t−T +j − v 2,t−T ,T )2
j =0
T
=
Côv(v2,t−T ,T , v3,t−T ,T )σ̂2,t−T ,T σ̂3,t−T ,T
(v3,t−T +j − v 3,t−T ,T )2
j =0
(12.26)where v i corresponds to the average rates of return.Instead of a single past correlation, a moving average of several past correlations can beused as a forecast of future correlation, attaching equal or different weights to each pastcorrelation. Equally weighted moving averages have the disadvantage of taking longer toreveal the impact of a shock to the market and to dissipate that impact.One of the most currently used weighting techniques is the exponentially weightedmoving averages (EWMA), where higher weights are attached to most recent observationsin the computation of the standard deviations and the covariance.24 This methodologyoffers some advantages over the traditional equally weighted moving averages, namelybecause volatilities and correlations react promptly to shocks and have less memory, giventhe higher weight attached to recent data.Using EWMA, the weighted returns (ṽ) are obtained by pre-multiplying the matrix ofreturns (v) by a diagonal matrix of weights (λ), as follows:ṽ = λvwith

λ=
1
0√λ
00√λ2
000..
·········
000
..
0...√λT −1
..

(12.27)
vt,1 vt−1,1 vt−2,1and v =  .. . . ..vt−T ,1
vt,2vt−1,2vt−2,2......vt−T ,2
vt,3vt−1,3vt−2,3......vt−T ,3
·········......···
···vt,k· · · vt−1,k · · · vt−2,k .... .. .... .. · · · vt−T ,k
where 0 ≤ λ ≤ 1.Each column of v corresponds to the returns of each asset price included in the portfolio,while each row corresponds to the time at which the return occurred. Consequently,standard deviations are calculated as:25Tσ̂i,t,T = (1 − λ)λT −j (vi,t−T +j − v i,t−T ,T )2(12.28)j =1
with i = 2, 3.24J.P. Morgan RiskMetrics methodology uses EWMA to calculate the variances and covariances of ﬁnancialasset prices, in order to compute the VaR of a ﬁnancial portfolio.25The term 1/(1 − λ) corresponds to the asymptotic limit of the sum of the weights, i.e., the sum of the termsof a geometric progression with ratio equal to λ (being λ < 1).
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Identically, the covariance corresponds to:Côv(v2,t,T , v3,t,T ) = (1 − λ)
T
λT −j (v2,t−T +j − v 2,t−T ,T )(v3,t−T +j − v 3,t−T ,T )
j =1
(12.29)Therefore, the correlation coefﬁcient is:26T
ρ̂t,T =  T
λT −j [(v2,t−T +j − v 2,t−T ,T )(v3,t−T +j − v 3,t−T ,T )]
j =1T2  T −j2λT −j (v2,t−T +j − v 2,t−T ,T )λ (v3,t−T +j − v 3,t−T ,T )
j =1
(12.30)
j =1
The formula in equation (12.30) can be represented in a recursive form. In fact, thestandard deviations and the covariance can be computed recursively, in the following way:22= (1 − λ)(vi,t − v i,t−T ,T )2 + λσ̂i,t−1,Tσ̂i,t,T
(12.31)
Côv(v2,t,T , v3,t,T ) = (1 − λ)(v2,t − v 2,t−T ,T )(v3,t − v 3,t−T ,T ) + λCôv(v2,t−1,T , v 3,t−1,T )(12.32)Nevertheless, EWMA measures are still past dependent. On the contrary, implied correlations computed from option prices are forward looking measures that should quicklyreﬂect any perceived structural breaks in the data generating process. These correlationsmay be obtained directly from volatility quotes. Both techniques will be implemented inSection 12.6.
12.5 INDICATORS OF CREDIBILITY OF AN EXCHANGERATE BANDAnother relevant exercise for portfolio management when ﬁnancial assets are denominatedin currencies managed within target zones is the assessment of the credibility of theexchange rate policy. Several indicators have been used in the past to perform theseexercises, namely spot interest rate spreads and spot and forward exchange rates.27 Morerecently, information from option prices started to be used for this purpose.The simplest analysis using currency option prices is that consisting of deriving theRND function and quantifying the probability of reaching the band limits. When there isonly information on at-the-money forward options, the RND may be approached usingimplied volatilities to calculate the standard deviation of the distribution,28 assuming thelog-normality of the exchange rate.26
In Campa and Chang (1996), the weights are directly attached to correlations in moving periods with a ﬁxednλj −1 ρt−T −(j −1),T .length: ρ̂t,T = (1 − λ)27
j =1
See Svensson (1991).28In the OTC market, the volatilities (in annual percentages) are quoted instead of the option prices.
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Applied Quantitative Methods for Trading and Investment
The credibility of exchange rate bands can be analysed through a more rigorous method,building indicators based on the constraints on option prices implicit in the perfect credibility assumption (see Malz (1996), Campa and Chang (1996) and Campa et al. (1997)).Consider a ﬂuctuation band with upper and lower bounds denoted by S and S, respectively. First, consider the extreme case of strike prices of a call-option at the upper boundof the band or above it (X ≥ S). Under perfect credibility the call-option is worthless,since it will never expire in-the-money. Conversely, if the strike price of a put-option isat the lower edge of the band or below it (X ≤ S), the put-option is also worthless, sincethe probability of expiring in-the-money is nil.It can be shown that the intrinsic value of a call-option – the maximum between zeroand the option price if immediately exercised – equals:29∗
C(X) = S exp (−iτf τ ) − X exp (−iτf τ )
(12.33)
For strike prices within the band, the maximum value of the call-option price isgiven by:C(X ≤ S) = (S − X) exp (−iτf τ )(12.34)This value is reached only when the exchange rate is expected to get to the band ceilingat the expiry date with certainty. This implies that the perfect credibility of the band isrejected whenever the current value of the call-option price, with strike price X ≤ S,exceeds the maximum value that the option price may take according to equation (12.34),assuming that the future exchange rate is within the band:C(X ≤ S) > (S − X) exp (−iτf τ )
(12.35)
A tighter constraint on option prices in a fully credible exchange rate band can bedeﬁned from the convexity relationship between the option price and the strike price. Theargument can be exposed in two steps. First, each unitary increase in the strike price yieldsa maximum reduction of the value of the call options expiring in-the-money that equals thecurrent value of that unit (when the probability of expiring in-the-money is 1).30 Second,the higher the strike price, the less probable it is that the call-option expires in-the-money,and hence the smaller the reduction in the call-option price for each unitary increase infthe strike price.31 Thus we have − exp (−iτ T ) ≤ ∂C(X)/∂X ≤ 0 and ∂ 2 C(X)/∂X2 ≥ 0.Figure 12.7 illustrates the constraints on the behaviour of options prices in a perfectlycredible exchange rate band. Straight line (1) represents the line of maximum slopethat passes through C(S), when the unit rise in the strike price cuts the option priceby the present value of that unit. This slope is the symmetric of the discount factorf− exp (−iτ τ ).32 Straight line (2) joins the prices of the options with strike prices S andf∗fS, being C(S) and C(S) respectively equal to S exp (−iτ τ ) − S exp (−it τ ) and zero.3329
Using compounded interest in continuous time. Campa and Chang (1996) consider discrete compounding.This happens when the absolute value of the slope of the call-option price function is one, i.e. the cumulativeprobability is zero.31The minimum reduction is obtained when C(X) is horizontal, i.e. when the slope of the curve is zero andthe cumulative probability is one.32The straight line crosses the point F , since its slope equals −C(S)/x − S, with x denoting the horizontalfintercept. Therefore, equalling that ratio to the value of the slope of line − exp (−iτ τ ) yields x = F .33The value of C(S) is found by substituting X by S in equation (12.33).30
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Optionprice
C (S )
(2)(1)
f
Slope = − exp(−it t)
S
F
Strikeprice
S
Figure 12.7 Option prices in a perfectly credible band (convexity test)
Since the relationship between the option price and the strike price is strictly convex,if the exchange rate band is fully credible, the call-option price should stand in a convexcurve between (1) and (2), that contains points (S, C(S)) and (S, 0) for any strike priceranging between S and S. Therefore, the perfect credibility of an exchange rate bandcan always be rejected provided that a given point in the call-option price function (forS ≤ X ≤ S) is higher than the corresponding point in line (2). This being the case, giventhat the call-option price function is strictly convex, the price of a call-option with strikeprice greater than S is positive, which means that the probability the exchange rate exceedsthe band upper bound is not zero.It can be shown that the assessment of the rejection of perfect credibility correspondsto checking whether the following convexity condition holds:34C(X) >
S−XS−S
∗
[S exp (−iτf τ ) − S exp (−iτf τ )]
(12.36)
As mentioned in Campa and Chang (1996), condition (12.36) is more restrictive thancondition (12.35) whenever the forward rate is within the band.35 In the case of wide3435
See Campa and Chang (1996).fMultiplying and dividing the right-hand term of equation (12.36) by exp (iτ τ ) yields:
C(X) ≥
S−XS−S
∗
exp (iτf τ ) exp (−iτf τ )[S exp (−iτf τ ) − S exp (−iτf τ )] ⇔ C(X) ≥
⇔ C(X) ≥ [(S − X) exp (−iτf τ )]

F −S

S−XS−S
exp (−iτf τ )(F − S)
S−S
The ﬁrst factor of the right-hand term in this inequality is the right-hand term of inequality (12.35), so (12.36)is a more restrictive condition when the second factor is smaller than 1 (i.e., when S ≤ F ≤ S), as in this casethe full credibility will be rejected from a lower value.
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Applied Quantitative Methods for Trading and Investment
exchange rate bands – as happened with the ERM of the EMS before the launch of theeuro – the results should be expected to point in general to the non-rejection of perfectcredibility, so the exercise provides no additional relevant information. However, moreinteresting exercises can be attempted in these cases – e.g. using equation (12.36) tocalculate the smaller possible bandwidth that would have permitted the non-rejection ofits perfect credibility – i.e., the minimum size of a perfectly credible band.Bearing in mind the process of monetary integration in the EU, this exercise allows usto use option prices to identify the band in which a currency could have ﬂoated withoutimposing a realignment within the term to maturity of the option. With the beginning ofthe EMU becoming closer, the currency would then be expected to ﬂoat in a progressivelynarrower interval.Transforming (12.36) into an identity and using the principle of a band’s symmetryaround the central parity yields S = Sc /(1 + α) and S = (1 + α)Sc (where Sc is the centralparity). Substituting S and S for these expressions, the equation resulting from (12.36)can be solved in order to α, given Sc :36∗
α = {[−SX exp (−iτf τ ) − Sc2 exp (−iτf τ ) −
√
∗
{[SX exp (−iτf τ ) + Sc2 exp (−iτf τ )]2
∗
− 4Sc2 [C(X) − S exp (−iτf τ )][−C(X) − X exp (−iτf τ )]}]/2Sc [C(X)∗
− S exp (−iτf τ )]} − 1
(12.37)
One can also build a measure of realignment intensity (G(T )) between t and T . Intuitively, this measure corresponds to the weighted average of the size of the possiblerealignments when the exchange rate overcomes the band ceiling:37G(T ) =
 ∞
(ST − S)q(ST )dST
(12.38)
S
Given that the upper ceiling of the band after a realignment may be higher than thespot rate, this must be considered as a measure of the minimum intensity of realignment.Comparing (12.38) to the call-option deﬁnition in (12.1) and making X = S, the intensityfof realignment is equivalent to exp (iτ τ )C(S), i.e., the capitalised value at the maturitydate of the call-option with strike corresponding to the upper bound of the exchange rateband.38Even when there are no options for the strike price S, it is possible to compute the minimum value of a theoretical option for that strike price, based on the convexity relationshipbetween option prices and strike prices. A lower bound for the intensity of realignmentwill then be computed according to:39C(S) ≥ C(X) −36
S−X∗[S exp (−iτf τ ) − S exp (−iτf τ ) − C(X)]X−S
(12.39)
This corresponds to the lowest value of the solutions to the equation resulting from (12.36).Therefore, this measure does not consider all possible realignments, as these may occur even when the spotrate is inside the band.38Notice that, in the interval [S, +∞], the maximum of the function is ST − X.39See Campa and Chang (1996).37
Portfolio Management and Information
365f∗
f
For at-the-money forward options (when X = S exp [(iτ − iτ )τ ] = F ), it can beshown that the intensity of realignment will be given as:40G(T ) ≥ C(X) exp (iτf τ ) ·
S−S+F −SF −S
(12.40)
12.6 EMPIRICAL APPLICATIONS12.6.1 The estimation of RND functions using option spreads: an application tothe US dollar/euro exchange rate
Oct-00
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According to the evolution of the EUR/USD spot exchange rate and risk-reversals(Figure 12.8), ﬁve main periods may be identiﬁed. The ﬁrst one is between the eurolaunch and mid-July 1999, during which the euro depreciated vis-à-vis the USD fromaround 1.17 to 1.02 and the risk-reversals got closer to zero. The second period, untilmid-October, was marked by the euro appreciation, up to around 1.08, with the riskreversals becoming positive, i.e. suggesting expectations of further appreciation of theeuro vis-á-vis the USD. During the third period, the risk-reversals returned to negativevalues with the depreciation of the euro until mid-May 2000, achieving a level around0.895. In the fourth period, the euro recovered to a level around 0.95 at the end of June2000. Lastly, the European currency started a downward movement until mid-October,only interrupted after the joint exchange rate intervention on 22 September, held by theECB, the Federal Reserve, the Bank of Japan and the Bank of England. This interventionmoved the risk-reversals sharply upwards, again to positive values.
2.0
1.201.151.101.051.000.950.900.850.80
1.51.00.50.0−0.5
EUR / USD
risk-reversals
Figure 12.8 Risk-reversals for δ = 0.2540
See also Campa and Chang (1996).
Oct-00
Sep-00
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−1.0
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Applied Quantitative Methods for Trading and Investment
Sub Full Estimation()’’ Estimated Vols Macro’ Macro recorded on 7-12-2002 by Jorge Barros Luı́s’Row = InputBox(Prompt:="First row for the estimation (e.g. 2)",Title:="1st Estimation Row")Row2 = InputBox(Prompt:="Last row for the estimation (e.g. 3)",Title:="Last Estimation Row")NoRows = Row2 - RowSheets("Strike Call").SelectLastRow = Range("a1").End(xlToRight).AddressRange("a1", LastRow).SelectFor m = 0 To NoRowsSheets("Strike Call").SelectRange("a1").Select’Estimation of the strikes corresponding to the deltasFor n = 1 To 19SolverResetSolverAdd CellRef:=Range(Cells(Row + m, 23 + n - 1), Cells(Row +m, 23 + n - 1)), Relation:=1, FormulaText:=Range(Cells(Row + m, 23+ n - 2), Cells(Row + m, 23 + n - 2)) - 0.001SolverAdd CellRef:=Range(Cells(Row + m, 22 + n), Cells(Row + m, 22+ n)), Relation:=3, FormulaText:="0"Range(Cells(Row + m, 1 + n), Cells(Row + m, 1 + n)).SelectSolverOk SetCell:=Range(Cells(Row + m, 1 + n), Cells(Row + m, 1 +n)), MaxMinVal:=2, ByChange:=Range(Cells(Row + m, 22 + n),Cells(Row + m, 22 + n))SolverSolve TrueNext n"For n = 21 To NoColumns = 1SolverResetSolverAdd CellRef:=Range(Cells(Row + m, 23 + n - 1), Cells(Row + m, 23+ n - 1)),Relation:=1, FormulaText:=Range(Cells(Row + m, 23 + n - 2), Cells(Row +m, 23 + n - 2)) - 0.001SolverAdd CellRef:=Range(Cells(Row + m, 22 + n), Cells(Row + m, 22 +n)),Relation:=3, FormulaText:="0"Range(Cells(Row + m, 1 + n), Cells(Row + m, 1 + n)).SelectSolverOk SetCell:=Range(Cells(Row + m, 1 + n), Cells(Row + m, 1 + n)),MaxMinVal:=2, ByChange:=Range(Cells(Row + m, 22 + n),Cells(Row + m, 22 + n))SolverSolve TrueNext n’’ActiveWindow.LargeScroll ToRight:=-2’Copy strikes to new sheetRange(Cells(Row + m, 23), Cells(Row + m, 41)).SelectSelection.Copy
Figure 12.9
Visual Basic macro: the estimation of strike prices
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Sheets("Sheet3").SelectRange("A1").SelectSelection.PasteSpecial Paste:=xlAll, Operation:=xlNone,SkipBlanks:=False, Transpose:=True’Copy call option prices to new sheetSheets("Call").SelectActiveWindow.LargeScroll ToRight:=-2Range(Cells(Row + m, 2), Cells(Row + m, 20)).SelectApplication.CutCopyMode = FalseSelection.CopySheets("Sheet3").SelectRange("B1").SelectSelection.PasteSpecial Paste:=xlValues, Operation:=xlNone, SkipBlanks:=False, Transpose:=True’Ordering by strikeColumns("A:B").SelectApplication.CutCopyMode = FalseSelection.Sort Key1:=Range("A1"), Order1:=xlAscending, Header:=xlNo,OrderCustom:=1, MatchCase:=False, Orientation:=xlTopToBottom’Copy of the ordered strikesRange("A1:A19").SelectSelection.CopySheets("Strikes").SelectRange(Cells(Row + m, 2), Cells(Row + m, 2)).SelectSelection.PasteSpecial Paste:=xlAll, Operation:=xlNone,SkipBlanks:=False, Transpose:=True’Copy of the ordered call-option pricesSheets("Sheet3").SelectRange("B1:B19").SelectApplication.CutCopyMode = FalseSelection.CopySheets("Strikes").SelectRange(Cells(Row + m, 23), Cells(Row + m, 23)).SelectSelection.PasteSpecial Paste:=xlAll, Operation:=xlNone,SkipBlanks:=False, Transpose:=True
Figure 12.9
(continued )
The estimations of the RND functions assuming a linear combination of two log-normaldistributions were performed in the ﬁles “OTC EUR USD.xls”, using a Visual Basicmacro (“Full Estimation”).41 Figure 12.9 presents the block of the macro concerning theestimation of strike prices.The macro starts by asking the user to insert the number of the ﬁrst and last row tobe estimated. After computing the number of rows and positioning in the ﬁrst cell ofthe sheet “Strike Call”, the strike prices corresponding to the option deltas are estimated.41This macro is run directly from the Excel menu and only demands the user to copy previously the ﬁrst rowin each sheet to the rows containing new data to be used for estimation.
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Applied Quantitative Methods for Trading and Investment
Volatility (% / year)
14.013.5
Observed
13.0
Estimated
12.512.011.511.010.510.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Delta
Figure 12.10
Observed and estimated 1-month volatilities at 13/01/99
This step follows equations (12.3) and (12.10) and is carried out for all the call-optiondeltas between 0.05 and 0.95, with an interval equal to 0.05.Figure 12.10 illustrates the results obtained in this step for the ﬁrst sample day, withthe larger markers representing the three volatility–delta pairs computed from the riskreversal, the strangle and the at-the-money volatility ﬁgures, with the remaining markersbeing those resulting from the curve in equation (12.10).42 Afterwards, the volatility–deltafunction is transformed into a volatility–strike (smile) curve, by estimating the strikes thatminimise the squared differences to the observed volatilities. With the volatility smile andusing the pricing formula in (12.1), call-option prices are obtained.The observed values in Figure 12.10 are computed in the “Delta Vol” sheet, while theestimated values are obtained in the “Strike Call” sheet. In this sheet, the ﬁrst set ofcolumns (up to column T) contains the squared difference between the observed volatility(in order to the delta) and the estimated volatility (in order to the strike price). Theestimated strike prices are in the second set of columns of the same sheet (betweencolumns V and AO). The third set of columns in the same sheet (between AQ and BJ)contains the estimated volatility, which results from equation (12.10), being the call-optiondelta obtained from inserting in equation (12.3) the estimated strike prices in the secondset of columns in the sheet “Strike Call”. Lastly, the sum of the squared residuals ispresent