Uncertainty in pricing multi-currency options
Martin K. de Voogd1 Thesis MSc. Finance Supervisor: A.A. Tsvetkov, PhD January 2016, University of Groningen
ABSTRACT
In the OTC market of exotic multi-currency instruments there is no unique market price for a given instrument. Vanilla options on currencies contain too little information on correlations, and an instrument is priced differently by different models. This paper compares the price of best-of options, dual-digital options, and basket options between a SABR Monte Carlo procedure and a model using a risk neutral density. We find that under extreme market conditions the difference in price between models can take up half of the option price itself.
1 University of Groningen
1 Introduction
In order to value derivative contracts on multiple foreign exchange (FX) rates, one needs to model both the volatility smile of all rates, and the correlation between the rates. There is however no one unique market price, as the market for these exotic instruments is illiquid and not transparent. The market for vanilla options on currencies do not provide sufficient information on correlations, and as a consequence different model result in different prices for the same instruments. Although model uncertainty has been discussed in existing literature, there is little literature trying to determine the size of model uncertainty. The size of the uncertainty is of interest as it can erode traders' profits or leave the risks of derivative contracts partly unhedged. This paper attempts to model the uncertainty in the price of several classes of multi-currency options under different market situations. By generating markets using the SABR model, we can compare the price of an option between a SABR Monte Carlo procedure and a different model based on implied risk neutral density. The rest of the paper is organized as follows. Section 2 gives the methodology used tot determine the uncertainty. Section 3 describes the found results. Section 4 will conclude on the findings.
2 Methodology
The uncertainty in the price of the contracts is modeled using the following steps. Each step is then explained in detail hereafter. The first step is to generate a market for two currency pairs using a SABR process. The paths the arise from the SABR process are used to determine the prices of a range of vanilla call options in accordance with the Black Scholes model. Using the analytical Black Scholes model the implied volatilities are then reverted from the generated call prices.
basket options are calculated using both the SABR paths and the risk neutral density. The differences between these prices is the model and market uncertainty.
This method is repeated for a selected ranges of initial market parameters to asses how the differences between the two models are affected by these parameters. After assessing how the differences develop for isolated market parameters, we will run the setup for a large range of market parameters to asses how wide the differences between the models can grow. In the remainder of this section the various steps in the described process are further explained.
2.1 Generating the market
The SABR model is a stochastic volatility model developed by Hagan, Kumar, and Lesniewski (2002). They developed the model to improve the ability to manage smile risk, an ability local volatility models lacked at the time. The model is used in this paper for two reasons: first to generate markets for two currency pairs, and second to be used as one of the models to price the selected exotic options. The SABR model allows for generating paths that result in a volatility smile with skew, which we use to stress the differences in price between the models.
The original SABR stochastic differential equations as described in 2002 are 𝑑𝐹 = 𝜎𝐹%𝑑𝑊
'
𝑑𝜎 = 𝜈𝜎𝑑𝑊*
1
Here 𝐹 is the currency’s forward rate, and the two Brownian motions 𝑑𝑊' and 𝑑𝑊* are correlated
𝑑𝑊'𝑑𝑊* = 𝜌𝑑𝑡 2
The initial conditions are
𝐹 0 = 𝑓 𝜎 0 = 𝛼
3
near the forward rate, however we require a method that is accurate for wide strike ranges. We will therefore generate paths using the stochastic differential equation and use Monte Carlo to calculate vanilla option prices, from which we can determine the implied volatility. Because we want to model exotics options on two currencies we need to adjust the basic SABR model in order two to generate two spot rates. We will only consider the 𝛽 = 1 case, which simplifies the SDE. First the differential equations in Eq. 2 are rewritten from the forward rate to the spot rate for one currency
𝑑𝑆 = 𝑟4− 𝑟6 𝑆𝑑𝑡 + 𝜎𝑑𝑊'𝑆 𝑑𝜎 = 𝜈𝜎𝑑𝑊*
4
Next we consider the SABR model for two currencies. This results in four stochastic differential equations in the following order
𝑑𝜎' = 𝜎'𝜈'𝑑𝑊' 𝑑𝜎* = 𝜎*𝜈*𝑑𝑊* 𝑑𝑆' = 𝑟4− 𝑟6' 𝑆 '𝑑𝑡 + 𝜎'𝑆'𝑑𝑊8 𝑑𝑆* = 𝑟4− 𝑟6* 𝑆 *𝑑𝑡 + 𝜎*𝑆*𝑑𝑊9 5
We switch to the logarithm function for the spot rate and the volatility, because they may not become negative in the Monte-Carlo simulation. Using Ito’s lemma, we get the four stochastic differential equations for the logarithm of both spot rates and both volatilities
𝑑ln 𝜎' = − 𝜈'* 2 + 𝜈'𝑑𝑊' 𝑑ln 𝜎* = −𝜈** 2 + 𝜈*𝑑𝑊* 𝑑ln 𝑆' = (𝑟4 − 𝑟6'−𝜎'* 2)𝑑𝑡 + 𝜎'𝑑𝑊8 𝑑ln 𝑆* = (𝑟4 − 𝑟6* −𝜎** 2)𝑑𝑡 + 𝜎*𝑑𝑊9 6
where 𝑆? is the spot rate, and the Brownian motions are correlated with the correlation matrix
Using a Monte Carlo (MC) simulation we generate 50.000 independent paths with 365 steps to simulate 𝑆', 𝑆*, σ', and σ*. Each set of four independent paths is multiplied with the Cholesky decomposition of the correlation matrix to generated four correlated paths. Using antithetic
sampling for variance reduction we end up with 100.000 paths. We use an Euler scheme to generate the paths for S' 𝑇 and S* 𝑇 .
The Black Scholes price of a vanilla call option on a foreign exchange rate is given by (Black and Scholes, 1973) 𝑐 = 𝑆I𝑒KLMN𝑁 𝑑' − 𝐾𝑒KLQN𝑁 𝑑* 8 where 𝑑' =ln 𝑆I 𝐾 + 𝑟4 − 𝑟6+ 𝜎* 𝑇 𝜎 𝑇 𝑑* = 𝑑'− 𝜎 𝑇 9
The market price of a vanilla call option is determined using the generated paths for the currencies 𝑆', 𝑆*, and for the cross currency 𝑆8. This is done using the following method
𝐶' 𝑆', 𝐾' = 𝑒KLQN 1 𝑁 𝑆'T− 𝐾' U V TW' 𝐶* 𝑆*, 𝐾* = 𝑒KLQN 1 𝑁 𝑆*T− 𝐾* U V TW' 𝐶8 𝑆'/𝑆*, 𝐾' = 𝑒KLQN 1 𝑆* 0 ⋅ 𝑁 (𝑆'T− 𝐾8⋅ 𝑆*T)U V TW' 10
Given these market prices of the call options, we can use the Black Scholes formula to back out the implied volatilities from the market prices by letting an objective function 𝑓 be
and applying a root finding method on it. The moment 𝑓changes sign is when the Black Scholes formula returns the market price. The input volatility in the objective function is the implied volatility for the given strike. The calculation of a risk neutral density, as discussed in the next section, requires the volatility function to be smooth. We therefore fit a second-degree polynomial to smoothen the volatility function. We fit the polynomial to the volatilities in delta space, as this prevents that any call price calculated using the interpolated smile is directly arbitragable. Delta is defined as the first partial derivative of the Black Scholes call price with regard to the spot rate, which is
𝛥?,_`aa(𝐾?) = 𝑒KLQN𝑁(𝑑1) 12
We now have a smooth function for the volatility for each of the currencies 𝜎? 𝐾? = 𝜎?∗(𝛥
?,_`aa 𝐾? ) 13
where 𝜎?∗ is the fitted smoothing function.
2.2 Calculating the risk neutral density
Austing (2011) proposes a analytic join density (PDF) to value contracts with multiple currencies. We have chosen this method over other multi-assets methods, because it correctly reprices vanilla contracts on the drivers of the PDF. Other methods depend on a correlation parameter between the two spot rates, which is unlikely to correctly reprice a vanilla option on the cross currency rate 𝑆'/𝑆*. Other authors have tackled this issue, for example by using copulas that are also fitted to the
cross rate (Bennett and Kennedy, 2003; Salmon and Schleicher, 2009). The PDF method is
appealing as it can also be used to analytically value certain multi-currency contracts. Although not applied in this paper, the analytical value of certain contracts could be used to determine the
At the base of the PDF calculation is the value of a best-of two currencies option. The payoff function of this option is defined as
𝑃(𝑆?, 𝑆T, 𝐾?, 𝐾T) = 𝑚𝑎𝑥((𝑆?(𝑇)/𝐾? − 1)U, (𝑆
T(𝑇)/𝐾T− 1)U) 14
The density is constructed using the undiscounted value function of best-of contract under the Black Scholes model
𝐵(𝐾', 𝐾*, 𝜎', 𝜎*, 𝜎8) = 𝐹'
𝐾'𝑁(𝑑'U, 𝑑8U, 𝜌'8) + 𝐹*
𝐾*𝑁(𝑑*U, 𝑑8K, 𝜌*8) + 𝑁(𝑑'K, 𝑑*K, 𝜌'*)
15
where N is the bivariate normal cumulative probability function and di is given by
𝑑?± =𝑙𝑜𝑔(𝐹?/𝐾?) ± 𝜎?
*𝑇/2
𝜎? 𝑇 , ∀𝑖 = 1,2,3
16
with the correlations defined by
𝜌'*= 𝜎'*+ 𝜎**− 𝜎8* 2𝜎'𝜎*
17
Using the smooth volatility functions, as generated in the previous section the PDF can be computed by the following formula
𝜙(𝐾', 𝐾*) = ∂* ∂𝐾'∂𝐾*[𝐾' ∂ ∂𝐾'+ 𝐾* ∂ ∂𝐾*+ 1]𝐵(𝐾', 𝐾*) 18
where the best-of function is rewritten to incorporate the implied volatility functions and create a density that depends only on the strikes
(𝐾', 𝐾*) = 𝐵(𝐾', 𝐾*, 𝜎'(𝐾'), 𝜎*(𝐾*), 𝜎8(𝐾'/𝐾*)) 19
𝐵
2.3 Valuing multi-FX options
To determine how different payoff functions suffer from model uncertainty, the prices of a best-of call option, a dual digital call option, and a basket call option are computed. We have selected these three option types for their different payoff structures.
The best-of call option gives one the right, but not the opportunity to buy a call option on one of two currencies. This allows an investor to profit from the best performing option, without the costs of buying all the vanilla call options. The payoff of a best-of option is given in Eq. 14.
The dual-digital call option only has a payout when both underlying currencies are above their respective strike prices. Investors who feel strongly about a a rise in two currencies at the same time can buy a dual-digital call option at a large discount compared to buying two vanilla options. The payoff of dual digital call option is
𝑃 𝑆?, 𝑆T, 𝐾?, 𝐾T = 1 𝑖𝑓 𝑆? 𝑇 > 𝐾? 𝐴𝑁𝐷 𝑆T 𝑇 > 𝐾T
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
20
The basket call option is a portfolio of call option on a portfolio of assets, in our case two
currencies. The basket call option we consider has only one strike 𝐾, which is the ratio the portfolio must cross for a payout. Investors can buy a basket option if they need to hedge their exposure to multiple currencies at the same time. Due to the correlation between the currencies, buying a basket is much cheaper than buying individual calls for all exposures. We selected a basket option with equal weights on two currencies. Its payout function is defined as
𝑃 𝑆?, 𝑆T, 𝐾 = 0.5 ⋅𝑆? 𝑇
𝑆? 0 + 0.5 ⋅ 𝑆T 𝑇 𝑆T 0 − 1
U 21
𝑝\•€ 𝐾?, 𝐾T = 𝑒KLQN 𝑃 S•, S‚, 𝐾?, 𝐾T ⋅ 𝜙?T(𝑆?, 𝑆T)𝑑𝑆?𝑑𝑆T ƒ
I
3 Results
The initial conditions of the model will be set without volatility of volatility (𝜈? = 0) , which is simply the Black Scholes model with constant volatility. In the remainder of this paper we will refer to the different spot rates as currency pairs; 𝑆' = 𝐸𝑈𝑅/𝑈𝑆𝐷, 𝑆* = 𝐽𝑃𝑌/𝑈𝑆𝐷, and 𝑆8 = 𝐸𝑈𝑅/𝐽𝑃𝑌. After the constant volatility market, the model is set to approximately match the smiles of the currency triangle USD-EUR-JPY, as seen in the real market at the time of writing. Different scenarios are then tested to see how the uncertainty develops under different market parameter values. The conditions of the model under constant volatility are
𝜎'(0) = 0.11 𝑆'(0) = 1.075 𝜈' = 0 𝑟6' = 0.0 𝑟4 = 0.0
𝜎*(0) = 0.09 𝑆*(0) = 0.008 𝜈* = 0 𝑟6* = 0.0 Σ = 0
Fig. 1 shows that these parameters result in a market with constant volatility, and a symmetric PDF which is expected for these parameters. The best-of, dual digital, and basket options are priced using both the PDF and the MC paths. Fig. 2 shows the option prices of the three types of options we consider in this paper. These option prices will also serve as a benchmark for the size of the absolute difference we will calculate hereafter.
Figure 1. Volatility smiles and risk neutral density, under constant volatility.
Figure 1 shows the volatility smiles that are generated using a SABR process. The risk neutral density was calculated using a method by Austing. The SABR process generates a constant volatility when correlations and volatility of volatility is set to zero.
1. Volatility smile 2. Risk neutral density Figure 2. Option prices using PDF, under constant volatility.
Figure 2 shows the option prices of a best-of call option, a dual digital call option, and a basket call option on the currencies JPYUSD and EURUSD. The option prices are calculated using a risk neutral density.
Figure 3. Absolute price differences of options on EURUSD & JPYUSD, under constant volatility.
Figure 3 shows the price differences of a best-of call option, a dual digital call option, and a basket call option on the currencies JPYUSD and EURUSD. These differences are calculated under a constant volatility market model, and should therefore be zero.
1. Best-of option 2. Dual digital option 3. Basket option
The absolute difference of the best-of option is at it maximum near the spot rate of both
currencies. We see a similar effect at the basket option, where the price difference peaks at 𝐾 = 1. The dual digital option shows a more discontinuous surface, but this is expected as the payoff function of this option is discontinuous itself. The average price of the best-of is 0,1249 for both the PDF and the MC model. The average price of the dual digital is 0,2472 using the PDF model and 0, 2479 using the MC model. The basket option has an average price of 0,585 for both methods. The absolute difference is the largest for the dual digital option, where the difference relative to the PDF price is at its maximum 7.91%. The average relative difference across all calculated strikes is 0.14% for the best-of option, 1.64% for the dual digital option, and 0.70% for the basket option.
The following initial model conditions are used as the basis for the remaining tests. These conditions are a rough approximation of the real market of the currency triangle EUR-USD-JPY at the time of writing. The model parameters are
𝜎'(0) = 0.11 𝑆'(0) = 1.075 𝜈' = 0.50 𝑟6' = 0.009 𝑟4 = 0.004
𝜎*(0) = 0.09 𝑆*(0) = 0.008 𝜈* = 1.00 𝑟6* = 0.020
Σ' = 𝜌A = 0.5 𝜌' = −0.3 𝜌BCAD = 0
𝜌BDAC = 0.4 𝜌* = 0 𝜌B = 0.4
Fig. 4 shows the implied volatilities and the resulting risk neutral density of the simulated currency pairs. Switching to the current realistic market scenario, reveals that adding the volatility of volatility and correlation adds skew and smiles to the previously constant volatility. The resulting risk neutral density also appears more peaked and skewed towards certain areas.
One of the main advantages of the Austing method is that is should correctly reprice all the vanillas on the driver. We can test this by integrating the two-dimensional density against one of its strikes and compare the result with a one-dimensional density generated according to the method described by Breeden and Litzenberger (1978). This method involves differentiating a call option price twice, with respect to its strike price, in order to get the discounted risk neutral density. Comparing the integrated two-dimensional density with the one-dimensional density reveals that Austing’s method is accurate, as shown in Fig. 5. Only the two-dimensional density for the cross currency is slightly off. This could be caused by the need to interpolate the two-dimensional density of the main currencies in order to scale the density to match the cross currency.
We re-price the options using the realistic market scenario and calculate the difference for each of the option classes. Fig. 6 shows the absolute differences in the prices for the three option classes. The average price of the best-of option is now 0,1141 under the PDF model, and 0,1160 under the MC model. The average PDF price of the dual digital is 0,2452 and the average MC price is 0,2420. The average PDF price of the basket option has become 0,0575 and the average MC price 0,0571. All differences have significantly increased in comparison with the constant volatility market. The dual-digital option price difference becomes as high 44.41% of the PDF price, although the price of the option itself becomes quite small and it might exaggerate the difference. The MC price
ATM strikes. For strikes near at the money and strikes in the money, the prices of the PDF and MC converge. The average difference relative to the PDF price is 3.16% for best-of options, 13.11% for dual digital options, and 5.17% for the basket option. Switching from the constant volatility to realistic market smiles appears to greatly affect the accuracy of the models. In the next section we will adjust individual parameters in order to gain a better understanding of the size of uncertainty in different market situations.
Figure 4. Volatility smiles and risk neutral density, under realistic market conditions.
Figure 4 shows the volatility smiles that are generated using a SABR process. The risk neutral density was calculated using a method by Austing. The SABR process’ parameters are chosen to match the current market smiles for EUR-USD-JPY.
1. Volatility smile 2. Risk neutral density
Figure 5. Tests if the PDF correctly reprices the vanillas on the drivers.
Figure 5 compares the one-dimensional densities of the Breeden and Litzenberger method with the two-dimension densities of Austing’s method. The two-dimensional PDF of Austing correctly reprices the one-dimensional PDF on its drivers.
Figure 6. Absolute price differences of options on EURUSD & JPYUSD, under realistic scenario.
Figure 6 shows the price differences of a best-of call option, a dual digital call option, and a basket call option on the currencies JPYUSD and EURUSD. These differences are calculated under a SABR parameters which match the current EUR-USD-JPY market. The differences in prices are a proxy for uncertainty in the model used to generate the prices.
1. Best-of option 2. Dual digital option 3. Basket option
3.1 Changing the volatility smile and skew
In this section, two market parameters will be changed within a certain range. The results will then be summarized into a table and we will draw interferences on how the market parameters change the uncertainty and what the magnitude of those changes are.
The volatilities of 𝑆' and 𝑆* have the largest effect on the magnitude of the resulting volatility
smile, and it has the largest effect on the number of extreme paths generated by the MC simulation. Table 1 shows the uncertainty for a range of sigma of both generated currency pairs. The base market scenario is included as a benchmark. The results show that there is the best-of option and the dual-digital option have the largest differences in their prices. The differences of the basket option price appear to be relatively stable. There is a small, but apparent change in absolute price
differences between the change in parameters. The mean difference of best-of price increases with every increase of 𝜎* for every constant level of 𝜎'. We notice the opposite effect for the dual digital option.
We repeat the same procedure for the volatility of volatility parameter 𝝂𝒊. This parameter
increases slightly as the both volatility of volatility parameters increase. The dual digital option is however again most affected by the change of parameters.
Table 1. Option price differences for adjustments in 𝝈𝟏 and 𝝈𝟐.
This table contains the absolute difference between options priced using a risk neutral density and using a Monte Carlo simulation. The differences are calculated under different market scenarios, through parameters 𝜎' and 𝜎*. * PDF
cannot be determined using Austings’ method, because 𝜌 is out of range.
𝝈𝟏 𝝈𝟐 Best-of Dual digital Basket
Mean Max Mean Max Mean Max
11% 9% 0,003 0,007 0,013 0,043 0,001 0,002 10% 10% 0,003 0,007 0,013 0,038 0,001 0,002 10% 30% * 10% 50% * 30% 10% 0,005 0,010 0,018 0,054 0,003 0,004 30% 30% 0,006 0,016 0,009 0,025 0,005 0,006 30% 50% 0,015 0,030 0,006 0,020 0,004 0,005 50% 10% 0,006 0,014 0,023 0,054 0,004 0,006 50% 30% 0,006 0,016 0,015 0,029 0,004 0,007 50% 50% 0,014 0,028 0,009 0,016 0,004 0,007
Table 2. Option price differences for adjustments in 𝝂𝟏 and 𝝂𝟐.
This table contains the absolute difference between options priced using a risk neutral density and using a Monte Carlo simulation. The differences are calculated under different market scenarios, through parameters 𝜈' and 𝜈*.
𝝂𝟏 𝝂𝟐 Best-of Dual digital Basket
Mean Max Mean Max Mean Max
The price difference of the dual digital option changes most with changes in 𝜈', but less with changes in 𝜈*. The difference reaches its maximum at 𝜈' = 120%, 𝜈* = 0%. For this scenario the mean MC price of the option is 0,241 and the mean PDF price of the option is 0,245. The maximum difference is thus more than half of the option price. These large differences occur again for dual digitals that are far out of the money on one of the strikes or on both of the strikes. This is visualized in Fig. 7.
Figure 7. Absolute price differences of options on EURUSD and JPYUSD, under realistic market
scenario with adjusted parameters 𝜈' = 120%, 𝜈* = 0%.
Figure 7 shows the absolute differences in prices between the PDF method and the MC method. Under this extreme market condition the differences grow large.
1. Best-of option 2. Dual digital option 3. Basket option
Next the correlation matrix is adjusted the see how the differences develop for different values in the correlation matrix. Not all possible values for each correlation parameter are possible, as the correlation matrix needs to be positive and definitive in order to perform the Cholesky
decomposition on the matrix. Table 3 contains the results for different values of 𝜌(𝑆', 𝑆*) and 𝜌(𝜎', 𝜎*). Table 4 contains the results for 𝜌(𝑆', 𝜎') and 𝜌(𝑆*, 𝜎*) and Table 5 contains the results
Table 3. Option price differences for adjustments in 𝝆(𝑺𝟏, 𝑺𝟐) and 𝝆(𝝈𝟏, 𝝈𝟐).
This table contains the absolute difference between options priced using a risk neutral density and using a Monte Carlo simulation. The differences are calculated under different market scenarios, through parameters 𝜌(𝑆', 𝑆*) and 𝜌(𝜎', 𝜎*).
* The resulting correlation matrix is not positive definitive, therefore a Cholesky decomposition is not possible.
𝝆(𝑺𝟏, 𝑺𝟐) 𝝆(𝝈𝟏, 𝝈𝟐) Best-of Dual digital Basket
Mean Max Mean Max Mean Max
0,40 0,50 0,003 0,007 0,013 0,043 0,001 0,002 -0,80 -0,80 0,002 0,005 0,015 0,072 0,001 0,003 -0,80 0,00 0,002 0,005 0,017 0,070 0,001 0,003 -0,80 0,50 * -0,40 -0,80 0,002 0,005 0,013 0,060 0,001 0,002 -0,40 0,00 0,002 0,005 0,014 0,057 0,001 0,002 -0,40 0,50 0,003 0,005 0,015 0,053 0,001 0,003 0,00 -0,80 0,003 0,008 0,014 0,052 0,001 0,002 0,00 0,00 0,003 0,006 0,014 0,050 0,001 0,002 0,00 0,50 0,003 0,005 0,013 0,046 0,001 0,002 0,40 -0,80 0,003 0,012 0,016 0,049 0,001 0,002 0,40 0,00 0,003 0,009 0,014 0,047 0,001 0,002 0,40 0,50 0,003 0,007 0,013 0,043 0,001 0,002 0,80 -0,80 0,004 0,019 0,019 0,051 0,001 0,003 0,80 0,00 0,003 0,013 0,016 0,049 0,001 0,002 0,80 0,50 *
The effects of the changing the correlation between the two assets and the correlation between their stochastic volatility is quite small for all three option types. Even the dual digital option price difference is not that different across the several scenarios. The difference of the dual digital option is decreasing as 𝜌(𝑆', 𝑆*) increases and keeping 𝜌(𝜎', 𝜎*). The difference increases for increases in 𝜌 𝜎', 𝜎* , while keeping 𝜌(𝑆', 𝑆*) constant. The best-of option and the basket option are nearly unaffected by the changes in correlation.
Table 4. Option price differences for adjustments in 𝝆(𝑺𝟏, 𝝈𝟏) and 𝝆(𝑺𝟐, 𝝈𝟐).
This table contains the absolute difference between options priced using a risk neutral density and using a Monte Carlo simulation. The differences are calculated under different market scenarios, through parameters 𝜌(𝑆', 𝜎') and 𝜌(𝑆*, 𝜎*).
* The resulting correlation matrix is not positive definitive, therefore a Cholesky decomposition is not possible.
𝝆(𝑺𝟏, 𝝈𝟏) 𝝆(𝑺𝟐, 𝝈𝟐) Best-of Dual digital Basket
Mean Max Mean Max Mean Max
-0,30 0,00 0,003 0,007 0,013 0,043 0,001 0,002 -0,40 0,40 * -0,40 0,00 0,003 0,008 0,017 0,061 0,001 0,002 -0,40 -0,40 0,005 0,010 0,028 0,148 0,001 0,002 0,00 0,40 0,003 0,008 0,027 0,138 0,001 0,002 0,00 0,00 0,002 0,006 0,010 0,024 0,001 0,001 0,00 -0,40 0,004 0,008 0,025 0,138 0,002 0,003 0,40 0,40 0,005 0,012 0,033 0,127 0,001 0,001 0,40 0,00 0,004 0,011 0,021 0,098 0,002 0,003 0,40 -0,40 0,004 0,011 0,035 0,130 0,003 0,005
It is clear from this table that skew adds to the uncertainty of the model and resulting prices. The case where both parameters are zero is also the case where all absolute differences are minimal. The presence of either a left or right skew in 𝑆* has a greater effect on the differences than an equal equal sized skew in 𝑆'. The uncertainty is at its maximum when both currency pairs are skewed.
Figure 8. Effect of 𝜌(𝑆*, 𝜎*) on the skew of the volatility. 𝜌 𝑆', 𝜎' is zero in all cases.
Figure 8 shows how the 𝜌 𝑆*, 𝜎* parameter of the SABR process adds a skew to the volatility function when non-zero
values.
1. 𝜌 𝑆*, 𝜎* = −0.4 2 𝜌 𝑆*, 𝜎* = 0 3. 𝜌 𝑆*, 𝜎* = 0.4
The final scenarios can be found in Table 5. It contains the differences for the option price under different values of 𝜌(𝑆', 𝜎*) and 𝜌(𝑆*, 𝜎'). These correlations measure how a change in one
changes in volatility of volatility and smile skew. The difference in all three option classes become larger as 𝜌(𝑆', 𝜎*) grows larger and smaller as 𝜌(𝑆*, 𝜎') grows, and the effect of the latter
correlation is stronger.
Table 5. Option price differences for adjustments in 𝝆(𝑺𝟏, 𝝈𝟐) and 𝝆(𝑺𝟐, 𝝈𝟏).
This table contains the absolute difference between options priced using a risk neutral density and using a Monte Carlo simulation. The differences are calculated under different market scenarios, through parameters 𝜌(𝑆', 𝜎*) and 𝜌(𝑆*, 𝜎').
* The resulting correlation matrix is not positive definitive, therefore a Cholesky decomposition is not possible.
𝝆(𝑺𝟏, 𝝈𝟐) 𝝆(𝑺𝟐, 𝝈𝟏) Best-of Dual digital Basket
Mean Max Mean Max Mean Max
0,00 0,40 0,003 0,007 0,013 0,043 0,001 0,002 -0,80 -0,80 0,004 0,010 0,022 0,071 0,002 0,004 -0,80 -0,40 0,004 0,010 0,024 0,071 0,003 0,005 -0,80 0,00 0,003 0,008 0,019 0,058 0,002 0,004 -0,80 0,40 0,003 0,006 0,016 0,044 0,001 0,003 -0,80 0,80 0,003 0,009 0,017 0,058 0,001 0,002 -0,40 -0,80 0,004 0,012 0,028 0,084 0,003 0,005 -0,40 -0,40 0,004 0,010 0,022 0,071 0,002 0,004 -0,40 0,00 0,003 0,008 0,018 0,058 0,002 0,003 -0,40 0,40 0,003 0,006 0,014 0,043 0,001 0,002 -0,40 0,80 * 0,00 -0,80 0,004 0,013 0,026 0,087 0,003 0,004 0,00 -0,40 0,003 0,010 0,021 0,072 0,002 0,003 0,00 0,00 0,003 0,009 0,017 0,058 0,001 0,002 0,00 0,40 0,003 0,007 0,013 0,043 0,001 0,002 0,00 0,80 * 0,40 -0,80 0,004 0,013 0,026 0,090 0,002 0,004 0,40 -0,40 0,003 0,011 0,020 0,073 0,002 0,003 0,40 0,00 0,003 0,010 0,016 0,058 0,001 0,002 0,40 0,40 *
4 Conclusion
We can conclude that uncertainty in pricing multi-currency options arises from markets where the volatilities show large smiles, combined with a clear skew. We have tested the model for some extreme conditions, but these can occur when valuing contracts on developed and pegged currencies. Under these conditions the difference between the two models grow to up to half the option price itself.
It would be interesting for follow up studies to research to see if any deterministic model can be fitted. Using such a model one could price exotics using a confidence interval, based on known market parameters and calibrated market parameters. The price differences of the dual digital option show us that the pricing uncertainty is highest for exotic options with discontinuous payoffs. It would therefore also be worthwhile to research how price differences of discontinuous path-dependent options would develop under different models.
5 References
Austing, P., 2011. Repricing the Cross Smile: An Analytic Joint Density. Risk.
Bennett, M.N., Kennedy, J., 2003. Quanto pricing with copulas. University of Warwick Statistics Department 1–44. doi:10.2139/ssrn.412441
Black, F., Scholes, M., 1973. The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81, 637–654. doi:10.2307/1831029
Breeden, D.T., Litzenberger, R.H., 1978. Prices of State-Contingent Claims Implicit in Option Prices. The Journal of Business 51, 621–651. doi:10.2307/2352653
Hagan, P.S., Kumar, D., Lesniewski, A.S., Woodward, D.E., 2002. Managing Smile Risk. Wilmott Magazine 84–108. doi:10.1080/13504860500148672
