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Journal of Computational Finance

Volume 16/Number 2, Winter 2012/13 (85–118)

Estimating multiple option Greeks simultaneously using random parameter regression Haifeng Fu Business School, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Road, Suzhou 215123, China; email: [email protected]

Xing Jin Finance Group, Warwick Business School, University of Warwick, Coventry CV4 7AL, United Kingdom; email: [email protected]

Guangming Pan Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371; email: [email protected]

Yanrong Yang Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371; email: [email protected]

The derivatives of option prices with respect to underlying parameters are commonly referred to as Greeks, and they measure the sensitivities of option prices to these parameters. When the closed-form solutions for option prices do not exist and the discounted payoff functions of the options are not sufficiently smooth, estimating Greeks is computationally challenging and could be a burdensome task for high-dimensional problems in particular. The aim of this paper is to develop a new method for estimating option Greeks by using random parameters and leastsquares regression. Our approach has several attractive features. First, just like the finite-difference method, it is easy to implement and does not require explicit knowledge of the probability density function and the pathwise derivative of the underlying stochastic model. Second, it can be applied to options with discontinuous discounted payoffs as well as options with continuous discounted payoffs.

The authors thank two anonymous referees for their numerous constructive and insightful comments on a previous version of this paper. The authors have also benefited from the helpful comments and suggestions of Yeneng Sun and Hwee Huat Tan. 85

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H. Fu et al Third, and most importantly, we can estimate multiple derivatives simultaneously. The performance of our approach is illustrated for a variety of examples with up to fifty Greeks estimated simultaneously. The algorithm is able to produce computationally efficient results with good accuracy.

1 INTRODUCTION Greeks are widely used in financial risk management. Greeks are the derivatives of option prices with respect to underlying parameters such as the stock price, volatility, interest rate and time-to-maturity. In other words, Greeks represent the sensitivities of option prices to those parameters and each Greek measures a different dimension of risk in the option position. For example, the derivative of an option price with respect to the price of an underlying asset (known as delta) measures the sensitivity of the option price to a small change in the price of an underlying asset. The delta can be used to form a delta-neutral portfolio consisting only of cash, a position in an underlying asset and an offsetting position in the option written on the asset, where the delta gives the units of the underlying asset held in the portfolio. The corresponding second-order derivative (known as the gamma) of the option price with respect to the underlying asset price measures the rate of change of the delta. The gamma can be used to adjust the delta or the number of units of the underlying asset in the portfolio above to keep the portfolio delta-neutral. Another important application of Greeks in risk management is the simulationbased estimation of value-at-risk (VaR) of a portfolio containing options (see Glasserman et al (2000) for details). Traditional Monte Carlo simulation is one main technique for estimating VaR but it is computationally expensive since the entire portfolio is repriced on each simulation trial. To be more specific, the whole procedure for estimating portfolio VaR is very time consuming when the simple closed-form solutions for prices of some options in the portfolio do not exist and one has to resort to some simulation-based or numerical methods to price the options. In contrast with the full Monte Carlo simulation, the delta–gamma approximation is much less time consuming because it avoids repricing the entire portfolio on each simulation trial. The examples above highlight the importance of the accurate estimation of Greeks. The conventional approaches to estimating the option Greeks as well as the derivatives of mathematical expectations or stochastic systems consist of three methods: the finitedifference (FD) approximation method (see, for example, Cao (1985); Glasserman (2003); and Zazanis and Suri (1993)), the perturbation analysis (PA) method (see, for example, Cao (1985); Glasserman (1991); Ho (1987); and Suri (1987, 1989)) and the likelihood ratio (LR) method (also called the score function method; see, for example, Rubinstein et al (1993) and Glasserman (2003)). Journal of Computational Finance

Volume 16/Number 2, Winter 2012/13

Estimating multiple option Greeks

The FD method is mainly about approximating derivatives by differences. It is simple and easy to apply but it is also quite computationally demanding. Estimating p derivatives for each simulation run requires pC1 measurements of the option payoff function using forward differences and 2p measurements using central differences. This difficulty becomes even more severe for second-order derivatives, as emphasized by Glasserman (2003, p. 385). This makes the FD method less attractive in dealing with the multidimensional cases that involve a large number of derivatives. Furthermore, the asymptotic convergence rate of the FD method is also lower than that of the PA method (see L’Ecuyer and Perron (1994) and Zazanis and Suri (1993)), often leading to a large mean squared error. The PA method differentiates each simulated outcome with respect to the parameter of interest. The PA method is attractive since, when applicable, it gives an unbiased estimator with the canonical asymptotic rate of n1=2 . It has been empirically documented that the PA method, when applicable, is of higher computational efficiency and produces the best estimates of sensitivities. But it is not applicable to options that have discontinuous payoffs (for example, the binary option and barrier option), and hence it is also not applicable to estimating second-order derivatives of option prices, including the gamma (see Glasserman (2003, p. 420)). To circumvent the difficulty of discontinuity, Fu and Hu (1997) proposed an approach, called smoothed perturbation analysis (SPA), to smooth the discontinuous payoff function by conditioning on certain random variables. But the success of SPA depends on the availability of the appropriate random variables to be conditioned upon. Recently, Hong and Liu (2008) developed another approach for circumventing the difficulty of discontinuity. Their method is applicable to estimating gamma but it may be difficult to verify some technical conditions for more general processes. In contrast, the LR method differentiates a probability density rather than the discounted payoff function and, therefore, does not require smoothness of the discounted payoff. Like the PA method, when applicable, the LR method produces unbiased estimates. However, the LR method generally produces estimates with large variances (see, for example, Glasserman (2003)). The LR method also requires explicit knowledge of the relevant probability density, which may not be easily obtained for more general models. The LR method has recently been extended by another Monte Carlo approach using ideas from Malliavin calculus (see, for example, Mrad et al (2006)). In particular, when the relevant densities used in the LR method are not available, the score is replaced with a Skorohod integral, which is estimated through simulation. As pointed out by Glasserman (2003), the evaluation of the Skorohod integral, however, is often computationally demanding. The purpose of this paper is to develop a new method for estimating multiple Greeks simultaneously for options with complex payoffs by using random parameters and the least-squares regression. Our approach has several attractive features. First, Research Paper

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just like the finite-difference method, it is easy to implement and does not require explicit knowledge of the probability density function and pathwise derivative of the underlying stochastic model used in LR and PA methods, respectively. Second, it can be applied to estimating Greeks for options with discontinuous discounted payoffs as well as options with continuous discounted payoffs. Third, and most importantly, it can estimate multiple derivatives simultaneously. Our method is related to the simultaneous perturbation stochastic approximation (SPSA) methods. Just like our method, SPSA methods require only two objective function measurements per iteration for the underlying gradient approximation regardless of the dimension of the optimization problem.1 Hence, SPSA methods are usually much more efficient than FD methods in high-dimensional cases. Recently, Spall (2009) proposed two enhancements, called feedback and optimal weighting methods, to improve the quality of the estimates of Hessian matrices by using the SPSA methods. More specifically, the feedback approach improves the quality of the estimate of a Hessian matrix by using the previous estimates of the Hessian matrix to reduce error, and the optimal weighting method improves the quality via an optimal weighting of per-iteration estimates. These methods are very powerful in error reduction and, thus, have the potential to be efficient techniques for estimating the second-order financial Greeks simultaneously. Fu (2002, 2006) and Spall (2003, 2009) are good references for SPSA methods. Although our algorithm offers some advantages, it also has some disadvantages. Perhaps one of the key drawbacks of our algorithm is that, like the FD method, our estimators are biased and the accuracy hinges on the trade-off between bias and variance. Another drawback is that the convergence rate of our algorithm is lower than the canonical asymptotic rate of n1=2 attained by PA and LR methods. We test this method in a range of applications. First, we apply the approach to two examples, where we estimate one delta and one gamma simultaneously. The two examples are a cash-or-nothing call option and a barrier option with irregular payoffs. Second, we consider three geometric average options over five, twenty and fifty assets and evaluate our method by simultaneously estimating the corresponding five, twenty and fifty deltas, respectively. Finally, the empirical performance of the method is provided via numerical results on simultaneously estimating fifteen secondorder derivatives for a geometric average option over five assets. The rest of this paper is organized as follows. Section 2 introduces the random parameter regression method in its simplest form. Section 3 presents the random parameter regression method in high-dimensional cases. Section 4 includes six numerical examples estimating Greeks of different options intended to illustrate the empir-

1 We

thank an anonymous referee for bringing this point to our attention.




ical performance of the algorithm. Section 5 contains some concluding remarks. All the proofs are provided in Appendix A.

2 RANDOM PARAMETER REGRESSION IN ONE DIMENSION For generality, we introduce our method by considering a general mathematical expectation (or a stochastic system) with option prices as special cases. Let h./ be defined as the expected value of a random variable, g.S; /, ie: Z g.S; /P .d!/ h./ D EŒg.S; / D ˝

where S is a random vector defined over a probability space (˝; F ; P ), which represents the random effect involved in the system and WD f1 ; : : : ; s g 2 Rs is a set of parameters of the system.2 Here we know the exact expression of g.S; / but not h./. Our purpose is to estimate @h./=@i for i D 1; : : : ; s, or, more generally, the high-order derivatives of h./ with respect to one or more parameters. As we focus on estimating Greeks of options in this paper, g.S; / represents the option payoff. In this section, for illustrative purposes, we focus our attention on the simplest situation where there is only one system parameter in the stochastic system, ie, D 2 R, and we call it the one-dimensional case in this paper. The high-dimensional cases involving multiple parameters will be discussed separately in the next section because it is tedious to verify that the matrix X 0 X (X is defined later) is invertible when a higher-order (higher than two) Taylor expansion is used.

2.1 The Taylor expansion and the random perturbation parameters Suppose that h./ has continuous derivatives with respect to up to order m C 1 for all in a closed interval D R, ie, h. / 2 C mC1 .D/ for all 2 D R. We want to estimate h.k/ .0 /, the higher-order derivatives of h.0 /, for some positive integer k, 1 6 k 6 m, and some real number 0 2 D. Now let fi gniD1 be n distinct points chosen from D with n > m and i ¤ 0 for all i D 1; : : : ; n. By Taylor’s formula, we have: m X h.k/ .0 / .i 0 /k C Rm .0 ; i / h.i / D kŠ

for all i D 1; : : : ; n

(2.1)

kD0

2 Throughout

the paper, we use the boldfaced to denote a vector in Euclidean spaces and the usual letter to denote a real scalar. Research Paper


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where: Rm .0 ; i / D

h.mC1/ .Qi / .i 0 /.mC1/ .m C 1/Š

for some Qi lying somewhere between 0 and i . Note that, in (2.1), the coefficients of those polynomials, h.k/ .0 /=kŠ, k D 0; : : : ; m, are not dependent on i . Thus, if Rm .0 ; i / D 0 for all i , we would have a polynomial regression model of degree m for h.i / and i 0 . Then, by using the least-squares linear regression, we can find unbiased least-squares estimates of h.k/ .0 /, k D 0; : : : ; m, with the least variance. If Rm .0 ; i / is not zero, we can still use least-squares regression to obtain estimators of those coefficients. But the acquired estimators will be biased. However, by choosing a suitable set of fi gniD1 (the regression parameters) we hope to find some methods such that we can control the bias of the estimators and obtain desirable estimators, as elaborated below. In the remainder of the paper, we consider a general setting in which a simple closedform solution for h./ does not exist and h. / is evaluated using some simulationbased or numerical methods. We now describe our method. Note that g.Si ; i / can be expressed as: g.Si ; i / D h.i / C "i

for all i D 1; : : : ; n

(2.2)

where "i D g.Si ; i / h.i /, a random variable with mean 0 and variance varŒg.S; i /. In particular, "i ; i D 1; : : : ; n, are independent random variables provided that Si and i , i D 1; : : : ; n, are independent. Combining (2.1) and (2.2) gives: m X h.k/ .0 / .i 0 /k C Rm .0 ; i / C "i g.Si ; i / D kŠ

for all i D 1; : : : ; n (2.3)

kD0

For convenience, we introduce the following shorthand notation: h.m/ .0 / ˇ1 WD h .0 /; :::; ˇm WD ˇ0 WD h.0 /; mŠ xi WD i 0 ; yi WD g.Si ; i /; ei WD Rm .0 ; i /; i WD ei C "i 0

9 > > > =

> > > ; i D 1; : : : ; n (2.4)

Then (2.3) can be rewritten as: yi D ˇ0 C ˇ1 xi C C ˇm xim C i Journal of Computational Finance

(2.5)



for all i D 1; : : : ; n, where, given fxi gniD1 , i has conditional mean ei and conditional variance i2 D var.g.S; i / j xi /. The regression model above can be recast in the matrix form as follows: Y D Xˇ C e C " (2.6) where: 3 y1 6 7 6y2 7 7 Y D6 6 :: 7 ; 4:5 2

yn 3 ˇ0 6 7 6 ˇ1 7 7 ˇD6 6 :: 7 ; 4 : 5 2

ˇm

2

1 x1 6 61 x2 X D6 6 :: :: 4: : 1 xn 2 3 e1 6 7 6 e2 7 7 eD6 6 :: 7 ; 4:5 en

3 x1m 7 x2m 7 :: 7 7 : 5 xnm

2 3 "1 6 7 6 "2 7 7 "D6 6 :: 7 4:5 "n

Now we consider the problem of choosing a suitable set of the regression parameters or independent variables, fxi gniD1 . First of all, in order to implement the standard linear regression, the variables fxi gniD1 should be chosen such that the matrix X 0 X is invertible. We will show that choosing independent and identically distributed (iid) continuous random variables as regression parameters can fulfil this requirement. That is, fxi gniD1 are some iid random variables. For the distribution of fxi gniD1 , we can have many choices among which the normal distribution, symmetric Bernoulli distribution and uniform distribution are relatively easy to implement practically. However, since our system parameter is often restricted to a certain range of the real line, we choose fxi gniD1 to be uniform distributed variables, truncated normal random variables or symmetric Bernoulli random variables so that these variables can always be tailored to fit the range of our parameters (interested readers can try variables with other distributions). Hereafter, we shall call fxi gniD1 the random perturbation parameters, or simply random parameters (RPs). We call our method the random parameter regression (RPR) method. We remind readers that, after randomization of i : E."i j xi / D 0

(2.7)

because E."i / D 0 when i is fixed in (2.2). Moreover, from (2.2) we also see that, although, for each i , "i is dependent on i and hence xi , "i should be independent of j , j ¤ i . In other words, we may assume that ."i ; xi /, i D 1; : : : ; n, is a sequence of iid random vectors. Research Paper


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2.2 Convergence of random parameter regression estimators In this section, we establish some convergence results for the regression method introduced above. Before carrying out further analysis, we need to make a mild assumption on the moments of the option payoff function. Assumption 2.1

Suppose that: EŒg 8 .S; / < 1

Since is bounded, this assumption is very general and is satisfied by many option pricing models in practice. Now we consider the polynomial regression model in the form of (2.5). For i D 1; : : : ; m, let ˇOi be the least-squares estimate of ˇi and let bias.ˇOi / D EŒˇOi ˇi be the bias of ˇi . In the remainder of this section, let xi , i D 1; : : : ; n, be n iid random variables having support on the interval .c; c/. Two simple and popular distributions that meet this requirement are the uniform distribution U.c; c/ and the symmetric Bernoulli ˙c distribution. Then we have the following theorem. Theorem 2.2

If Assumption 2.1 is satisfied, then, with probability 1: bias.ˇOi j X / D O.c mi C1 / 1 O for all i D 1; : : : ; m var.ˇi j X / D O nc 2i

Proof See Appendix A.1.

One observation can be made from Theorem 2.2. The estimator of every Greek is biased and its accuracy depends on the choice of the value of perturbation parameter c. Similar to the classic FD method, if c is very small, the resulting estimator could have a large variance. On the other hand, a large c yields a small variance but the bias could be large, thereby swaying the simulated Greek from its true value. In the next section, based on Proposition 2.3, an algorithm is developed in search of an appropriate c. Now we are in a position to establish the convergence rates of our estimates, which are commonly measured by the root mean squared errors (RMSEs) of the estimates. First, we note that given fxi gniD1 , with probability 1: MSE.ˇOi j X / D bias.ˇOi j X /2 C var.ˇOi j X / 1 2.mi C1/ /CO D O.c nc 2i b2 D b1 c 2.mi C1/ C 2i nc Journal of Computational Finance

(2.8)



where b1 and b2 are some positive constants. Thus, the minimal conditions for convergence are c 2.miC1/ ! 0 and nc 2i ! 1, which can always be satisfied since we can choose to let c ! 0 and n ! 1. It is also easy to check that the solution to minimize (2.8) is c D O.n.1=2.mC1// /, and the minimum value is: MSE.ˇOi j X / D O.n1C.i=.mC1// / Thus: RMSE.ˇOi j X / D

q MSE.ˇOi j X / D O.n.1=2/C.i=.2.mC1/// /

is the convergence rate of the estimator. The above results are summarized in the following proposition. Proposition 2.3 Under the hypotheses of Theorem 2.2, with probability 1, the optimal convergence rate of ˇOi is RMSE.ˇOi j X / D O.n.1=2/C.i=.2.mC1/// / and the optimal choice of c is c D O.n.1=.2.mC1/// / for all i D 1; : : : ; m. Remark 2.4 Consider the estimate of the first order derivative, ˇO1 . For the FD method in its simple form, the optimal convergence rate for ˇO1 is O.n1=4 / (see Zazanis and Suri (1993)). For the PA method, this rate can reach O.n1=2 / (see L’Ecuyer and Perron (1994)). For our method this rate is between O.n1=4 / and O.n1=2 / depending on the regression order m and can be improved by increasing the value of m. This theoretical result may not be very useful in practice unless we know the optimal value of c. As Proposition 2.3 suggests, the higher-order Greeks, for example, the third-order Greek relative to gamma, gamma relative to delta, are estimated at a slower rate of convergence, which is commonly called the curse of differentiation. On the one hand, from (2.5), the estimates of ˇ1 ; : : : ; ˇm depend on the approximation error e D Rm .0 ; / in the Taylor expansion. On the other hand, the inaccurate estimators of higher-order Greeks adversely affect the approximation error e D Rm .0 ; / in the Taylor expansion and, hence, the precision of all Greeks involved. Remark 2.5 Our method enables us to estimate derivatives of different orders simultaneously and, hence, can be more efficient than the FD and LR methods, especially when option valuations are computationally expensive. Remark 2.6 By the properties of Taylor’s extension and linear regression, we know that our method always works as long as the corresponding derivatives exist. However, the PA method is generally inapplicable to options of which the underlying assets have discontinuous payoffs as well as second-order derivatives of option prices, including the gamma (see Glasserman (2003, p. 392)). Thus, our method outperforms the PA method in the above areas (that is, our method works for options whose underlying assets have discontinuous payoffs). Research Paper


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The following central limit theorem states the limiting behavior of our Greek estimators. Theorem 2.7

In addition to the hypotheses of Theorem 2.2, suppose that: max E"4i < 1

16i6n

If

p mC1 nc ! 0, then: p O D Q n.ˇQ ˇ/ ! N.0; A 1 CA 1 /

(2.9)

where ˇOQ D .c ˇO1 ; : : : ; c m ˇOm /, ˇQ D .cˇ1 ; : : : ; c m ˇm /, C D .cij /.mC1/.mC1/ with cij D E.x1iCj "21 /, and where A is an .mC1/.mC1/ matrix defined in Appendix A.1. Moreover, if the function h.mC1/ . / is continuous in a neighborhood of 0 , then, when p mC1 nc ! c0 > 0: p O D Q n.ˇQ ˇ/ ! N.0; A 1 CA 1 / C D

(2.10)

where D D .D0 ; D1 ; : : : ; Dm /0 with: Dj D

c0 h.mC1/ .0 /Ex1j ; .m C 1/Š

j D 0; 1; : : : ; m

Proof See Appendix A.2.

2.3 Algorithm for estimating c We now turn to searching for an optimal c for empirical implementation. Like the FD method, the success of our approach depends on an appropriate choice of c. Our algorithm works well in all the examples we tested. For expository purposes, we consider estimating only one Greek. From Proposition 2.3, the optimal c can be written as: c D h n.1=.2.mC1/// where h is a positive constant to be determined below. Given h and c D hn.1=2.mC1// , let ˇOih denote the corresponding estimator of ˇi . From Proposition 2.3, note that: bias.ˇOih / D O.c mC1i / ai hmC1i n.1=2/C.i=.2.mC1///

(2.11)

and: var.ˇOih / D O.c 2i n1 / bi h2i n1C.i=.mC1// Journal of Computational Finance

(2.12)



where ai and bi are positive constants and n is sufficiently large. Given ai and bi , the optimal h solves: min bias2 .ˇOih / C var.ˇOih / h

and is given by:

h D

i bi .m C 1 i/ai2

1=.2.mC1// (2.13)

We now estimate ai and bi as follows. From (2.12), bi can be estimated by: bi D var.ˇOih /h2i n1.i=.mC1// From (2.11): EŒˇOih D ˇi C ai h.mC1i/ n.1=2/C.i=.2.mC1/// Consequently, for h1 ¤ h2 : h.mC1i/ n.1=2/C.i=.2.mC1/// EŒˇOih1 EŒˇOih2 D ai Œh.mC1i/ 1 2 yielding: ai D

EŒˇOih1 EŒˇOih2 / Œh.mC1i h.mC1i/ n.1=2/C.i=.2.mC1/// 1 2

We are now in a position to present our algorithm.

Algorithm h

Step 1 Let hj D j; j D 1; : : : ; J , and, for each j , estimate ˇOi j for N times, h denoted by ˇOi;lj , l D 1; : : : ; N . Evaluate the mean and variance as: N 1 X O hj hj O ˇi;l EŒˇi D N lD1

N

h var.ˇOi j / D

1 X O hj h .ˇi;l EŒˇOi j /2 N 1 lD1

Step 2

For j D 1; : : : ; J 1, estimate ai;j and bi;j by: h

ai;j D

h

EŒˇOi j C1 EŒˇOi j Œhj.mC1i/ hj.mC1i/ n.1=2/C.i=.2.mC1/// C1

and: h bi;j D var.ˇOi j C1 /hj2iC1 n1.i=.mC1//

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Step 3

Estimate the optimal hj according to the formula (2.13) by letting: hj

D

ibi;j 2 .m C 1 i /ai;j

1=.2.mC1//

for j D 1; : : : ; J 1. Finally, the optimal h is evaluated by: h D

J 1 1 X hj J 1 j D1

Since, in step 3, each hj is an inaccurate estimate of the true h given by the formula (2.13), we take an average over J 1 estimates to improve the estimation accuracy. On the one hand, J cannot be very small, since an average over a small number of hj may lead to an inaccurate estimation for h . On the other hand, J cannot be very big since a large number of estimates for hj may take a lot of time. For each example in our numerical section, we varied J from 5 to 20 and found that the estimates for h and Greeks were all insensitive to the choice of J . In particular, we choose J D 10 in all examples.

3 RANDOM PARAMETER REGRESSION FOR THE MULTIDIMENSIONAL CASES In this section, we examine our method for the more general case of multiple system parameters by mainly focusing on the option Greeks. For the Greeks of options, only the first- and second-order derivatives are involved but there is more than one system parameter in the expectation. Thus, we generalize our analysis to the case where there are s parameters (that is, D f1 ; : : : ; s g, s > 1, s 2 N)3 and where the regression model is of degree 2. Let D be a closed subset in Rs . Suppose that h W D Rs ! R has continuous mixed partial derivatives with respect to all its parameters of order 3 (ie, h 2 C 3 .D/). Let xi D fxi1 ; : : : ; xis g, i D 1; : : : ; n, be the i th random perturbation parameter for D f1 ; : : : ; s g. Since different parameters may have different ranges,4 in the remainder of this section we assume xij 2 .cj ; cj / for some cj > 0 and for

3 Note that, here, we use to denote the ith parameter in the stochastic system or the option price, i not the i th regression parameter used in the last section. 4 This is also one of the reasons why we need to separate the multidimensional case from the one-dimensional case.




j D 1; : : : ; s, i D 1; : : : ; n. Then, by Taylor’s formula, we have: h.0 C xi / D h.0 / C

s s X 1 X @2 h @h .0 /xij C .0 /xij2 @j 2 @j2

j D1

C

j D1

s X

s X

2

kD1 lDkC1

@ h .0 /xik xil C R2;i .0 ; xi / @k @l

(3.1)

where xi 2 D 0 and: 1 R2;i .0 ; xi / D 6

s X k;l;mD1

@3 h .Q i /xik xil xim @k @l @m

for some Q i , which lies somewhere on the line joining 0 to 0 C xi . Here xij denotes the ith random perturbation parameter for the system parameter j . Furthermore, given j , fxij gniD1 are assumed to be iid random variables. Furthermore, we assume fxij1 gniD1 and fxij2 gniD1 are independent for j1 ¤ j2 2 f1; : : : ; sg. In (3.1), we have an expansion up to the second order, primarily due to the difficulty of verifying the invertibility of the regression matrix. Like the one-dimensional case, for all i D 1; : : : ; n, we have: g.Si ; 0 C xi / D h.0 C xi / C "i

(3.2)

where f"i gniD1 are independent of each other and E."i j xij ; 1 6 j 6 s/ D 0. Furthermore, we also need the following assumption, which is similar to the one we made in Section 2. Assumption 3.1 E.g 8 .S; // < 1 For convenience, we introduce the following shorthand notation: ˇ0 h.0 /;

ˇj

@h .0 /; @j

ˇj k

@2 h .0 /; @j @k

9 = j; k D 1; : : : ; s >

> ; i D 1; : : : ; n (3.3) By combining (3.1)–(3.3), we have the following regression model: yi g.Si ; 0 C xi /;

yi D ˇ0 C

s X j D1

ˇj xij C

ei R2;i .0 ; xi /;

s X j D1

ˇjj xij2

C

s s X X

i ei C "i ;

ˇkl xik xil C i ;

8i D 1; : : : ; n

kD1 lDkC1

(3.4) Research Paper


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The regression model above can be rewritten in the matrix form as follows: Y D Xˇ C where: Y D .y1 ; : : : ; yn /T D .1 ; : : : ; n /T ˇ D .ˇ0 ; ˇ1 ; : : : ; ˇs ; ˇ12 ; : : : ; ˇ.s1/s /T and: 2

1 x11

6 61 x21 6 X D 6: :: :: 6 :: : : 4 1 xn1

x1s

2 x11

2 x1s

x11 x12

x11 x13

x2s :: :

2 x21 :: :: : :

2 x2s :: :

x21 x22 :: :

x21 x23 :: :

:: :

xns

2 xn1

2 xns

xn1 xn2 xn1 xn3

x1s1 x1s

3

7 x2s1 x2s 7 7 7 :: 7 : 5 xns1 xns

Let ˇOl be the least-squares estimates of ˇl and let ˇOlk be the least-squares estimates of ˇlk for all l; k D 1; : : : ; s. Then we have the following theorem. Theorem 3.2

If Assumption 3.1 is satisfied, then, with probability 1: jbias.ˇOl j X/j 6 O jbias.ˇOlk j X/j 6 O

var.ˇOl j X/ D O

cN 3 cl

cN 3 cl ck 1 ncl2

and: var.ˇOlk j X/ D O for all 1 6 l, k 6 s, where cN D .1=s/

Proof See Appendix A.3. Journal of Computational Finance

1 ncl2 ck2

Ps

iD1 ci .



First, consider the simple case of c1 D D cs . Then we have the following proposition. Proposition 3.3 Under the hypotheses of Theorem 2.7, if c1 D D cn , then with probability 1, the optimal convergence rates are RMSE.ˇOl j X/ D O.n1=3 / and RMSE.ˇOlk j X/ D O.n1=6 /. Proof The proof is trivial and very similar to the proof of Proposition 2.3 so it is omitted here. As mentioned earlier, Spall (2009) proposed two enhancements, called feedback and optimal weighting mechanisms, to improve the quality of estimates for Hessian matrices. Moreover, it can be shown that the RMSE of the estimate for a Hessian matrix is of order O.n1=6 /, which is the same as our result in Proposition 3.3 above. This suggests that the feedback and optimal weighting mechanisms have the potential to be efficient methods for estimating the second-order sensitivities in finance. It is clear that, by using linear regression, all the first- and second-order derivatives can be estimated simultaneously by our method. This property can be an advantage when we need to estimate a large number of derivatives induced by multiple system parameters. In some numerical examples in the next section, we will estimate the first-order and the second-order Greeks separately for two reasons. The first reason is that, like the central-difference estimator used in the FD method, we can enhance the precision of estimation for the first-order Greeks by eliminating the second-order Greeks. The second reason is that, when estimating the first-order and the secondorder Greeks simultaneously, we cannot obtain a closed-form formula for the optimal h like (2.13) and we need to resort to a numerical method for a solution. We first consider estimating all deltas simultaneously by using the centraldifference method to eliminate all second-order terms in (3.4). To be more specific, using (3.4), we have: 1 .g.Si ; 0 2

s X @h C xi / g.Si ; 0 xi // D .0 /xij C RN 2;i .0 ; xi / @j

(3.5)

j D1

where: RN 2;i .0 ; xi / D 12 .R2;i .0 ; xi / R2;i .0 ; xi // Then we use the algorithm proposed in Section 2.3 to estimate the optimal h for each delta separately. Similarly, for the second-order Greeks, we have: 1 .g.Si ; 0 2

D

C xi / C g.Si ; 0 xi / 2g.Si ; 0 //

s s s 2 X X @2 h 1X@ h 2 . /x C .0 /xik xil C RO 2;i .0 ; xi / 0 ij 2 2 @k @l @j j D1 kD1 lDkC1

Research Paper

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where: RO 2;i .0 ; xi / D 12 .R2;i .0 ; xi / C R2;i .0 ; xi // Then we use the algorithm proposed in Section 2.3 to estimate the optimal h for each gamma separately. However, it should be noted that the optimal h obtained in this way may not be optimal for the estimates of: @2 h ; @k @l

16k K/ Journal of Computational Finance


Estimating multiple option Greeks TABLE 1 Delta and gamma of cash-or-nothing options. (a) ı D @V =@S0 ‚ 90 Exact RPR

0.0149 0.0151

Initial stock price S0 …„ RMSE 100 RMSE — 2.1e4

0.0768 0.0766

— 3.7e4

110

ƒ RMSE

0.0074 0.0075

— 1.4e4

110

ƒ RMSE

(b) D @2 V =@S02 ‚ 90 Exact RPR

0.0061 0.0061

Initial stock price S0 …„ RMSE 100 RMSE — 0.0042 3.1e4 0.0042

— 0.0029 6.5e4 -0.0030

— 2.6e4

The parameters are r D 0.05; K D 100; ı D 0; D 0.1; T D 0.25. Computational effort: first, 1000 (path) 1000 (runs) is used to estimate the optimal c, which takes around 15 seconds; 1 000 000 (path) 50 (runs) is used to estimate both the derivatives, which takes around 50 seconds.

The parameters are r D 0:05, K D 100, D 0:1, ı D 0 and T D 0:25. For the simulations we used n D 1 000 000. Table 1 shows the estimation results for an in-the-money, an at-the-money and an out-of-money cash-or-nothing call option. We also provide the true values of derivatives for comparison. Those true values are obtained from the explicit forms of derivatives derived in Appendix A.4.

4.2 Example 2: down-and-out call option The example used is a down-and-out call option, whose discounted payoff function can be written as: f .ST / D erT .ST K/C 1 inf S t > H 06t6T

We generate inf 06t 6T S t by using Brownian bridges (see Glasserman (2003)). The parameters are: r D 0:05;

K D 100;

D 0:2;

H D 90;

ı D 0;

T D 0:25

For the simulations, we used n D 50 000. Research Paper


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H. Fu et al TABLE 2 Delta and gamma of down-and-out call (barrier) options. (a) ı D @V =@S0 Initial stock price S0 …„ RMSE 100 RMSE 105

ƒ RMSE

— 0.0061

— 0.0021

‚ 95 Exact RPR

0.4364 0.4375

0.5889 0.5885

— 0.0028

0.7510 0.7515

(b) D @2 V =@S02 Initial stock price S0 …„ RMSE 100 RMSE 105

ƒ RMSE

— 0.0037

— 0.0028

‚ 95 Exact RPR

0.0235 0.0227

0.0341 0.0336

— 0.0048

0.0291 0.0294

The parameters are r D 0.05, K D 100, ı D 0, D 0.2, T D 0.25. Computational effort: first, 200 (path) 100 (runs) is used to estimate the optimal c, which takes around 30 seconds; 50 000 (path)20 (runs) is used to estimate both the derivatives, which takes around 80 seconds.

Table 2 shows the estimation results for the delta and gamma of an in-the-money, an at-the-money and an out-of-money down-and-out call option. The true values of those deltas and gammas are obtained from the calculator provided by the website FTSweb.5

4.3 Examples 3, 4, 5 and 6: geometric average option on multiple assets Consider a European geometric average call option on d underlying assets formulated as follows. The price of the ith asset at time t , denoted by S ti , is assumed to be a geometric Brownian motion with drift r ıi and volatility i , where r is the risk-free interest and ıi the dividend yield. The initial price S0i has been given. At exercise Q time T , the payoff of the option is given by maxŒ diD1 STi K; 0, where K is the strike price. We also suppose that all the underlying assets are independent of each other. The analytic solutions for all the Greeks can be solved and are presented in Appendix A.5. Thus, we can evaluate our method by comparing the numerical results obtained against the true values.6 5

See www.ftsweb.com/options/opbarr.htm. pointed out by Glasserman (2003, p. 99), although such options are seldom, if ever, found in practice, they are useful as test cases for computational procedures since they are mathematically convenient to work with. 6 As



Estimating multiple option Greeks TABLE 3 Delta of geometric average option on five assets, D @V =@Si0 .

2

Asset i …„ 3

4

ƒ 5

0.1251

0.1112

0.1001

0.0910

0.0834

0.1251 (8.1e4)

0.1113 (7.6e4)

0.1000 (7.3e4)

0.0910 (7.6e4)

0.0834 (6.5e4)

‚ 1 Exact RPR

The numbers in parentheses are the RMSE of the estimate. The parameters are r D 0.1, K D 100, ı D Œ0.02; 0.025; 0.03; 0.035; 0.04, D Œ0.1; 0.15; 0.2; 0.25; 0.3, T D 0.2, S0 D Œ80; 90; 100; 110; 120. Computational effort: first, 1000 (path) 500 (runs) is used to estimate the optimal c, which takes around 10 seconds; 100 000 (path) 50 (runs) is used to estimate each of the derivatives, which takes around 5 seconds.

By using the RPR method, we estimate all the deltas in Examples 3, 4 and 5 and all the gammas as well as other second-order mixed partial derivatives (fifteen in total) in Example 6. The results are presented in Table 3, Table 4 on the next page, Table 5 on page 105 and Table 6 on page 107. All parameters are specified in each table. From the tables, we can see that the results are quite accurate. The estimates of deltas and gammas are either equal to or very close to the true values with relatively small RMSEs.

5 CONCLUDING REMARKS By modeling the Greek estimation as a linear regression problem, our method has the advantage of simplicity. It does not require knowledge of sophisticated mathematics and hence is easily implemented by practitioners. Moreover, it can estimate multiple derivatives simultaneously and it is also applicable to options with discontinuous payoffs, for which the PA method may not be suitable. Another advantage of our method is that it involves very little programming effort beyond what is required for the pricing simulation itself. Our examples show that our method is able to produce estimation results with good accuracy.

APPENDIX A A.1 Proof of Theorem 2.2 Let fxi gniD1 be random variables distributed over the interval Œc; c. Set xO i D xi =c. Then fxO i gniD1 are distributed over the interval Œ1; 1. It follows that the model in (2.5) can be recast as: yi D ˇQ0 C ˇQ1 xO i C C ˇQm xO im C ei C "i Research Paper

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H. Fu et al TABLE 4 Delta of geometric average option on twenty assets, D @V =@Si0 . Asset i …„

‚

ƒ

1

2

3

4

5

Exact

0.0632

0.0613

0.0595

0.0578

0.0562

RPR

0.0632 (8.4e5)

0.0613 (6.5e5)

0.0595 (6.2e5)

0.0578 (7.4e5)

0.0562 (6.6e5)

Asset i …„

‚

ƒ

6

7

8

9

10

Exact

0.0547

0.0532

0.0519

0.0506

0.0493

RPR

0.0547 (7.0e5)

0.0532 (6.2e5)

0.0519 (6.4e5)

0.0506 (6.0e5)

0.0493 (5.1e5)

Asset i …„

‚

ƒ

11

12

13

14

15

Exact

0.0482

0.0470

0.0460

0.0450

0.0440

RPR

0.0482 (7.4e5)

0.0471 (6.2e5)

0.0460 (5.7e5)

0.0450 (6.1e5)

0.0440 (5.1e5)

Asset i …„

‚

ƒ

16

17

18

19

20

Exact

0.0430

0.0421

0.0413

0.0405

0.0397

RPR

0.0430 (6.4e5)

0.0422 (4.8e5)

0.0413 (5.4e5)

0.0405 (7.1e5)

0.0397 (5.1e5)

The numbers in parentheses are the RMSE of the estimate. The parameters are r D 0.1, K D 100, ı D Œ0.021 W 0.001 W 0.04, D Œ0.11 W 0.01 W 0.20; 0.11 W 0.01 W 0.20, T D 0.2, S0 D Œ80 W 2.5 W 127.5. Computational effort: first, 1000 (path) 500 (runs) is used to estimate the optimal c, which takes around one minute; 100 000 (path) 20 (runs) is used to estimate each of the derivatives, which takes around 50 seconds.

where: ˇQi D c i ˇi ; i D 0; 1; : : : ; m;

ei D Rm .0 ; i / D c mC1

h.mC1/ .Qi / mC1 xO .m C 1/Š i

We still use xi instead of xO i in what follows to simplify notation. However, readers should bear in mind that the xi are distributed over the interval Œ1; 1. Journal of Computational Finance



TABLE 5 Delta of geometric average option on fifty assets, D @V =@Si0 . [Table continues on next page]. Asset i …„

‚

ƒ

1

2

3

4

5

Exact

0.0015

0.0015

0.0015

0.0015

0.0015

RPR

0.0015 (1.0e4)

0.0015 (1.3e4)

0.0015 (1.0e4)

0.0014 (1.6e4)

0.0014 (1.2e4)

Asset i …„

‚

ƒ

6

7

8

9

10

Exact

0.0015

0.0014

0.0014

0.0014

0.0014

RPR

0.0015 (1.1e4)

0.0014 (1.0e4)

0.0014 (1.2e4)

0.0014 (1.2e4)

0.0014 (1.4e4)

Asset i …„

‚

ƒ

11

12

13

14

15

Exact

0.0014

0.0014

0.0014

0.0013

0.0013

RPR

0.0014 (1.1e4)

0.0013 (1.1e4)

0.0014 (1.2e4)

0.0013 (1.1e4)

0.0013 (1.0e4)

Asset i …„

‚

ƒ

16

17

18

19

20

Exact

0.0013

0.0013

0.0013

0.0013

0.0013

RPR

0.0013 (1.2e4)

0.0014 (1.4e4)

0.0013 (1.1e4)

0.0013 (1.2e4)

0.0013 (1.1e4)

Asset i …„

‚

ƒ

21

22

23

24

25

Exact

0.0013

0.0013

0.0013

0.0012

0.0012

RPR

0.0012 (1.0e4)

0.0013 (1.0e4)

0.0012 (8.6e5)

0.0012 (1.1e4)

0.0012 (1.0e4)

Research Paper


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H. Fu et al TABLE 5 Continued. Asset i …„

‚

ƒ

26

27

28

29

30

Exact

0.0012

0.0012

0.0012

0.0012

0.0012

RPR

0.0012 (1.2e4)

0.0012 (1.1e4)

0.0012 (1.1e4)

0.0012 (1.0e4)

0.0012 (1.0e4)

Asset i …„

‚

ƒ

31

32

33

34

35

Exact

0.0012

0.0012

0.0012

0.0011

0.0011

RPR

0.0012 (1.1e4)

0.0012 (1.2e4)

0.0011 (1.5e4)

0.0012 (1.2e4)

0.0011 (1.1e4)

Asset i …„

‚

ƒ

36

37

38

39

40

Exact

0.0011

0.0011

0.0011

0.0011

0.0011

RPR

0.0011 (1.1e4)

0.0011 (1.1e4)

0.0011 (1.0e4)

0.0011 (1.1e4)

0.0011 (1.2e4)

Asset i …„

‚

ƒ

41

42

43

44

45

Exact

0.0011

0.0011

0.0011

0.0011

0.0011

RPR

0.0011 (7.1e5)

0.0011 (1.1e4)

0.0011 (1.1e4)

0.0011 (1.1e4)

0.0010 (8.5e5)

Asset i …„

‚

ƒ

46

47

48

49

50

Exact

0.0011

0.0010

0.0010

0.0010

0.0010

RPR

0.0011 (1.0e4)

0.0011 (1.2e4)

0.0010 (1.0e4)

0.0010 (9.0e5)

0.0010 (1.0e4)

The numbers in parentheses are the RMSE of the estimate. The parameters are r D 0.06; K D 105; ı D Œ0.02 C .0.04 0.02/=50 W .0.04 0.02/=50 W 0.04, D Œ0.1 C .0.20 0.10/=50 W .0.20 0.10/=50 W 0.20, T D 1; S0 D Œ80C..12080/=50/ W ..12080/=50/ W 120. Computational effort: first, 500 (path)500 (runs) is used to estimate the optimal c, which takes around 2 minutes and 10 seconds; 100 000 (path) 20 (runs) is used to estimate each of the derivatives, which takes around 50 seconds.



Estimating multiple option Greeks TABLE 6 Gamma and mixed derivatives of geometric average option on five assets, D @2 V =@Si0 Sj0 . (a) True values Asset i …„

‚ 8 1 > > > > > < 2 Asset j 3 > > > 4 > > : 5

ƒ

1

2

3

4

5

0.0045 — — — —

0.0054 0.0036 — — —

0.0049 0.0043 0.0029 — —

0.0044 0.0039 0.0035 0.0024 —

0.0040 0.0036 0.0032 0.0029 0.0020

(b) RPR estimates Asset i …„

‚ 8 1 > > > > > > > > 2 > > > > > > > > < 3 Asset j > > > > > > > 4 > > > > > > > > > : 5

ƒ

1

2

3

4

5

0.0046 (0.0007)

0.0053 (0.0004)

0.0049 (0.0004)

0.0044 (0.0002)

0.0041 (0.0002)

— —

0.0037 (0.0006)

0.0043 (0.0003)

0.0039 (0.0003)

0.0037 (0.0003)

— —

— —

0.0028 (0.0004)

0.0035 (0.0002)

0.0033 (0.0002)

— —

— —

— —

0.0023 (0.0005)

0.0029 (0.0002)

— —

— —

— —

— —

0.0020 (0.0004)

The numbers in parentheses are the RMSE of the estimate. The parameters are r D 0.1, K D 100, ı D Œ0.02; 0.025; 0.03; 0.035; 0.04, D Œ0.1; 0.15; 0.2; 0.25; 0.3, T D 0.2, S0 D Œ80; 90; 100; 110; 120. Computational effort: first, 1000 (path) 100 (runs) is used to estimate the optimal c, which takes around 5 seconds; 100 000 (path)50 (runs) is used to estimate each of the derivatives, which takes around 250 seconds.

Let: 3 y1 6 7 6 y2 7 7 Y D6 6 :: 7 ; 4:5 2

yn Research Paper

2

1 x1 6 61 x2 X D6 6 :: :: : : : 4: : 1 xn

3 x1m 7 x2m 7 :: 7 7 : 5 xnm


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2

3 ˇQ0 6Q 7 6 ˇ1 7 7 ˇQ D 6 6 :: 7 ; 4 : 5 ˇQm

2 3 e1 6 7 6e2 7 7 eD6 6 :: 7 ; 4:5

2 3 "1 6 7 6 "2 7 7 "D6 6 :: 7 4:5

en

"n

Thus, the regression model of (A.1) can be written as: Y D X ˇQ C e C "

(A.2)

By applying the linear regression, the least-squares estimator for ˇQ in the above equation is: ˇOQ D

1 0 XX n

1

X 0Y D ˇQ C n

1 0 XX n

1

X 0e C n

1 0 XX n

1

X 0" n

(A.3)

Here we claim that ..1=n/X 0 X /1 exists with probability 1 when n > m and fxi g are continuous random variables. In fact, X is a square Vandermonde matrix when n D m C 1 and its determinant is given by: det.X / D

m Y

.xj xk /

(A.4)

16k m and fxi g are continuous random variables, implying the matrix X 0 X is invertible and ˇOQ is well-defined. Furthermore, by the strong law of large numbers we have: n

1 X k a.s. xi ! E.x k /; n

k D 1; : : : ; 2m

(A.5)

iD1

where a.s. stands for almost surely and x denotes the random variable on the interval Œ1; 1. This implies that: 1 0 a.s. X X ! A (A.6) n where: 2 3 1 E.x/ E.x 2 / E.x m / 6 E.x/ E.x 2 / E.x 3 / E.x mC1 /7 6 7 6 7 E.x 3 / E.x 4 / E.x mC2 /7 E.x 2 / AD6 6 7 :: :: :: 6 :: 7 :: : 4 : 5 : : : E.x m / E.x mC1 / E.x mC2 / E.x .2m/ / Journal of Computational Finance



By Corollary 2B of Lindsay (1989), the determinant of the matrix A is given by: Y m 1 2 E .uj uk / det.A/ D .m C 1/Š

(A.7)

j >k

where u1 ; u2 ; : : : ; um are independent and identical distributions over the interval Œ1; 1. Obviously, by (A.7), det.A/ > 0 when ui are continuous random variables. Thus, by (2.7), with probability 1: EŒˇOQ j X D ˇQ C .X 0 X /1 X 0 e C .X 0 X /1 X 0 EŒ" j X 1 0 Xe 1 0 Q XX DˇC n n Note that:

n n n X 1 0 1 X 1X m 0 ei ; xi e i ; : : : ; xi ei XeD n n n iD1

iD1

iD1

and, for j D 0; : : : ; m: ˇ n .mC1/.Q / ˇ ˇ ˇ X i ˇ ˇ 1 n j ˇ c mC1 ˇ X h j CmC1 ˇ ˇ ˇ ˇ x i ei ˇ 6 xi ˇ ˇ ˇn n .m C 1/Š iD1

iD1

n

c mC1 M X 6 jxi jj CmC1 n.m C 1/Š iD1

D O.c

mC1

/

(A.8)

where we make use of the fact that xi s are distributed on the interval Œ1; 1 and that h.mC1/ ./ is bounded by M in a closed interval. Here (and throughout this paper) M is a constant that may denote different constants on different occasions to facilitate statements. This, together with (A.6), implies that, with probability 1, the bias: EŒˇOQ j X ˇQ D O.c mC1 /

(A.9)

which is equivalent to: EŒˇOi j X ˇi D O.c mC1i /;

i D 1; : : : ; m

(A.10)

because ˇQi D c i ˇi . Here ˇOi is the least-squares estimator of ˇi . Research Paper


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For the variance we have, with probability 1: var.ˇOQ j X / D var..X 0 X /1 X 0 " j X / D .X 0 X /1 X 0 var." j X /X .X 0 X /1 D .X 0 X /1 X 0 diag.12 ; 22 ; : : : ; n2 /X .X 0 X /1

(A.11)

where diag.12 ; 22 ; : : : ; n2 / is a diagonal matrix. Note that, for each k D 1; : : : ; 2m, by the Burk–Holder inequality (see, for example, Petrov (1975, Problem 14, p. 59)) and by noticing that jxi j 6 1: ˇ4 ˇ X 2 X n ˇ ˇ1 n 2 k M 2 k i xi E.1 x1 /ˇˇ 6 4 E .i2 xik E.12 x1k //2 E ˇˇ n n iD1

iD1

6

n M X EŒi2 xik E.12 x1k /4 n3 iD1

M E.18 x14k / n2 M 6 2 EŒg.S; 1 /8 n M D 2 n 6

(A.12)

with the last inequality following from the Jensen’s inequality. Consequently, by Chebyshev’s inequality: Pn ˇ ˇ X 2 k 2 k 4 ˇ ˇ1 n 2 k Ej.1=n/ iD1 i xi E.1 x1 /j 2 k i xi E.1 x1 /ˇˇ > ı 6 P ˇˇ ı4 n iD1

6

M n2 ı 4

for any ı > 0. It follows from the Borel–Cantelli lemma that: n

1X 2 k a.s. i xi E.12 x1k / ! 0 n iD1

This leads to: 1 0 a.s. X diag.12 ; 22 ; : : : ; n2 /X C ! 0 n Journal of Computational Finance

(A.13)



where C D .cij / with: cij D E.12 x1iCj 2 / 6 E.12 / 0: ˇ2 ˇ ˇ ˇ n m m X X X ˇ ˇ 1 ˇ 1 ˇ 1 jˇ jˇ ˇ ˇ E ˇ p "i aj xi ˇ I ˇ p "i aj xi ˇ > 2 ı n j D0 n j D0 iD1 6 4ı

2

ˇ ˇ4 n m X X ˇ 1 ˇ M jˇ ˇ E ˇ p "i aj xi ˇ 6 n n iD1

j D0

because the xi are bounded. Thus, the Linderberg condition is satisfied, implying: p

na

0

2 X m 1 0 D j X" ! N 0; E aj x1 "1 n j D0

since E.1=n/X 0 " D 0 by (A.15). This, together with the Cramér–Wold theorem, implies that: 1 D ! N.0; C / p .X 0 "/ n completing the proof of (A.16). Furthermore, note that: X n n n 1 1X 1X m 0 1 0 XeD ei ; xi ei ; : : : ; x i ei n n n n iD1

iD1

iD1

and, for each j D 1; : : : ; m; by (A.8): ˇ n ˇ p mC1 ˇ p ˇ1 X nc M j ˇ ˇ nˇ x i ei ˇ 6 n .m C 1/Š iD1

Hence, when

p

nc mC1 ! 0: p

X n 1 a.s. xij ei ! 0 n

n

iD1

Consequently:

p 1 0 1 0 D n X "C X e ! N.0; C / n n

since the normal distribution is continuous, implying (2.9) by (A.3) and (A.6). Likewise, if: p mC1 nc ! c0 and h.mC1/ ./ Q is continuous at 0 , then c ! 0; and: a.s.

h.mC1/ .Qi / ! h.mC1/ .0 / Journal of Computational Finance



As a result:

n .mC1/ p 1X .0 /Ex1j a.s. c0 h n xij ei ! n .m C 1/Š iD1

since fxi g are bounded. Thus, by (A.3) and (A.6), (2.10) follows.

A.3 Proof of Theorem 3.2 Here we provide only the argument that the inverse of X0 X exists because the remaining arguments for Theorem 2.7 are similar to those for Theorem 2.2. In the multidimensional case, the matrix X is given in Section 3. If: n D 1 C 2s C 12 s.s 1/ then X is a square matrix and its determinant det.X/ is a polynomial of variables xij , i D 1; : : : ; n, j D 1; : : : ; s. In particular, it is not difficult to prove that det.X/ ¤ 0 with probability 1 provided that xij , i D 1; : : : ; n, j D 1; : : : ; s, are independent and continuous random variables. Therefore, X0 X is invertible when: n > 1 C 2s C 12 s.s 1/ Furthermore, as in Theorem 2.2, by assuming xij ; i D 1; : : : ; n; j D 1; : : : ; s, are uniformly distributed random variables and by the strong law of large numbers: n

n

1X a.s. xij ! 0; n

1X a.s. xij1 xij2 ! 0; n

iD1

j1 ¤ j2

iD1

n

1X 2 a.s. xij1 xij2 ! 0; n

j1 ¤ j2

iD1

n

1 X 3 a.s. 3 xij ! Ex1j D0 n iD1

and: n cjm 1 X m a.s. m ; xij ! Ex1j D n mC1

m D 2; 4

iD1

n

1 X 2 2 a.s. 1 2 2 xij1 xij2 ! 9 cj1 cj2 ; n

j1 ¤ j2

iD1

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It follows that, with probability 1: 1 0 a.s. X X ! B n where: 2

1 6 6 0 6 6 6 0 6 6 :: 6 : 6 6 6 0 61 6 c2 63 1 61 2 c BD6 63 2 6 :: 6: 6 61 2 6 3 cs 6 6 0 6 6 6 0 6 6 : 6 :: 4 0

0

0

1 2 c 3 1

1 2 c 3 2

1 2 c 3 1

0

0

0

0

0 :: :

1 2 c 3 2

:: :

:: :

0 :: :

0 :: :

0 :: :

:: :

0

0

1 2 c 3 s

0

0

1 4 c 5 1 1 2 2 c c 9 2 1

1 2 2 c c 9 1 2 1 4 c 5 2

:: :

:: :

:: :

0

0

0

0

0 :: :

0 :: :

:: :

0 :: :

0

0

0

1 2 2 c c 9 s 1

1 2 2 c c 9 s 2

0

0

0

0

0

0 :: :

0 :: :

:: :

0 :: :

0 :: :

0 :: :

:: :

0

0

0

0

0

1 2 c 3 s

0

0

0

0

0

0

0

0 :: :

0 :: :

0 :: :

:: :

0 :: :

0

0

0

0

1 2 2 c c 9 2 s 1 2 2 c c 9 2 s

0

0

0

0 :: :

0 :: :

:: :

0 :: :

1 4 c 5 s

0

0

0

0

1 2 2 c c 9 1 2

0

0

0 :: :

0 :: :

1 2 2 c c 9 1 3

:: :

:: :

0 :: :

0

0

0

1 2 c c2 9 s1 s

:: :


3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5



Furthermore, by the cofactor expansion: ˇ ˇ 1 ˇ ˇ1 2 s s s ˇ c Y Y Y 1 2 1 2 2 ˇ3 1 c c c jBj D ˇ : i j k 3 9 ˇ :: iD1 j D1 kDj C1 ˇ ˇ1 2 ˇ cs 3

1 2 c 3 2 1 2 2 c c 9 1 2

:: :

:: :

:: :

1 2 2 c c 9 s 1

1 2 2 c c 9 s 2

ˇ ˇ1 ˇ ˇ Y s s s s Y Y Y ˇˇ 31 2 1 2 1 2 2 c c c cl ˇ : D 3 i 9 j k ˇ :: iD1 j D1 kDj C1 ˇ lD1 ˇ1 ˇ 3

D

iD1

1 6 c 3 i

s Y

ˇ ˇ ˇ ˇ 1 4 ˇ c 5 s :: :

1 2 c 3 2 1 2 c 9 2

:: :

:: :

:: :

1 2 c 9 1

1 2 c 9 2

3

s Y

1 2 ˇ c 3 s ˇ ˇ 1 2 2ˇ c c 9 2 sˇ

1 2 c 3 1 1 2 c 5 1

ˇ ˇ1 ˇ ˇ1 Y s s s s 2 ˇ Y Y Y ˇ3 2 1 2 1 2 2 c c c c D ˇ: l 3 i 9 j k ˇ :: iD1 j D1 kDj C1 ˇ lD1 ˇ1 ˇ s Y

ˇ

1 2 c 3 1 1 4 c 5 1

1 3 1 5

1 3 1 9

:: :

:: :

:: :

1 9

1 9

ˇ

1 2ˇ c 3 sˇ ˇ 1 2ˇ c 9 sˇ

:: ˇˇ : ˇ ˇ 1 2ˇ cs 5

ˇ

1ˇ 3ˇ ˇ 1ˇ 9ˇ

:: ˇˇ :ˇ ˇ 1ˇ 5

1 2 2 4 s c c . / 9 j k 45

j D1 kDj C1

Before concluding this section, we consider the case where we estimate all deltas simultaneously by using (3.5). In this case, the matrix X can be expressed as: 2 3 x11 x1s 6 7 6x21 x2s 7 6 XD6 : :: :: 7 7 : : 5 4 :: xn1 xns Hence, with probability 1: 1 0 a.s. X X ! B n where:

2

2 Ex11 0 6 2 Ex12 6 0 BD6 :: :: 6 :: : : 4 : 0 0

0 0 :: : 2 Ex1s

3 7 7 7 7 5

Therefore, .X0 X/1 is well-defined for sufficiently large n. Note, particularly, that this conclusion still holds true for symmetric Bernoulli distributions. Research Paper


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A.4 The delta and gamma of a cash-or-nothing call option Let S0 , K, T , r, and ı denote the initial stock price, the strike price of the option, the time to maturity, the interest rate, the volatility of the stock and the dividend yield of the stock, respectively. The value of a cash-or-nothing call option is V D exp.rT /N.d2 /, where N./ is the cumulative distribution function of a standard normal random variable: d1 D

ln.S0 =K/ C .r ı C 12 2 /T p T

p and d2 D d1 T . Then we have: exp.rT /N 0 .d2 / @V D p @S0 S0 T ı d2 @2 V D 1 C D p S0 @S02 T ıD

A.5 The delta and gamma of a geometric average option Since Si GBM.r ıi ; i2 / with initial value Si0 , we have: Si .t/ D Si0 exp.Œr ıi 12 i2 t C i Wi .t // Then, at time t , we have: G.t / D .

d Y

Sj .t//1=d

j D1

D.

d Y

Sj0 /1=d exp

j D1

r

d d X 1X i .ıi C 12 i2 / t C Wi .t / d d iD1

(A.18)

iD1

Since we assume that the Si are independent of each other, we have that all the P fWi .t/g16i 6d are mutually independent. Thus, diD1 .i =d /Wi .t / has the same distribution as: X d i2 1=2 W .t/ d2 iD1

where W .t / is a standard Brownian motion. Thus, (A.18) can be written as: G.t / D G0 exp.Œr ıQ 12 Q 2 t C Q W .t // where:

d X .d 1/i2 Qı D 1 ; ıi C d 2d iD1


Q 2 D

d X 2 i

iD1

d2



Q and G0 . jdD1 Sj0 /1=d . That is, G.t / is again a geometric Brownian motion with an initial value G0 . By the Black–Scholes formula, we know that the price of a European call option based on G is given by: Q /N.d1 / exp.rT /KN.d2 / V D G0 exp.ıT where K is the strike price: d1 D

log.G0 =K/ C .r ıQ C 12 Q 2 /T p Q T

p and d2 D d1 Q T . It is not difficult to calculate that: 1 G0 @G0 D 0 d @Si Si0 and: @2 G0 2 @Si0

D

1 d G0 d 2 S 02 i

Therefore, we have: 1 @V G0 @V D 0 d @G0 Si0 @Si 1 @2 V G0 2 d 1 @V G0 @2 V D 2 d 2 @G02 Si0 d 2 @G0 S 0 2 @Si0 i and:

2 @2 V 1 @V G0 1 G0 @ V @ @V D G0 C @G0 @Si0 @Sj0 @Sj0 d @G0 Si0 d 2 Si0 Sj0 @G02

where: @V @G0

and

@2 V @G02

are the delta and gamma, respectively, of G.

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