# Option Pricing by Probability Distortion Operator Based on the Quantile Function

Authors: Luogen Yao; Gang Yang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

A new class of distortion operators based on quantile function is proposed for pricing options. It is shown that option prices obtained with our distortion operators are just the prices under mean correcting martingale measure in exponential Lévy models. In particular, Black-Scholes formula can be recuperated by our distortion operator. Simulation analysis shows that our distortion operator is superior to normal distortion operator and NIG distortion operator.


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## Body

## 1. Introduction

With the convergence of financial and insurance markets, there has been a drastic increase in the trading and securitisation of insurance. The techniques adopted in insurance pricing are being considered for the option pricing. This has aroused wide interest in insurance pricing and how these techniques relate to financial theory.In the actuarial literature, many probability transforms for pricing financial and insurance risks have been developed, such as the variance loading, the standard deviation loading, and the Esscher transform. Wang [1] and Wang [2] have proposed the following class of distortion operators based on normal distribution (1)gαu=ΦΦ-1u+α,where Φ denotes the standard normal cumulative distribution function (CDF) and α is a constant. The transform is now called the Wang transform, and the parameter α is considered to be a risk premium. The Wang transform not only possesses various desirable properties as a pricing method but also has a reasonable economic interpretation. However, Hamada and Sherris [3] have concluded that the Wang transform does not work well under non-Gaussian assumptions. Pelsser [4] also points out that, for general stochastic processes, Wang transform does not lead to a price which is consistent with the arbitrage free price.It is well known that the returns of most financial assets have semiheavy tails and the actual kurtosis is higher than that of a normal distribution. The pricing principle approach of Wang must be somehow modified in order to capture the non-Gaussian feature of market prices. Many scholars have studied these issues from various perspectives. For example, Wang [2] proposes the two-parameter transformation (2)gν,λu=TνΦ-1u+λand reports that the transform (2) is much better than (1) to fit the market data, where Tν denotes the CDF of t-distribution with ν degree of freedom. Kijima and Muromachi [5] study the following transform: (3)gν,λu=Pν,-λTν-1u,where Pν,δ denotes the CDF of noncentral t-distribution with ν degree of freedom and noncentrality parameter δ. Kijima and Muromachi [6] consider the transform(4)gλu=EΦG-1uY+λ∣Y,where Y is a positive random variable and the expectation is taken with respect to Y. Here, G(x) denotes the CDF of random variable U/Y and U represents a standard normal random variable, independent of Y. Godin et al. [7] introduce a new distortion based on Normal Inverse Gaussian (NIG) distribution (5)gα,β,δ,θu=ΦNIGΦNIG-1u+θ,where ΦNIG denotes the NIG cumulative distribution function and show that the NIG-based distortion function performs better than the normal-based distortion function in Wang [1].The price of an option obtained by any method should be arbitrage free. However, transforms (1)-(5) do not necessarily satisfy this property. We hold that we should make full use of the yield information to get the distortion functions. Inspired by the above works, we propose a new distortion operator based on the quantile function. It is shown that the celebrated Black-Scholes formula can be recuperated through our distortion operator. For a Lévy model, option price obtained with our distortion operator is just the price under mean correcting martingale measure (MCMM). Simulation analysis shows that our distortion operator is superior to normal distortion operator and NIG distortion operator.


## 2. A New Class of Distortion Operators Based on Quantile Function

Wang [1] recognizes that the Proportional Hazards probability transform cannot be applied consistently to assets and liabilities simultaneously. He proposes the so-called Wang transform aiming at integrating financial and actuarial insurance pricing theories. Let FX and SX be the distribution function and survival function of a risk X, respectively. Wang [1] defines the risk-adjusted premium for a risk X by the Choquet integral representation (6)HX;α=∫-∞0gαSXx-1dx+∫0+∞gαSXxdx,where gα is a distortion function. When X is a positive random variable, the Choquet integral representation has a simpler form (7)HX;α=∫0+∞gαSXxdx.In Wang transform, the tail probability distribution SX(x) is distorted by the function (1). One of the prominent properties of Wang transform is that it can recuperate the Black-Scholes formula in the standard Black-Scholes model.The CDF of a real-valued random variableX is the function given by FX(x)=P(X≤x),x∈R. It is well known that FX(x) is nondecreasing and right-continuous, which makes it a càdlàg function. Furthermore, limx→-∞FX(x)=0,limx→+∞FX(x)=1. If the CDF FX is strictly increasing, then FX-1(y) is the unique real number x such that FX(x)=y. In such a case, this defines the inverse distribution function. However, the CDF does not, in general, have the ordinary inverse as it may not be strictly increasing.Definition 1 (See [8]).
For a nondecreasing functionh:R→R with h(-∞)=limx↓-∞h(x) and h(∞)=limx↑∞h(x), the generalized inverse h-:R→R¯=[-∞,∞] of h is defined by (8)h-y=inf⁡x∈R:hx≥y,y∈R, with the convention that inf{ϕ}=∞. If h:R→[0,1] is a CDF, h-:[0,1]→R¯ is also called the quantile function of h.Ifh is continuous and strictly increasing, h- coincides with h-1, the ordinary inverse of h on ran h={h(x):x∈R}, the range of h. If h is continuous and nondecreasing, h- is continuous, strictly increasing and (9)hx>y⇔x>h-y,(10)hh-y=y,∀y∈ranh∪infranh,supranh.The following theorem will be applied in this work.Theorem 2.
Ifh is strictly increasing, then ∀x∈R,h-(h(x-0))=x, where h(x-0) is the left limit of h at x.Proof.
x ≥ h - ( h ( x - 0 ) ) since h(x)≥h(x-0). On the other hand, ∀ε>0, there must exist some x′∈R such that (11)x-ε<x′<x.By the strictly monotonicity of h, we have (12)hx-ε<hx′≤hx-0.This implies h-(h(x-0))=x.AsFX may not be continuous in Theorem 2, we can only use the quantile function FX-, not the ordinary inverse FX-1. In this work, we will introduce a new class of distortion operators based on quantile function rather than on normal distribution.Definition 3.
LetX be a random variable representing a financial (insurance) risk. We define the quantile distortion operator as (13)gλXu=F-XF-X-u+λ.Note that unlike the distortion operators in (1)-(5), our distortion operator is completely determined by the distribution of the risk X. Essentially, our distortion operator shifts the uth quantile of -X by a positive or negative value λ and reevaluates the probability for the shifted quantile. We start our discussion with the following theorem to show the effect of the quantile distortion operator on the risk.Theorem 4.
LetX be a risk and Z=h(X) be a transformation through a continuous, positive, and nondecreasing function h. If FX is strictly increasing, then we have (14)HZ;X,λ≜∫gλXSZxdx=EhX+λ.Proof.
By definition, (9), and (10), we have (15)HZ;X,λ=∫gλXSZxdx,SZx=PZ>x=PhX>x=PX>h-x=P-X<-h-x=F-X-h-x-0.As FX is strictly increasing and F-X(x)=1-FX(-x-0), we can conclude that F-X is also strictly increasing. Due to Theorem 2, if we now apply the distortion gλ on SZ, we have (16)gλXSZx=F-XF-X-F-X-h-x-0+λ=F-Xλ-h-x=P-X≤λ-h-x=PX+λ≥h-x=PhX+λ≥x,from which we conclude that (17)∫gλXSZxdx=EhX+λ.Remark 5.
Under assumption thatX is a random variable with the CDF FX whose probability density function is symmetric around 0, Hamada and Sherris [3] also derive the formula (14). In their proofs, the strict monotonically increasing property of the CDF is also adopted. However, Theorem 4 only needs the assumption that FX is strictly increasing. Hence, Theorem 4 generalizes the equivalent result found in Hamada and Sherris [3].


## 3. Option Pricing

Let us consider a market model where the stock price processS={St,t∈[0,T]} is a stochastic process of the following form: (18)St=S0eXt,0≤t≤T,where X={Xt,t∈[0,T]} is a stochastic process defined on a stochastic basis (Ω,F,F=(Ft)t∈[0,T],P) which satisfies the usual conditions. Moreover, in our market we have a riskless asset B={Bt,t∈[0,T]}, evolving as (19)Bt=exp⁡rt,where S0>0,r>0 are constants.Under the assumption that the CDFFXT of XT is strictly increasing, we now study how this distortion affects model (18) for asset prices and in particular if there is a value of λ such that the discounted asset prices behave like risk-neutral asset prices. Note that ST can be expressed as ST=h(XT), where h(u)=S0eu. Applying Theorem 4, we have (20)HST;XT,-λ=EhXT-λ=ES0eXT-λ=S0e-λEeXT.If we set (21)λ∗=ln⁡EeXT-rT,then the expression for H[ST;XT,-λ] simplifies to (22)HST;XT,-λ∗=S0erT.In other words, under our distortion with a value of λ∗ given in (21), the price ST evolves like a risk-neutral asset.Next, we will expound how our distortion operator is consistent with the option pricing theory. We need the following lemma.Lemma 6.
LetX={Xt,t∈[0,T]} be a P-Lévy process with the generating triplet (α,σ2,ν)P. One possible arbitrage free price of a European call option C(T,K)=(ST-K)+ is given by (23)C=EQe-rTCT,K, where Q is the so-called MCMM under which X={Xt,t∈[0,T]} is a Lévy process with the generating triplet (α+r-φ(1),σ2,ν)Q, and φ is given by (24)φu≜ln⁡EPeuX1=uα+12σ2+∫eux-1-xIx<1νdx.Proof.
We refer to Schoutens [9] for a proof of this result which is now standard in the literature.In fact, (23) immediately yields the following formula: (25)C=S0e-Tφ1∫y+∞exFXTdx-e-rTKPXT>y,where y=(φ(1)-r)T-ln⁡S0/K.Theorem 7.
LetX={Xt,t∈[0,T]} be a P-Lévy process with the generating triplet (α,σ2,ν)P. If FXT is strictly increasing, then we have (26)e-rTHCT,K;XT,-λ∗=EQe-rTCT,K,where λ∗=ln⁡E[eXT]-rT=(φ(1)-r)T. Namely, option price under our distortion operator is just the price under MCMM.Proof.
It is clear thatC(T,K) can be expressed as C(T,K)=h(XT) for a function h(u)=(S0eu-K)+. By Theorem 4 we have that (27)HCT,K;XT,-λ∗=EhXT-λ∗=ES0eXT+r-φ1T-K+.Let Y={Yt,t∈[0,T]}, where Yt=Xt+(r-φ(1))t, and then (28)HCT,K;XT,-λ∗=ES0eYT-K+.As X={Xt,t∈[0,T]} is a P-Lévy process with the generating triplet (α,σ2,ν)P, Y={Yt,t∈[0,T]} is also a P-Lévy process, and the corresponding generating triplet is (α+r-φ(1),σ2,ν)P. Note that, by Lemma 6, X={Xt,t∈[0,T]} is a Q- Lévy process with the generating triplet (α+r-φ(1),σ2,ν)Q. Thus, the CDF of YT under probability measure P is the same as the CDF of XT under probability measure Q. We then have (29)HCT,K;XT,-λ∗=EQS0eXT-K+=EQCT,K.In other words, (30)e-rTHCT,K;XT,-λ∗=EQe-rTCT,K.Theorem7 implies that the price of a European call option which is evaluated with the pricing kernel associated with our distortion (13) with a parameter λ∗ is just this call price under MCMM Q. Therefore, our distortion operator is consistent with the option pricing theory.Remark 8.
Many popular market models based on Lévy processes in option pricing, such as Gaussian process, Normal Inverse Gaussian process, Meixner process, Generalized Hyperbolic process, Variance Gamma process, and CGMY process, can meet the condition under which the CDF of financial asset returns is strictly increasing. Thus, as for above Lévy processes, option price under our distortion operator is arbitrage free.Remark 9.
IfX is a NIG process, the NIG-distorted pricing kernel in (5) reduces to the Black-Scholes-type formula under the MCMM Q. As for other Lévy processes, we do not know whether the price under (5) is arbitrage free or not. Therefore, our distortion operator is more flexible.Theorem 10.
LetXT=(μ-1/2σ2)T+σWT and Z=h(XT) be a transformation through a continuous, positive, and nondecreasing function h. We have (31)F-XTF-XT-PZ>x-λ∗=ΦΦ-1PZ>x-α∗,where λ∗=(μ-r)T and α∗=(μ-r)T/σ.Proof.
Note that(32)PZ>x=PhXT>x=PXT>h-x=1-PXT≤h-x=1-Φh-x-μ-1/2σ2TσT=Φμ-1/2σ2T-h-xσT.So (33)ΦΦ-1PZ>x-α∗=Φμ-1/2σ2T-h-xσT-μ-rTσ=Φr-1/2σ2T-h-xσT.On the other hand, (34)PZ>x=PhXT>x=P-XT<-h-x=F-XT-h-x,F-XTF-XT-PZ>x-λ∗=F-XT-h-x-μ-rT=P-XT≤-h-x-μ-rT=Φr-1/2σ2T-h-xσT.Theorem10 shows that the effect of our distortion operator with λ∗=(μ-r)T is the same as that of Wang’s distortion operator with α∗=(μ-r)T/σ under Gaussian distribution assumptions.Theorem 11.
LetXT be a NIG distribution with parameters (α,β,δT,μT) and Z=h(XT) be a transformation through a continuous, positive, and nondecreasing function h. We have (35)F-XTF-XT-PZ>x-λ∗=ΦNIGΦNIG-1PZ>x-θ∗,where ΦNIG is the cdf of NIG distribution with parameters (αδT,-βδT/α,αδT,0), i.e., ΦNIG(x)=NIG(x;αδT,-βδT/α,αδT,0) and (36)λ∗=μ-r-δα2-β+12-α2-β2T,θ∗=μ-r-δα2-β+12-α2-β2Tδ/α.Proof.
ForZ=h(XT), (37)PZ>x=PhXT>x=1-PXT≤h-x=1-PXT-μTδT/α≤h-x-μTδT/α.As XT-μT/δT/α is a NIG distribution with parameters (αδT,βδT/α,αδT,0), we have (38)PZ>x=1-NIGh-x-μTδT/α;αδT,βδTα,αδT,0.Because of the symmetry property of the parameter β, we have (39)PZ>x=NIG-h-x-μTδT/α;αδT,-βδTα,αδT,0=ΦNIGμT-h-xδT/α.So(40)ΦNIGΦNIG-1PZ>x-θ∗=ΦNIGμT-h-xδT/α-θ∗=ΦNIGr+δα2-β+12-α2-β2T-h-xδT/α.On the other hand, (41)F-XTF-XT-PZ>x-λ∗=F-XTF-XT-F-XT-h-x-λ∗=F-XT-h-x-λ∗=1-PXT-μTδT/α≤h-x+λ∗-μTδT/α=1-NIGh-x+λ∗-μTδTα;αδT,βδTα,αδT,0=NIG-h-x+λ∗-μTδT/α;αδT,-βδTα,αδT,0=ΦNIGr+δα2-β+12-α2-β2T-h-xδT/α.Theorem11 implies that the effect of our distortion operator is the same as that of NIG distortion operator under NIG distribution assumptions.It is well known that Wang’s distortion can recuperate the Black-Scholes formula in the standard Black-Scholes model. Now we show that our distortion can also recuperate the Black-Scholes formula.In the standard Black-Scholes model, the stock prices follow a geometric Brownian motion with(42)St=S0exp⁡μ-12σ2t+σWt,0≤t≤T,where W={Wt,0≤t≤T} is a standard Brownian motion. Let XT=(μ-1/2σ2)T+σWT, then XT~N((μ-1/2σ2)T,σ2T), where N(μ,σ2) denotes the normal distribution with mean μ and variance σ2. By Theorem 4, we have (43)HCT,K;XT,-λ∗=EhXT-λ∗=∫-∞+∞S0ex-λ∗-K+12Tπσexp⁡-x-μ-1/2σ2T22σ2Tdx=∫-∞+∞S0eσTy+μ-1/2σ2T-λ∗-K+12πexp⁡-y22dy.The values of y for which S0eσTy+(μ-1/2σ2)T-λ∗≥K determine the region of integration. This region is [ymin,+∞) where (44)ymin=ln⁡K/S0+λ∗-μ-1/2σ2TσT.Hence (45)HCT,K;XT,-λ∗=S0eμT-λ∗∫ymin+∞12πe-1/2y-σT2dy-K1-Φymin=S0eμT-λ∗1-Φymin-σT-KΦ-ymin=S0eμT-λ∗ΦσT-ymin-KΦ-ymin.By (21), (46)λ∗=ln⁡EeXT-rT=μ-rT,from which we conclude that (47)e-rTHCT,K;XT,-λ∗=S0ΦlnS0/K+r+1/2σ2TσT-e-rTKΦlnS0/K+r-1/2σ2TσT.This demonstrates that our distortion operator (13) can also recuperate the Black-Scholes price of a European call option with λ∗=(μ-r)T.


## 4. Simulation Analysis

On the basis of (13) under different types of data sets, we carry out an empirical analysis of the performance of our pricing kernel in this section. Our aim is twofold. First, we want to show that our distortion operator works well as Wang distortion operator when it is used in B-S model, and our distortion operator also works well as NIG distortion operator when it is used in NIG model. Our second aim is to show that our distortion operator performs better than Wang and NIG distortion operator in other Lévy option pricing models.We use simulated data to test Wang, NIG, and our distortion operators, which provides a controlled setting for our analysis. These pricing kernels are tested on five different simulated data sets. Asset prices are simulated according to five well-known models for which theoretical option price formulae are available: geometric Brownian model, NIG model, Jump diffusion model, VG model, and Meixner model. In order to compare how the three distortions perform when evaluating option on underlying assets that do not have Gaussian log-returns, these pricing kernels were tested on these data sets. The valuation procedure of European call options is as follows.Wang Distortion. (1) Simulate the stock price ST=S0eXT and compute the empirical survival function S^ST of ST;(2) Using Wang distortion (1), the parameter α^ is then calibrated to verify the risk-neutral condition such that (48)HST;-α^=S0erT;(3) Using the simulated data, compute the empirical survival functionS^C(T,K) of European call option C(T,K)=(ST-K)+;(4)α^ is used to compute the call option price(49)e-rTHCT,K;-α^=e-rT∫0+∞gα^S^CT,Kdx.NIG Distortion. (1) Simulate the stock price ST=S0eXT and compute the empirical survival function S^ST and S^C(T,K).(2) Estimate the parametersα,β,δ by maximum likelihood method from log-returns of simulated data.(3) Using NIG distortion (5), the parameter θ^ is then calibrated to verify the risk-neutral condition such that (50)HST;-θ^=S0erT.(4)θ^ is used to compute the call option price (51)e-rTHCT,K;-θ^=e-rT∫0+∞gα,β,δ,θ^S^CT,Kdx.Our Distortion. (1) Simulate the stock price ST=S0eXT and compute the empirical survival function S^ST and S^C(T,K).(2) Using our distortion (13), the parameter λ^ is then calibrated to verify the risk-neutral condition such that (52)HST;XT,-λ^=S0erT.(3)λ^ is then used to compute the call option price (53)e-rTHCT,K;XT,-λ^=e-rT∫0+∞gλ^XTS^CT,Kdx.


### 4.1. Geometric Brownian Motion Model

In the standard Black-Scholes model, the stock prices follow a geometric Brownian motion with(54)St=S0exp⁡μ-12σ2t+σWt,0≤t≤T,where W={Wt,0≤t≤T} is a standard Brownian motion. Note that the cdf of XT=(μ-1/2σ2)T+σWT is strictly increasing; option prices obtained with our distortion are just the Black-Scholes theoretical prices by Theorem 7. In order to verify this, we simulate 3000 log-normal prices ST with drift μ=0.1, volatility σ=0.2, and an initial price of S0=50. The maturity is set at T=0.5; the risk free rate is set at r=0.05. The strike prices K range from 47 to 53, which is used to calculated the option price.The theoretical option prices are shown and compared with the option prices in Table1, and the latter is obtained through procedure (49), (51), and (53). The prices obtained with Wang’s distortion and our distortion are numerically equivalent to the Black-Scholes theoretical prices as expected. This first test shows that all three distortions perform well under Gaussian conditions.Table 1
Numerical European call option prices from a sample of asset prices simulated from B-S model.
Strike price 47 48 49 50 51 52 53 Black-Scholes price 5.2746 4.6111 4.0006 3.4444 2.9428 2.4950 2.0994 Wang’s price 5.2727 4.6126 4.0057 3.4558 2.9554 2.5052 2.1057 NIG price 5.2710 4.6121 4.0066 3.4582 2.9593 2.5106 2.1125 Our price 5.2727 4.6126 4.0057 3.4558 2.9554 2.5052 2.1057 Wang’s relative error 0.0004 0.0003 0.0013 0.0033 0.0043 0.0041 0.0030 NIG relative error 0.0007 0.0002 0.0015 0.0040 0.0056 0.0062 0.0062 Our relative error 0.0004 0.0003 0.0013 0.0033 0.0043 0.0041 0.0030


### 4.2. Exponential NIG Model

The exponential NIG Lévy model for asset prices is of the form(55)St=S0eXt,0≤t≤T,where X={Xt,0≤t≤T} is a NIG Lévy process with parameters (α,β,δ,μ); see Barndorff-Nielsen [10] and Rydberg [11]. The density function of X1 is given by (56)fNIGx;α,β,δ,μ=αδeδγπK1αδ2+x-μ2δ2+x-μ2eβx-μ,x∈R,where Kρ is the modified Bessel function of the third kind with index ρ given by (57)Kρx=12∫0+∞uρ-1exp⁡-12xu-1+udu,x>0,and γ=α2-β2. The parameter domain is δ>0,α>0,α2>β2, and μ∈R.As the cdf ofXT is also strictly increasing, option prices with our distortion are just the prices under MCMM by Theorem 7. To verify this, we simulate 3000 stock prices ST with α=6,β=2,μ=-0.5,δ=1, and an initial price of S0=20. The risk free interest and the maturity of the call are, respectively, set at r=0.08 and T=0.5 years. The strike prices K range from 17 to 23, which is used to calculated the option price.The theoretical option prices are shown and compared with the option prices in Table2, and the latter is obtained through procedure (49), (51), and (53). The NIG price and our price are expected to be very accurate because of Godin et al. [7] and Theorem 7. The results of the simulation show that NIG distortion prices and our distortion prices overperform Wang’s distortion prices for every strike price. The relative error for NIG prices and our prices remains smaller than 4% while it can climb up from 11% to 30% for Wang’s distortion prices. The NIG distortion and our distortion have been proved to be more efficient in this case as well.Table 2
Numerical European option prices from a sample of asset prices simulated from NIG model.
Strike price 17 18 19 20 21 22 23 Price under MCMM 5.4240 4.9783 4.5854 4.2383 3.9308 3.6575 3.4137 Wang’s price 4.7464 4.2030 3.7277 3.3165 2.9603 2.6538 2.3884 NIG price 5.4267 4.9849 4.5935 4.2476 3.9401 3.6673 3.4233 Our price 5.4267 4.9849 4.5935 4.2476 3.9401 3.6673 3.4233 Wang’s relative error 0.1422 0.1727 0.2008 0.2270 0.2504 0.2709 0.2906 NIG relative error 0.0005 0.0013 0.0018 0.0022 0.0024 0.0027 0.0028 Our relative error 0.0005 0.0013 0.0018 0.0022 0.0024 0.0027 0.0028


### 4.3. Jump Diffusion Model

In Merton [12], the prices of the asset are given by (58)St=S0exp⁡μ-σ22-λeμJ+σJ2/2-1t+σWt+∑i=1NtYi,0≤t≤T,where μ∈R,σ>0,λ>0,μJ∈R,σJ>0 are constants; Wt is a standard Brownian motion; Nt is a Poisson process with intensity λ; and {Yi}i≥1 is a sequence of Gaussian random variables with mean μJ and standard deviation σJ. Variables Nt,Wt, and Yi are assumed to be all independent. Under MCMM, the price of a European call option with strike K is given by Merton [12] as (59)CT=e-rT∑n=0+∞e-λTλTnn!CBSSn,K,r,T,σn,where σn2=σ2+nσJ2/T, (60)Sn=S0exp⁡nμJ+n2σJ2-λTeμJ+0.5σJ2-1,CBSS,K,r,T,σ=SΦln⁡S/K+r+1/2σ2TσT-e-rTKΦln⁡S/K+r-1/2σ2TσTis the value of a European call option with time to maturity T and volatility σ in a Black-Scholes model.Once again, 3000 asset pricesST are simulated with μ=0.25,σ=0.15,λ=1,μJ=0.2,σJ=0.1 and an initial price of S0=30. The maturity and the risk free rate are, respectively, set at T=0.5 and r=0.1. The strike prices K range from 27 to 33, which is used to calculated the option price.The theoretical option prices are shown and compared with the option prices in Table3, and the latter is obtained through procedure (49), (51), and (53). Our distortion prices are expected to be very accurate because of Theorem 7. The results of the simulation show that our distortion prices overperform Wang’s distortion prices and NIG distortion prices for every strike price.Table 3
Numerical European call option prices from a sample of asset prices simulated from Merton model.
Strike price 27 28 29 30 31 32 33 Price under MCMM 4.7345 4.0541 3.4620 2.9553 2.5254 2.1611 1.8515 Wang’s price 4.6560 3.9455 3.3164 2.7865 2.3465 1.9628 1.6437 NIG price 4.9895 4.2927 3.6724 3.1437 2.6974 2.3028 1.9683 Our price 4.7573 4.0844 3.4906 2.9881 2.5645 2.1891 1.8686 Wang’ relative error 0.0166 0.0268 0.0421 0.0571 0.0708 0.0918 0.1122 NIG relative error 0.0539 0.0589 0.0608 0.0638 0.0681 0.0656 0.0631 Our relative error 0.0048 0.0075 0.0082 0.0111 0.0155 0.0130 0.0092


### 4.4. Exponential VG Model

Consider the following exponential VG stock price model:(61)St=S0eXt,0≤t≤T,where X={Xt,0≤t≤T} is a VG Lévy process with parameters (θ,ν,σ,μ). The density function of X1 is given by (62)fVGx;θ,ν,σ,μ=2eθx-μ/σ2ν1/νσΓ1/ν2πx-μ2σ2/ν+θ21/ν-0.5K1/ν-0.5x-μ2σ2/ν+θ2σ2.The parameter domain is ν>0,σ>0,θ∈R,μ∈R.We simulate 3000 stock pricesST with θ=-0.2,ν=0.1,σ=0.15,μ=0.2 and an initial price of S0=40. The maturity and the risk free rate are set at T=0.75 and r=0.05, respectively. The option prices are calculated for the strike prices K ranging from 37 to 43.In Table4, the theoretical option prices are shown and compared with option prices obtained through procedures (49), (51), and (53), respectively. Once again, our distortion prices are very accurate because of Theorem 7. The results of the simulation also show that our distortion prices overperform Wang’s distortion prices and NIG distortion prices for every strike price.Table 4
Numerical European call option prices from a sample of asset prices simulated from VG model.
Strike price 37 38 39 40 41 42 43 Price under MCMM 5.0259 4.2929 3.6154 2.9993 2.4487 1.9661 1.5516 Wang’s price 4.9340 4.1888 3.5035 2.8855 2.3333 1.8527 1.4424 NIG price 4.9768 4.2366 3.5548 2.9384 2.3858 1.9028 1.4883 Our price 4.9788 4.2395 3.5588 2.9432 2.3913 1.9086 1.4939 Wang’ relative error 0.0183 0.0242 0.0310 0.0379 0.0471 0.0577 0.0704 NIG relative error 0.0098 0.0131 0.0168 0.0203 0.0257 0.0322 0.0408 Our relative error 0.0094 0.0124 0.0157 0.0187 0.0234 0.0293 0.0372


### 4.5. Exponential Meixner Model

The last model on which all three distortions are tested is the exponential Meixner model. The prices of the asset are given by(63)St=S0eXt,0≤t≤T,where X={Xt,0≤t≤T} is a Meixner Lévy process with parameters (a,b,m,d). The density function of X1 is given by (64)fMXx;a,b,m,d=2cos⁡b/22d2aπΓ2dexp⁡βx-maΓd+ix-ma2,where x∈R,a>0,|b|<π,d>0,m∈R,i=-1.Once more, 3000 asset pricesST are simulated with a=0.3,b=-0.1,m=0.15,d=0.2 and an initial price of S0=50. The maturity and the risk free rate are, respectively, set at T=0.5 and r=0.08. The option prices are calculated for the strike prices K ranging from 47 to 53.In Table5 the theoretical option prices are shown and compared with option prices obtained through procedures (49), (51), and (53), respectively. It can be seen that our distortion and the NIG distortion show an overall better performance than the normal distortion. We can also observe that, relatively speaking, our distortion performs better than the NIG distortion.Table 5
Numerical European call option prices from a sample of asset prices simulated from Meixner model.
Strike price 47 48 49 50 51 52 53 Price under MCMM 4.9900 4.0883 3.2148 2.3889 1.6504 1.0714 0.7010 Wang’s price 5.1825 4.3165 3.4769 2.6839 1.9535 1.3087 0.7852 NIG price 4.9958 4.0922 3.2148 2.3951 1.6737 1.1094 0.7299 Our price 4.9951 4.0895 3.2086 2.3822 1.6524 1.0874 0.7184 Wang’ relative error 0.0386 0.0558 0.0816 0.1235 0.1836 0.2215 0.1201 NIG relative error 0.0012 0.0010 0.0000 0.0026 0.0141 0.0355 0.0413 Our relative error 0.0010 0.0003 0.0019 0.0028 0.0012 0.0150 0.0249


## 4.1. Geometric Brownian Motion Model

In the standard Black-Scholes model, the stock prices follow a geometric Brownian motion with(54)St=S0exp⁡μ-12σ2t+σWt,0≤t≤T,where W={Wt,0≤t≤T} is a standard Brownian motion. Note that the cdf of XT=(μ-1/2σ2)T+σWT is strictly increasing; option prices obtained with our distortion are just the Black-Scholes theoretical prices by Theorem 7. In order to verify this, we simulate 3000 log-normal prices ST with drift μ=0.1, volatility σ=0.2, and an initial price of S0=50. The maturity is set at T=0.5; the risk free rate is set at r=0.05. The strike prices K range from 47 to 53, which is used to calculated the option price.The theoretical option prices are shown and compared with the option prices in Table1, and the latter is obtained through procedure (49), (51), and (53). The prices obtained with Wang’s distortion and our distortion are numerically equivalent to the Black-Scholes theoretical prices as expected. This first test shows that all three distortions perform well under Gaussian conditions.Table 1
Numerical European call option prices from a sample of asset prices simulated from B-S model.
Strike price 47 48 49 50 51 52 53 Black-Scholes price 5.2746 4.6111 4.0006 3.4444 2.9428 2.4950 2.0994 Wang’s price 5.2727 4.6126 4.0057 3.4558 2.9554 2.5052 2.1057 NIG price 5.2710 4.6121 4.0066 3.4582 2.9593 2.5106 2.1125 Our price 5.2727 4.6126 4.0057 3.4558 2.9554 2.5052 2.1057 Wang’s relative error 0.0004 0.0003 0.0013 0.0033 0.0043 0.0041 0.0030 NIG relative error 0.0007 0.0002 0.0015 0.0040 0.0056 0.0062 0.0062 Our relative error 0.0004 0.0003 0.0013 0.0033 0.0043 0.0041 0.0030


## 4.2. Exponential NIG Model

The exponential NIG Lévy model for asset prices is of the form(55)St=S0eXt,0≤t≤T,where X={Xt,0≤t≤T} is a NIG Lévy process with parameters (α,β,δ,μ); see Barndorff-Nielsen [10] and Rydberg [11]. The density function of X1 is given by (56)fNIGx;α,β,δ,μ=αδeδγπK1αδ2+x-μ2δ2+x-μ2eβx-μ,x∈R,where Kρ is the modified Bessel function of the third kind with index ρ given by (57)Kρx=12∫0+∞uρ-1exp⁡-12xu-1+udu,x>0,and γ=α2-β2. The parameter domain is δ>0,α>0,α2>β2, and μ∈R.As the cdf ofXT is also strictly increasing, option prices with our distortion are just the prices under MCMM by Theorem 7. To verify this, we simulate 3000 stock prices ST with α=6,β=2,μ=-0.5,δ=1, and an initial price of S0=20. The risk free interest and the maturity of the call are, respectively, set at r=0.08 and T=0.5 years. The strike prices K range from 17 to 23, which is used to calculated the option price.The theoretical option prices are shown and compared with the option prices in Table2, and the latter is obtained through procedure (49), (51), and (53). The NIG price and our price are expected to be very accurate because of Godin et al. [7] and Theorem 7. The results of the simulation show that NIG distortion prices and our distortion prices overperform Wang’s distortion prices for every strike price. The relative error for NIG prices and our prices remains smaller than 4% while it can climb up from 11% to 30% for Wang’s distortion prices. The NIG distortion and our distortion have been proved to be more efficient in this case as well.Table 2
Numerical European option prices from a sample of asset prices simulated from NIG model.
Strike price 17 18 19 20 21 22 23 Price under MCMM 5.4240 4.9783 4.5854 4.2383 3.9308 3.6575 3.4137 Wang’s price 4.7464 4.2030 3.7277 3.3165 2.9603 2.6538 2.3884 NIG price 5.4267 4.9849 4.5935 4.2476 3.9401 3.6673 3.4233 Our price 5.4267 4.9849 4.5935 4.2476 3.9401 3.6673 3.4233 Wang’s relative error 0.1422 0.1727 0.2008 0.2270 0.2504 0.2709 0.2906 NIG relative error 0.0005 0.0013 0.0018 0.0022 0.0024 0.0027 0.0028 Our relative error 0.0005 0.0013 0.0018 0.0022 0.0024 0.0027 0.0028


## 4.3. Jump Diffusion Model

In Merton [12], the prices of the asset are given by (58)St=S0exp⁡μ-σ22-λeμJ+σJ2/2-1t+σWt+∑i=1NtYi,0≤t≤T,where μ∈R,σ>0,λ>0,μJ∈R,σJ>0 are constants; Wt is a standard Brownian motion; Nt is a Poisson process with intensity λ; and {Yi}i≥1 is a sequence of Gaussian random variables with mean μJ and standard deviation σJ. Variables Nt,Wt, and Yi are assumed to be all independent. Under MCMM, the price of a European call option with strike K is given by Merton [12] as (59)CT=e-rT∑n=0+∞e-λTλTnn!CBSSn,K,r,T,σn,where σn2=σ2+nσJ2/T, (60)Sn=S0exp⁡nμJ+n2σJ2-λTeμJ+0.5σJ2-1,CBSS,K,r,T,σ=SΦln⁡S/K+r+1/2σ2TσT-e-rTKΦln⁡S/K+r-1/2σ2TσTis the value of a European call option with time to maturity T and volatility σ in a Black-Scholes model.Once again, 3000 asset pricesST are simulated with μ=0.25,σ=0.15,λ=1,μJ=0.2,σJ=0.1 and an initial price of S0=30. The maturity and the risk free rate are, respectively, set at T=0.5 and r=0.1. The strike prices K range from 27 to 33, which is used to calculated the option price.The theoretical option prices are shown and compared with the option prices in Table3, and the latter is obtained through procedure (49), (51), and (53). Our distortion prices are expected to be very accurate because of Theorem 7. The results of the simulation show that our distortion prices overperform Wang’s distortion prices and NIG distortion prices for every strike price.Table 3
Numerical European call option prices from a sample of asset prices simulated from Merton model.
Strike price 27 28 29 30 31 32 33 Price under MCMM 4.7345 4.0541 3.4620 2.9553 2.5254 2.1611 1.8515 Wang’s price 4.6560 3.9455 3.3164 2.7865 2.3465 1.9628 1.6437 NIG price 4.9895 4.2927 3.6724 3.1437 2.6974 2.3028 1.9683 Our price 4.7573 4.0844 3.4906 2.9881 2.5645 2.1891 1.8686 Wang’ relative error 0.0166 0.0268 0.0421 0.0571 0.0708 0.0918 0.1122 NIG relative error 0.0539 0.0589 0.0608 0.0638 0.0681 0.0656 0.0631 Our relative error 0.0048 0.0075 0.0082 0.0111 0.0155 0.0130 0.0092


## 4.4. Exponential VG Model

Consider the following exponential VG stock price model:(61)St=S0eXt,0≤t≤T,where X={Xt,0≤t≤T} is a VG Lévy process with parameters (θ,ν,σ,μ). The density function of X1 is given by (62)fVGx;θ,ν,σ,μ=2eθx-μ/σ2ν1/νσΓ1/ν2πx-μ2σ2/ν+θ21/ν-0.5K1/ν-0.5x-μ2σ2/ν+θ2σ2.The parameter domain is ν>0,σ>0,θ∈R,μ∈R.We simulate 3000 stock pricesST with θ=-0.2,ν=0.1,σ=0.15,μ=0.2 and an initial price of S0=40. The maturity and the risk free rate are set at T=0.75 and r=0.05, respectively. The option prices are calculated for the strike prices K ranging from 37 to 43.In Table4, the theoretical option prices are shown and compared with option prices obtained through procedures (49), (51), and (53), respectively. Once again, our distortion prices are very accurate because of Theorem 7. The results of the simulation also show that our distortion prices overperform Wang’s distortion prices and NIG distortion prices for every strike price.Table 4
Numerical European call option prices from a sample of asset prices simulated from VG model.
Strike price 37 38 39 40 41 42 43 Price under MCMM 5.0259 4.2929 3.6154 2.9993 2.4487 1.9661 1.5516 Wang’s price 4.9340 4.1888 3.5035 2.8855 2.3333 1.8527 1.4424 NIG price 4.9768 4.2366 3.5548 2.9384 2.3858 1.9028 1.4883 Our price 4.9788 4.2395 3.5588 2.9432 2.3913 1.9086 1.4939 Wang’ relative error 0.0183 0.0242 0.0310 0.0379 0.0471 0.0577 0.0704 NIG relative error 0.0098 0.0131 0.0168 0.0203 0.0257 0.0322 0.0408 Our relative error 0.0094 0.0124 0.0157 0.0187 0.0234 0.0293 0.0372


## 4.5. Exponential Meixner Model

The last model on which all three distortions are tested is the exponential Meixner model. The prices of the asset are given by(63)St=S0eXt,0≤t≤T,where X={Xt,0≤t≤T} is a Meixner Lévy process with parameters (a,b,m,d). The density function of X1 is given by (64)fMXx;a,b,m,d=2cos⁡b/22d2aπΓ2dexp⁡βx-maΓd+ix-ma2,where x∈R,a>0,|b|<π,d>0,m∈R,i=-1.Once more, 3000 asset pricesST are simulated with a=0.3,b=-0.1,m=0.15,d=0.2 and an initial price of S0=50. The maturity and the risk free rate are, respectively, set at T=0.5 and r=0.08. The option prices are calculated for the strike prices K ranging from 47 to 53.In Table5 the theoretical option prices are shown and compared with option prices obtained through procedures (49), (51), and (53), respectively. It can be seen that our distortion and the NIG distortion show an overall better performance than the normal distortion. We can also observe that, relatively speaking, our distortion performs better than the NIG distortion.Table 5
Numerical European call option prices from a sample of asset prices simulated from Meixner model.
Strike price 47 48 49 50 51 52 53 Price under MCMM 4.9900 4.0883 3.2148 2.3889 1.6504 1.0714 0.7010 Wang’s price 5.1825 4.3165 3.4769 2.6839 1.9535 1.3087 0.7852 NIG price 4.9958 4.0922 3.2148 2.3951 1.6737 1.1094 0.7299 Our price 4.9951 4.0895 3.2086 2.3822 1.6524 1.0874 0.7184 Wang’ relative error 0.0386 0.0558 0.0816 0.1235 0.1836 0.2215 0.1201 NIG relative error 0.0012 0.0010 0.0000 0.0026 0.0141 0.0355 0.0413 Our relative error 0.0010 0.0003 0.0019 0.0028 0.0012 0.0150 0.0249



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*Source: 5831569-2019-06-16.xml*