Performance of tail hedged portfolio with third moment variation swap
PPerformance of tail hedged portfolio with third moment variationswap
Kyungsub Lee ∗† and Byoung Ki Seo ‡§ Abstract
The third moment variation of a financial asset return process is defined by the quadraticcovariation between the return and square return processes. The skew and fat tail risk of anunderlying asset can be hedged using a third moment variation swap under which a prede-termined fixed leg and the floating leg of the realized third moment variation are exchanged.The probability density function of the hedged portfolio with the third moment variationswap was examined using a partial differential equation approach. An alternating directionimplicit method was used for numerical analysis of the partial differential equation. Underthe stochastic volatility and jump diffusion stochastic volatility models, the distributions ofthe hedged portfolio return are symmetric and have more Gaussian-like thin-tails.
The distribution of a financial asset return is negatively skewed and has fat tails compared to anormal distribution. For risk management, asset pricing and hedging purposes, it is importantto consider the high moments of the return distribution. Despite their importance, the thirdand fourth moments of asset returns are difficult to measure precisely by averaging the thirdand fourth powers of the sample returns due to the large deviations in the estimators.One of the methods for estimating the third moment of the return distribution is to usethe third moment variation process of the return based on high-frequency data. The thirdmoment variation is defined as the quadratic covariation between the return and its squaredprocesses over a fixed time period (Choe and Lee, 2014). This approach is an extension of thegrowing literature regarding the realized variance of return, including Barndorff-Nielsen andShephard (2002),Andersen et al. (2003), Barndorff-Nielsen and Shephard (2004), Hansen andLunde (2006), Mykland and Zhang (2009) and Wang and Mykland (2014). Similarly, withthe realized variance, the third moment variation has good properties as an estimator such asconsistency, relative efficiency and unbiasedness under a martingale assumption (Lee, 2015). ∗ Department of Statistics, Yeungnam University, Gyeongsan, Gyeongbuk 38541, Korea † This work was supported by the 2015 Yeungnam University Research Grant. ‡ School of Business Administration, UNIST, Ulsan 44919, Korea § Byoung Ki Seo was supported by the 2012 Research Fund(1.120071.01) of UNIST(Ulsan National Instituteof Science and Technology). a r X i v : . [ q -f i n . P R ] A ug urthermore, the third moment variation can play an important role in hedging the skewand fat tail risks of return distributions. An investor who wants to hedge the skew and tailrisk may contract the third moment variation swap under which a predetermined fixed leg andthe negative value of the realized third moment variation, the floating leg, are exchanged atmaturity. The basic trading mechanism of the third moment variation swap is similar to thevariance swap. If the underlying asset price plunges, then the floating leg is likely to have apositive value, since the third moment variation itself is likely to have a negative value in theturmoil, and the floating leg of the swap is defined as the negative value of the third momentvariation. Therefore, an investor can be compensated for the loss of an underlying asset by thefloating leg of the swap. As a result, the return of the total portfolio follows a more Gaussian-likethin-tail distribution than the underlying asset alone.Similar studies of trading the skew risk have been reported. Schoutens (2005) defined themoment swaps based on the finite sum of k -th powers of log-return and showed that the hedgingperformance of the variance swap can be enhanced by a third moment swap, where the thirdmoment swap is defined as being different from our approach. Neuberger (2012) constructed asimilar approach for a skew swap but this was also based on different definition of the measureof skewness focused on the aggregation property. With this definition of the skew swap, Kozhanet al. (2013) examined the skew risk premium in equity index markets. For more informationon financial studies about skewness in asset returns, see Kraus and Litzenberger (1976), Harveyand Siddique (1999), Harvey and Siddique (2000), Bakshi et al. (2003) and Christoffersen et al.(2006).This study examine the return distribution of the portfolio hedged by the third momentvariation swap based on partial differential equations. With the definition of the third momentvariation swap and under the stochastic volatility and jump diffusion model, the joint proba-bility density function of the hedged portfolio and underlying return is represented by partialdifferential equations. The probability density function is calculated using the alternating direc-tion implicit methods (ADI). The ADI scheme is an efficient algorithm for a numerical solutionto partial differential equations and there are financial applications such as In’t Hout and Foulon(2010), Haentjens and In’t Hout (2012), Jeong and Kim (2013) and Haentjens and In’t Hout(2015). The numerical results show that the third moment variation swap hedges the fat tailsof the underlying asset return distributions in terms of skewness and kurtosis under both thestochastic volatility and stochastic volatility with jump models.The remainder of the paper is organized as follows: Section 2 explains the structure of thethird moment swap and examine the empirical performance based on the S&P 500 return series.In Section 3, the probability density function is computed under the stochastic volatility model.Section 4 extends the result to a jump diffusion stochastic volatility model. Section 5 concludesthe paper. 2 Third moment variation swap
This subsection briefly reviews the structure of the third moment swap to hedge the skew andtail risk, as introduced by Choe and Lee (2014), and examines the hedging performance usingthe S&P 500 index. The swap is based on the quantity called the third moment variation ofthe return process. (More precisely, it is a covariation but for simplicity, it is called the thirdmoment variation.) The third moment variation of a semimartingale return process R is definedby the quadratic covariation between the return and its square processes as follows:[ R, R ] t = lim (cid:107) π n (cid:107)→ N (cid:88) i =1 ( R t i − R t i − )( R t i − R t i − ) in probabilitywhere π n is a sequence of partitions 0 = t < · · · < t N = t and (cid:107) π n (cid:107) is the mesh of the partition.In addition, [ R, R ] t = [ R, R ] ct + (cid:88)
60 and 250 work-days. All plotsshow a significant negative fat tail compared to the normal distribution. The right panel showsthe QQ-plots of the hedged portfolio by the third moment swaps with previously mentionedmaturities, and the plots represent thinner return distributions.In this analysis, for simplicity, the fixed values of the swaps are assumed to be zeros. Indeed,the fixed value of the swap is not necessarily zero and its fair price would be determined by themarket participants. The theoretical values of the swap are expected to be based on Europeanoption prices under the assumption that the option prices properly reflect the risk-neutralmeasure. For more information on the pricing issues of the swap, see Choe and Lee (2014).Note that the fixed value of the swap, i.e., the price of the swap, is predetermined at themoment of contract and only affects the mean of the portfolio’s return distribution, but notother distributional properties such as the variance, skewness or kurtosis. Therefore, withouta loss of generality, the fixed value of the swap is set to zero because this study focus on theshape of the return distribution of the hedged portfolio.The hedge numbers are determined to minimize the squares of the differences between thequantiles of the realized portfolio return, hypothetically hedged by the swap, and the normaldistribution function with the same mean and standard deviation with the portfolio distribu-tion. In other words, the L -norm of the difference between the empirical portfolio returnand the corresponding normal distribution was minimized. The amounts of the swap posi-tions are 242 . , . , . . T = 5 , ,
60 and 250 days, respectively. For example, if the index is 1,000 at time zero and thematurity of the swap is 20 work days, then the notional amounts of the swap position for a unitindex is 80 . × ,
000 = 80 , Q uan t il e s o f unde r l y i ng r e t u r n QQ plot of return versus standard normal, T = 5 days (a) −4 −3 −2 −1 0 1 2 3 4−0.100.1 Standard Normal Quantiles Q uan t il e s o f unde r l y i ng r e t u r n QQ plot of return versus standard normal, T = 5 days (b) −4 −3 −2 −1 0 1 2 3 4−0.2−0.100.10.2 Standard Normal Quantiles Q uan t il e s o f unde r l y i ng r e t u r n QQ plot of return versus standard normal, T = 20 days (c) −4 −3 −2 −1 0 1 2 3 4−0.2−0.100.10.2 Standard Normal Quantiles Q uan t il e s o f unde r l y i ng r e t u r n QQ plot of return versus standard normal, T = 20 days (d) −4 −3 −2 −1 0 1 2 3 4−0.3−0.2−0.100.10.20.3 Standard Normal Quantiles Q uan t il e s o f unde r l y i ng r e t u r n QQ plot of return versus standard normal, T = 60 days (e) −4 −3 −2 −1 0 1 2 3 4−0.3−0.2−0.100.10.20.3 Standard Normal Quantiles Q uan t il e s o f unde r l y i ng r e t u r n QQ plot of return versus standard normal, T = 60 days (f) −4 −3 −2 −1 0 1 2 3 4−0.4−0.200.20.4 Standard Normal Quantiles Q uan t il e s o f unde r l y i ng r e t u r n QQ plot of return versus standard normal, T = 250 days (g) −4 −3 −2 −1 0 1 2 3 4−0.4−0.200.20.4 Standard Normal Quantiles Q uan t il e s o f unde r l y i ng r e t u r n QQ plot of return versus standard normal, T = 250 days (h)
Figure 1: QQ plots of the underlying asset returns (left) and the hedged portfolios’ returns(right) with various maturities 5
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Figure 2: The dynamics of S&P 500 index (left) and the hedged portfolio’ value (right) from1990 to 2007dynamics, the return of the hedged portfolio shows more steady growth all over time, evenduring the dot-com bubble crash in the early 2000’s. The relatively large excess return in thehedged portfolio compared to the S&P 500 index is due to the assumption that the fixed leg ofthe swap is zero.
Consider the effect of the transaction cost such as bid-ask spread due to the illiquidity of the thirdmoment variation swap. The transaction cost does not affect the shape of the hedged portfolio’sconditional distributions upon the contract date, since the transaction cost is predetermined atthe time of the contract. The total return of the hedged portfolio over the long run period, e.g.,three years, will be diminished, when one repeatedly contract the third moment variation swap,e.g., every month as in the previous example. The dynamics of the hedged portfolios’ valuesare plotted along with presumed transaction costs with 0.2% and 0.5% of the underlying assetprice in Figure 3 based on the S&P 500 index. As expected, with large amounts of transactioncosts, the total returns have been diminished from 1990 to 2007.One way to avoids the risk associated with the transaction costs is to contract a long-termthird moment swap with multiple legs similar to the interest rate swap. For example, one cancontract the third moment variation swap with three years maturity and the legs with a onemonth interval. The swap then has 36 floating and fixed legs to be exchanged and the exchangeof each leg is performed in the same way as the single third moment variation swap explainedin the previous subsection. In this way, the buyer of the swap can be compensated for the lossby the skew distribution every month for up to three years without taking the risks associatedwith future possible transaction costs.The third moment variation swap can transfer the skew and tail risk from one party toanother but cannot remove the risk entirely. In addition, in market turmoil, the swap seller whohas an obligation to pay the floating leg to the buyer might have difficulty in making payment.6
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Figure 3: The dynamics of the hedged portfolios’ values with transaction cost 0.2% (left) and0.5% (right)The swap is associated with a wrong-way risk as the buyer’s exposure to the counterparty iscorrelated with the seller’s credit risk. If the underlying asset price plunges, then the swapseller’s payment to the buyer tends to increase but in such an asset price crash, the seller is alsolikely to be exposed to severe market and credit risk. One way to minimize the wrong-way andcounterparty risk is central clearing. Central counterparties (CCP) bear the counterparty creditrisk of the bilateral trades, such as the interest rate and credit default swaps, and the role ofCCP becomes increasing worldwide. The third moment variation swap also can be standardizedand hence expected to be traded via CCPs.
In this section, the probability density function of the tail hedged portfolio with the thirdmoment variation swap under a stochastic volatility model is computed. Since the analyticalformula of the distribution of the tail hedged portfolio is not known, this paper proposes aPDE and a numerical approach to calculate the probability density function of the tail hedgedportfolio. Assume that the asset return process R follows the square root stochastic volatilitymodel (as reported by Heston (1993)):d R t = (cid:18) µ − V t (cid:19) d t + (cid:112) V t d W st d V t = κ ( θ − V t )d t + γ (cid:112) V t (cid:16) ρ d W st + (cid:112) − ρ d W vt (cid:17) where W s and W v are independent standard Brownian motions. Then the third momentvariation process is represented by[ R, R ] t = 2 (cid:90) t R s d[ R ] s = 2 (cid:90) t R s V s d s where we use d[ R ] s = V t d s . Note that we do not assume the zero or risk-neutral drift but usethe drift µ under the physical probability and µ is implicit in the integrand R s V s .7s in the previous section, consider an investor holding a hedged portfolio composed of anunderlying asset S and βS numbers of the third moment variation swap, i.e., receiving thefloating leg − βS [ R, R ] T , with maturity T . For simplicity, assume that the fixed leg of theswap is zero. The log-return of the hedged portfolio over [0 , T ] is approximated by X T = log (cid:18) S T − βS [ R, R ] T S (cid:19) ≈ R T − β [ R, R ] T = R T − β (cid:90) T R s V s d s. Now we explain the backward and forward approaches to compute the distribution of the hedgedportfolio.
To compute the probability density function of the portfolio return distribution, first, consider abackward approach based on the Feynman-Kac theorem. The time t conditional characteristicfunction of the portfolio’s return is u ( r, v, t ) = E (cid:104) e i φX T (cid:12)(cid:12)(cid:12) R t = r, V t = v (cid:105) = E (cid:20) exp (cid:18) − φβ (cid:90) Tt R s V s d s (cid:19) e i φR T (cid:12)(cid:12)(cid:12)(cid:12) R t = r, V t = v (cid:21) . Then, by the theorem, u ( r, v, t ) satisfies the following partial differential equation (PDE) ∂u∂t + (cid:18) µ − v (cid:19) ∂u∂r + κ ( θ − v ) ∂u∂v + 12 v ∂ u∂r + 12 γ v ∂ u∂v + ργv ∂ u∂r∂v = 2i φβrvu with the terminal condition u ( r, v, T ) = e i φr .We can derive the characteristic function with sufficiently enough numbers of grid points ∈ [ φ min , φ max ] by the above PDE with initial points r = 0 and v = v , and compute theprobability density function of the hedged portfolio by the discrete Fourier transform. Thedrawback of this approach is that the boundary conditions of the PDE is not well-defined andthis cause errors when we apply numerical procedure on the PDE. Therefore, we apply theforward approach to compute the probability density function. We have three dimensional stochastic differential equations for the dynamics of the portfolioreturn: d X t d R t d V t = µ − V t − βR t V t µ − V t κ ( θ − V t ) d t + √ V t √ V t γρ √ V t γ (cid:112) − ρ √ V t d W xt d W st d W vt where W x is a dummy variable which is not used elsewhere. The variance-covariance matrix isrepresented by √ V t √ V t γρ √ V t γ (cid:112) − ρ √ V t √ V t √ V t γρ √ V t γ (cid:112) − ρ √ V t = V t V t γρV t V t V t γρV t γρV t γρV t γ V t .
8y the forward Kolmogorov equation, also known as the Fokker-Planck equation, we derive thePDE for the joint probability density function f ( x, r, v, t ) with x = X t , r = R t and v = V t attime t of the three dimensional random vectors ( X t , R t , V t ): ∂f∂t = − (cid:18) µ − v − βrv (cid:19) ∂f∂x − (cid:18) µ − v (cid:19) ∂f∂r − ∂∂v κ ( θ − v ) f + v ∂ f∂x + v ∂ f∂r + γ ∂ ∂v vf + v ∂ f∂x∂r + ργ ∂ ∂x∂v vf + ργ ∂ ∂r∂v vf with the initial condition f ( x, r, v,
0) = δ ( x ) δ ( r ) δ ( v − v ) where δ denotes the Dirac deltafunction.The spatial domain of the PDE is three dimensional. For the numerical procedure, it isconvenient to reduce the dimension of the space. Since x appears in the PDE only in thederivative operators, we apply the Fourier transform of f ( x, r, v, t ) with respect to x :ˆ f ( r, v, t ; φ ) = (cid:90) ∞−∞ f ( x, r, v, t )e − i φx d x. The Fourier transforms of the partial derivatives with respect to x are F (cid:26) ∂f∂x (cid:27) = i φ ˆ f , F (cid:26) ∂ f∂x (cid:27) = − φ ˆ f . Applying the above Fourier transforms to the PDE, we have ∂ ˆ f∂t = (cid:18) − µ + 12 v + i φv + ργ (cid:19) ∂ ˆ f∂r + v ∂ ˆ f∂r + (cid:8) − κ ( θ − v ) + γ + i ργφv (cid:9) ∂ ˆ f∂v + γ v ∂ ˆ f∂v + ργv ∂ ˆ f∂r∂v + (cid:26) i φ (cid:18) − µ + 12 v + 2 βrv (cid:19) − φ v + i ργφ + κ (cid:27) ˆ f (1)with the initial condition ˆ f ( r, v, φ ) = δ ( r ) δ ( v − v ). When φ = 0, the solution of the PDEis reduced to the joint probability density function of ( R t , V t ) under the square root stochasticvolatility model as plotted in Figure 4.For brevity, let µ r ( r, v ) = − µ + 12 v + i φv + ργ, σ r ( r, v ) = v ,µ v ( r, v ) = − κ ( θ − v ) + γ + i ργφv, σ v ( r, v ) = γ v,α ( r, v ) = i φ (cid:18) − µ + 12 v + 2 βrv (cid:19) − φ v + i ργφ + κ. Then we can rewrite ∂ ˆ f∂t = µ r ( r, v ) ∂ ˆ f∂r + σ r ( r, v ) ∂ ˆ f∂r + µ v ( r, v ) ∂ ˆ f∂v + σ v ( r, v ) ∂ ˆ f∂v + ργv ∂ ˆ f∂r∂v + α ( r, v ) ˆ f . We also construct a PDE for the joint probability density function g ( y, r, v, t ) of the thirdmoment variation with y = Y t := [ R , R ] t , r = R t and v = V t at time t of the three dimensionalrandom vectors ( Y t , R t , V t ): ∂g∂t = − rv ∂g∂y − (cid:18) µ − v (cid:19) ∂g∂r − ∂∂v κ ( θ − v ) g + v ∂ g∂r + γ ∂ ∂v vg + ργ ∂ ∂r∂v vg (2)9 v -0.25 0.2 r
020 0.25 0.30.540
Figure 4: When φ = 0, the solution of the Eq. (1) is the joint probability density function ofthe return and variance in Heston’s modelA transformed PDE with respect to y is represented by ∂ ˆ g∂t = (cid:18) − µ + 12 v + ργ (cid:19) ∂ ˆ g∂r + v ∂ ˆ g∂r + {− κ ( θ − v ) + γ } ∂ ˆ g∂v + γ v ∂ ˆ g∂v + ργv ∂ ˆ g∂r∂v + ( − φrv + κ )ˆ g. Now we explain the details of the numerical method to solve the PDE (1). The finitedifference method for the transformed PDE is employed to compute the numerical solution ofthe joint distribution. The spatial domain is restricted to a bounded region [ r min , r max ] × [0 , v max ]where r min = − r max . The numbers of the grid points of the return r and the volatility v spaces are equal and N denotes the number such that r min = r < · · · < r N = r max and0 = v < · · · < v N = v max . The difference sizes of the grid points of the return and volatilityspaces are denoted by ∆ r and ∆ v , respectively.The derivatives in r and v directions are computed using the central difference scheme when r < r i < r N and v < v j < v N . Under the scheme, the partial derivatives are approximated by ∂ ˆ f∂r ( r i , v j ) ≈ ˆ f i +1 ,j − ˆ f i − ,j r∂ ˆ f∂r ( r i , v j ) ≈ ˆ f i +1 ,j − f i,j + ˆ f i − ,j (∆ r ) ∂ ˆ f∂r∂v ( r i , v j ) ≈ ˆ f i +1 ,j +1 − ˆ f i − ,j +1 − ˆ f i +1 ,j − + ˆ f i − ,j − v ∆ r , where, for simplicity, the time notation t is omitted and ˆ f i,j is an approximation of ˆ f ( r i , v j )under our numerical procedure. Similarly, the derivatives with respect to v are approximated.When r i or v j has the boundary value of the spatial grid, we use the one-sided difference scheme,10or example, ∂ ˆ f∂r ( r , v j ) ≈ ˆ f ,j − ˆ f ,j ∆ r∂ ˆ f∂r ( r , v j ) ≈ ˆ f ,j − f ,j + ˆ f ,j (∆ r ) . For the time discretization, the alternating direction implicit (ADI) method of Peacemanand Rachford (1955) is used. The direction of r is first treated implicitly and the next, thedirection of v is treated implicitly. The error due to the explicit scheme is reduced by thedecreased error in the next implicit step. The mixed derivative term is calculated explicitly.The time points are distributed equally with the difference ∆ t . Between two discrete time points t n and t n +1 , there is an intermediate point t n +1 / .To apply the ADI scheme, we have a finite difference formula for the intermediate step. For1 < i < N ,ˆ f n +1 / i,j − ˆ f ni,j ∆ t/ µ r ( r i , v j ) ˆ f n +1 / i +1 ,j − ˆ f n +1 / i − ,j r + σ r ( r i , v j ) ˆ f n +1 / i +1 ,j − f n +1 / i,j + ˆ f n +1 / i − ,j ∆ r + µ v ( r i , v j ) ˆ f ni,j +1 − ˆ f ni,j − v + σ v ( r i , v j ) ˆ f ni,j +1 − f ni,j + ˆ f ni,j − (∆ v ) + ργv j ˆ f ni +1 ,j +1 − ˆ f ni − ,j +1 − ˆ f ni +1 ,j − + ˆ f ni − ,j − r ∆ v + α ( r i , v j ) ˆ f n +1 / i,j . We rewrite (cid:18) µ r r − σ r (∆ r ) (cid:19) ˆ f n +1 / i − ,j + (cid:18) t + 2 σ r (∆ r ) − α (cid:19) ˆ f n +1 / i,j + (cid:18) − µ r r − σ r (∆ r ) (cid:19) ˆ f n +1 / i +1 ,j = b i,j (3)where b i,j = 2 ˆ f ni,j ∆ t + µ v ˆ f ni,j +1 − ˆ f ni,j − v + σ v ˆ f ni,j +1 − f ni,j + ˆ f ni,j − (∆ v ) + ργv j ˆ f ni +1 ,j +1 − ˆ f ni − ,j +1 − ˆ f ni +1 ,j − + ˆ f ni − ,j − r ∆ v and without confusion, let µ r = µ r ( r i , v j ) and similarly for µ v , σ r , σ v and α .When i = 1, one-sided difference schemes are used and we haveˆ f n +1 / ,j − ˆ f n ,j ∆ t/ µ r ˆ f n +1 / ,j − ˆ f n +1 / ,j ∆ r + σ r ˆ f n +1 / ,j − f n +1 / ,j + ˆ f n +1 / ,j (∆ r ) + µ v ˆ f n ,j +1 − ˆ f n ,j − v + σ v ˆ f n ,j +1 − f n ,j + ˆ f n ,j − (∆ v ) + ργv j ˆ f n ,j +1 − ˆ f n ,j +1 − ˆ f n ,j − + ˆ f n ,j − r ∆ v + α ˆ f n +1 / ,j and (cid:18) t + µ r ∆ r − σ r (∆ r ) − α (cid:19) ˆ f n +1 / ,j + (cid:18) − µ r ∆ r + 2 σ r (∆ r ) (cid:19) ˆ f n +1 / ,j − σ r (∆ r ) ˆ f n +1 / ,j = b ,j . (4)11here b ,j = 2 ˆ f n ,j ∆ t + µ v ˆ f n ,j +1 − ˆ f n ,j − v + σ v ˆ f n ,j +1 − f n ,j + ˆ f n ,j − (∆ v ) + ργv j ˆ f n ,j +1 − ˆ f n ,j +1 − ˆ f n ,j − + ˆ f n ,j − r ∆ v . Similarly, when i = N , we have − σ r (∆ r ) ˆ f n +1 / N − ,j + (cid:18) µ r ∆ r + 2 σ v (∆ r ) (cid:19) ˆ f n +1 / N − ,j + (cid:18) t − µ r ∆ r − σ v (∆ r ) − α (cid:19) ˆ f n +1 / N,j = b N,j (5)where b N,j = 2 ˆ f nN,j ∆ t + µ v ˆ f nN,j +1 − ˆ f nN,j − v + σ v ˆ f nN,j +1 − f nN,j + ˆ f nN,j − (∆ v ) + ργv j ˆ f nN,j +1 − ˆ f nN − ,j +1 − ˆ f nN,j − + ˆ f nN − ,j − r ∆ v . Let ˆ f n +1 / j and b j denote the j -th column vectors that consist of ˆ f n +1 / i,j and b i,j , respectively.Then, for each j , we have a matrix multiplication form A ( j ) r ˆ f n +1 / j = b j (6)where A ( j ) r is a tridiagonal matrix except the first and last row: A ( j ) r = a ( r,j )1 , a ( r,j )1 , a ( r,j )1 , · · · a ( r,j )2 , a ( r,j )2 , a ( r,j )2 , a ( r,j ) N − ,N − a ( r,j ) N − ,N − a ( r,j ) N − ,N · · · a ( r,j ) N,N − a ( r,j ) N,N − a ( r,j ) N,N where the entries are determined by Eqs. (3),(4) and (5). For example, if 1 < < N , then, byEq. (3), we have a ( r,j ) i,i = 2∆ t + 2 σ r ( r i , v j )(∆ r ) − α ( r i , v j ) a ( r,j ) i,i − = µ r ( r i , v j )2∆ r − σ r ( r i , v j )(∆ r ) a ( r,j ) i,i +1 = − µ r ( r i , v j )2∆ r − σ r ( r i , v j )(∆ r ) . The matrix is sparse and the solution of the Eq. (6) can be solved efficiently.For the next step, we apply the finite difference scheme implicitly on v -direction. For12 < j < N , we haveˆ f n +1 i,j − ˆ f n +1 / i,j ∆ t/ µ r ˆ f n +1 / i +1 ,j − ˆ f n +1 / i − ,j r + σ v ˆ f n +1 / i +1 ,j − f n +1 / i,j + ˆ f n +1 / i − ,j ∆ r + µ v ˆ f n +1 i,j +1 − ˆ f n +1 i,j − v + σ v ˆ f n +1 i,j +1 − f n +1 i,j + ˆ f n +1 i,j − (∆ v ) + ργv j ˆ f n +1 / i +1 ,j +1 − ˆ f n +1 / i − ,j +1 − ˆ f n +1 / i +1 ,j − + ˆ f n +1 / i − ,j − r ∆ v + α ˆ f n +1 i,j . We rewrite (cid:18) µ v v − σ v (∆ v ) (cid:19) ˆ f n +1 i,j − + (cid:18) t + 2 σ v (∆ v ) − α (cid:19) ˆ f n +1 i,j + (cid:18) − µ v v − σ v (∆ v ) (cid:19) ˆ f n +1 i,j +1 = c i,j (7)where c i,j = 2 ˆ f n +1 / i,j ∆ t + µ v ˆ f n +1 / i +1 ,j − ˆ f n +1 / i − ,j r + σ v ˆ f n +1 / i +1 ,j − f n +1 / i,j + ˆ f n +1 / i − ,j ∆ r + ργv j ˆ f n +1 / i +1 ,j +1 − ˆ f n +1 / i − ,j +1 − ˆ f n +1 / i +1 ,j − + ˆ f n +1 / i − ,j − r ∆ v . Similarly with the previous step, when j = 1 or j = N , we use the one-sided different schemesand we have (cid:18) t + µ v ∆ v − σ v (∆ v ) − α (cid:19) ˆ f ni, + (cid:18) − µ v ∆ v + 2 σ v (∆ v ) v j (cid:19) ˆ f ni, − σ v (∆ v ) ˆ f ni, = c i, (8)and − σ v (∆ v ) ˆ f ni,N − + (cid:18) µ v ∆ v + 2 σ v (∆ v ) (cid:19) ˆ f ni,N − + (cid:18) t − µ v ∆ v − σ v (∆ v ) v j − α (cid:19) ˆ f ni,N = c i,N (9)where c i, and c i,N are defined by the one-sided schemes as in the previous step. Thus, for each i , we have a matrix multiplication form with raw vectors of ˆ f ni and c ni A ( i ) v (cid:16) ˆ f ni (cid:17) T = ( c ni ) T where T denotes the non-conjugate transpose and A ( i ) v = a ( v,i )1 , a ( v,i )1 , a ( v,i )1 , · · · a ( v,i )2 , a ( v,i )2 , a ( v,i )2 , a ( v,i ) N − ,N − a ( v,i ) N − ,N − a ( v,i ) N − ,N · · · a ( v,i ) N,N − a ( v,i ) N,N − a ( v,i ) N,N where the entries are determined by Eqs (7),(8) and (9).The boundary conditions are imposed asˆ f ( r min , v, t ; φ ) = ˆ f ( r max , v, t ; φ ) = ˆ f ( r, , t ; φ ) = ˆ f ( r, v max , t ; φ ) = 0under the assumption that the parameters in the model satisfy the Feller condition to guaranteethe positiveness of the variance process: 2 κθ > γ .13
40 −20 0 20 40−0.200.20.40.60.811.2
Hestonhedged −40 −20 0 20 40−0.2−0.100.10.2
Hestonhedged
Figure 5: The real (left) and imaginary (right) parts of the characteristic functions with β = 0(dashed) and 40 (solid) By performing the numerical procedure, we get the characteristic function of X T as plottedin Figure 5 with parameter settings κ = 18 , θ = 0 . , γ = 1 , ρ = − .
62 and T = 0 .
1. In thefigure, the real (left) and imaginary (right) parts of the characteristic functions are presentedfor underlying asset (dashed) and hedged portfolio with β = 40 (solid). Taking the transform tothe characteristic functions, we compute the joint probability density function of ( X T , R T , V T ).By applying the ADI scheme, the numerical procedure takes much less time compared tothe explicit scheme as the time step needed to ensure the numerical stability is much larger thanin the case of the explicit scheme. In the ADI scheme, with a spatial grid of [ R min , R max ] =[ − . , . V min , V max ] = [0 , . r = 0 .
025 and ∆ v = 0 . t =0 . t needs be around 2 × − when the same spatialgrid is used.Figure 6 shows the probability density functions of the underlying asset (left) and the hedgedportfolio (right) with hedge number β = 40 and T = 0 . . Figure 7 presents the probabilitydensity functions of the hedged portfolio returns (solid) compared to the return of the underlyingasset (dashed) with various hedge numbers β = 10 , , , ,
50 and 60. The hedged portfolioshave more Gaussian-like thin-tail distribution compared to the distribution of the underlyingasset.Table 2 lists the numerically computed mean, standard deviation, skewness and kurtosis ofthe return distributions of the portfolios with various hedge numbers β = 0 , , . . . ,
60. Thetable suggests that with β between 30 and 40, the skewness of the portfolio return is aroundzero and has minimal kurtosis. This result is consistent with the simulation study where theoptimal hedge number is reported to be 38 .
42. A simulation study is performed with the samemethod for the empirical analysis explained in Section 2.Figure 8 shows the probability density functions of the annualized third moment variation14
Figure 6: Probability density functions and histograms of simulated data : underlying asset(left) and hedged portfolio (right)Table 1: Numerically computed standardized moments with various hedge number β withparameter setting µ = 0 . κ = 18 , θ = 0 . , γ = 1 , ρ = − .
62 and T = 0 . β mean std.dev. skewness kurtosis0 0.0041 0.0964 -0.4281 3.374110 0.0071 0.0861 -0.3097 3.226620 0.0100 0.0766 -0.1875 3.157430 0.0129 0.0683 -0.0671 3.156340 0.0158 0.0618 0.0600 3.158650 0.0188 0.0575 0.2296 3.218460 0.0216 0.0560 0.4275 3.2131derived by PDE (2). As expected, the distribution of the third moment variation is left skewed.The local truncation error of the Peaceman and Rachford two-dimensional ADI scheme isknown as O ((∆ t ) + (∆ r ) + (∆ v ) ). In other words, the error caused by one iteration of thenumerical procedure is bounded by C ((∆ t ) + (∆ r ) + (∆ v ) ) for some constant C . Since theclosed form formula for the probability distribution of the hedged portfolio does not exist, itis difficult to demonstrate the global truncation error for the entire iteration by comparing theexact solution and the numerical solution. On the other hand, for the underlying asset alone,the numerically computed p.d.f. of the return under the present procedure can be comparedwith the p.d.f. retrieved from the known characteristic function of the Heston-type model: c ( ψ ) = exp (cid:26) − (i ψ + ψ ) θξ coth ( ξT ) + κ − i γρψ + κθT ( κ − i γρφ ) γ + i ψµT (cid:27)(cid:16) cosh ξT + κ − i γρψξ sinh ξT (cid:17) κθγ (10)where ξ = (cid:112) γ ( ψ + i ψ ) + ( κ − i γρψ ) . In the above formula, V = θ for simplicity as in thenumerical procedure.The result is presented in Figure 9, the global errors measured by the root mean squared15 Hestonhedged (a) β = 10 −0.4 −0.2 0 0.2 0.40246 Hestonhedged (b) β = 20 −0.4 −0.2 0 0.2 0.40246 Hestonhedged (c) β = 30 −0.4 −0.2 0 0.2 0.40246 Hestonhedged (d) β = 40 −0.4 −0.2 0 0.2 0.40246 Hestonhedged (e) β = 50 −0.4 −0.2 0 0.2 0.40246 Hestonhedged (f) β = 60 Figure 7: Probability density functions with various hedge numbers β (solid) compared to theunderlying asset under Heston’s model (dashed)16 Figure 8: Probability density function of the third moment variation
Figure 9: Global truncation error with respect to partition size ∆ r errors (RMSE) between the numerical PDE solutions and the p.d.f. retrieved from Eq. (10) areexamined. For the analysis, the spatial grids are set over r = [ − . , .
8] and v = [0 , . T = 0 .
1. The global error, which is the total errorfrom whole procedure, is calculated for ∆ r = 0 . , , , · · · , . , . v , also changes accordingly such that the number of partitions in r is equal to the number ofpartitions in v . The result shows that as the partition size decrease, the RMSE converges tozero. For example, when ∆ r = 0 . . × − , which is very close to zero. Sincethe numerical analysis on the p.d.f. of the hedged portfolio is based on the same method, thenumerical solution for the hedged portfolio’s return distribution has the same level of accuracy.17 Jump diffusion stochastic volatility
In this section, we consider a stochastic volatility jump diffusion model with jump in returnprocess: R t = (cid:90) t (cid:18) µ − V s (cid:19) d s + (cid:90) t (cid:112) V s d W ss + (cid:88)
R, R ] t = 2 (cid:90) t R s V s d s + (cid:88) over L ( R ), by L ∗ f ( x, r, v, t ) = − ∂∂x (cid:18) µ − v − βrv (cid:19) f − ∂∂r (cid:18) µ − v (cid:19) f − ∂∂v κ ( θ − v ) f + ∂ ∂x vf + ∂ ∂r vf + ∂ ∂v γ vf + ∂ ∂x∂v ργvf + ∂ ∂r∂v ργvf + ∂ ∂x∂r vf + λ (cid:18)(cid:90) R f ( x − z X ( t ) , r − z, v, t ) ψ ( z )d z − f ( x, r, v, t ) (cid:19) . (12)It is well known that the parts involving derivatives in Eqs. (11) and (12) are adjoint to eachother. For the integration part, we show that (cid:28)(cid:90) R h ( x + z X ( t ) , r + z, v, t ) ψ ( z )d z, f ( x, r, v, t ) (cid:29) = (cid:90) R (cid:90) R h ( x + z X ( t ) , r + z, v, t ) f ( x, r, v, t ) ψ ( z )d z d x d r d v = (cid:90) R (cid:90) R h ( x (cid:48) , r (cid:48) , v, t ) f ( x (cid:48) − z X ( t ) , r (cid:48) − z, v, t ) ψ ( z )d z d x (cid:48) d r (cid:48) d v = (cid:28) h ( x (cid:48) , r (cid:48) , v, t ) , (cid:90) R f ( x (cid:48) − z X ( t ) , r (cid:48) − z, v, t ) ψ ( z )d z (cid:29) . h ( X t , R t , V t , t ) = (cid:90) t ∂h ( X s − , R s − , V s , s ) ∂s + L h ( X s − , R s − , V s , s )d s + (cid:90) t (cid:18)(cid:112) V s ∂h ( X s − , R s − , V s , s ) ∂x + (cid:112) V s ∂h ( X s − , R s − , V s , s ) ∂r (cid:19) d W st + (cid:90) t γ (cid:112) V s ∂h ( X s − , R s − , V s , s ) ∂v d W vs + (cid:88)
0) = δ ( x ) δ ( r ) δ ( v − v ). For further and rigorous informationabout the Fokker-Planck or forward equation for jump diffusion model, see Pappalardo (1996),Andersen and Andreasen (2000), Hanson (2007), Bentata and Cont (2009), Bentata and Cont(2015).As in the previous section, we use the PDE (13) to compute the joint probability densityfunction of ( X t , R t , V t ). To reduce the dimension, we apply the Fourier transform with respectto x , and the transformed PDE is ∂ ˆ f∂t = (cid:18) − µ + 12 v + i φv + ργ (cid:19) ∂ ˆ f∂r + v ∂ ˆ f∂r + (cid:8) − κ ( θ − v ) + γ + i ργφv (cid:9) ∂ ˆ f∂v + γ v ∂ ˆ f∂v + ργv ∂ ˆ f∂r∂v + (cid:26) i φ (cid:18) − µ + 12 v + 2 βrv (cid:19) − φ v + i ργφ + κ − λ (cid:27) ˆ f + λ (cid:90) R e − i z X ( t ) φ ˆ f ( r − z, v, t ; φ ) ψ ( z )d z. By applying the numerical procedure explained in the previous section, the probability densityfunctions are calculated under the stochastic volatility jump diffusion model.In this study, λ = 20 and the standard deviation of jump size σ j = 0 .
01. For the returnand volatility parameters, µ = 0 . , κ = 18 , θ = 0 . , γ = 1 , ρ = − .
62. Figure 10 shows theprobability density functions of the underlying asset (left) and the hedged portfolio (right) withthe hedge number β = 45 and T = 0 . . In Figure 11 presents the probability density functions ofthe hedged portfolio returns (solid) compared to the return of the underlying asset (dashed)with various hedge numbers β = 15 , ,
45 and 60. As in the previous section, the hedgedportfolios have more Gaussian-like thin-tail distributions compared to the distribution of theunderlying asset. 20
Figure 10: Probability density functions and histograms of simulated data under stochasticvolatility and jump diffusion : underlying asset (left) and hedged portfolio (right)Table 2: Numerically computed standardized moments with various hedge number β withparameter setting µ = 0 . κ = 18 , θ = 0 . , γ = 1 , ρ = − . λ = 20, σ j = 0 .
02 and T = 0 . β mean std.dev. skewness kurtosis0 0.0012 0.0759 -0.5955 3.975715 0.0039 0.0682 -0.4245 3.558930 0.0065 0.0612 -0.2663 3.288945 0.0091 0.0552 -0.1289 3.143360 0.0117 0.0506 0.0107 3.1072Table 2 list the numerically computed mean, standard deviation, skewness and kurtosisof the return distributions of the portfolios with various hedge numbers β = 0 , , , , β between 45 and 60, the skewness of the portfolio return is approximately zero and hasminimum kurtosis. This result is consistent with the simulation study, where the optimal hedgenumber is reported to be 45 . The probability density functions of the tail hedge portfolio with the third moment variationswap were calculated. The method is based on numerical analysis of alternating direction im-plicit for the partial differential equations of the joint density functions. The computed densityfunctions show that the swap properly eliminates the skew and fat tail risk of an underlyingasset under Heston’s stochastic volatility and jump diffusion stochastic volatility models. Infuture work, a faster method to calculate the probability density function will be needed be-cause the partial differential equation approach has time complexity. Therefore, the computedprobability function can be used to find the optimal hedge number of the swap to eliminate the21
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