Fairing the Gamma: An Engineering Approach to Sensitivity Estimation Wanmo Kang∗, Kyoung-Kuk Kim†, and Hayong Shin‡ April 1, 2012

Abstract In financial industry, obtaining stable estimates for sensitivities of derivatives to the price changes of the underlying asset is very important from a practical point of view. However, this aim is often hindered by the absence of closed form expressions for Greeks or the requirement of an excessive computational workload due to the complexities of various exotic derivatives structures. But, ad-hoc numerical schemes to produce stable Greeks such as nonlinear regression can result in nonsensical values. In this article, we propose a fairing algorithm designed for the computation of gamma values of exotic derivatives. We present some examples of exotic derivatives to which the algorithm is applied and provide some analytical and numerical results that show its usefulness in reducing the mean square error of gamma estimates.

Keywords: Sensitivities; Monte Carlo estimation; Fairing

1

Introduction

To any participant in the financial derivatives markets, the sensitivities of derivative products in her portfolio are indispensable data for hedging the market risks that the portfolio is exposed to. Moreover, it is important to have these so called “Greeks” that are reasonably good and stable in a sense to be made more precise below. There have been several methods to address this issue with mixed degrees of successes depending on payoff structures of derivatives. To name just a few of those, a finite difference scheme (FD) is well summarized in Glasserman (2004), a pathwise method and a likelihood ratio method were proposed by Broadie and Glasserman (1996), and a Malliavin calculus based method was developed in Fourni´e et al. (1999) and Fourni´e et al. (2001). And it is well known that, in many cases, FDs are less reliable than other methods when applicable. However, depending on particular products of interest, we often have to resort to FD because of failed regularity conditions (on payoff functions or on underlying stochastic processes), or because of final formulae which might be formidable to draw or difficult to implement even when all conditions are satisfied. ∗

Department of Mathematical Sciences, KAIST, Dajeon, South Korea, Email:[email protected] Department of Industrial and Systems Engineering, KAIST, Daejeon, South Korea, Email:[email protected] ‡ Corresponding author, Department of Industrial and Systems Engineering, KAIST, Daejeon, South Korea, Email:[email protected] †

1

For example, let us consider the following equity linked security of which underlying assets are S 1 and S 2 . We call this product ELS2 to emphasize its dependence on two equity prices. The payoff is basically a function of the minimum of two ratios Sti /S0i , i.e., { 1 2} St St ˆ , , t ∈ [0, T ] St = min S01 S02 where Sti is the price of asset i at time t and T indicates the maturity, and we have six early redemption dates ti = iT /6, i = 1, . . . , 6 with redemption thresholds li > 0. The early redemption means that the contract ends at time ti if a certain condition on Sˆ is met at that time. More specifically, 1. if Sˆti > li for some i for the first time, then the contract expires at time ti with payoff 1 + ri to the investor, 2. if there was no early redemption and min{Sˆt : 0 ≤ t ≤ T } > b , the payoff is 1, 3. the final payoff is SˆT otherwise. Here min{Sˆt : 0 ≤ t ≤ T } is the minimum of daily closing prices until maturity and b is the value of the lower nock-in barrier. The computation of Greeks for this product is non-trivial by any of the existing methods for sensitivity estimation. ELS2 is a typical kind of equity derivatives actively traded in Korean financial markets. Many financial contracts, indeed, contain payoffs as complex structured as this example. There are many payoff features that can be easily incorporated into any derivative contract. Simple examples include digital and barrier features as in ELS2. If derivative products in a portfolio consist of various combinations of such payoff features, then each combination might easily require a separate Greek calculation code to employ non-FD methods. (In this paper, we focus on sensitivities with respect to the underlying stock price up to the second degree.) In addition, one needs to run the code at each parameter value in a certain range where the user wants to see Greeks for the purpose of risk management. (One exception is Giles and Glasserman (2006) where the authors provide a variant of the pathwise method that gives multiple Greeks at once.) However, if one wants to implement FD, a notable practical difficulty arises. Namely, the sensitivity estimates become quite unstable especially when we take higher order sensitivities such as gammas. Figure 1 shows the prices, deltas, and gammas for ELS2 computed by FD in a rectangular range. Geometric Brownian motions were assumed as the underlying asset price processes. It is clear that the gammas fluctuate much wider than the prices and the deltas. This instability has been one of the big barriers in the implementation of Monte Carlo methods in practice because it is essential to have a robust way of calculating Greeks in a financial firm where computations of Greeks need to be automated and various derivatives portfolios are managed. This article is an attempt to address some computational issues that appear in applying FD to sensitivity estimates for derivatives that involve severe costs in implementing non-FD methods. For a related work by the authors, we refer the reader to Kang et al. (2012). Now, we briefly summarize our approach to delta and gamma computations using Monte Carlo simulation. First, we assume that a Monte Carlo method provides price estimates at a set of initial asset values. They are given in the form of confidence intervals or pairs of sample means and sample variances from the

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Figure 1: Price, delta, and gamma surfaces for ELS2 by FD

Monte Carlo runs. Thus, we work with a sequence of intervals in the case of a derivative on a single underlying asset or an array of intervals for two underlying assets. We can safely assume that model parameters except asset prices stay at their estimated values during a few hours or a single trading day. Hence, once we compute Greeks in a range of asset prices, hedging operations can be performed based on these values. The “circuit breaker” in each exchange provides a natural range for this preparatory Greek computation. Starting with a set of sample means, we apply fairing iterations to these values within fixed intervals which, throughout this paper, we set as being centered at sample means with the half width equal to one or 1/2 times sample standard errors. Then, we apply the ordinary FD for delta and gamma computations after the fairing algorithm stops according to a certain stopping condition. This stopping condition will be detailed in a later section. Hence, fairing iterations can be uniformly applied to derivatives with any type of payoff functions or to any specification of stochastic models. Fairing is a computer-aided design (CAD) technology developed originally in automotive industry to obtain a surface with a smoother curvature distribution for high quality machining and aesthetic bodyshapes of vehicles. Under the condition that the value function under consideration is twice continuously differentiable with respect to underlying assets prices, our new fairing approach is applicable. We provide some motivating examples in Section 2. A detailed procedure is presented in Section 3. Section 4 exhibits numerical test results while Section 5 deals with some analysis results for the method. We conclude in Section 6.

3

2

Motivation

Let Y (θ) be the payoff of interest where θ is a parameter. The price is given by E[Y (θ)] under some probability measure P. We set α(θ) = E[Y (θ)].

(1)

We are looking for α′ (θ) and α′′ (θ) where θ varies in a certain range. In this paper, θ is set to be the underlying stock price. In addition to the difficulties of applying existing methods for those mathematical derivatives, it is often out of practical concerns that we are interested in computing discrete delta and discrete gamma at given points {θ1 , . . . , θn }. See, e.g., Taleb (1997) about discrete re-balancing. We assume that the points are equally-spaced, say h = θi+1 − θi for each i. The discrete delta (∆i ) and gamma (Γi ) for i = 1, . . . , n − 1 are, then, defined by ∆i = Γi =

α(θi+1 ) − α(θi−1 ) , 2h α(θi+1 ) − 2α(θi ) + α(θi−1 ) . h2

We denote the Monte Carlo estimate at θ with m simulation trials by Y m (θ). Then, the corresponding estimates of discrete delta and gamma become bi = ∆ bi = Γ

Y m (θi+1 ) − Y m (θi−1 ) , 2h Y m (θi+1 ) − 2Y m (θi ) + Y m (θi−1 ) . h2

(2) (3)

These estimates can be regarded as the FD approximations to the instantaneous delta and gamma values. To motivate our approach, let us consider some examples which illustrate the difficulties in obtaining reliable Greek estimates via FD. Throughout the paper, we consider the following four contracts with varying degrees of exoticness: CO : Vanilla European call option (strike K = 100 and maturity T = 4) DO : Cash-or-nothing digital option (K = 100, T = 4, payoff = 0 or 10) ELS1 : Structured security linked to the performance of an underlying equity with early redemption features and down barrier feature. This product can be characterized by many discontinuities in the payoff structure. Details of the payment plan of this product are same as those of ELS2 introduced in Section 1. The only difference is that the worst performer is replaced by the normalized price of the single underlying equity. In numerical experiments, we work with parameters T = 1 year, l1 = l2 = 0.95, l3 = l4 = 0.9, l5 = l6 = 0.85, ri = 0.05 × i, and finally b = 0.7. ELS2 : Explained in Section 1 and the parameters are the same as in ELS1. Black-Scholes model is used in the experiments, however, it should be noted that our approach is independent of any model specification as it becomes clear in the subsequent section. As to CO and DO, 4

we use the Black-Scholes formulae to compute the true prices. Since there is no analytic or semi-analytic formula available for ELS1 or ELS2, Monte Carlo estimates with 1 billion simulation trials are used as the true prices. The price curves from simulation with 105 paths as well as the true curves are plotted in Figures 2–4 for comparison. Volatility levels in those figures are fixed at 30%. Also, in Figure 4, the time-to-maturity is 4 weeks which is the same as CO and DO. Lastly, we note that the antithetic variable and the common random technique are employed. As a measure of the goodness of estimates, we consider root mean squared error (RMSE) and root mean squared relative error (RMSRE). If α ˆ i and αi denote the estimated value and the true value, respectively, then these measures are defined as v v u n u n ( ) u∑ u∑ α ˆ i − αi 2 2 t t RMSE = (ˆ αi − αi ) and RMSRE = (4) α ˜i i=0

i=0

∑ where α ˜ i = max{|αi |, 10−3 (n+1)−1 nj=0 |αj |}. The reason for the introduction of α ˜ i is that, otherwise, it may exaggerate the error when the true value is too small. But, still, there is a possibility to have large RMSRE values when αi values are very small. Indeed, this is what we observe in some of numerical tests reported below. However, usually small true values do not cause problems in actual trading operations. For the standard European call option (Figure 2) whose payoff function is continuous, all of three curves (price, delta, and even gamma curve) are quite close to the true ones. For the digital option with cash or nothing payoff (Figure 3), the gamma curve reveals jig-saw shape. For ELS1 (Figure 4), it is highly unstable. Actually, even the delta curve deviates from the true one under different conditions such as fewer samples, longer maturity, or higher asset volatility.

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Figure 2: Price/Delta/Gamma of CO

Certainly, high fluctuations in gamma estimates are undesirable in terms of dynamic hedging of financial derivatives. Now, recall that, as said in the introduction, a risk manager or a trader might want to pre-compute relevant Greek values and retrieve appropriate numbers during the day to enhance the operational efficiency. Our approach is based on the utilization of price estimates at different parameter values

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Figure 3: Price/Delta/Gamma of DO

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Figure 4: Price/Delta/Gamma of ELS1

as well as the interval estimates (i.e., (1 − β) confidence intervals), ( ) sˆm (θi ) sˆm (θi ) Y m (θi ) − zβ/2 √ , Y m (θi ) + zβ/2 √ , n m √ )2 ∑ ( where sˆm (θ) = (m − 1)−1 m Y (θ) − Y (θ) . Basically, we use this extra information by ali m i=1 lowing some flexibility in choosing the values for FD. Let us fix a positive constant δ to determine the width of intervals from which input values, say pi , for finite differencing are chosen: ) ( sˆm (θi ) sˆm (θi ) , Y m (θi ) + δ √ . (5) Ii = Y m (θi ) − δ √ m m We, then, put these pi ’s in place of Y m (θi ) in (2) and (3) to fair the Greek curves and obtain b i (p) = ∆

pi+1 − pi−1 , 2h 6

(6)

b i (p) = Γ

pi+1 − 2pi + pi−1 h2

(7)

for i = 1, . . . , n − 1, and call them as the Greek estimates for p = (p0 , . . . , pn ). We often call p a curve, meaning the piecewise linear curve connecting {(θi , pi ) : i = 0, . . . , n}. Before we move onto the next section, let us discuss some numerical implications of Ii . In the case of ELS1, 104 sample paths yield 0.01 to 0.11 USD of standard errors for the price range taken in the test. This is less than 0.12% of the share price. (The number is reduced to 0.04% for 105 paths.) Actually, the modified pi via the algorithm below are only within −0.1% ∼ 0.03% (for 104 runs) and −0.011% ∼ 0.018% (for 105 runs) from the original price estimates.

3

Curve/Surface Fairing

Facing the question in Section 2 about how to enhance the gamma estimates, the following techniques seem to be two natural approaches as the main problem arises due to the undesirable heavy fluctuations of Greek estimates; first, signal processing techniques based on Fourier analysis, and second, curve fairing techniques developed in geometry processing community, such as CAD. And in this paper, we explore the avenue of curve fairing techniques. In general, curve fairing is a method of changing the original curve by only a small amount so as to obtain a new curve with more pleasing characteristics. A fast growing research area in CAD is “digital shape reconstruction” (formerly called reverse engineering) field, which deals with constructing mathematical representation of the shape from the measured data of a physical object possibly with measurement noise, e.g., Salvi and Varadi (2005). And an essential step in digital shape reconstruction is fairing, which is also referred to as de-noising. (Some researchers argue that de-noising is different from fairing in its problem formulation, e.g., Sun et al. (2008).) The goal of fairing in digital shape reconstruction is to remove possible noises under the assumption that original curve (or surface) is smooth. Although this goal is perfectly in suit with our needs, to the best of our knowledge, there is no literature applying this technique to the computation of prices and hedging parameters of financial products. Among various fairing algorithms, we start with a simple one proposed by Cho and Choi (2001) and modify it to fit with our problem. For the variety of alternative fairing algorithms, we refer the reader to Eck and Jaspert (1994), Sapidis and Farin (1990), Yamada et al. (1999), and Zhang et al. (2001). Recall that the true function is α(θ) = E[Y (θ)] and we assume that α(θ) is a smooth function of parameter θ. At each θi , Ii defined by (5) is a modified confidence interval for α(θi ). Greek estimates such as delta or gamma are defined by (6) and (7) for original Monte Carlo estimates or values obtained via fairing. As it will be seen later, delta estimates after fairing exhibit smoother shapes even though our primary focus is on gamma estimates.

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3.1

Local fairing formula for internal points

ˆ i (p) in i In order to obtain a curve with smooth gamma values, it is desirable to have gradual changes of Γ . So, we define the local fairness measure of a price curve p at θi as : ( ) ( ) p − 4p + 6p − 4p + p i+2 i+1 i i−1 i−2 ˆ ˆ i (p) − Γ ˆ i (p) − Γ ˆ i−1 (p) = εi (p) = Γi+1 (p) − Γ . (8) h2 (Note that εi (p) is actually a measure of “unfairness”. However, we will follow the terminology in the curve fairing literature such as Cho and Choi (2001), Eck and Jaspert (1994), and many others.) Minimizˆ i (p) to the average of Γ ˆ i−1 (p) and Γ ˆ i+1 (p) (see Figure 5), ing εi (p) becomes optimal when we move Γ which in turn yields the optimal position of pi as : 1 1 1 p∗i = (4m1 − m2 ), where m1 = (pi+1 + pi−1 ), m2 = (pi+2 + pi−2 ). (9) 3 2 2 Hence the optimal position of pi can be interpreted as the 1:4 extrapolation point between the mid-point of the first order neighbors pi±1 and the mid-point of the second order neighbors pi±2 , as shown in Figure 6.

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Figure 6: Optimal position among 5 points.

Figure 5: Optimal position among 3 points.

3.2 Considerations for boundary points There are various ways of imposing boundary conditions, including local approximations such as cubic curve fitting. However, extensive numerical tests suggested that there is no single method that performs well when compared with the naive one which fixes four boundary points p0 , p1 , pn−1 , pn at the original price estimates. Hence, in our numerical implementation, we fix those boundary points.

3.3 Fairing algorithm Now, we are ready to describe the algorithm for curve fairing. We begin with the initial discrete price curve p0 = {pi : i = 0, . . . , n}. Let pk = {pi,k : i = 0, . . . , n} denote the discrete curve obtained after the k-th iteration. We compute pk by applying the local fairing formula (9) to pk−1 , i.e., pi,k+1 = p∗i,k with pi,0 = pi . However, this simple setting is modified to reflect the following two concerns. Damping factor: Blind application of the local fairing scheme may result in oscillations as shown in Figure 7. This can be explained by an analogy to a swinging pendulum without any resistance, which will 8

swing forever. To avoid this situation, a damping factor is introduced. Let vi = p∗i,k − pi,k . Then the new position becomes pi,k+1 = pi,k + τ vi , where τ ∈ [0, 1] is the damping factor. If we use a large value for τ (near 1), resulting curves from each iteration become unstable, whereas small τ (near 0) results in slow convergence. From numerical tests, we found that τ = 0.5 gives a reasonable convergence speed. More detailed analysis is done in Section 5. The reader is referred to, for example, Cho and Choi (2001) for additional discussions regarding damping factor. Fairing tolerance: Equation (9) may give the optimal points far from the original ones. Since it is desirable that altered price estimates from fairing are still good reference values for true prices, we confine √ the movement of each data point pi to Ii where δ is the fairing tolerance, i.e. |pi,k − pi | < δˆ sn (θi )/ n. In our numerical tests, we use δ = 1 so that each pi,k stays within one standard error from pi . pk,i+1

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Figure 7: Oscillations in fairing iterations.

Combining the above two issues, we get the following iterative algorithm :

Algorithm Gamma Curve Fairing do { for i = 2 to n − 2 { compute p∗i,k using (9); vi = p∗i,k − pi,k ; wi = pi,k + τ vi − pi : √ √ if wi > δˆ sn (θi )/ n, then wi = δˆ sn (θi )/ n; √ √ if wi < −δˆ sn (θi )/ n, then wi = −δˆ sn (θi )/ n; pi,k+1 = pi + wi ; } } while (stopping condition is not met)

3.4

Stopping condition

The remaining question is when to stop the iteration in the algorithm. Figure 8 shows the change of gamma curves of DO as the number of iterations increases. Notice that too many fairing iterations could be harmful, resulting in somewhat flattened gamma curves. 9

Fairness of a curve may have different meanings in different applications, depending on the requirements as discussed in Salvi and Varadi (2005). For the discretely represented curve as we have in our problem, we will follow the suggestion in Eck and Jaspert (1994), where the global fairness of a curve ∑ is defined by ε = (κ′′i )2 where κi is the curvature of the curve at θi . Since it is not straightforward to define the term κ′′i for a discrete curve, we approximate κ′′i in the same way that we approximate κi by ˆ i (p), the second order difference of price estimates. In other words, using Γ ˆ i (p), we set Γ ε(p) =

∑ ((

) ( )) 2 ∑ ˆ i+1 (p) − Γ ˆ i (p) − Γ ˆ i (p) − Γ ˆ i−1 (p) Γ = εi (p)2

i

i

where εi (p) is the local fairness defined in (8). With this global fairness measure, we continue the fairing algorithm in Section 3.3 while ε(p) is decreasing or while the ratio of measurements from two consecutive iterations is larger than some threshold value. Table 1 shows the global fairness measurements for the original price curve, the output of the fairing algorithm, and the true price curve. All necessary parameter specifications are explained in Section 4.

(a) Fairing Iteration = 1

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Figure 8: Evolution of gamma curves

3.5 Extension to surface fairing The curve fairing algorithm above can be easily modified for a price surface that consists of two-dimensional array of data points. The relevant parameters are, then, the underlying asset prices (θ1 , θ2 ). Since each true price α(θi1 , θj2 ) on the price surface is located on two cross-sectional curves α(θi1 , ·) and α(·, θj2 ), we can apply the curve fairing technique independently, then the two resulting positions are averaged. We will omit the details of surface fairing since the reader can easily follow the main idea. 10

Table 1: Reduction of fairness measures for ELS2, number of sample paths = 105 Maturity 1M 3M 5M 7M 9M 11M

4

None

Fairness measure Fairing True

0.701971 1.208936 1.448335 1.670309 1.436607 1.722928

0.128094 0.110894 0.095314 0.086091 0.061357 0.036982

0.089540 0.023177 0.013826 0.018881 0.010141 0.012825

Numerical Tests

We conduct numerical experiments on the performance of the fairing algorithm, focusing on four contingent claims (CO, DO, ELS1, and ELS2) of which payoff structures are described in Section 2. The asset price dynamics of interest is assumed to follow the geometric Brownian motion. Other parameters used in the tests are as follows; for CO and DO, we fix the maturity equal to 4 weeks and vary the level of asset volatility from 10% to 50% with step size 10% while for the other two contracts we fix volatility parameters, correlation level (volatility 30% for ELS1, 30% and 40% for the two assets in ELS2 with correlation 0.6), and vary the maturities from 1 month to 11 months. Even though we do not report here, the performance of the algorithm for ELS1 and ELS2 turns out to be very similar for different volatility and correlation specifications. For the benchmark values to be compared with algorithm outputs, analytical closed form formulae for CO and DO are used, but for the other two exotic securities, we use the values obtained from 1 billion simulation trials. For all contracts, 104 , 105 , and 106 number of simulated paths are used to produce price and Greek estimates. Results are exhibited in Tables from 2 to 13. Recall that we construct delta, gamma curves or surfaces by applying FD to price estimates from the usual Monte Carlo computations. The range of the parameter θ is {84, . . . , 116}. Therefore, we compute delta and gamma estimates for θ ∈ {85, . . . , 115}. Through fairing, new price estimates, or put differently, new price curves or surfaces are generated, which in turn yield new delta, gamma curves or surfaces. The performance measures employed are RMSE and RMSRE as in (4). The effectiveness of fairing is essentially captured by the ratios of these measures from the original FD estimates and the “faired” estimates. There are three main observations that can be made. Firstly, we do not see much improvement for CO where the ratios for price/delta/gamma are greater than 1 in several places and close to 1 even when they are less than 1. However, the results improve quite a bit as discontinuities start to kick in. Secondly, we see much better performance for gammas rather than prices or deltas. Intuitively, this can be understood as a result of fairness measure by which we are purposely smoothing gamma estimates. Finally, if we closely look at the RMSEs of gamma values from the faired estimates with 104 sample paths, then the numbers mostly lie somewhere between the RMSEs of gamma values from the original Monte Carlo estimates with 105 (Table 11) and 106 (Table 13) sample paths. Consequently, at least for securities covered in this paper,

11

it is seen that the algorithm performs better for more exotic securities and for gamma estimates. In addition, the computational savings seem to be between 10 and 100 times, which is remarkable considering that the processing time for the algorithm is negligible, not to mention the ease of coding. Now, Figures 14–16 present how the original gamma curves for CO, DO, ELS1 are changed after fairing. We do this only for gamma curves as the improvement for price/delta estimates is not as much as gamma estimates. For Figure 14, the asset volatility is set equal to 30%. We note that the fluctuations of gammas are mitigated after fairing. Two additional figures depict price/Greek estimates corresponding to ELS2 with 105 trials. To see the changes from the original values, Figure 17 needs to be compared with Figure 1. Figure 18 amplifies this for Γ11 for the reader’s convenience. Remark We observe very large RMSRE values in some parts of tables below. See, e.g., the gamma values of Tables 5–7. Even though RMSRE works fine most of the time, it can be absurdly large when a true value is very close to zero. Such feature of RMSRE can lead to somewhat seemingly strange behavior in the outcomes, but, such small values do not lead to practical concerns.

5

Analysis of Fairing

In this section, we give some analytical and numerical results that support the use of fairing in the Monte Carlo simulation context. To this end, let us assume that we have point estimates pi for the unknown function values αi = α(i/n) for i = 0, . . . , n, i.e., θi = i/n where α(·) is defined as in (1). We deliberately focus on analyzing the effects of the first iteration of fairing on the mean square error of gamma estimates. This is to get some intuition about the method without getting lost in the analysis which becomes extremely complicated as further iterations are performed. For notational convenience, we set pi = pi,0 = Y m (i/n). In Section 5.1, we deal with the case where the pi are independent of each other. This happens when we do not employ the common random number technique. Even though this technique is frequently used and thus the analysis in the subsection does not apply in such a case, it greatly simplifies computations and yields analytical results. In Section 5.2, we investigate how a correlation structure between the pi is generated via simple examples, and conduct a numerical experiment to see how correlation structures affect the performance of the algorithm.

5.1

Independent case

Throughout this subsection, we assume that Zi := pi − αi are independent and identically distributed as N (0, σ 2 ). The assumption of normal distribution is not harmful as long as the number of simulation trials is large enough. However, the same variance σ 2 at all points is not guaranteed because the variance E[(Y (i/n) − αi )2 ] depends on the Y (·) which could be quite different from point to point. Nevertheless, we can consider this σ 2 as a sort of maximal possible variance that point estimates pi have, so that we can see how fairing works even in the worst case. Regarding the independence, we deal with the correlated pi case in the subsequent section. However, at least it should be noted that this correlation structure of the pi is quite different from a perfect correlation case as long as the payoff function has exotic features. For this 12

reason and for the analytical tractability, the independence assumption on the Zi is imposed. Let us denote the vector (−1/6, 2/3, −1, 2/3, −1/6) by φ. Then, from the definition of p∗i in (9), we obtain − → 4(pi+1 + pi−1 ) pi+2 + pi−2 p∗i − pi = − − pi = −Λi h4 + φ · Z i 6 6 → − −1 for the interior points (i = 2, . . . , n − 2) where h = n , Z i = (Zi−2 , . . . , Zi+2 ), and ( ) 1 1 2 2 1 Λi = 4 αi+2 − αi+1 + αi − αi−1 + αi−2 . h 6 3 3 6 If we write Γi for the approximate value of the second-order derivative of the function α(·) using finite difference, then Λi can be written as h−2 (Γi+1 − 2Γi + Γi−1 ), which is an approximate value for the fourth-order derivative of α(·) at θi . After the first iteration, we get pi,1 = pi + g(τ (p∗i − pi )),

where g(x) = x1{|x|≤δσ} + δσ1{x>δσ} − δσ1{x<−δσ} .

Recall that τ and δ stand for the damping factor and the fairing tolerance, respectively. Then, finally we have the estimates for the Γi from fairing: Γi,1 = h−2 (pi+1,1 − 2pi,1 + pi−1,1 ). As a measure of goodness of the estimates Γi,1 , we use the mean square error (MSE), 1 ∑ E[(Γi,1 − Γi )2 ], n−1 n−1

(10)

i=1

as n increases. Note that this MSE can be re-written as 1 ∑ E[(Γi,1 − Γi )2 ] n−1 n−3 i=3

=

=

n−3 ] ∑ [ 1 2 E (p − 2p + p − (α − 2α + α )) i+1,1 i,1 i−1,1 i+1 i i−1 (n − 1)h4

1 (n − 1)h4

i=3 n−3 ∑

[ ] E (A˜i+1 − 2A˜i + A˜i−1 )2

i=3

− → plus the sum of the summands at i = 1, 2, n − 2, n − 1, where A˜i = Zi + g(−τ Λi h4 + τ φ · Z i ). The first two and the last two terms are treated separately because they involve pi,1 obtained from the fairing algorithm at boundary points. The next proposition gives us an upper bound that is more amenable to analysis than the MSE itself. Lemma 1 The mean square error (10) is bounded by n−3 ∑ [ ] 1 E (Ai+1 − 2Ai + Ai−1 )2 , 4 (n − 1)h i=3

− → where Ai = Zi + g(τ φ · Z i ), plus extra terms [ (√ ) ] [ ] √ n−3 2 ∑ 3 4σ 3 1 4σ + 2δ ζi + ζi2 + 16δ 2 + 16δ +6 (n − 1)h4 π (n − 1)h4 π i=3

with ζi = |τ Λi+1 h4 | ∧ (δσ) + 2|τ Λi h4 | ∧ (δσ) + |τ Λi−1 h4 | ∧ (δσ). 13

b i which is the estimate for Γi with no fairing We aim to compare the MSE (10) with the MSE for the Γ algorithm applied. The latter is easily shown to be 6σ 2 /h4 . Then, the extra terms in the above lemma can be considered negligible for all large values of n when divided by 6σ 2 /h4 . Indeed, if maxt∈[0,1] |α′′ (t)| is well defined and finite, then each |Λi h4 | = h2 |Γi+1 − 2Γi + Γi−1 | is bounded by Kh2 for some constant K because (αi+1 − 2αi + αi−1 )/h2 converges as h decreases. Hence, the first term is O(h2 ). Moreover, if α(·) has bounded fourth derivatives on [0, 1], then by a similar observation we have that this term is O(h4 ). The second one in the extra terms is clearly O(h). Consequently, it is enough to consider n−3 ∑ [ ] 1 2 E (A − 2A + A ) i+1 i i−1 6(n − 1)σ 2

(11)

i=3

to measure the performance of fairing algorithm on the MSE of gamma estimates approximately. Proposition 1 The quantity (11) is given by (n − 5)/(n − 1) times √ ] 2 70δ 35τ (τ − 2) [ 2 2 1 + δ P(|Y | > x) − E[Y ; Y > x] + E Y ; |Y | ≤ x 3 18 √ 2 2δ 2 70τ δ 2δ 2 ′ ′ P(Y > x, Y > x) − P(Y > x, Y < −x) + E[Y ; |Y | ≤ x, Y ′ > x] + 3 3 9 (12) 35τ 2 8δ 2 8δ 2 ′ ′ ′′ ′′ + E[Y Y ; |Y | ≤ x, |Y | ≤ x] − P(Y > x, Y > x) + P(Y > x, Y < −x) 54 3 3 √ 70τ 2 8 70τ δ E[Y ; |Y | ≤ x, Y ′′ > x] − E[Y Y ′′ ; |Y | ≤ x, |Y ′′ | ≤ x] − 9 27 √ where x = 6δ/( 70τ ) and Y , Y ′ , Y ′′ are correlated standard normal random variables with Corr(Y, Y ′ ) = 0.4 and Corr(Y, Y ′′ ) = −0.8. Numerical implementation of (12) can be done using softwares such as Matlab which contains probability density functions (PDF) and cumulative distribution functions (CDF) for standard normal random variables and multivariate normal random variables. For the terms that involve Y only, we can further simplify them using the following formulae: E[Y ; Y > x] = ϕ(x),

E[Y 2 ; |Y | ≤ x] = −2xϕ(x) + 2Φ(x) − 1

where ϕ(·) and Φ(·) are the PDF and the CDF of a standard normal. Results of numerical experiments are shown in Figure 9. The graphs exhibit the behavior of this function depending on δ and τ . We can see the effectiveness of fairing algorithm in reducing the MSE of gamma estimates for appropriately chosen damping factor τ . This ratio of the MSEs seems to be minimized when τ is in a neighborhood of 0.5, which coincides with our choice of τ = 0.5 in the previous sections.

5.2

Dependent case

Now, let us look at two examples to illustrate how payoff functions affect the resulting correlation structure of the estimates pi , and thus that of the Zi . The first example is when the payoff function is 1{S(x)≥0.5} and 14

Effects on Gammas 1 δ=1 δ = 1.5 δ=2 δ = 2.5

approximate ratio of MSEs

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

1

damping factor

bi . Figure 9: The effects of δ and τ on the ratio of MSEs for Γi,1 and Γ S(x) = xU where U is a uniform random variable with values in [0, 1]. Thus, the true price (or expected payoff) is α(x) = (1 − 1/(2x))+ and the corresponding Monte Carlo estimates are given by 1 ∑ 1{Uk ≥n/(2i)} , m m

pi =

i = 0, . . . , n

k=1

after m simulation trials where the Uk are i.i.d. uniform random variables. It is also easy to compute the variance and covariance of the pi . With i < j, Var(pi ) =

1 (2i/n − 1)+ αi (1 − αi ) = , m 4i2 m/n2

Cov(pi , pj ) =

(2i/n − 1)+ 1 αi (1 − αj ) = , m 4ijm/n2

from which we notice that the correlation matters only when i/n > 0.5 and √ 2i − n Corr(pi , pj ) = . 2j − n For comparison, consider the second payoff function given by S(x) = xU itself. Then, α(x) = x/2 and the correlation is always 1. Therefore, when the payoff function involves more exotic features with discontinuities, we expect that the correlation structure would exhibit more apparent departure from the perfect correlation case. This leads to greatly fluctuating discrete gamma estimates and consequently the common random number technique becomes non-satisfactory in producing stable gammas. For the rest of this section, we investigate how the algorithm performs in the presence of nontrivial correlation structures. Let τ be fixed. For a given correlation structure, we [ generate correlated normal ] random vectors of length 7. Using those random vectors, we compute E (Ai+1 − 2Ai + Ai−1 )2 with and without fairing applied, where the expectations are set as the averages of 104 trials. Finally, the approximate ratio of MSEs is computed for this fixed τ and fixed correlation structure. To see the average performance of the algorithm based on many different correlation structures, we use 1,000 random instances of correlation structures. Then, for each fixed τ , the mean and standard deviation of approximate ratios of MSEs are obtained. Figure 10 shows those values for damping factor τ varying 15

Effects on Gammas 1 mean +1 std dev −1 std dev

approximate ratio of MSEs

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

1

damping factor

Figure 10: The effects of τ on the ratio of MSEs, using 1,000 randomly generated correlation structures. in [0, 1] together with ±1 standard deviations. We observe a similar performance as in Figure 9. There could be many different ways to produce various instances of correlation structures. One simple way we take here is to generate a 7 × 7 matrix such that entries are independent standard normals. If we divide each row by its magnitude and denote the resulting matrix by F , then Z = F W with W standard normal vector of size 7 becomes correlated normals. Remark So far, we have discussed the effectiveness of fairing algorithm in terms of MSEs by focusing on the first iteration of the algorithm. Instead, we can also analyze the algorithm by looking at some specific examples. Figures 11 and 13 present the ratios of MSEs of faired estimates and the original FD estimates when we vary τ and δ, respectively. The contingent claim used is DO from Section 4, but we observe similar behaviors for other examples. Even though the analysis of the first iteration shows that the optimal τ seems to be around 0.5, the MSE ratio flattens as the number of iterations increases, making the fairing result insensitive to τ as shown in Figure 11. However, for small τ values, the algorithm requires a large number of iterations, slowing down the convergence. On the other hand, for the examples used in this paper, we found that the fairing algorithm may fail to converge for large τ values as in Figure 12. (The algorithm converges for every τ < 0.75 in all numerical tests conducted.) Therefore, τ = 0.5 can be regarded as a reasonable choice that achieves convergence and efficiency. Regarding fairing tolerance, the ratio of MSEs decreases as δ increases, as one can easily expect. But, we observe from curves of the MSE ratio that there are very little improvements by increasing δ when δ is greater than 1. See Figure 13 for an illustration of a typical case. On the other hand, it would not be desirable to alter final price estimates too much from the original Monte Carlo estimates. Hence, we suggest to use δ between 1 and 1.5 as a practical choice for algorithm effectiveness, i.e., we change price estimates within one or a bit larger standard error range. Note that we used δ = 1 for the numerical tests in Section 4.

16

τ vs MSE ratio of Γ

τ vs MSE ratio of Γ 0.2

solid : MSTτ / MSE0 dashed : # of iteration for fairing 0.1

500

ratio of MSEs (MSEτ / MSE0)

MSE0 : MSE of Γ before fairing MSEτ : MSE of Γ after fairing using damping factor τ

0.15

0.05

0

0

0.1

0.2

0.3

0.4

τ

0.5

0.6

0 0.7

MSE0 : MSE of Γ before fairing MSEτ : MSE of Γ after fairing using damping factor τ

0.15

300

solid : MSTτ / MSE0 dashed : # of iteration for fairing 200 0.1

100

0.05

0

# of iterations

1000

# of iterations

ratio of MSEs (MSEτ / MSE0)

0.2

0

0.1

0.2

0.3

τ

0.4

0.5

0.6

0 0.7

Figure 11: The effects of τ on the ratio of MSEs for DO with δ = 1: (left) 104 paths (right) 106 paths

τ=0.5

τ=0.8

1.5

1.5 before fairing analytic after fairing

1

1

Gamma

0.5

0

0

−0.5

−0.5

−1

−1

90

95

100 S

105

110

−1.5 85

115

90

95

100 S

105

Figure 12: Gamma curves for DO with δ = 1: (left) τ = 0.5 (right) τ = 0.8

δ vs MSE ratio of Γ 1 0.9 MSE0 : MSE of Γ before fairing MSEδ : MSE of Γ after fairing using tolerance δ

0.8 ratio (MSEδ / MSE0) of Γ

Gamma

0.5

−1.5 85

before fairing analytic after fairing

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.5

1

1.5 δ

2

2.5

3

Figure 13: The effects of δ on the ratio of MSEs for DO with τ = 0.5

17

110

115

6

Concluding Remarks

Fairing is a collection of techniques that have been used in digital shape reconstruction. We introduced a simple curve/surface fairing algorithm to achieve reliable gamma estimates for complex financial derivatives. Instead of smoothing gamma curves or surfaces themselves, the algorithm varies the price estimates pi within a small interval [pi − δσ, pi + δσ] so that the new price estimates are still good candidates for the true prices. This small interval is determined by the fairing tolerance δ. Another important parameter in the algorithm is the damping factor τ which controls the oscillation of estimates from one iteration to the next. Pictorial and numeric examples are provided to demonstrate the performance of the method for complex exotic equity derivatives that are currently traded in Korean financial markets with great popularity. With no extra cost of applying the algorithm (as it is simple and fast), we observed that more stable gamma values are produced. Since Monte Carlo simulation is often the only method to rely on when it comes to the pricing and hedging of exotic derivatives, the importance of reliable and stable gamma values cannot be overemphasized. Finally, we showed the performance of fairing algorithm in terms of relative mean square errors of the b i without fairing. It is gamma estimates Γi,1 after the first iteration with respect to the gamma estimates Γ shown that it achieves more than 80% decrease in the mean square error with the damping factor τ ≈ 0.5. Also, through examples, we demonstrated that parameter values τ = 0.5 and δ between 1 and 1.5 are a reasonable choice in implementing the algorithm.

Acknowledgement Kang’s work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (Ministry of Education, Science and Technology, MEST) (No. 2009-0068043, 20100015465). Kim’s work was supported by Basic Science Research Program through the NRF funded by the MEST (No. 2011-0007651). Part of this project was completed while the second-named author was visiting the Fields Institute in Toronto, Canada. Their support and hospitality are greatly appreciated. Shin’s work was supported by DAPA and ADD (No. UD110006MD).

References Broadie, M. N. and P. Glasserman (1996). Estimating security price derivatives using simulation. Management Science 42, 269–285. Cho, S. and B. Choi (2001). Analysis of difference fairing based on DFT filter. Computer-Aided Design 33, 45–56. Eck, M. and R. Jaspert (1994). Automatic fairing of point sets. In N. Sapidis (Ed.), Designing Fair Curves and Surfaces, pp. 45–60. PA, USA: SIAM. Fourni´e, E., J.-M. Lasry, J. Lebuchoux, and P.-L. Lions (2001). Applications of malliavin calculus to Monte Carlo methods in finance II. Finance and Stochastics 5, 201–236. 18

Fourni´e, E., J.-M. Lasry, J. Lebuchoux, P.-L. Lions, and N. Touzi (1999). Applications of malliavin calculus to Monte Carlo methods in finance. Finance and Stochastics 3, 391–412. Giles, M. B. and P. Glasserman (2006). Smoking adjoints: fast Monte Carlo greeks. Risk 19, 88–92. Glasserman, P. (2004). Monte-Carlo Methods in Financial Engineering. NY, USA: Springer. Kang, W., K.-K. Kim, and H. Shin (2012). Denoising Monte Carlo sensitivity estimates. Operations Research Letters 40, 195–202. Salvi, P. and T. Varadi (2005). Local fairing of freeform curves and surfaces. In E. Yucesan, C.-H. Chen, J. Snowdon, and J. Charnes (Eds.), Proceedings of the 3rd Hungarian Conference on Computer Graphics and Geometry, pp. 1493–1501. Sapidis, N. and G. Farin (1990). Automatic fairng algorithm for B-spline curves. Computer-Aided Design 22, 121–129. Sun, X. F., P. Rosin, R. Martin, and F. Langbein (2008). Random walks for feature-preserving mesh denoising. Computer Aided Geometric Design 25, 437–456. Taleb, N. (1997). Dynamic Hedging: Managing Vanilla and Exotic Options. NY, USA: Wiley. Yamada, A., T. Furuhata, K. Shimada, and K. Hou (1999). A discrete spring model for generating fair curves and surfaces. In E. Yucesan, C.-H. Chen, J. Snowdon, and J. Charnes (Eds.), Seventh Pacific Conference on Computer Graphics and Applications, pp. 270–279. Zhang, C., P. Zhang, and F. Cheng (2001). Fairing spline curves and surface by minimizing energy. Computer-Aided Design 33, 913–923.

Appendix A: Proofs Proof of Lemma 1: For notational convenience, we denote A˜i+1 − 2A˜i + A˜i−1 , Zi+1 − 2Zi + Zi−1 by ˜ and F , respectively. We define G in a similar way so that Ai+1 − 2Ai + Ai−1 = F + G. Then, in F +G a straightforward manner, we get [ ] E (A˜i+1 − 2A˜i + A˜i−1 )2 − (Ai+1 − 2Ai + Ai−1 )2 [ ] ˜ 2 − (F + G)2 = E (F + G) [ ] ˜ − G) + (G ˜ − G)2 . = E 2(F + G)(G On the other hand, for any real numbers a and b, we have |g(a + b) − g(b)| ≤ |a| ∧ (δσ). This leads us to √ ˜ − G| ≤ ζi . Also, from |g(a)| ≤ δσ, |G| ≤ 4δσ. Since E[|F |] = E[|Zi+1 − 2Zi + Zi−1 |] = 6σE[|Z|] |G for a standard normal random variable Z, we get [ ] [ √ ] E (A˜i+1 − 2A˜i + A˜i−1 )2 − (Ai+1 − 2Ai + Ai−1 )2 ≤ E (2 6σ|Z| + 8δσ)ζi + ζi2 . 19

As for E[(Γi,1 − Γi )2 ] when i = 1, 2, n − 2, n − 1, we note that ] [ ] [√ ] 1 [ ˜ 2 ≤ 1 E (|F | + |G|) ˜ 2 ≤ 1 E ( 6σ|Z| + 4δσ)2 . E (F + G) h4 h4 h4 √ Combining above two observations and E[|Z|] = 2/π gives the result. E[(Γi,1 − Γi )2 ] =

Proof of Proposition 1: Let us fix i and compute E[A2i ], E[Ai+1 Ai−1 ], and E[Ai Ai−1 ] separately. For the first one, we proceed as follows: [( ] → )2 − 2 E[Ai ] = E Zi + g(τ φ · Z i ) [ − → − → ] = E Zi2 + 2Zi g(τ φ · Z i ) + g(τ φ · Z i )2 [ ] = σ 2 + 2σE [Xg(τ |φ|σY )] + E g(τ |φ|σY )2 − → where X = σ −1 Zi and Y = (σ|φ|)−1 φ · Z i . Then, it is easy to see that X, Y are two standard normal √ random variables with the correlation ρ1 = −1/|φ|. Since we can write ρ1 Y + 1 − ρ21 Z in place of X for an independent standard normal Z, we are led to [ ] E[A2i ] = σ 2 + 2σρ1 E [Y g(τ |φ|σY )] + E g(τ |φ|σY )2 . The second one can be computed in a similar fashion. [( )( )] → − → − E[Ai+1 Ai−1 ] = E Zi+1 + g(τ φ · Z i+1 ) Zi−1 + g(τ φ · Z i−1 ) [ ] − → − → − → − → = E Zi+1 g(τ φ · Z i−1 ) + Zi−1 g(τ φ · Z i+1 ) + g(τ φ · Z i+1 )g(τ φ · Z i−1 ) [ ] = E 2σXg(τ |φ|σY ) + g(τ |φ|σY )g(τ |φ|σY ′ ) where X, Y , and Y ′ are standard normal random variables with correlations ρ2 = Corr(X, Y ) = −1/(6|φ|) √ and ρ3 = Corr(Y, Y ′ ) = 7/(9|φ|2 ). Again, using ρ2 Y + 1 − ρ22 Z instead of X, we get [ ] E[Ai+1 Ai−1 ] = 2σρ2 E [Y g(τ |φ|σY )] + E g(τ |φ|σY )g(τ |φ|σY ′ ) . Likewise, we obtain [(

)] → )( − → − Zi + g(τ φ · Z i ) Zi−1 + g(τ φ · Z i−1 ) [ ] − → − → − → − → = E Zi g(τ φ · Z i−1 ) + Zi−1 g(τ φ · Z i ) + g(τ φ · Z i )g(τ φ · Z i−1 ) [ ] = E 2σXg(τ |φ|σY ) + g(τ |φ|σY )g(τ |φ|σY ′′ ) [ ] = 2σρ4 E [Y g(τ |φ|σY )] + E g(τ |φ|σY )g(τ |φ|σY ′′ )

E[Ai Ai−1 ] = E

where X, Y , and Y ′′ are correlated standard normals with ρ4 = Corr(X, Y ) = 2/(3|φ|) and ρ5 = Corr(Y, Y ′′ ) = −14/(9|φ|2 ). Combining all of three computations, we arrive at [ ] E (Ai+1 − 2Ai + Ai−1 )2 20

[ ] 70σ = 6σ 2 − E [Y g(τ |φ|σY )] + 6E g(τ |φ|σY )2 3|φ| [ ] [ ] +2E g(τ |φ|σY )g(τ |φ|σY ′ ) − 8E g(τ |φ|σY )g(τ |φ|σY ′′ ) . The next step is to represent each of these terms in their simplest forms so that numerical implementation becomes straightforward. First of all, ] [ ] [ ] [ δ δ δ 2 − E δσY ; Y < − + E τ |φ|σY ; |Y | ≤ E [Y g(τ |φ|σY )] = E δσY ; Y > τ |φ| τ |φ| τ |φ| [ ] ] [ δ δ = 2δσE Y ; Y > + τ |φ|σE Y 2 ; |Y | ≤ τ |φ| τ |φ| where we used the symmetric property of Y . For notational convenience, we write x for δ/(τ |φ|). Then, secondly, [ ] [ ] E g(τ |φ|σY )2 = (δσ)2 P (|Y | > x) + (τ |φ|σ)2 E Y 2 ; |Y | ≤ x . For the third term, we have [ ] E g(τ |φ|σY )g(τ |φ|σY ′ ) = 2(δσ)2 P(Y > x, Y ′ > x) − 2(δσ)2 P(Y > x, Y ′ < −x) [ ] [ ] +4τ |φ|δσ 2 E Y ; |Y | ≤ x, Y ′ > x + (τ |φ|σ)2 E Y Y ′ ; |Y | ≤ x, |Y ′ | ≤ x . Similar computations are made for the last term. We sum them up and divide it by 6σ 2 , obtaining the formula in the statement.

21

Appendix B: Numerical Results Table 2: Fairing performance for CO, number of sample paths = 104 Volatility

RMSE None Fairing

Ratio Fairing/None

RMSRE (%) None Fairing

Ratio Fairing/None

Price

0.1 0.2 0.3 0.4 0.5

0.008431 0.009865 0.004483 0.043476 0.019260

0.010312 0.011016 0.011717 0.030359 0.039004

1.223 1.117 2.614 0.698 2.025

6.056399 6.495915 0.754664 2.532658 1.286652

3.440069 2.188127 1.288579 2.064095 1.877689

0.568 0.337 1.707 0.815 1.459

Delta

0.1 0.2 0.3 0.4 0.5

0.001725 0.001146 0.000861 0.001667 0.002209

0.002754 0.001505 0.001845 0.001906 0.002030

1.596 1.313 2.142 1.143 0.919

6.227193 5.744987 1.863818 0.708596 0.603340

4.892254 2.004467 1.034105 0.548860 0.744392

0.786 0.349 0.555 0.775 1.234

Gamma

0.1 0.2 0.3 0.4 0.5

0.001304 0.000818 0.000809 0.000790 0.001006

0.001431 0.000498 0.000509 0.000504 0.000421

1.097 0.609 0.629 0.639 0.419

31.206356 4.687777 4.073068 3.101856 4.345946

43.310057 2.877317 2.623079 2.255516 1.778993

1.388 0.614 0.644 0.727 0.409

22

Table 3: Fairing performance for CO, number of sample paths = 105 Volatility

None

RMSE Fairing

Ratio Fairing/None

RMSRE (%) None Fairing

Ratio Fairing/None

Price

0.1 0.2 0.3 0.4 0.5

0.000625 0.001281 0.012215 0.011412 0.002801

0.000499 0.001031 0.014139 0.007892 0.006546

0.799 0.804 1.158 0.692 2.337

1.198545 1.528602 1.558366 0.720879 0.074107

2.165792 0.540757 1.243107 0.573542 0.279549

1.807 0.354 0.798 0.796 3.772

Delta

0.1 0.2 0.3 0.4 0.5

0.000367 0.000433 0.000717 0.000566 0.000423

0.000238 0.000394 0.001187 0.000686 0.000821

0.648 0.910 1.655 1.212 1.941

2.593194 0.590155 0.819275 0.242859 0.118531

4.000983 0.978936 0.701442 0.256121 0.320519

1.543 1.659 0.856 1.055 2.704

Gamma

0.1 0.2 0.3 0.4 0.5

0.000431 0.000366 0.000208 0.000294 0.000196

0.000401 0.000241 0.000269 0.000207 0.000184

0.931 0.657 1.295 0.706 0.938

9.481634 1.928106 0.746287 1.135561 0.874006

22.353851 0.932435 1.275929 0.977121 0.817375

2.358 0.484 1.710 0.860 0.935

Table 4: Fairing performance for CO, number of sample paths = 106 Volatility

None

RMSE Fairing

Ratio Fairing/None

RMSRE (%) None Fairing

Ratio Fairing/None

Price

0.1 0.2 0.3 0.4 0.5

0.000275 0.000511 0.001030 0.001802 0.005781

0.000184 0.000918 0.001618 0.001087 0.007900

0.669 1.797 1.571 0.603 1.366

0.121964 0.685632 0.103982 0.050889 0.189364

0.431934 1.197275 0.108245 0.062015 0.273682

3.541 1.746 1.041 1.219 1.445

Delta

0.1 0.2 0.3 0.4 0.5

0.000097 0.000138 0.000103 0.000250 0.000187

0.000103 0.000244 0.000226 0.000211 0.000415

1.058 1.773 2.197 0.846 2.220

0.249694 0.507179 0.068867 0.070394 0.069341

0.639947 0.939763 0.189072 0.073581 0.189217

2.563 1.853 2.745 1.045 2.729

Gamma

0.1 0.2 0.3 0.4 0.5

0.000076 0.000107 0.000082 0.000086 0.000118

0.000214 0.000096 0.000074 0.000070 0.000064

2.822 0.900 0.902 0.812 0.543

3.434934 0.696168 0.313318 0.325209 0.525378

8.077912 0.543899 0.468134 0.301834 0.298024

2.352 0.781 1.494 0.928 0.567

23

Table 5: Fairing performance for DO, number of sample paths = 104 Volatility

None

RMSE Fairing

Ratio Fairing/None

RMSRE (%) None Fairing

Ratio Fairing/None

Price

0.1 0.2 0.3 0.4 0.5

0.177991 0.144163 0.093758 0.168825 0.227867

0.204037 0.155648 0.059618 0.097084 0.176645

1.146 1.080 0.636 0.575 0.775

6.204482 5.732985 1.999274 0.595954 0.719249

7.161622 4.669140 1.495868 0.237579 0.610910

1.154 0.814 0.748 0.399 0.849

Delta

0.1 0.2 0.3 0.4 0.5

0.091601 0.075925 0.068504 0.066104 0.078895

0.085297 0.052138 0.029330 0.023914 0.033109

0.931 0.687 0.428 0.362 0.420

21.590139 4.473775 3.308067 2.232643 3.397535

24.636857 1.926412 1.777163 1.232896 1.621266

1.141 0.431 0.537 0.552 0.477

Gamma

0.1 0.2 0.3 0.4 0.5

0.184289 0.197668 0.133909 0.196291 0.145305

0.070915 0.026891 0.022409 0.012310 0.010608

0.385 0.136 0.167 0.063 0.073

136.011529 83.801240 248.176466 667.814406 23289.509582

59.310041 6.686550 27.313975 16.996175 212.832827

0.436 0.080 0.110 0.025 0.009

Table 6: Fairing performance for DO, number of sample paths = 105 Volatility

None

RMSE Fairing

Ratio Fairing/None

RMSRE (%) None Fairing

Ratio Fairing/None

Price

0.1 0.2 0.3 0.4 0.5

0.048711 0.051435 0.066943 0.058959 0.041649

0.031598 0.045635 0.099913 0.037956 0.029571

0.649 0.887 1.493 0.644 0.710

2.839865 0.359608 0.747506 0.268482 0.135095

4.757498 0.874123 0.992980 0.174729 0.074141

1.675 2.431 1.328 0.651 0.549

Delta

0.1 0.2 0.3 0.4 0.5

0.038277 0.032428 0.014477 0.026170 0.015758

0.026569 0.025313 0.017056 0.016340 0.006374

0.694 0.781 1.178 0.624 0.404

8.542883 1.414315 0.594527 1.020187 0.692426

13.584750 1.802560 0.573660 0.747576 0.244048

1.590 1.275 0.965 0.733 0.352

Gamma

0.1 0.2 0.3 0.4 0.5

0.085309 0.042971 0.031946 0.047244 0.040049

0.063838 0.017349 0.005791 0.006809 0.002171

0.748 0.404 0.181 0.144 0.054

21.708945 24.398032 42.658058 76.938641 155.816662

19.606209 4.846873 2.606274 14.543675 166.204966

0.903 0.199 0.061 0.189 1.067

24

Table 7: Fairing performance for DO, number of sample paths = 106 Volatility

None

RMSE Fairing

Ratio Fairing/None

RMSRE (%) None Fairing

Ratio Fairing/None

Price

0.1 0.2 0.3 0.4 0.5

0.009760 0.014863 0.009818 0.025664 0.018673

0.006458 0.022572 0.011665 0.019397 0.017536

0.662 1.519 1.188 0.756 0.939

0.333251 0.511282 0.055361 0.078652 0.071071

0.881242 0.896094 0.100499 0.052960 0.074206

2.644 1.753 1.815 0.673 1.044

Delta

0.1 0.2 0.3 0.4 0.5

0.006364 0.007903 0.006528 0.007930 0.010076

0.005007 0.007984 0.003615 0.005385 0.004172

0.787 1.010 0.554 0.679 0.414

2.887798 0.406854 0.258660 0.291070 0.460961

2.893965 0.579385 0.144399 0.174859 0.209221

1.002 1.424 0.558 0.601 0.454

Gamma

0.1 0.2 0.3 0.4 0.5

0.010948 0.014575 0.012971 0.018651 0.013359

0.012300 0.005231 0.003025 0.002106 0.002584

1.123 0.359 0.233 0.113 0.193

10.685524 9.889703 14.849271 52.548515 661.434011

9.064668 1.933308 3.415856 5.453844 150.584595

0.848 0.195 0.230 0.104 0.228

Table 8: Fairing performance for ELS1, number of sample paths = 104 Maturity

None

RMSE Fairing

Ratio Fairing/None

RMSRE (%) None Fairing

Ratio Fairing/None

Price

1M 3M 5M 7M 9M 11M

0.026772 0.043370 0.091541 0.060849 0.124374 0.106448

0.015176 0.035100 0.116871 0.087711 0.154804 0.111599

0.567 0.809 1.277 1.441 1.245 1.048

0.024156 0.040728 0.090537 0.063189 0.131593 0.113904

0.013683 0.032964 0.116250 0.092160 0.164284 0.120635

0.566 0.809 1.284 1.458 1.248 1.059

Delta

1M 3M 5M 7M 9M 11M

0.008870 0.013627 0.018166 0.026126 0.025667 0.027759

0.004243 0.007820 0.010850 0.009932 0.014513 0.011158

0.478 0.574 0.597 0.380 0.565 0.402

45.042267 11.125575 3.791569 6.885227 5.461150 6.378829

29.009132 2.836769 5.074421 3.272720 3.087952 2.920302

0.644 0.255 1.338 0.475 0.565 0.458

Gamma

1M 3M 5M 7M 9M 11M

0.013089 0.030933 0.036285 0.042971 0.043630 0.039148

0.001605 0.004728 0.002407 0.003351 0.004873 0.004551

0.123 0.153 0.066 0.078 0.112 0.116

170.944067 94.916265 83.672159 119.386012 142.049137 158.717747

61.484846 7.814702 8.892477 10.844639 19.154051 22.222170

0.360 0.082 0.106 0.091 0.135 0.140

25

Table 9: Fairing performance for ELS1, number of sample paths = 105 Maturity

RMSE None Fairing

Ratio Fairing/None

RMSRE (%) None Fairing

Ratio Fairing/None

Price

1M 3M 5M 7M 9M 11M

0.009988 0.015732 0.028741 0.023301 0.026116 0.034854

0.009041 0.012036 0.023026 0.018377 0.017394 0.028540

0.905 0.765 0.801 0.789 0.666 0.819

0.009172 0.015271 0.028781 0.023292 0.026662 0.036485

0.008248 0.011556 0.023004 0.018151 0.017613 0.029925

0.899 0.757 0.799 0.779 0.661 0.820

Delta

1M 3M 5M 7M 9M 11M

0.003476 0.005888 0.007681 0.007702 0.006119 0.006672

0.001551 0.003144 0.002870 0.001995 0.001078 0.001437

0.446 0.534 0.374 0.259 0.176 0.215

25.007081 3.023069 3.397839 2.768374 1.544639 1.788126

26.501342 2.804295 2.278815 1.624420 0.551511 0.802062

1.060 0.928 0.671 0.587 0.357 0.449

Gamma

1M 3M 5M 7M 9M 11M

0.006361 0.010540 0.011789 0.014614 0.010928 0.014552

0.001189 0.001209 0.000906 0.000600 0.000322 0.000454

0.187 0.115 0.077 0.041 0.029 0.031

77.965766 37.974661 35.792781 40.307293 37.409181 64.440685

68.387683 4.815578 3.783252 3.452645 1.460261 2.231438

0.877 0.127 0.106 0.086 0.039 0.035

Table 10: Fairing performance for ELS1, number of sample paths = 106 Maturity

None

RMSE Fairing

Ratio Fairing/None

RMSRE (%) None Fairing

Ratio Fairing/None

Price

1M 3M 5M 7M 9M 11M

0.002658 0.004025 0.004892 0.005648 0.003963 0.003954

0.003407 0.003688 0.005633 0.004725 0.004648 0.005322

1.282 0.916 1.151 0.837 1.173 1.346

0.002420 0.003808 0.004695 0.005663 0.004114 0.004195

0.003102 0.003525 0.005434 0.004631 0.004826 0.005490

1.282 0.926 1.157 0.818 1.173 1.309

Delta

1M 3M 5M 7M 9M 11M

0.000981 0.001475 0.001713 0.002042 0.002009 0.001970

0.000581 0.001174 0.000904 0.001163 0.000596 0.000744

0.593 0.796 0.528 0.570 0.297 0.377

29.978766 1.235948 0.832191 0.810540 0.684755 0.642783

27.903137 0.779192 0.692663 0.808872 0.275490 0.309150

0.931 0.630 0.832 0.998 0.402 0.481

Gamma

1M 3M 5M 7M 9M 11M

0.001526 0.003849 0.003209 0.003910 0.004715 0.004420

0.000708 0.000497 0.000273 0.000388 0.000269 0.000167

0.464 0.129 0.085 0.099 0.057 0.038

69.692625 11.540481 8.402842 11.863576 15.575413 17.578385

52.667074 4.395740 1.490872 1.949523 1.062526 0.780319

0.756 0.381 0.177 0.164 0.068 0.044

26

Table 11: Fairing performance for ELS2, number of sample paths = 104 Maturity

RMSE None Fairing

Ratio Fairing/None

RMSRE (%) None Fairing

Ratio Fairing/None

Price

1M 3M 5M 7M 9M 11M

0.033084 0.057145 0.130751 0.082856 0.131875 0.108610

0.026732 0.059545 0.140888 0.092738 0.150059 0.109098

0.808 1.042 1.078 1.119 1.138 1.004

0.029835 0.054356 0.131126 0.088028 0.141872 0.118138

0.024098 0.056950 0.141777 0.098868 0.162125 0.119415

0.808 1.048 1.081 1.123 1.143 1.011

Delta 1

1M 3M 5M 7M 9M 11M

0.008793 0.012892 0.017210 0.021428 0.022969 0.024212

0.006660 0.006206 0.006946 0.009625 0.011262 0.010417

0.757 0.481 0.404 0.449 0.490 0.430

59.427882 10.095031 6.033619 6.950013 6.221762 6.621264

42.770070 4.211589 2.280254 3.524935 3.063781 3.728343

0.720 0.417 0.378 0.507 0.492 0.563

Delta 2

1M 3M 5M 7M 9M 11M

0.005573 0.013558 0.013138 0.019124 0.012211 0.019139

0.002251 0.008629 0.010743 0.011733 0.006995 0.010155

0.404 0.636 0.818 0.614 0.573 0.531

453.964648 35.546945 16.355078 11.406066 8.956580 9.777811

409.756625 19.091001 10.186645 8.654134 6.393383 6.217412

0.903 0.537 0.623 0.759 0.714 0.636

Gamma 11

1M 3M 5M 7M 9M 11M

0.012409 0.030306 0.033010 0.035884 0.040377 0.035805

0.002848 0.004581 0.001925 0.003199 0.003873 0.005519

0.230 0.151 0.058 0.089 0.096 0.154

2042.119060 117.069119 111.551933 128.218000 156.904086 166.655311

338.539397 14.322664 6.343803 13.422504 15.582539 28.578097

0.166 0.122 0.057 0.105 0.099 0.171

Gamma 22

1M 3M 5M 7M 9M 11M

0.009358 0.031598 0.025977 0.037298 0.032570 0.035285

0.001493 0.005422 0.003748 0.008772 0.003434 0.004192

0.160 0.172 0.144 0.235 0.105 0.119

1070.117668 593.728970 149.792054 129.832723 145.192843 175.920220

478.007650 86.114763 29.734086 26.034627 14.843527 22.410546

0.447 0.145 0.199 0.201 0.102 0.127

Gamma 12

1M 3M 5M 7M 9M 11M

0.000523 0.002116 0.002964 0.003465 0.003291 0.003568

0.000305 0.000535 0.000547 0.000965 0.000711 0.001188

0.582 0.253 0.184 0.278 0.216 0.333

4046.771682 5106.211309 3091.679720 3302.801124 977.581637 1009.590933

1923.908560 1395.730774 515.772550 389.754606 298.692771 905.007315

0.475 0.273 0.167 0.118 0.306 0.896

27

Table 12: Fairing performance for ELS2, number of sample paths = 105 Maturity

RMSE None Fairing

Ratio Fairing/None

RMSRE (%) None Fairing

Ratio Fairing/None

Price

1M 3M 5M 7M 9M 11M

0.009949 0.016027 0.026624 0.019818 0.021299 0.029558

0.008403 0.013773 0.024659 0.018349 0.017138 0.026948

0.845 0.859 0.926 0.926 0.805 0.912

0.009134 0.015579 0.026776 0.020201 0.022602 0.032072

0.007698 0.013329 0.024846 0.018702 0.017972 0.029229

0.843 0.856 0.928 0.926 0.795 0.911

Delta 1

1M 3M 5M 7M 9M 11M

0.003495 0.005577 0.006993 0.006380 0.006041 0.005968

0.001772 0.003095 0.003174 0.002032 0.002043 0.002216

0.507 0.555 0.454 0.319 0.338 0.371

31.377765 3.070638 3.189354 2.445714 1.754497 1.724396

26.813794 2.154082 1.516771 1.044479 0.654452 0.803891

0.855 0.702 0.476 0.427 0.373 0.466

Delta 2

1M 3M 5M 7M 9M 11M

0.001233 0.002786 0.004393 0.004874 0.004672 0.004564

0.000573 0.001759 0.002458 0.003427 0.002375 0.002850

0.465 0.631 0.560 0.703 0.508 0.624

278.992852 14.675797 6.467751 3.801238 3.784935 3.612032

180.654766 9.898480 3.897348 2.649542 2.126802 2.288814

0.648 0.674 0.603 0.697 0.562 0.634

Gamma 11

1M 3M 5M 7M 9M 11M

0.006239 0.010013 0.011197 0.012261 0.010620 0.013112

0.001082 0.001414 0.001236 0.000880 0.000696 0.000756

0.173 0.141 0.110 0.072 0.066 0.058

547.459024 38.291820 43.173254 42.045901 42.605912 62.077639

214.146369 6.622703 5.720355 4.588565 3.431557 4.599280

0.391 0.173 0.132 0.109 0.081 0.074

Gamma 22

1M 3M 5M 7M 9M 11M

0.001819 0.006808 0.009748 0.011297 0.009781 0.010467

0.000712 0.001286 0.000922 0.001035 0.000918 0.000896

0.391 0.189 0.095 0.092 0.094 0.086

511.204917 315.426466 53.359035 46.575628 41.572930 51.193592

205.932286 111.532618 9.989504 8.202007 7.708663 7.626556

0.403 0.354 0.187 0.176 0.185 0.149

Gamma 12

1M 3M 5M 7M 9M 11M

0.000176 0.000636 0.000926 0.001036 0.001030 0.001071

0.000142 0.000179 0.000292 0.000290 0.000265 0.000325

0.810 0.281 0.315 0.280 0.258 0.304

2443.485241 1326.216089 720.166137 956.262255 443.069192 693.343928

1800.273969 520.126928 531.307996 250.915086 88.899924 198.121479

0.737 0.392 0.738 0.262 0.201 0.286

28

Table 13: Fairing performance for ELS2, number of sample paths = 106 Maturity

RMSE None Fairing

Ratio Fairing/None

RMSRE (%) None Fairing

Ratio Fairing/None

Price

1M 3M 5M 7M 9M 11M

0.002911 0.006726 0.008274 0.006961 0.007541 0.009645

0.003525 0.005754 0.008704 0.007053 0.007990 0.010000

1.211 0.856 1.052 1.013 1.060 1.037

0.002651 0.006346 0.008114 0.007018 0.008049 0.010537

0.003209 0.005449 0.008606 0.007093 0.008484 0.010814

1.211 0.859 1.061 1.011 1.054 1.026

Delta 1

1M 3M 5M 7M 9M 11M

0.000965 0.001274 0.001666 0.001833 0.001582 0.001854

0.000643 0.001021 0.000919 0.001115 0.000780 0.001047

0.666 0.802 0.552 0.608 0.493 0.565

39.251085 1.158340 0.782159 0.758948 0.572684 0.567359

39.555403 0.702197 0.436444 0.445795 0.238785 0.301094

1.008 0.606 0.558 0.587 0.417 0.531

Delta 2

1M 3M 5M 7M 9M 11M

0.000420 0.001024 0.001665 0.001657 0.001802 0.001682

0.000511 0.000776 0.000938 0.001002 0.000993 0.000816

1.218 0.757 0.563 0.605 0.551 0.485

148.585265 5.628705 1.518567 1.104430 1.029175 0.893341

137.128162 4.719485 1.410389 0.640570 0.564742 0.561947

0.923 0.838 0.929 0.580 0.549 0.629

Gamma 11

1M 3M 5M 7M 9M 11M

0.001493 0.003803 0.003293 0.003645 0.003767 0.003833

0.000712 0.000481 0.000354 0.000398 0.000306 0.000439

0.477 0.126 0.108 0.109 0.081 0.114

216.048600 14.561589 10.747362 12.615216 14.507076 17.604703

205.834303 4.020437 1.881142 1.882688 1.416406 2.437236

0.953 0.276 0.175 0.149 0.098 0.138

Gamma 22

1M 3M 5M 7M 9M 11M

0.000869 0.002140 0.003074 0.005245 0.003441 0.003329

0.000507 0.000316 0.000440 0.000396 0.000457 0.000303

0.584 0.148 0.143 0.076 0.133 0.091

241.660508 95.880964 16.351848 15.688571 14.335122 15.133258

162.651063 80.676162 2.938703 1.922473 3.430490 2.420689

0.673 0.841 0.180 0.123 0.239 0.160

Gamma 12

1M 3M 5M 7M 9M 11M

0.000106 0.000221 0.000305 0.000324 0.000369 0.000380

0.000099 0.000096 0.000112 0.000129 0.000114 0.000118

0.934 0.435 0.369 0.399 0.308 0.310

1857.373371 373.803350 262.351788 254.714631 125.985431 164.644789

1626.926020 218.324963 200.695015 136.117780 28.771846 48.020995

0.876 0.584 0.765 0.534 0.228 0.292

29

Before Fairing, # of Paths = 104

4

After Fairing, # of Paths = 10 0.05

0.04 Gamma

Gamma

0.05

0.03 RMSE = 0.0008 RMSRE = 0.0407

0.02 0.01 85

90

95

100 S

105

110

0.04 0.03

0.01 85

115 5

0.04

Gamma

Gamma

0.05

0.04 0.03 RMSE = 0.0002 RMSRE = 0.0075 90

95

100 S

105

110

0.01 85

115

0.04

Gamma

Gamma

0.04 0.03 RMSE = 0.0001 RMSRE = 0.0031 100 S

105

110

115

90

95

100 S

105

110

115

6

0.05

95

110

After Fairing, # of Paths = 10

0.05

90

105

RMSE = 0.0003 RMSRE = 0.0128

0.02

6

0.01 85

100 S

0.03

Before Fairing, # of Paths = 10

0.02

95

After Fairing, # of Paths = 10

0.05

0.01 85

90

5

Before Fairing, # of Paths = 10

0.02

RMSE = 0.0005 RMSRE = 0.0262

0.02

0.03

0.01 85

115

RMSE = 0.0001 RMSRE = 0.0047

0.02 90

Figure 14: Gamma curves for CO

30

95

100 S

105

110

115

Before Fairing, # of Paths = 104

4

After Fairing, # of Paths = 10

0.6

0

0.2 0 −0.2

−0.2 −0.4 85

90

95

100 S

105

110

−0.4 85

115 5

RMSE = 0.0319 RMSRE = 0.4266

105

110

115

RMSE = 0.0058 RMSRE = 0.0261

0.4 Gamma

Gamma

100 S

After Fairing, # of Paths = 10

0.2 0 −0.2

0.2 0 −0.2

90

95

100 S

105

110

−0.4 85

115 6

90

95

100 S

105

110

115

6

Before Fairing, # of Paths = 10

After Fairing, # of Paths = 10

0.6

0.6 RMSE = 0.0130 RMSRE = 0.1485

RMSE = 0.0030 RMSRE = 0.0342

0.4 Gamma

0.4 Gamma

95

0.6

0.4

0.2 0 −0.2 −0.4 85

90

5

Before Fairing, # of Paths = 10 0.6

−0.4 85

RMSE = 0.0224 RMSRE = 0.2731

0.4

0.2

Gamma

Gamma

0.6

RMSE = 0.1339 RMSRE = 2.4818

0.4

0.2 0 −0.2

90

95

100 S

105

110

−0.4 85

115

90

Figure 15: Gamma curves for DO

31

95

100 S

105

110

115

Before Fairing, # of Paths = 104

4

After Fairing, # of Paths = 10

0

−0.1

Gamma

Gamma

0 −0.05 RMSE = 0.0355 RMSRE = 0.7253

90

95

100 S

105

110

−0.1

85

115 5

−0.05

Gamma

Gamma

−0.05

85

RMSE = 0.0134 RMSRE = 0.2818 90

95

100 S

105

110

115

85

−0.05

Gamma

Gamma

−0.05 RMSE = 0.0038 RMSRE = 0.0718 95

100 S

105

110

115

90

95

100 S

105

110

115

6

0

90

110

After Fairing, # of Paths = 10

0

85

105

RMSE = 0.0023 RMSRE = 0.0512

−0.15

6

−0.15

100 S

−0.1

Before Fairing, # of Paths = 10

−0.1

95

After Fairing, # of Paths = 10 0

−0.15

90

5

Before Fairing, # of Paths = 10 0

−0.1

RMSE = 0.0039 RMSRE = 0.1028

−0.15

−0.15 85

−0.05

−0.1

RMSE = 0.0005 RMSRE = 0.0232

−0.15 115

85

90

Figure 16: Gamma curves for ELS1

32

95

100 S

105

110

115

100

2

80

1

60 120

0 120

0

S2

0 120

80

120

100

100 80

80

80

120

100

100

S2

S1

100 80

S2

S1

0

0.1

4 2

120

100

80

S1

0.5

Γ22

−0.1

Γ12

−0.2

0

Γ11

2

∆2

3

∆1

Price

120

0.2

0

0

−0.2 120

−0.2 120 120

100

S2

100

100 80

80

−0.5 120 120

S2

S1

80

80

120

100

100

100

S2

S1

80

80

S1

Figure 17: Price, delta, gamma surfaces for ELS2 after fairing

Γ11

Γ11

0.2

0.2

0.1

0.1

0

Fairing −0.1

0

−0.1

−0.2 120 110

−0.2 120

120 100

90

S2

110

110

100

120

90 80

80

110

100 100

90

S1

S2

90 80

Figure 18: Changes of Γ11 surfaces for ELS2

33

80

S1