Nonparametric Predictive Inference for Asian options
NNonparametric Predictive Inference for Asian options
Ting HeSchool of Finance, Capital University of Economics and Business, Beijing, [email protected]
ABSTRACT
Asian option, as one of the path-dependent exotic options, is widely traded in the energy market, either forspeculation or hedging. However, it is hard to price, especially the one with the arithmetic average price. Thetraditional trading procedure is either too restrictive by assuming the distribution of the underlying asset orless rigorous by using the approximation. It is attractive to infer the Asian option price with few assumptionsof the underlying asset distribution and adopt to the historical data with a nonparametric method. Inthis paper, we present a novel approach to price the Asian option from an imprecise statistical aspect.Nonparametric Predictive Inference (NPI) is applied to infer the average value of the future underlying assetprice, which attempts to make the prediction reflecting more uncertainty because of the limited information.A rational pairwise trading criterion is also proposed in this paper for the Asian options comparison, as arisk measure. The NPI method for the Asian option is illustrated in several examples by using the simulationtechniques or the empirical data from the energy market.Key words: Asian Option; Imprecise Probability; Nonparametric Predictive Inference; Uncertainty1 a r X i v : . [ q -f i n . M F ] A ug Introduction
Asian options, as one kind of the exotic options, are strongly path-dependent and widely traded in thecommodity and foreign exchange market (Klassen, 2001). The main advantages of the Asian option are thatits usage of avoiding the risk of market manipulation of the underlying instrument at maturity, and it holdsa cheaper price compared to European or American options. The Asian option payoff is contingent on theaverage value of the underlying asset price, either arithmetic or geometric. For the Asian option settled onthe basis of the geometric average price, there are closed formulae by the Black-Scholes model under theassumption that the underlying asset price is the lognormal distributed, so the geometric average price alsofollows the lognormal distribution with different mean and variance. Although the geometric Asian optionsare easily priced they are rarely used in practice (Milevsky and Posner, 1998). While the Asian option withthe arithmetic average price is very hard to be evaluated since the density function of the arithmetic averageprice is unknown(Vecer, 2014).Many scholars try to develop and improve the method for the Asian option with the arithmetic averageprice. One study direction is by assuming a lognormal diffusion process of the underlying asset price andapproximating the density function of the arithmetic average price. The moment matching is used to do theapproximation of the option payoff presented by Turnbull (1991); Levy (1992). Curran (1994) approximatesthe payoff of the option by conditioning on the geometric mean price. Another method is to use the numericalmethod to obtain the solution of the PDE of the Asian option. The problem of this method is that whenthe explicit finite difference method is used in PDE of a path-dependent option pricing, it is numericallyunstable. The implicit method is stable referring to the Asian option pricing, but it only provides the resultfor some specific volatility structure. Vecer (2001) improves the instability problem by presenting a numericalone dimensional PDE for the Asian option pricing which is stable under the finite difference method. MonteCarlo simulation (Boyle) as a very effective way to price the path-dependent option that has been developedfor the Asian option pricing (Hull, 2009). Kemna and Vorst (1990) present a Monte Carlo strategy of pricingthe option with the arithmetic average price with the variance reduction elements. Ballotta and Kyriakou(2014) study the Asian option pricing problem by presenting a joint Monte Carlo-Fourier transform samplingscheme under the CGMY process. The concern of Monte Carlo simulation of the option pricing is to estimatean accurate result is very time-consuming. Another popular method is the discrete lattice method. Hulland White (1993) propose the first tree pricing model for Asian options, which has some drawbacks of theapproximation precision and the convergence to the continuous value. Klassen (2001) and Dai et al. (2008)improve Hull and White’s tree model considering the representative average prices to limit the approximation2rror. Liu et al. (2014) present a binomial approach for the Asian option pricing leading to the upper andlower bounds of the approximation result reducing the interpolation error. The study discussed above isbased on the assumption that the future pattern of the Asian option is well known. When the informationof the future market is limited, imprecise probability allows us to predict the Asian option price with theobserved information.Imprecise probability as a generalization of classical probability theory enables various less restrictiverepresentations of uncertainty (Augustin et al., 2014). Nonparametric predictive inference (NPI) is one of thestatistical inference methods for imprecise probability, which is a frequentist statistics framework with strongconsistency properties (Augustin and Coolen, 2004); (Coolen, 2011). The NPI method provides the approachto calculate the upper and lower probabilities of the interested event aiming to do the prediction by makingfew assumptions in addition to observed data. One property of the NPI method is when multiple futureobservations are predicted, the observations are interdependent, meaning after one prediction, this predictedvalue is added to the observed data together forecasting the next future observations. Therefore, the NPImethod reflects more uncertainty by increasing the variability if the multiple future observations are assumedto be conditionally independent. The NPI method has been applied to the finance area, predicting futurestock returns when little further information is available and providing a way of the pairwise comparisonbetween stock returns (Baker et al., 2017). Coolen et al. (2018) presented a new approach to quantify thecredit risk by using the NPI method based on the ROC analysis. The implements of the NPI method forthe vanilla option pricing well perform when there are less certain information of the underlying asset (Heet al., 2019);(He et al., 2020).In this paper, we present the NPI method for Asian option pricing, which attempts to evaluate the Asianoption price based on the historical data and offering a rational pairwise trading scenario. Some relevantbackground about the NPI method is summarized in Section 2. The Asian option pricing model based onthe NPI method is proposed in Section 3 along with a rational trading criterion of the comparison of twoAsian options. In Section 4, we illustrate the NPI method by using the simulation as well as the empiricalexamples of the energy market. Some conclusions and extensions are written in Section 5.
Nonparametric Predictive Inference (NPI) is an inferential framework based on the assumption A ( n ) (Hill,1968), which directly provides probabilities for future observations by using few model assumptions and3bserved values of related random quantities. A ( n ) assumption makes sure that the future observation isequally likely to fall in the interval of a real value line created by n observed random quantities, whichkeeps the consistency of the exchangeability. Based on the A ( n ) assumption, NPI offers the upper and lowerprobabilities for one or multiple future random quantities when n observed random quantities are available,which follows De Finetti’s fundamental theorem of probability (De Finetti, 1974). NPI is a frequentiststatistical method which has strong consistency properties (Augustin and Coolen, 2004).NPI for m multiple future random quantities is concerned in this paper, which is based on A ( n + m − .The NPI method predicts the future observation based on historical data and keeps updating the data, whichmeans the prediction of m future data is identical to the prediction of one future data. After the first one ispredicted, the prediction is used as the historical information to forecast the next observation. NPI assumesthat each future data is equally likely in the interval I j , where j = 1 , , ..., n + 1 , created by n observed data,which also means that all possible orderings of n observed data and m future data are equally likely. Totally, (cid:0) n + mm (cid:1) possible orderings can be derived, so for each ordering the probability is equal to (cid:0) n + mm (cid:1) − .To inference the arithmetic average price of the underlying asset based on the NPI method, we firstforcast the return of the underlying asset. Baker (Baker et al., 2017) predicts the stock future return withfew information by applying the NPI method, but the predicted return is under the frame of simple interest.In this paper, we follows the same idea and extend it to compound interest. Through the prediction of theunderlying asset return, the imprecise arithmeric price of the underlying asset can inferred and utilized inthe Asian option pricing procedure. In this section, an Asian option pricing method is presented based on the NPI method, which reflects theuncertainty not only from the stochastic environment but also from the limited prior information. And atrading criterion by comparing the Asian options contingent on two different product is shown as a riskmeasure.
Define S t is the underlying asset price at time t . By assuming there are n historical underlying asset price S t = s t , t = 1 , , . . . , n available in the market, where the time intervals between these historical data are4dentical to each other. Then the continuous compounding rates of return of the underlying asset price r t is, r t = ln (cid:18) S t S t +1 (cid:19) , t = 1 , , . . . , n To predict future return of underlying asset price based on the NPI method, the exchangeability is assumed inour model meaning the order of the underlying asset return is irrelevant. After we calculate the compoundingreturn, we rank these values from the lowest value to the highest value, r (1) , . . . , r ( n ) . Then on this realvalue line created by r (1) , . . . , r ( n ) , there are n + 1 intervals. To avoid the influences of ∞ and −∞ , we needto find the lowest and the highest returns, r (0) and r ( n +1) , which can be the extreme returns in a long-termhistorical period or the extreme values referring to the user’s preference. On the basis of the historicalinformation, we assume that the future data randomly falls in any interval on this real value line. From theassumption of multiple future data prediction through the NPI method, totally there are (cid:0) n + mm (cid:1) orderingsof m future compounding returns, which are equally likely. Investor can infer the future returns by countingthe orderings fitted in one’s investment criterion.As the aim of this paper is to study the Asian option with arithmetic average and do the prediction, theaggregate compounding return is concerned. The general formula of the aggregate compounding return is ˆ R i = (cid:80) n + it = n +1 R t i This presents the aggregate compounding return for i future cumulative time, where i = 1 , . . . , m . Forexample, when i is equal to one, ˆ R represents the aggregate return during the first time step, and when i = 2 , ˆ R represents the aggregate return during the first and second time steps and so on. By applying thisformula to the NPI framework, the upper bound ˆ R ui and the lower bound ˆ R li of the aggregate compoundingreturn for i future period can be calculated. The fundamental idea is that for a specific ordering, R t randomlyfalls in the interval I j , j = 1 , . . . , n + 1 , which defines the upper bound R ut of R t equal to r ( j ) and the lowerbound R lt equal to r ( j − . By putting the upper and lower bounds of R t into the aggregate calculation, theupper and lower bounds of ˆ R i can be obtained. As we mentioned, the Asian option’s payoff depends on two type of average value, the arithmetic averageprice and the geometric average price. In this paper, the Asian option with arithmetic average price is inconsideration. According to the different type of the strike price , there are two types of Asian options, with5he fixed strike price and the float price. The Asian option with fixed strike price is discussed in this paper.Therefore, if a m period Asian option with fixed strike price K is priced, the general pricing formula is, V = B (0 , m )[ S mµ − K ] + (1)where V is the initial expected price of this Asian option, S mµ is the arithmetic average price of the underlyingasset during m period, and B (0 , m ) is the discount factor during m period. From Equation (1), we canconclude that the payoff of this type Asian option is the positive value of the subtraction between theaverage underlying asset price and the predetermined strike price.To calculate the arithmetic average price of the underlying asset during m period, the aggregate com-pounding returns for every i ∈ (1 , . . . , n ) period are needed. S mµ = 1 m m (cid:88) k =0 S e i ˆ R i (2)where S is the initial underlying asset price and R is set to be zero. By the definition of the Asian option,the exact value of the future underlying asset is less important. Rather than the explicit value of eachtime step S t , the average behavior of the underlying asset is considered, where the aggregate return is theappropriate value to represent the asset behavior during a period. Thus, based on the NPI method, we do notconcern about the exact value of S , . . . , S m . Instead, the upper and lower bounds of aggregate compoundingreturns for every i ∈ (1 , . . . , n ) time-steps are calculated. Putting the bounds of the compounding returnsin Equation (2), we get the upper and lower bounds of the arithmetic average underlying asset price. For m future time steps, the minimum average underlying asset price is, S mµ = 1 m m (cid:88) k =0 S e i ˆ R li (3)The maximum average underlying asset is, S mµ = 1 m m (cid:88) k =0 S e i ˆ R ui (4)Thus, according to the definition of the Asian option payoff, we can calculate the upper and lower expectedoption price based on the NPI method, which is called the minimum selling price and the maximum buyingprice according to the trading intention. 6 he minimum selling price for the call option V = B (0 , m )[ S mµ − K ] + = B (0 , m ) (cid:34) m m (cid:88) k =0 S e i ˆ R li − K (cid:35) + (5) The maximum buying price for the call option V = B (0 , m )[ S mµ − K ] + = B (0 , m ) (cid:34) m m (cid:88) k =0 S e i ˆ R ui − K (cid:35) + (6) The minimum selling price for the put option V = B (0 , m )[ K − S mµ ] + = B (0 , m ) (cid:34) K − m m (cid:88) k =0 S e i ˆ R ui (cid:35) + (7) The maximum buying price for the put option V = B (0 , m )[ K − S mµ ] + = B (0 , m ) (cid:34) K − m m (cid:88) k =0 S e i ˆ R li (cid:35) + (8)The upper and lower bounds of the Asian option indicate the buying and selling thresholds of the investor.The investor who trades according to the result from the NPI method would not like to be in the game whenthe quoted price is in the interval of the minimum selling price and the maximum buying price. However,if the quoted price is higher than the minimum selling price, the investor prefers to sell the option. Orthe longing position is triggered when the maximum buying price is greater than the quoted price. Theadvantage of this method is we do not assume any distribution of the underlying asset distribution. Theprediction is based on the information from the historical data. Different from calculating the average priceof historical data directly, this method considers the randomness of the stock price and its outcome is aninterval, which avoids the error of the historical data bias and reflects more uncertainty of the underlyingasset. Other than pricing the Asian option by using the NPI method, this method also offers a way to make areasonable decision in the Asian option trade. The NPI method provides the upper and lower probabilitythat the investor can get a positive profit in this investment. Suppose there is a sequence of historical datawith an amount of n , which is continuous, consistent and exchangeable. Same as what we did for the option7ricing procedure, we calculate the historical aggregate compounding returns and rank them from the lowestone to the highest one r (1) , r (2) ..., r ( n ) . And to avoid the influence from the infinity values, we set the newhistorical sequence started with r (0) , and end with r ( n +1) , which these two values can be determined by usingthe minimum historical price and the maximum historical price in a long-term time. By having the aggregatecompounding returns, we can calculate the average price of the underlying asset S mµ . Next, the NPI lowerand upper probabilities of the positive payoff are derived for the Asian option involving the average stockprice S mµ and the strike Price K . The investor can use these probabilities to compare the Asian options andset their trading criteria. The formulae are listed below. The upper and lower probability of a positive payoff P ( Payoff >
0) = ( n + mm ) (cid:80) O { S mµ > K } call option ( n + mm ) (cid:80) O { S mµ < K } put option (9) P ( Payoff >
0) = ( n + mm ) (cid:80) O { S mµ > K } call option ( n + mm ) (cid:80) O { S mµ < K } put option (10)where S mµ , S mµ are calculated by Equations (3) and (4). (cid:80) O is the summation over all the ( n + mm ) possibleorderings of the m future returns within the n + 1 intervals, and { A } is an indicator function which is equalto 1 if A is true or 0 otherwise.This interval probabilities can help an investor to choose the better underlying asset in the Asian optioninvestment as either a speculator or a hedger. As in the commodity market, especially in the crude oilmarket. there are a variety of underlying assets correlated to each other, so it is hard to choose whichunderlying asset is a better investment. By the NPI method, an investor is offered an indicator that canbe referred according to the investor’s risk aversion and character, speculator or hedger. Suppose there aretwo underlying assets A and B that have similar price values and trends. A speculator is a risk-taker whosepurpose of an investment is to seek the opportunities to earn some profit. If the speculator would like toinvest in either of these two assets, then the indicator below suggests the speculator invest in asset A , thelower probability of a positive payoff for asset A , P ( Payoff A ) , is greater than the lower probability of apositive payoff for asset B , P ( Payoff B ) , or the upper probability of a positive payoff for asset A , P ( Payoff A ) ,is greater than the upper probability of a positive payoff for asset B , P ( Payoff B ) . For a hedger, the purposeinvolving in the option trading is to hedge the risk in the trade of the underlying asset, so the hedger has8 high level of risk aversion. An absolute strength of asset A needs to be revealed to instruct this hedger’saction. Thus, when P ( Payoff A ) > P ( Payoff B ) , the investment in asset A is appealing to the hedger. Several examples are discussed in this section to illustrate the NPI method for the Asian option. We firststudy the performance of the NPI method for Asian option pricing by the simulation techniques. Then aperformance study of the energy market is developed to assess the empirical value of this method.
As acknowledged, the Geometric Brownian Motion (GBM) is widely used to model the stock price behavior.Therefore, to start the illustration, an example based on the GBM is presented in this section. By utilizingthe R program, we first simulate 100 paths of stock prices following the GBM with the return equal to0.02 and the volatility equal to 0.02 as well. The simulated stock paths are displayed in Figure 1. In eachpath, the initial price is 50, and the program simulated the stock price movement for 110 time steps. Inour example, the time period 0 to 100, is assumed as the historical time period calling the correspondingdata the historical data, while assuming the time period 100 to 110, to be the predictive time period callingthe corresponding data the future data. The idea is using the NPI method for the Asian call option pricingformulae, Equations (5) and (6) to forecast the option price, and using the future data to calculate theoption price based on payoff definition as the benchmark. In the following example, we predict the price ofan at-the-money (ATM) call option where the strike price equals to the initial price 50.Figure 2 discloses that the NPI method can provide an interval that includes the benchmark price inmost of the cases. In these cases, the NPI method’s prediction includes the benchmark value, but this doesnot mean that NPI method can predict the price accurately. If the result from the NPI method is an intervalwith a large value gap, the benchmark price could be in the interval for sure. Some further discussions ofthe accuracy based on the NPI method are illustrated in the next paragraph. Figure 2 also reveals thatthe fluctuations of NPI option prices have a similar pattern to that of the GMB option prices. In these100 paths, for the GBM option price with a higher value, the maximum buying and minimum selling pricescorresponding to this path also have a higher value, and the same conclude can be derived from the figurefor the path with a lower GBM option price as well. However, it is also clear that the boundary prices fromthe NPI method are less fluctuated than the GBM prices. It is easy to understand from the perspective of9
20 40 60 80 100
Geometric Brownian
Time S t o ck P r i c e Figure 1: Simulated stock price paths path op t i on p r i c e s GBMNPI
Figure 2: Asian option prices predicted by the NPI method with the GBM option price as the benchmark E [ | median NPI − benchmark GBM | ] , which reflects the deviation between the NPI outcomes and the benchmark. Precision isto calculate the mean value of the interval length from the NPI method. Including the precision in our studyis because if the precision is very large, the result of the coverage percentage is supposed to be better thanthe case when the precision is very small. Herein, we study the influence of the varying volatilities on thesethree factors in order to estimate the performance of the NPI prediction result for the same ATM option asthat in the last example. In this study, the volatility is in the range from 5% to 10% to simulate the dailyvolatility in the market. Three factors are monitored to assess the NPI results.Precision is large [3,4] Precision is small[1,1.5]volatility percentage accuracy percentage accuracy0.5% 1 0.5856104 0.9653 1.3917951% 0.9999 0.6747436 0.9625 1.4019271.5% 0.9963 0.8498949 0.9576 1.4213812% 0.9828 1.047153 0.9597 1.5017272.5% 0.951 1.262822 0.9345 1.6097553% 0.9208 1.425547 0.8975 1.723993.5% 0.8797 1.590992 0.8744 1.8668144% 0.8481 1.752489 0.8289 1.9698984.5% 0.8271 1.937935 0.8001 2.0823825% 0.7876 2.064087 0.7875 2.1880285.5% 0.7638 2.174651 0.7576 2.3582336% 0.7387 2.301093 0.7376 2.4712411.5% 0.7169 2.459455 0.7137 2.5540127% 0.6921 2.543524 0.6897 2.70547.5% 0.6832 2.706887 0.6722 2.8191318% 0.6808 2.821586 0.6625 2.9805328.5% 0.6514 2.913285 0.6504 3.0594329% 0.6403 3.012923 0.64 3.167129.5% 0.6364 3.167188 0.6242 3.29657410% 0.6153 3.283862 0.6147 3.356551 Table 1: The study of volatility influence
Table 1 displays the outcomes of three factors with the varying volatility in two simulations. Whenwe calculate the precision with different volatilities, we find that as the volatility increases, the precisiondecreases. This result does not mean that high volatility has a positive effect on the precision. The reasonwhy high volatility causes a small precision is when the underlying asset is more volatile, there are more timesof the simulated paths with a zero payoff either from NPI method or from the GBM model. As the averageprecision value of all simulated paths is calculated as the estimator, the average result is getting smalleras the more zeros appearing in the simulation. Therefore, the value of precision with varying volatility isless instructive in this study. Based on this, we categorize the outcomes in two parts according to two sizesof precision, the large precision with the value from 3 to 4 and the small precision with the value from 1to 1.5. No matter in the simulation with large or small precision, the result indicates that the percentageof the NPI interval including the benchmark value gets lower, and the NPI results are less accurate alongwith the increasing volatility. If the results are compared horizontally, it is not difficult to conclude thatthe NPI prediction presents a better result with a larger precision than that with a smaller precision. Thus,inputting a larger precision is a safer choice. From the finance perspective, it illustrates that a conservativeinvestor who uses NPI method can do the prediction with a larger precision to behave safely, but at the sametime he may miss a lot of trading chances in the market. In the simulation with a large precision, when thevolatility is lower than 4.5%, the NPI method can offer a good prediction with the percentage greater than80%, and accuracy less than 2. To get a better result with a percentage greater than 90% and accuracy lessthan 1.5, the volatility needs to be restricted within 3%. In the simulation with small precision, the NPI’sresult is good when the volatility is also lower than 4.5%, and the corresponding accuracy is less than 2.0812 igure 3: NPI predictions with a ten year historical data worse than the one with large precision. But if a better result is required, the volatility should be lower than2.5% in order to make the percentage greater than 90%, then the accuracy under these circumstances is lessthan 1.61. Overall, we can draw the conclusion that the NPI method performance is better with an optionbased on an underlying asset at a lower volatility less than 3% daily. An Asian option is normally used incommodity and foreign exchange markets where the underlying asset is less volatile than the equity in thestock market. This allows the NPI method to provide a relatively good result for the Asian option pricing inthese markets. To support the statement, an empirical example of the Asian option in the crude oil marketis investigated.
The crude oil commodity market is considered in this example. The set of data is the New York MercantileExchange (NYMEX) daily closed price of the WTI crude oil normally used as a benchmark in the oil pricing.The Asian option price is the Chicago Mercantile Exchange (CME group) the WTI average price call optionstarted on 23/10/2019 and expired on 29/11/2019 with the strike price 54($). By the end of the trading13 igure 4: NPI predictions with a one year historical data time on 23/10/2019, the settlement price of this call option is 2.83($), which is a reference price provided bythe CME group. The NPI method, an imprecise statistical framework based on the historical data, controlsthe precision of the prediction by managing the historical data size. By large historical data, the degree ofprior information dispersion is more significant than a small historical data leading to a less precise intervalresult. In the following example, we forecast the average price option based on ten years of historical data,from 23/10/2009 to 23/10/2019, and the plotted result is shown in Figure 3. We first calculate the dailyvolatility based on the historical data, which equals to 2.1%. According to our volatility study by simulation,the NPI prediction is supposed to provide some valuable results under this volatility. After 10000 trails, weget the expected NPI maximum buying price is equal to 2.6152 and the expected NPI minimum selling priceis equal to 3.29124, which are shown as the two horizontal lines in orange and green in Figure 3. The dotsin Figure 3 the NPI results in each trail. To make the graph clear and well recognized, only 1000 trailsresults are plotted in the figure. Figure 3 indicates that the NPI method provides a relatively good resultthat includes the real price in its interval.Next, we improve the precision of the NPI prediction by limiting the historical data size to one year,14
Date W T I p r i c e Figure 5: WTI crude oil price from 23/10/2018 to 23/10/2019 from 23/10/2018 to 23/10/2019. From Figure 4, the trial outcomes are more concentrated leading to a moreprecise result with a smaller interval from the NPI method. The maximum buying price is 2.447181, andthe minimum selling price is 2.538937. The daily volatility during this historical period is 2.4%. Althoughthe NPI result is more precise, the interval deviates distinctly from the real market price 2.83 comparing tothe result from a large historical data. This means we scarify the accuracy in order to gain a more preciseresult. But this is not a good deal since the investor referring to the NPI result would lose money when hetrades WTI in the market.The examples above are a rigid investigation of the NPI performance by controlling the size of historicaldata. To study the historical data more clearly, we display the WTI crude oil price in this recent one yearin Figure 5. It is obvious that there is a deep drop that started in October 2018 ended in December 2018,which is the worst performance in nearly three years. The price is down to 44.48 on 27/12/2018 the lowestclosing price since January 2016. There exist multiple reasons causing this drop, global oversupply keepingthe investors away, investors with less confidence of economic recovery in the next year and the longestUS government shutdown on 22 December 2018. Through the comprehensive consideration, the data from23/04/2019 to 23/10/2019 has a better reference value to do the prediction. But it is an arbitrary decisionto cut off the data of early half year crudely. What we do here is to adjust the sampling procedure makingit focus more on the latter half years’ data than the earlier one. To achieve this, we use the maximum andminimum one year historical prices as the boundary values, but the main sampling data is the historical data15 igure 6: NPI predictions after adjustment from 23/04/2019 to 23/10/2019. By doing this, the pricing procedure not only considers the probabilitiesof the unexpected event but also places emphasis on the historical information in a relatively stable marketenvironment. The adjusted result is plotted in Figure 6. It is obvious that after adjustment, the accuracy ofthe NPI result gets better, the maximum buying price at 2.757054 and the minimum selling price 2.936124.This interval covers the real market price which is a better investment indicator than the NPI result withoutadjustment. In addition, the precision of the NPI result is nearly as same as that without adjustment. Thisexample manifests that the NPI method performs better combining with the assessment of historical data.The NPI method as discussed in Section 3.3 can be used as the market director for an investor. Asacknowledged, in the crude oil market, WTI from the American and Brent from the North Sea are twobenchmark prices of the crude oil market that are both sweet and normally track one another. Their pricestrend and pattern are similar to each other making the investor hard to compare these two values directlyfrom the market price. According to the NPI method illustrated in Section 3.3, an investor can get anindicator of the trading action according to the investor’s risk aversion. In the following example, we assumethe investor wants to buy an Asian call option on either WTI or Brent that the average underlying asset16rading Date P WTI P WTI P Brent P Brent
Table 2: NPI probabilities for WTI vs Brent with K = 0 . ∗ S Trading Date P WTI P WTI P HO P HO Table 3: NPI probabilities for WTI vs Heating Oil (HO) with K = 0 . ∗ S price during the option life period is not less than 95% of its spot price meaning the strike price K is equalto 95% of the spot price S . The call option’s trading day is from 2019-11-22 to 2019-11-30, and the expireday is 2019-11-30, so the option period is from 9 days to 0 days. Then the upper and lower probabilitiesof both WTI and Brent average prices greater than strike price are calculated getting the results displayedin Table 2. From Table 2, it is obvious that the lower probabilities of the WTI price are greater than theupper probability of the Brent price. WTI definitely has a higher possibility to earn a positive payoff in thecall option market. For the investor either as a speculator or a hedger, it is optimal to invest in WTI. Theresult also plotted in Figure 7 showing that the probability pattern of the WTI price is similar to that of theBrent price but with greater values. Also, from the figure, we can tell the best time to get in the market,which is 2019-11-23 in this example, since the NPI probabilities of Nov 23rd are the greatest value amongthese dates except the one of Nov 30th. The WTI and Brent oil price returns from 2019-11-22 to 2019-11-30are also calculated to assess our prediction. The WTI return equals to 4.102% higher than the Brent return,1.935%, which confirms that the trading strategy based on NPI is profitable.To end this study, we calculate the upper and the lower probabilities of the WTI price comparing to17 .000.250.500.751.00 Nov 23 Nov 25 Nov 27 Nov 29 Trading Date (K=0.95*S0) N P I P r obab ili t i e s BrentWTI
Figure 7: NPI probabilities for WTI and Brent
Trading Date (K=0.95*S0) N P I P r obab ili t i e s HOWTI
Figure 8: NPI probabilities for WTI and the heating oil S µ ended by Nov 30th is greater than the 95% of the spot price S . From the perspective of the Asian option,we are interested in the probability of a call option with K = 0 . ∗ S end up with a positive payoff. We plotthe NPI upper and lower probabilities in Figure 8. The decision of the option selection is harder to makein this comparison group, because there are intersections and overlapping of the NPI probabilities betweenthese two products. Unlike the result of WTI versus Brent that WTI always dominates, in this example,there are overlapping and intersections in Figure 8. The different underlying asset is picked according to thetrading date. To specify the underlying asset selection based on the trading data, the exact value of NPIupper and lower probabilities are listed in Table 3. Between 2019-11-22 to 2019-11-24, the lower probabilityof S µ > K for the heating oil dominates the upper probability of S µ > K for WTI. During this period, aspeculator or a hedger is better to invest in the heating oil. On Nov 25th, the upper and lower probabilitiesof WTI are greater than the corresponding value of the heating oil. So on this day, a speculator is supposedto choose the call option based on the WTI oil price, but a hedger would wait. On Nov 26th, the NPIprobability intervals of WTI and the heating oil are overlapped with each other, so there is no indicationwhich underlying asset is better. The next day’s upper probabilities of these two oil price are still the samevalue, while the lower probability of WI is less than the lower probability of the heating oil. Thus, on Nov27th, a speculator is better to get in the game of the call option based on the heating oil, and a hedger stillwaits for the sign of a more determined trading indicator. This indicator appears on Nov 28th and lasts untilNov 30th, the lower probability of the heating advantages over the upper probability of the WTI, leading tothe trade for both a speculator or a hedger in the Asian call option contingent on the heating oil. This paper presents a novel approach to evaluate the Asian option with the arithmetic price from theimprecise probability aspect through the NPI method, which forecasts the option price on the basis of thehistorical data with few assumptions. This approach provides an interval of prices as the result, whichnot only contains the uncertainty from the probability perspective but also the uncertainty from limitedprior information. This property makes it more advanced than the traditional method for the Asian option,19specially the one in the energy market because of the less liquidity of the Asian option in the energy market.The NPI method also gives a risk measure by comparing two energy products inspiring the investor witha trading criterion. We study the performance of the NPI method first by the simulation using the GBMprediction as the benchmark. Three factors, precision, coverage percentage and accuracy are defined andinvestigated to help us assess the performance. It turns out the NPI forecasting has more reference value forthe less volatile product. Then we predict the WTI crude oil price based on the NPI method comparing tothe real market price. With a long period of historical data, the NPI forecasting interval contains the realmarket price, but the precision of the result is not entirely satisfactory. To get a more precise interval result,narrowing the size of the historical data is going to scarify the accuracy of the result. After the investigation,we found that using the extreme value of the historical data to control the precision and considering thehistorical event of adjusting the sampling period of the historical data can offer a better outcome. We alsoillustrate the risk measure, the NPI trading criterion, by two examples, the trade of WTI and Brent andthe trade of WTI and the heating oil. An investor is guided according to their risk aversions by using thiscriterion.In order to get a better result from the NPI method, there are several aspects we need to consider.The time period of the predictive data should be considered discreetly since we assume the exchangeabilityof all data including the historical data and the future data in Section 2. If the prediction period is toolong, it challenges the reasonableness of the exchangeability assumption, because some significant fluctuationmay happen in the market. To inference these fluctuations, a large historical data is needed to infer thesituation, which as we discussed in the last paragraph, this will reduce the level of accuracy of the result.The extreme value of the historical data, r (0) and r n +1 , is also an important consideration that will affect theNPI prediction as we illustrated in the example. These two values play a very important role in balancingthe precision of the result and the inference ability of the historical data for the significant fluctuations.To avoid the effect from the unexpected historical event or seasonal effect, the sampling historical data s (1) , ..., s n has been picked in the time period with a more stable market. 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