Correcting the Hedge of Brazilian Currency Options Jo˜ao Amaro de Matos ∗

Jorge C. Kapotas †

Pedro Paulo Schirmer ‡

Final Draft Sept. 6, 2004

Abstract We consider the impact on the pricing and hedging of brazilian currency options of an average procedure done by the Brazilian monetary authorities. Currency options in Brazil are denominated in terms of the averaged PTAX800 U.S.Dollar/Brazilian Real exchange rate. Practitioners typically use Garman-Kohlhagen model to hedge their positions incurring in errors. The computations suggest that brazilian PTAX800 options are some kind of hybrid instruments, behaving like a plain vanilla option for long maturities and as a particular type of asian option as maturity approaches. As a result of the wrong hedging, these contracts become illiquid close to maturity. The asian-like model described here corrects for this misbehavior, allowing for liquid contracts.

∗

Faculdade de Economia da Universidade Nova de Lisboa Rua Marquˆes de Fronteira 20, 1099-038, Lisboa, Portugal. † Octaplus Financial Analytics. Rua Arandu 205 -cj1003, CEP 04562-030, Brooklin, S˜ao Paulo, SP, Brazil ‡ Instituto de Matem´atica e Estat´ıstica da Universidade de S˜ao Paulo, Rua do Mat˜ao 1010, Cidade Universit´aria, CEP 05508-090, Butant˜a, S˜ao Paulo, SP, Brazil.

1

1 Introduction The purpose of this work is to study a particular class of asian-like currency options traded in the Brazilian market, where the averaging of the underlying is performed solely on the last day before expiration. Such contracts appear naturally due to local regulatory constraints aiming to protect the Brazilian economy against currency speculation. In Brazil, currency contracts are denominated in terms of the official PTAX800 U.S.Dollar/Brazilian Real exchange rate, an average value computed internally by Brazil’s Central Bank1 and disclosed daily at late market hours. The exact averaging procedure is not revealed by the central bank authorities. As a matter of fact, this is done intentionally to avoid price fixing schemes by market agents. The payoff of Brazilian FX options are based on the PTAX800 rate at maturity, reflecting the average exchange rate of the day before.2 Because average is done only in the last day, practitioners try to hedge such contracts as plain vanilla during the whole life of the option. In fact, they hedge the PTAX800 options using the standard Garman-Kohlhagen formula ([GK]), assuming that the FX rate follows a Geometric Brownian motion. However, due to the subtle asian nature of these contracts, such hedging becomes inadequate as maturity approaches, leading to market illiquidity. In order to correct for this problem, market players started to specify common hedging strategies, recovering partially some of the market liquidity close to maturity.3 This common hedging strategies, however are also Garman-Kohlhagen-based, and do not correctly address the fundamental modelling issues underlying the problem. Under the very same original assumptions underlying the asset dynamics, we explain how hedging should be done in order to keep markets liquid up to maturity. In particular, we show in this paper that, under the Geometric Brownian motion assumption, such common hedging strategies lead to significant mispricings as maturity approaches. In fact, our computations suggest that Brazilian PTAX800 options are some kind of hybrid instrument behaving like a plain vanilla option at the time they are issued and then as an asian option 4 as maturity approaches. This feature has a significant impact in the pricing of these options, which we measure as a function of time to expiration. Brazilian currency options are settled on the next business day after the release of the PTAX rate. The value of the contract is known one day before its effective settlement. 1

The methodology adopted for the calculation of this average daily rate can be found in the brazilian central bank market communique number 8507 of 06/04/2001. 2 The specific terms and conditions of brazilian dollar exchange traded options can be viewed in the Bolsa de Mercadorias e Futuros website: www.bmf.com.br/contratos1 3 This practice is described in terms of the VTC contracts of the Bolsa de Mercadorias e Futuros. 4 There are various works characterizing asian options, e.g. [L, KV, TW]. See [H] and references therein.

2

This paper is organized as follows. In the next section we model Brazilian Currency options (PTAX options) by considering the unweighted geometric average of continuously quoted FX rates in the last day before maturity, leading to a “last day average” (LDA) asian option. In section three we cover pricing of such contracts, comparing with the GarmanKohlhagen model. Section four compares the hedging results of both models. Section five provides a simple numerical example illustrating monetary losses due to incorrect hedging, followed by a conclusion.

2

Last Day Average Options Model (LDA)

We model our economy in continuous time t ∈ [0, +∞) and consider a log-normal model for the dynamics of the USD/BRL exchange rate St in a risk-adjusted world as follows: dSt = (r − ri )dt + σdWt . St

(2.1)

Here r is the domestic (BRL) risk-free interest rate, ri is the foreign (USD) risk-free interest rate, σ is the volatility of the spot FX rate and {W t }t>0 is the standard Wiener process relative to the risk-adjusted measure. We assume for the purposes of this paper that all rates and volatilities are constant. The process solution is: St = S0 e(r−ri −

σ2 )t+σWt 2

(2.2)

To simulate the averaging procedure done by the monetary authorities we introduce a high-frequency time scale δ > 0 representing, in this case, the period of one business day. Here T − t represents the time to expiration of the option at time t, and can be written as a multiple of n business days, T − t = δn. Although the real average procedure used by the monetary authorities is not made available to the public, we take advantage of the fact that the averaging time is a single day. In fact, as the averaging time tends to zero, any two averages become arbitrarily close. For this reason, we use a geometric average as a proxy for the PTAX and follow Kemna and Vorst ( cf. [KV]) to get closed formulas to solve our pricing problem. We introduce a proxy for the PTAX800 FX rate, Pt , valid at time t ≥ δ according to the following formula: 1 Pt = exp( δ




[ δt ]δ ([ δt ]−1)δ

3

log(Sτ )dτ )

(2.3)

This means that Pt performs a geometric average of the foreign exchange rate during the period of one business day5 . Let us compute this quantity for the log-normal model and for some time t in the n-th day:

Pt

1 = exp( δ = = = =




[ δt ]δ

log(Sτ )dτ )

([ δt ]−1)δ


1 nδ log(Sτ )dτ ) exp( δ (n−1)δ  
nδ 1 σ2 log S0 + (r − ri − )τ + σWτ dτ exp δ (n−1)δ 2  
σ 2 τ 2 nδ σ nδ + Wτ dτ exp log S0 + (r − ri − )  2 2δ (n−1)δ δ (n−1)δ  
σ2 δ σ nδ Wτ dτ S0 exp (r − ri − ).(nδ − ) + 2 2 δ (n−1)δ

(2.4)

One can now compute the price CLDA (K, T ) of an european option with srike K and effective settlement in T = nδ business days, with n ≥ 2 , using the risk-adjusted expectations: CLDA (K, T ) = e−rT E[(PT −δ − K)+ ]

(2.5)

Note that at any time before markets close on the settlement day, that is to say, for t ∈ [(n − 1)δ, nδ], the value of the PTAX is already known and given by P T −δ = P(n−1)δ . Inserting the expression (2.4) with t = (n − 1)δ into the formula (2.5), we compute the expectation as follows: CLDA (K, T ) = e−rT E[(P(n−1)δ − K)+ ] −rT

= e

2

(r−ri − σ2 ).((n−1)δ− δ2 )

E[(S0 e

σ . exp( δ




(n−1)δ

(n−2)δ

Wτ dτ ) − K)+ ]

The integral inside the expectation operator, which arises due to the geometric nature of the average, can be handled in an straightforward way, using the fact thatthe space of gaussian T variables is closed under addition. Let X t,T denote the random variable t Wτ dτ . We already 5

The expression [x] denotes the integer part of the real number x.

4

know that Xt,T is a gaussian variable of mean zero. We show now that the variance of Xt,T is given by: T 3 2t3 + − t2 T σ 2 [Xt,T ] = (2.6) 3 3 Indeed, using the well-known fact that E[Ws Wu ] = min(s, u), we have:


2

T




2

T




T

Wu du) ] = E[( Wu du).( Ws ds)] σ [Xt,T ] = E[( t t t
T
T
T
T = E[ Wu Ws duds] = E[Wu Ws ]duds t t t t
T
T = min(u, s)duds t t
u
T
T du[ min(u, s)ds + min(u, s)ds] = t t u
T
u
T = du[ sds + uds] t t u
T u2 t2 = du[− + uT − ]du 2 2 t 3 3 T 2t = + − t2 T 3 3

(2.7)

Applying this expression with t = (n−2)δ and T = (n−1)δ , we find out that the random variable X(n−2)δ,(n−1)δ is a gaussian variable with mean zero and standard deviation equal to  δ( 3n−5 ). Therefore: 3

δ

−rT




CLDA (K, T ) = e

+∞

−∞

√ √ 3n−5 2 (r−ri − σ2 ).(T − 3δ )+σ T x 2 3n

(S0 e

x2

e− 2 − K) √ dx 2π +

(2.8) Now, calling ψ(n) = the constants d∗2 (n) by:



3n−5 3n

the adjustment factor of the volatility 6 and defining as usual

Notice that ψ(n) → 1 as n → +∞ showing that asymptotically, the effect of the central bank averaging is mathematically negligible. 6

5

2

d∗2 (n)

log(S0 /K) + (r − ri − σ2 ).(T − √ = σ T ψ(n)

3δ ) 2

(2.9)

√ and d∗1 (n) = d∗2 (n) + σ T ψ(n), we obtain the pricing formula of our LDA model: CLDA (K, T ) = e−rT [S0 e(r−ri )T +

σ2 T 2

2

(ψ(n)2 −1)− 3δ (r−ri − σ2 ) 2

N(d∗1 (n)) − KN(d∗2 (n))]

which can be simply written as: CLDA (K, T ) = S0 F (n)e−ri T N(d∗1 (n)) − Ke−rT N(d∗2 (n))

(2.10)

where F (n) is the correction factor which accounts for the impact of the δ- average on a period of n days:   σ2T 3δ 3δ 2 (ψ(n) − 1 + ) . F (n) = exp − (r − ri ) + 2 2 2T Rewriting the above formula with δ = T /n we obtain the equivalent expression for the correction factor:   3T σ2T 3 2 F (n) = exp − (r − ri ) + (ψ(n) − 1 + ) (2.11) 2n 2 2n Recall that, holding T = nδ fixed in the limit δ → 0 and n → +∞, formula (2.10) reduces to the usual Garman-Kohlhagen formula7 for pricing currency options in the lognormal setting: CGK (K, T ) = S0 e−ri t N(d1 ) − Ke−rt N(d2 ), √ log(S0 /K)+(r−ri − 21 σ2 )t √ where d2 = and d1 = d2 + σ t. σ t In the next sections we shall demonstrate numerically the impact of the correction factor F (n) over options premia and their hedging parameters.

3

Numerical Examples and the LDA Option Premia

In this part we will numerically demostrate the effect of the averaging procedure when determining the daily PTAX rate, on european PTAX call options. On Table 1 we display both, GK and LDA european call option prices, for different strikes (ITM, ATM, OTM) and increasing maturities. 6

Table 1 GK and LDA European PTAX call options values with the following parameters: K = 2.50, r = 18%, ri = 5%, σ = 20% and δ = 1/252 (one business day). ITM ATM OTM Time to expiration S0 = 2.55 S0 = 2.50 S0 = 2.45 GK LDA GK LDA GK LDA 2 days 0.0552 0.0506 0.0191 0.0076 0.0032 0.0000 3 days 0.0586 0.0531 0.0237 0.0155 0.0059 0.0015 4 days 0.0619 0.0565 0.0277 0.0208 0.0086 0.0041 5 days 0.0649 0.0598 0.0314 0.0252 0.0113 0.0068 6 days 0.0679 0.0630 0.0347 0.0291 0.0138 0.0095 8 days 0.0736 0.0690 0.0408 0.0358 0.0187 0.0147 10 days 0.0789 0.0746 0.0463 0.0418 0.0234 0.0196 15 days 0.0909 0.0871 0.0585 0.0547 0.0341 0.0307 1 month 0.1038 0.1004 0.0714 0.0680 0.0457 0.0426 2 months 0.1422 0.1394 0.1092 0.1065 0.0809 0.0783 4 months 0.2049 0.2027 0.1706 0.1684 0.1394 0.0137 6 months 0.2590 0.2569 0.2236 0.2216 0.1906 0.1886 1 year 0.3964 0.3945 0.3589 0.3571 0.3229 0.3211 2 years 0.6111 0.6093 0.5720 0.5703 0.5337 0.5320 3 years 0.7735 0.7719 0.7345 0.7328 0.6958 0.6943 5 years 0.9879 0.9864 0.9507 0.9492 0.9137 0.9122 10 years 1.1367 1.1355 1.1066 1.1054 1.0765 1.0754

We observe, as our intuition would indicate, that long maturity PTAX options do not present large deviations from standard GK prices. On the other hand, short maturity european call options, do differ substantially from GK standard call prices. This clearly indicates that PTAX options are a sort of hybrid derivative, which start their life as a plain vanilla european call option at long maturities to gradually transform itself into an asian option as the life of the option decreases. This can be easily observed on Figure 1, when the GK and LDA european call option values are plotted as a function of time to expiration. In Table 2 and Figure 2, we present the percentual relative differences of two european PTAX call option values (GK & LDA), clearly showing that the differences increase as the 7

Garman-Kolhagen formula is equivalent to the Black & Scholes equity call option with dividends expression.

7

Figure 1 GK and LDA European PTAX call options curves, using the parameters of table 1 ATM. Call 0.2 0.175 0.15 0.125 0.1 0.075 0.05 0.025

GK BCO

20

40

60

T

80

option approaches expiration time.

Figure 2 Percentual absolute relative differences curves between the GK and LDA european PTAX call options prices using Table 2 entries. % ITM ATM OTM

40 30 20 10

20

40

60

80

T

The results shown in Table 2 and Figure 2 demonstrate that the use of a “plain vanilla” pricing framework to value PTAX options can generate a considerable pricing difference, specially when valuing short maturity european PTAX call options. On the other hand, long maturity options, which are very rare in the brazilian markets, could be reasonably priced even 8

Table 2 Percentual relative differences between the GK and LDA European PTAX call options values with the following scenario: K = 2.50, r = 18%, ri = 5%, σ = 20% and δ = 1/252 (one business day). Relative differences (%) Time to expiration ITM ATM OTM S0 = 2.55 S0 = 2.50 S0 = 2.45 2 days -8.30 -60.39 -99.48 3 days -9.34 -34.86 -74.25 4 days -8.74 -25.08 -52.48 5 days -8.01 -19.71 -39.39 6 days -7.34 -16.29 -31.11 8 days -6.27 -12.16 -21.55 10 days -5.48 -9.74 -16.31 15 days -4.20 -6.58 -10.00 1 month -3.32 -4.78 -6.78 2 months -1.99 -2.52 -3.19 4 months -1.17 -1.36 -1.59 6 months -0.85 -0.96 -1.08 1 year -0.50 -0.54 -0.58 2 years -0.30 -0.31 -0.32 3 years -0.22 -0.22 -0.23 5 years -0.15 -0.16 -0.16 10 years -0.10 -0.11 -0.11

without the last-day-average adjustment effect produced by our LDA model, as errors for one year and longer options are smaller than one percent.

4

Calculating Sensitivity Measures of LDA European Call Options

In this section we compute the effects of the last day averaging procedure on some of the partial derivatives8 (“Greeks”) of European LDA call options. Firstly, we will consider the effect of the average on the Delta. Using formula (2.10) and 8

A classic treatment of the “Greeks” can be found in [CR].

9

differentiating relative to S we obtain the LDA option Delta: ∆LDA =

∂CLDA ∂S

= F (n)e−ri T N(d∗1 (n)) + SF (n)e−ri T N  (d∗1 (n))

∂d∗1 ∂d∗ − Ke−rT N  (d∗2 (n)) 2 ∂S ∂S

1 [SF (n)e−ri T N  (d∗1 (n)) − Ke−rT N  (d∗2 (n))] Sσ T ψ(n) (4.12) √ Using now the fact that d∗1 (n) = d∗2 + σ T ψ , we find that: = F (n)e−ri T N(d∗1 (n)) +

√

2 2 (d∗ (d∗ 1 1 1) 2) . √ [SF (n)e−ri T e− 2 − Ke−rT e− 2 ] Sσ T ψ(n) 2π 1 1 √ .√ = F (n)e−ri T N(d∗1 ) + Sσ T ψ(n) 2π

∆LDA = F (n)e−ri T N(d∗1 ) +

[SF (n)e−ri T e−

√

2 2 2 2 (d∗ 2 ) − σ T ψ −log( S )−(r−r − σ )(T − 3T ) i 2 2 K 2 2n

2 (d∗ 1 1 2) √ . √ e− 2 = F (n)e−ri T N(d∗1 ) + Sσ T ψ(n) 2π 2 2 σ Tψ σ2 3T K [SF (n)e−ri T − 2 −(r−ri − 2 )(T − 2n ) . − Ke−rT ] S 1 √ = F (n)e−ri T N(d∗1 ) + .N  (d∗2 )K Sσ T ψ(n)

[F (n)e−ri T −

2 σ2 T ψ2 −(r−ri − σ2 )(T − 3T ) 2 2n

= F (n)e−ri T N(d∗1 ) + e−ri T −

2 (d∗ 2) 2

]

− e−rT ]

3T σ2 T 3 K 2 √ .N  (d∗2 )[e− 2n (r−ri )+ 2 (ψ(n) −1+ 2n ) . Sσ T ψ(n)

2 σ2 T ψ2 −(r−ri − σ2 )(T − 3T ) 2 2n

= F (n)e−ri T N(d∗1 ) +

− Ke−rT e−

− e−rT ]

K √ .N  (d∗2 )[e−rT − e−rT ] Sσ T ψ(n)

= F (n)e−ri T N(d∗1 ).

(4.13)

This way, the delta measure, ∆LDA , is simply given by a correction to the corresponding Garman-Kohlhagen ∆GK :   N(d∗1 ) ∆LDA = F (n) ∆GK N(d1 ) 10

We can now compute and compare the ∆GK and ∆LDA values of european PTAX call options for different strikes and maturities. We summarize these values in Table 3 and Figure 3.

Table 3 GK and LDA European PTAX Call Option Delta values for K = 2.50, r = 18%, ri = 5%, σ = 20% and δ = 1/252 (one business day). ITM ATM OTM Time to expiration S0 = 2.55 S0 = 2.50 S0 = 2.45 ∆GK ∆LDA ∆GK ∆LDA ∆GK ∆LDA 2 days 0.8803 0.9959 0.5264 0.5143 0.1429 0.0031 3 days 0.8382 0.9210 0.5323 0.5230 0.1992 0.0918 4 days 0.8100 0.8640 0.5372 0.5294 0.2395 0.1648 5 days 0.7898 0.8276 0.5415 0.5347 0.2702 0.2147 6 days 0.7747 0.8025 0.5454 0.5392 0.2946 0.2512 8 days 0.7535 0.7703 0.5523 0.5469 0.3312 0.3022 10 days 0.7395 0.7506 0.5583 0.5535 0.3590 0.3374 15 days 0.7199 0.7245 0.5709 0.5670 0.4074 0.3940 1 month 0.7091 0.7109 0.5833 0.5799 0.4451 0.4360 2 months 0.7025 0.7019 0.6151 0.6127 0.5191 0.5146 4 months 0.7152 0.7135 0.6565 0.6546 0.5919 0.5893 6 months 0.7302 0.7289 0.6847 0.6831 0.6347 0.6328 1 year 0.7629 0.7618 0.7356 0.7344 0.7056 0.7043 2 years 0.7880 0.7871 0.7742 0.7732 0.7589 0.7579 3 years 0.7854 0.7846 0.7772 0.7765 0.7683 0.7675 5 years 0.7456 0.7449 0.7424 0.7417 0.7388 0.7382 10 years 0.6016 0.6011 0.6012 0.6007 0.6007 0.6002

The results obtained in Table 3 and Figure 3 shown a behaviour analogous to the one observed for the option call values. As expected, the ∆LDA values approach unity faster than the ∆GK values for ITM options as T decreases. For OTM options the ∆ LDA values approach zero faster than their equivalent GK values. These differences are caused by the last day averaging effect that tends to decrease the volatility of underlying rate as we approach expiration. The second order price sensitivity measure of the PTAX call option, Γ LDA , is computed

11

Figure 3 Percentual absolute relative difference curves between the GK and LDA European PTAX call option delta values for Table 3 entries. % ITM ATM OTM 3 2 1

20

40

60

80

T

in a similar manner: ΓLDA

∂ 2 CLDA = ∂S 2 ∂∆LDA , = ∂S

Differentiating equation (4.13) with respect to S we arrive at: ∂d∗1 ∂S (d∗ )2 F (n) exp(−ri T − 12 ) √ . = 2πT Sσψ(n)

ΓLDA = F (n)e−ri T N  (d∗1 )

(4.14)

As before, this relation can be rewritten in terms of a correction factor to the GK gamma: ΓLDA =

F (n) 1 (d21 −(d∗1 )2 ) e2 ΓGK ψ(n)

Having calculated ΓLDA , we then perform the same analysis as the one done for the ∆LDA . The results are shown in Table 4 and Figure 4 below. The results in Table 4 and Figure 4 confirm our suspition that LDA deltas are in general more sensitive to price changes than plain vanilla call option deltas (GK deltas). Both gammas increase as T approaches expiration, but the rate of increase is much higher for the ATM 12

Table 4 GK and LDA European PTAX Call Option Gamma values for K = 2.50, r = 18%, ri = 5%, σ = 20% and δ = 1/252 (one business day). ITM ATM OTM Time to expiration S0 = 2.55 S0 = 2.50 S0 = 2.45 ΓGK ΓLDA ΓGK ΓLDA ΓGK ΓLDA 2 days 4.3843 0.4768 8.9327 21.8970 5.1699 0.5237 3 days 4.3923 3.9142 7.2839 10.9345 5.2230 4.6209 4 days 4.2108 4.4082 6.2998 8.2549 5.0282 5.2579 5 days 4.0009 4.3223 5.6273 6.8975 4.7898 5.1769 6 days 3.8008 4.1166 5.1303 6.0416 4.5578 4.9533 8 days 3.4564 3.7276 4.4313 4.9843 4.1535 4.4849 10 days 3.1801 3.3976 3.9530 4.3338 3.8262 4.0928 15 days 2.6879 2.8221 3.2065 3.4037 3.2395 3.4045 1 month 2.3075 2.3936 2.6887 2.8044 2.7838 2.8897 2 months 1.6346 1.6670 1.8494 1.8888 1.9743 2.0142 4 months 1.1099 1.1379 1.2375 1.2698 1.3415 1.3540 6 months 0.8618 0.8677 0.9561 0.9628 1.0418 1.0488 1 year 0.5189 0.5209 0.5729 0.5751 0.6274 0.6298 2 years 0.2641 0.2647 0.2909 0.2916 0.3194 0.3201 3 years 0.1549 0.1553 0.1705 0.1709 0.1874 0.1877 5 years 0.0619 0.0620 0.0681 0.0682 0.0749 0.0749 10 years 0.0083 0.00837 0.0092 0.0092 0.0101 0.0101

ΓLDA , making this option delta very sensitive to price changes close to maturity and therefore more costly to hedge. On the other hand, the hedge ratio sensitivity to price changes (Γ LDA ) decreases dramatically in the last two days before expiration, for the ITM and OTM options. This is due to the fact that near expiration, the deltas are practically constant.

5 Hedging Differences To ilustrate the hedging error that may be induced by a naive use of the GK formula in the Brazilian U.S.Dollar/Brazilian Real PTAX market, we will consider a simple delta hedging strategy9 for a short position in BRL/USD PTAX european call options. In Table 5 below 9

See [W] and [H] on hedging options and dynamic hedging strategies.

13

Figure 4 Percentual relative differences curve between the GK and LDA European PTAX call option Gamma values for Table 4 entries. % 80

ITM ATM OTM

60 40 20 20

40

60

80

T

-20 -40 -60 -80

we implement the hedge and compute the differences between the two models for a given scenario until expiration. Table 5 Example of a hedging error induced by a naive use of GK model. Considering an European PTAX Call option with notional value USD 50,000.00, K = 2.50, r = 18%, ri = 5%, σ = 20% and δ = 1/252. (Currencies in thousands) T = 15 T =8 T =1 St = 2.50 St = 2.47 ST = 2.45 CLDA = 0.0547 CLDA = 0.021680 CLDA = 0.0 ∆GK |∆LDA 0.5709 0.5670 0.4181 0.3967 – – USDGK | USDLDA 28,545 28,350 20,905 19,835 – – BRLGK | BRLLDA 71,362 70,875 51,635 48,992 – – MtM USD – 70,506 70,024 51,217 48,596 Differences (BRL) – 481.65 2,621.50 Total dif. (BRL) – – 3,103.15

In the scenario we created above, an additional total loss of BRL 3,103,150.00 was generated exclusively by the failure to adjust the hedge ratio, ∆ GK to the last-day-average effect of the underlying.

14

Conclusion In this work we have proposed a very simple Brazilian Currency Option Model (LDA) to analyse and measure the impact of the last-day-average effect on the pricing and hedging of local US$ dollar options (PTAX options). We compared the results obtained by our model to the ones generated by the commonly used Garman Kolhagen formula. The results indicate a significant difference in the european call prices and its hedging parameters (“Greeks”) for options maturities shorter than one year. Our calculations have shown that local PTAX option should behave as a hybrid derivative with plain vanilla and exotic (asian) components as its two main pricing factors. The asian component prevails very noticebly during the last days before expiration, whereas the plain vanilla component accounts overwhelmingly for the long maturity LDA option values, specially the ones longer than one year. The effect of hedging PTAX options without taking into account the important asian component of its pricing elements can generate misleading hedging ratios, specially near expiration date.

Acknowledgements The authors would like to thank Mariela Fern´andez for its valuable assistance in the preparation and organization of this material. All mistakes are, needless to say, ours.

References [CR]

J. C. Cox and M. Rubinstein, Options Markets. Englewood Cliffs, N.J.:Prentice Hall, 1985.

[GK]

M. B. Garman and S.W. Kohlhagen, Foreign Currency Option Values. Journal of International Money and Finance, 2, 231-37, December 1983.

[H]

J. C. Hull, Options, Futures and Other Derivatives, 5 th edition. Prentice Hall, 2003.

[KV]

A. Kemna, and A. Vorst, A pricing Method for Options Based on Average Asset Values. Journal of Banking and Finance, 14, 113-29, March 1990.

15

[L]

E. Levy, Pricing European Average Rate Currency Options. Journal of International Money and Finance, 11, 474-491, 1992.

[TW]

S.M. Turnbull and L.M. Wakeman, A quick algorithm for pricing European Average Options. Journal of Financial and Quantitative Analysis, 26, 377-389, 1991.

[W]

P. Wilmott, Derivatives: The Theory and Practice of Financial Engineering. John Wiley & Sons, 1998.

[email protected] [email protected] [email protected]

16