Mean Square Optimal Hedges Using Higher Order Moments

4 Multinomial lattices with momentsWe will present a general description of a random walkon a multinomial lattice with moments (or cumulants). Supposethat u n and d n satisfy u n > d n > 0. Then a multinomialtree with L branches at each node is given byS n+1 = u L−ln d l−1n S n , l = 1,...L, (15)where p l , l = 1,...L are the corresponding probabilitieswhich satisfyp 1 + · · · + p L = 1. (16)To make the multinomial tree recombine, we further assumethat u n /d n = c for all n = 0,...,N − 1 for some constantc (> 1). One can verify that the process in (15) consistsof a lattice (or a recombining multinomial tree), where thestock may achieve n(L − 1) + 1 possible prices at time t =t n , n = 0,...,N. For example, in the case of u n = u andd n = d for all n = 1,...,N − 1, the price of the stock atthe k-th node from the top of the lattice is given byS (k)n = u n(L−1)+1−k d k−1 S 0 , k = 1,...,n(L − 1) + 1.(17)4.1 Parameterization for Multinomial Lattices withCumulantsWe will provide a parameterization of multinomial latticerandom walks which take cumulants into account. Supposethat ν n τ is the mean of X n , i.e., the first cumulant (mean)of X n isC (1)n = E(X n ) = ν n τ. (18)and let(νn τu n := expd n := exp)L − 1 + α√ τ ,(νn τL − 1 − α√ τ), (19)where α > 0 is some constant. One can readily see thatu n /d n is constant for all n = 0,...,N − 1 if α is fixed.With these choices for u n and d n , X n may be computed asX n = lnS n+1 − lnS n = ν n τ + (L − 2l + 1) α √ τ.In this case,L∑p l (L − 2l + 1) = 0 (20)l=1must hold, and the k-th central moment ˆM(k) nˆM n (k) = E[(X n − ν n τ) k]is given by= ( α √ τ ) k∑L p l (L − 2l + 1) k , k ≥ 2. (21)l=1Note that the second through fourth cumulants are computedby C (2) (2)n = ˆM n and the formulas in (25).4.1.1 Binomial Lattice Case:We first consider the case of L = 2, i.e., the binomial latticecase. Since there are already two constraints for the probabilitiesp 1 and p 2 , i.e.,p 1 + p 2 = 1,2∑p l (L − 2l + 1) = p 1 − p 2 = 0,l=1we obtain p 1 = p 2 = 1/2. Suppose that the variance of X nis given by σ 2 nτ. This condition restricts α = 1 and σ n to beconstant, i.e., σ n = σ (n = 0,...,N − 1), and we have thewell-known binomial lattice formula provided in [14] (seealso the original work of [3]):u n = exp(ν n τ + σ √ τ) , d n = exp (ν n τ − σ √ τ)4.1.2 Trinomial Lattice Case:p 1 = p 2 = 1/2. (22)In the case of a trinomial lattice, i.e., L = 3, we have onemore parameter p 3 , and this allows us to take local volatilityinformation into account, i.e., the second cumulant. Supposethat the second cumulant (i.e., variance) of X n is givenby σ 2 nτ. In this case, we havep 1 + p 2 + p 3 = 1,2p 1 − 2p 2 = 0,4p 1 + 4p 2 = σ2 nα 2 , (23)where the second and third equations are obtained from(20) and (21), respectively. By solving (23) with respectto p 1 , p 2 and p 3 , we find[ σ2[p 1 , p 2 , p 3 ] = n8α 2 , 1 − σ2 n4α 2 , σn2 ]8α 2 .To guarantee that these probabilities are positive, α mustsatisfy σ/2 < α. If σ n is constant, i.e., σ n = σ (n =0,...,N − 1), one may use α = √ 3σ/2, which provides atrinomial lattice formula whose up, middle, and down ratesand corresponding probabilities are given byu 2 n = exp ( ν n τ + σ √ 3τ ) , d 2 n = exp ( ν n τ − σ √ 3τ ) ,u n d n = exp (ν n τ) , [p 1 , p 2 , p 3 ] = [1/6, 2/3, 1/6].This also corresponds to a well known finite differencescheme (see e.g., [13]).If σ n is a function of (S n , n), i.e., σ n = σ(S n , n), theabove formula can be modified by writing σ n in terms of anominal value ˆσ asσ n = (1 + δ n )ˆσ.Let α be chosen as α = √ 3ˆσ/2. Then the up, middle, anddown probabilities are given as[ (1 + δn ) 2[p 1 , p 2 , p 3 ] = , 1 − (1 + δ n) 2, (1 + δ n) 2 ].6 3 6Note that the probabilities are positive as long as − √ 3−1

4.1.3 Multinomial Lattice Case:Now, we show a general case where we have any given moments.Given the first m moments, there are m + 1 constraintsfor L plus 1 unknown parameters, p 1 ,...,p L , andα. If α > 0 is fixed a priori, p 1 ,...,p L can be computedby solving m + 1 linear equations. Therefore, we need atleast L = m + 1 branches to guarantee the existence of afeasible solution. In this case, p 1 ,...,p L can be parameterizedas a function of α given below. Finally, α > 0 maybe adjusted such that all the probabilities are positive. Herewe provide a parameterization for up to the 4-th moment (orcumulant), but the extension to the cases with even higherorder moments is straightforward.Suppose that we have third cumulant information correspondingto skewness, in addition to the first and the secondcumulants. This imposesC (3)n= s n τ ( √ ) 3 ( √ ) 3∑ Lσ n τ = α τ p l (L − 2l + 1) 3 ,l=1where s n τ is the skewness of X n . Let L = 4, and solve fourlinear equations for the probabilities p 1 , p 2 , p 3 , p 4 . Then,we obtain[p 1 , p 2 , p 3 , p 4 ] = 1 16 × [−1 + σ2 nα 2 (9 − σ2 nα 2 (9 + σ2 nα 2 (−1 + σ2 nα 2 (1 + s nτσ nα1 + s nτσ n3α),−1 + s )nτσ n,α1 − s )nτσ ] n.3α),If σ n is constant, i.e., σ n = σ (n = 0,...,N − 1), thechoice α = σ/2 results in the following formulas:(u 3 n = exp ν n τ + 3σ )√ (τ , u 22nd n = exp ν n τ + σ √ )τ ,2(u n d 2 n = exp ν n τ − σ √ ) (τ , d 3 n = exp ν n τ − 3σ )√ τ ,22[p 1 , p 2 , p 3 , p 4 ] =[ 316 + 1 6 s nτ,516 − s nτ8 ,516 + s nτ8 , 3 16 − 1 6 s nτIf we would like to match the 4th cumulant or “kurtosis,”we may introduce a multinomial lattice with five branches,i.e., L = 5. Let κτ denote the kurtosis of X n . Then we haveC (4)4 = κ n τ ( σ n√ τ) 4].5∑= α 4 τ 2 p l (6 − 2l) 4 − 3 ( √ ) 4σ n τ ,l=1as an additional constraint. In this case, the probabilitiesp 1 , p 2 , p 3 , p 4 , p 5 can be calculated through the solution offive linear equations, and are given by[p 1 , p 2 , p 3 , p 4 , p 5 ] = 1 96 ×[ ( σ2nα 2 −1 + s )nτσ n+ σ2 nα 4α 2 (3 + κ nτ) ,σn2 (α 2 16 − 2s )nτσ n− σ2 nα α 2 (3 + κ nτ) ,{ ()}364 + σ2 n2 α 2 −20 + σ2 nα 2 (3 + κ nτ) ,σn2 (α 2 16 + 2s )nτσ n− σ2 nα α 2 (3 + κ nτ) ,σn2 (α 2 −1 − s )]nτσ n+ σ2 nα 4α 2 (3 + κ nτ) .To understand the effect of kurtosis, assume that s n = 0 andσ n = σ (n = 0,...,N − 1), then we obtain[p 1 , p 2 , p 3 , p 4 , p 5 ]= 1 [ ()σ296 α 2 −1 + σ24α 2 (3 + κ nτ) ,σ 2 ()α 2 16 − σ2α 2 (3 + κ nτ) ,()}3{64 + σ22 α 2 −20 + σ2α 2 (3 + κ nτ) ,σ 2 ()α 2 16 − σ2α 2 (3 + κ nτ) ,σ 2 ()]α 2 −1 + σ24α 2 (3 + κ nτ) .In this case, all the probabilities are positive ifσ √ 3 + κ n τ45 − κ n τ24, 9 + κ nτ16< α < σ√ 3 + κ n τ.2Furthermore, if we choose α = σ/ √ 2, then the above probabilitiesreduce to[ 1 + κn τ[p 1 , p 2 , p 3 , p 4 , p 5 ] = ,96]. (24), 5 − κ nτ, 1 + κ nτ24 96The up-down rates corresponding to five branches can becalculated as(u 4 n = exp ν n τ + 2σ √ )2τ ,(u 3 nd n = exp ν n τ + σ √ )2τ ,u 2 nd 2 n = exp (ν n τ) ,(u n d 3 n = exp ν n τ − σ √ )2τ ,(d 4 n = exp ν n τ − 2σ √ )2τ .We first notice that the probabilities are symmetric, i.e.,p 1 = p 5 and p 2 = p 4 . In this formulation, p 1 , p 3 and p 5

increase with larger kurtosis. On the other hand, p 2 and p 4decrease if kurtosis increases. Therefore, this confirms thatthe probability distribution of X n becomes heavy tailed underpositive kurtosis.If skewness is not zero, the formulation in (24) becomes[p 1 , p 2 , p 3 , p 4 , p 5 ][=1 + κ n τ + 2 √ 2s n τ969 + κ n τ16, 5 − κ nτ − √ 2s n τ,24, 5 − κ nτ + √ 2s n τ, 1 + κ nτ − 2 √ 2s n τ2496with the choice of α = σ/ √ 2. In this case, we readily seethat the probabilities are not symmetric if s n ≠ 0. Moreover,positive (negative) skewness causes p 1 and p 4 to increase(decrease), and the corresponding probabilities p 5and p 2 to decrease (increase) by an equal amount.5 Illustrative ExamplesHere we provide some numerical examples to demonstratethe effect of higher order moments on the impliedvolatility smile. We use stock index data downloaded fromthe Chicago Mercantile Exchange. Fig. 1 shows the cumulativedistribution of daily log-returns of the stock indexfrom January 1998 to December 2000. The sample mean,standard deviation, skewness and kurtosis are computed asfollows:Mean Standard Deviation Skewness Kurtosis0.0007 0.0089 -0.3923 3.8207250200150100500−0.05 0 0.05Figure 1: Cumulative distribution of daily log-returnWith these statistics, we solved the mean square optimalhedging problem to compute the price of a Europeancall option. We first note that the standard model (i.e., theBlack-Scholes model) uses the information up to the secondmoment (i.e., standard deviation) only. We comparethe MSOH solution with higher order cumulants with thestandard Black-Scholes solution. To understand the differencebetween the two, we computed the implied volatility]for European call options with different maturities and strikeprices. Figs. 2–5 are our numerical results, where the impliedvolatilities are plotted versus the strike prices normalizedby the initial price of stock index (i.e., K/S(0)). Eachfigure has a different time to expiration: Fig. 2 has an expirationof 10 days, Fig. 3 20 days, Fig. 4 40 days, and Fig. 580 days. Note that the dashed line in each figure is a constantvolatility corresponding to the Black-Scholes solution.First, we note that the smile effect is most clearly observedwhen the maturity is shortest. The smile effect slowlydisappears as we have longer maturities, but a smirk effectremains. This can be explained using the term structure ofskewness vs. smirk and the term structure of kurtosis vs.smile as follows.Let c k be the k-th order cumulant of daily log-returnsof the stock index, and let s and κ be the correspondingskewness and kurtosis, respectively. Note that the first andsecond order cumulants are the mean and variance, respectively.Moreover, skewness and kurtosis are functions ofcumulants, which are given as follows:s = c 3c 3/22, κ = c 3c 2 2(25)Now, assume that the log-returns of the stock index at eachday are independent. Since the cumulants have the additiveproperty when independent random variables are summed,the k-th order cumulant of ∑ N−1n=0 X n, is just the sum of them-th cumulants of X n for n = 0,...,N − 1, i.e.,Nc k , k = 1,2,.... (26)If we substitute equation (26) to (5), we obtain“N day skewness” =“N day kurtosis” =Nc 3(Nc 2 ) = s , 3/2 N1/2 (27)Nc 3(Nc 2 ) 2 = κ N . (28)Equations (27) and (28) provide the term structure of skewnessand kurtosis for N days with respect to daily skewnessand kurtosis.In the Black-Scholes setting where the distribution of logstockreturn is given by a Gaussian distribution, skewnessand kurtosis are both zero and the implied volatility is constant.On the other hand, non-zero skewness and kurtosismay increase the smirk and the smile effects as observedfrom our numerical experiments. Note that this result isconsistent with the one described in [4] using the risk neutralprobability measure, whereas our numerical experiments indicatethat the similar effect is observed in the MSOH problemsetting.6 ConclusionIn this paper, we posed and solved a mean square optimalhedging problem that takes higher order moments (or cumulants)into account. We first showed a discrete stochasticdynamics model using a general multinomial lattice, where

Implied volatility0.160.1550.150.1450.140.135the first m cumulants are matched at each step. We thenanalyzed the effect of these higher order moments in the underlyingasset process on the price of derivative securities.The relationship between the term structure of the volatilitysmile and smirk and higher order cumulants was illustratedthrough numerical experiments.REFERENCESImplied volatilityImplied volatilityImplied volatility0.130.1250.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12K/S 0Figure 2: 10 days expiration0.160.1550.150.1450.140.1350.130.1250.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12K/S 0Figure 3: 20 days expiration0.160.1550.150.1450.140.1350.130.1250.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12K/S 0Figure 4: 40 days expiration0.160.1550.150.1450.140.1350.130.1250.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12K/S 0Figure 5: 80 days expiration[1] F. Black and M. Scholes, “The Pricing of Options andCorporate Liabilities,” Journal of Political Economy,81:637–654, 1973.[2] J.C. Cox and S.A. Ross, “The valuation of options foralternative stochastic processes,” Journal of FinancialEconomics, 3:145–166, 1976.[3] J.C. Cox, S.A. Ross, and M. Rubinstein, “Option pricing:A simplified approach,” Journal of Financial Economics,7:229–263, 1979.[4] S.R. Das and R.K. Sundaram, ”Of Smiles and Smirks:A Term-Structure Perspective,” Research report, 1998.[5] E. Derman and I. Kani, “Implied Trinomial Treesof the Volatility Smile,” Goldman Sachs QuantitativeStrategies Research Notes, February, 1994.[6] E. Derman and I. Kani, “Riding on a Smile,” Risk,7:18–20, 1994.[7] D. Duffie and J. Pan, “An Overview of Value at Risk,”Journal of Derivatives, Spring:7-49, 1997. “Riding ona Smile,” Risk, 7:18–20, 1994.[8] D. Duffie and H.R. Richardson, “Mean-variance hedgingin continuous time.” Annals Appl. Probability, 1,1-15, 1991.[9] S. Fedotov and S. Mikhailov, “Option Pricing for IncompleteMarkets via Stochastic Optimization: TransactionCosts, Adaptive Control, and Forecast,” Int. J. ofTheoretical and Applied Finance, 4(1):179–195, 2001.[10] H. Follmer and M. Schweizer, “Hedging of contingentclaims under incomplete information”, In AppliedStochastic Analysis (M.H.A. Davis and R.J. Elliott,eds.), Stochastics Monographs 5, 389-414. Gordonand Breach, New York, 1991.[11] D.T. Gillespie, Markov process: an introduction forphysical scientists, Academic Press, 1992.[12] O. Hammarlid, “On minimizing risk in incompletemarkets option pricing models,” Int. J. Theoretical andApplied Finance, 1(2):227–233, 1998.[13] J. Hull, Options, Futures, and Other Derivative Securities,4th edition. Englewood Cliffs: Prentice-Hall,1999.[14] R. Jarrow and A. Rudd, Option Pricing. McGraw-HillProfessional Book Group, 1983.

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