European Option Pricing Formula in Risk-Aversive Markets

In this study, using the method of discounting the terminal expectation value into its initial value, the pricing formulas for European options are obtained under the assumptions that the financial market is risk-aversive, the risk measure is standard deviation, and the price process of underlying asset follows a geometric Brownian motion. In particular, assuming the option writer does not need the risk compensation in a risk-neutral market, then the obtained results are degenerated into the famous Black–Scholes model (1973); furthermore, the obtained results need much weaker conditions than those of the Black–Scholes model. As a by-product, the obtained results show that the value of European option depends on the drift coefficient 
 
 μ
 
 of its underlying asset, which does not display in the Black–Scholes model only because 
 
 μ
 =
 r
 
 in a risk-neutral market according to the no-arbitrage opportunity principle. At last, empirical analyses on Shanghai 50 ETF options and S&P 500 options show that the fitting effect of obtained pricing formulas is superior to that of the Black–Scholes model.


Introduction
e option pricing theory began in 1900 when the French mathematician Louis Bachelier deduced an option pricing formula under the assumption that underlying asset prices follow a Brownian motion with zero drift. Since then, lots of researchers have contributed to the theory. Black and Scholes [1] present the very famous option pricing formula (i.e., Black-Scholes model) in a risk-neutral market and according to the no-arbitrage opportunity principle. Merton [2] shows the Black-Scholes-type model can be derived from weaker assumptions than in their original formulation and present some pricing methods for non-European options. Bakshi et al. [3] first derive an option pricing model that allows volatility, interest rates, and jumps to be stochastic. Gârleanu et al. [4] model demand-pressure effects on option prices. e model shows that demand pressure in one option contract increases its price by an amount proportional to the variance of the unhedgeable part of the option. Cai and Kou [5] propose a jump diffusion model for asset prices whose jump sizes have a mixed-exponential distribution, which is a weighted average of exponential distributions but with possibly negative weights, and then they extend the analytical tractability of the Black-Scholes model to alternative models. Bernarda and Czadob [6] investigate the pricing of basket options and more generally of complex exotic contracts depending on multiple indices. eir approach assumes that the underlying assets evolve as dependent GARCH (1,1) processes. e dependence among the assets is modeled using a copula based on pair copula constructions. Bandi and Bertsimas [7] combine robust optimization and the idea of ε−arbitrage to propose a tractable approach to price a wide variety of options. Bao et al. [8] present a method that there is a possibility to get statistical arbitrage from Black-Scholes's option price.
In the last five years, there are still many researchers contributing to the theory of option pricing. Moretto et al. [9] study option pricing under deformed Gaussian distributions. Leippold and Scharer [10] develop a stochastic liquidity model, and they investigate discrete-time option pricing with stochastic liquidity. Hoka and Chanb [11] develop an option pricing method based on Legendre series expansion of the density function, and approximation formulas for pricing European type options are derived. Davison and Mamba [12] obtain a solution of the Black--Scholes equation with a nonsmooth boundary condition using symmetry methods. Willems [13] derives a series expansion for the price of a continuously sampled arithmetic Asian option in the Black-Scholes setting. e expansion is based on polynomials that are orthogonal with respect to the log-normal distribution. More literature studies can refer to Liu et al. [14], Friz et al. [15], Dubinsky et al. [16], Huh [17], Liu et al. [18], Siddiqi [19] and their studies.
Although there are a huge number of literature studies on option pricing, they almost assume the financial markets are risk-neutral and complete, especially since Black and Scholes [1]. However, according to the theory and empirical analysis of risk, real financial markets are risk-aversive and incomplete. at is, investors need risk compensation for risky assets, and many risky assets cannot be duplicated by any portfolio constructed in real financial markets.
In this study, using the method of discounting the terminal expectation value into its initial value, we obtain European option pricing formula under the assumptions that the financial market is risk-aversive, the risk measure is standard deviation, and the price process of underlying asset follows a geometric Brownian motion. In particular, if the option writer does not need the risk compensation in a riskneutral market, then our obtained results are degenerated into the Black-Scholes model [1]; furthermore, our obtained results need much weaker conditions than those in the Black-Scholes model. At last, we take the Shanghai 50 ETF options, the first floor option in the Chinese financial market, and S&P 500 options as samples to compare the fitting effect. e empirical analyses show that the fitting effect of our pricing formulas is superior to that of the Black-Scholes model.

The Black-Scholes Formula
In this study, we will investigate European option pricing and compare our results with those of Black and Scholes [1].
us, we first retell the main results of Black & Scholes [1]. Black and Scholes [1] present nine assumptions in the market for the security and for the option and then obtain their famous option pricing formula. Assumption 1. Security price satisfies a geometric Brownian motion (GBM) model, where its drift coefficient and diffusion coefficient are constant through time.
at is, the security price satisfies stochastic differential equation: where μ and σ are constant and σ > 0.
Assumption 2. e short-term interest rate r is known and is constant through time.
at is, the risk-free bond price satisfies ordinary differential equation: where r is constant.
Assumption 3. e security pays no dividends or other distributions.

Assumption 4.
ere are no transaction costs in buying or selling the security or the option.

Assumption 5.
e security can be continuously transacted.
Assumption 6. e amount of security can be arbitrarily divided.

Assumption 7.
It is possible to borrow any fraction of the price of a security to buy it or to hold it, at the short-term interest rate.

Assumption 8.
ere are no penalties to short selling. A seller who does not own a security will simply accept the price of the security from a buyer and will agree to settle with the buyer on some future date by paying him an amount equal to the price of the security on that date.

Assumption 9.
ere is no-arbitrage opportunity. When the above assumptions all hold, Black and Scholes [1] derived the pricing formula for European options, which is the Black-Scholes model. Theorem 1 (see [1]). If Assumptions 1 to 9 hold, then the values of European call option and European put option follow as where S 0 is the initial price of underlying asset, K is the strike price of option, r is the short-term interest rate, σ is the diffusion coefficient of underlying asset, τ is the left expiration time of option, Φ(·) is the cumulative density function of standard normal distribution, and

European Option Pricing in Risk-Aversive Markets
Black-Scholes model and its modified versions have some defects. In fact, because real financial markets are incomplete, an option may not be duplicated constantly, so its value deduced by the asset duplication method and noarbitrage principle may lose the deductive basis. On the other hand, real financial markets are risk-aversive. Option seller undertakes the total risk and option buyer has no any risk, so option seller needs a reasonable risk compensation according to the theory of risk. In this section, we will deduce the option pricing formula in risk-aversive markets only under three assumptions, i.e., Assumptions 1, 2, and 4 in Section 2, which is reasonably far more than the Black--Scholes model and its modified versions. In risk-aversive markets, assume the price process of some risky asset X by X t , t ≥ 0 . en the value of European call option at expiration time T with underlying asset X and strike price K follows as here and in the sequel the operator (·) + � max 0, · { }. e value of European put option at expiration time T with underlying asset X and strike price K follows as Note that no matter call option or put option, it is always its seller undertakes the total risk and its buyer has no any risk. According to risk theory, the seller reasonably requires some risk compensation λρ(X), where λ ≥ 0 is the riskcompensation coefficient, ρ(·) is the risk measure, and X is the risk size. After the seller has received the reasonable risk compensation, the seller takes risky asset as equivalent riskfree bond, so it follows from (6) that the value at time t of European call option with underlying asset X, strike price K, and expiration time T follows as where r ≥ 0 is the risk-free rate during [t, T]. Analogically, it follows from (7) that the value of European put option with underlying asset X, strike price K, and expiration time T follows as In conclusion, we obtain the following proposition from (8) and (9). Proposition 1. In risk-aversive market, assume that the risk measure is ρ(·) and the risk-compensation coefficient is λ ≥ 0, and assuming a European option with underlying asset X, strike price K, and expiration time T, then its call-option value at time t follows as and its put-option value at time t follows as where r ≥ 0 is the risk-free rate during [t, T] and t ≤ T.
In order to obtain a closed-form solution to Proposition 1, in the following, we always assume the price process of underlying asset follows some geometric Brownian motion model (1), and the risk measure is the standard deviation, i.e., ρ(Z) � std(Z) for any risk variable Z.
Using Proposition 1, we can deduce the value of European call option in risk-aversive markets and under Assumptions 1, 2, and 4.

Theorem 2.
In a risk-aversive market, assuming that underlying asset follows a geometric Brownian motion with drift coefficient μ and diffusion coefficient σ > 0, the current price of underlying asset is S 0 , risk-free interest rate is r through the time, the risk-compensation factor is λ ≥ 0, and the risk measure is standard deviation, then the value of European call option with strike price K and left expiration time τ follows as where e Proof of eorem 2 refers to Appendix A. If a financial market is risk-neutral, then investors treat expected return and deterministic return equally, so expectation yield μ equals risk-free yield r, that is, μ � r. Otherwise, if μ ≠ r, there will exist arbitrage opportunity. Furthermore, in a risk-neutral financial market, the risk-compensation factor equals zero, that is, λ � 0. us, it yields from eorem 1 that we have the following corollary.

Corollary 1.
In a risk-neutral market, assuming that underlying asset follows a geometric Brownian motion with drift coefficient μ and diffusion coefficient σ > 0, the current price of underlying asset is S 0 , and risk-free yield is r through the time, then the value of European call option with strike price K and left expiration time τ follows as where Mathematical Problems in Engineering In the following, we will deduce the pricing formula of European put option. Using Proposition 1, we will construct the pricing formula of European put option in risk-aversive markets and under Assumptions 1, 2, and 4.

Theorem 3.
In a risk-aversive market, assuming that underlying asset follows a geometric Brownian motion with drift coefficient μ and diffusion coefficient σ > 0, the current price of underlying asset is S 0 , risk-free yield is r through the time, the risk-compensation factor is λ ≥ 0, and the risk measure is standard deviation, then the value of European put option with strike price K and left expiration time τ follows as where e Proof of eorem 3 refers to Appendix B. According to the analysis before Corollary 1, if a financial market is risk-neutral, then μ � r and λ � 0. Furthermore, it yields from eorem 3 that we have the following corollary.

Corollary 2.
In a risk-neutral market, assuming that underlying asset follows a geometric Brownian motion with drift coefficient μ and diffusion coefficient σ > 0, the current price of underlying asset is S 0 , and risk-free yield is r through the time, then the value of European put option with strike price K and left expiration time τ follows as where Remark 1. Although Corollaries 1 and 2 obtain the same values as those in eorem 1 for European call option and European put option, Corollaries 1 and 2 need much weaker conditions than those of eorem 1. at is, Corollaries 1 and 2 improve eorem 1.
In fact, if Assumptions 1, 2, and 4 in Section 2 hold, Corollaries 1 and 2 hold. However, the conditions that eorem 1 (i.e., the Black-Scholes model) holds are Assumptions 1 to 9 in Section 2.

Empirical Analysis of Shanghai 50 ETF Options
In the section, we will present empirical analysis on Shanghai 50 ETF options, the first floor option in the Chinese financial market, and use the data of September 3 and 4, 2018, to compare the fitting effect of our pricing model and the Black-Scholes pricing model. All used data come from the CSMAR Database, which includes actual option price C or P, trading date t, exercise date T, strike price K, and current price of underlying asset S 0 . In addition, the database also includes the historical volatility of Shanghai 50 ETF σ � 0.1994 on September 3, 2018, and σ � 0.2005 on September 4, 2018, and the 1-year deposit benchmark interest rate of Chinese Central Bank r � 1.5%, which is chosen as the reference level of risk-free interest rate in the Chinese financial market. All data analyses in the following are worked out by the software MATLAB R2018b.

Parameter Estimation.
ere are 48 call options on September 3, 2018, and the current price of underlying asset S 0 � 2.512. We take the annual average return rate of the last month as the drift coefficient of the underlying asset μ. According to simple computation, we obtain μ � 2.557% and the left expiration time τ � (T − t)/365 (years), where t � 2018/09/03, and then we work out d 1 , d 2 , and d 3 by eorem 2; see Table 1 for detailed data, where C is the actual closing price of call option, T is the expiration time of call option, K is the strike price of call option, S 0 is the initial price of underlying asset, τ is the left expiration time of call option, and d 1 , d 2 , and d 3 are the parameters in eorem 2. Furthermore, we obtain the estimated value of the riskcompensation factor λ � 0.0077963 by the least square method with R 2 � 0.7058.
ere are 60 put options on September 3, 2018, and the current price of underlying asset S 0 � 2.512. Similarly, we obtain μ � 2.93% and the left expiration time τ � (T − t)/365 (years), where t � 2018/09/03, and then we work out d 1 , d 2 , and d 3 by eorem 3; see Table 2 for detailed data, where P is the actual closing price of put option, T is the expiration time of put option, K is the strike price of put option, S 0 is the initial price of underlying asset, τ is the left expiration time of put option, and d 1 , d 2 , and d 3 are the parameters in eorem 3. Furthermore, we obtain the estimated value of the risk-compensation factor λ � 0.013306 by the least square method with R 2 � 0.6218.
In addition, we obtain the estimation of risk-free rate r � 1.768% by minimizing the mean square error of the Black-Scholes model on September 3, 2018. Note that r � 1.768% is a little higher than the 1-year deposit benchmark interest rate of Chinese Central Bank 1.5%, so it is very reasonable that we take r � 1.768% as the risk-free rate. Furthermore, it is far advantageous for the Black-Scholes model to improve its fitting effect.

Comparison of Pricing Effect.
In the section, we will compare the fitting effect of our pricing formulas with the Black-Scholes model.   Table 3 for detailed data, where T is the expiration time of call option, K is the strike price of call option, S 0 is the initial price of underlying asset, τ is the left expiration time of call option, d 1 , d 2 , and d 3 are the parameters in eorem 3, C 1 is the value of call option computed by the Black-Scholes model, C 2 is the value of call option computed by eorem 3, and C is the actual closing price of call option.
According to simple computing, the expectation and variance of absolute errors follow as where C is the actual closing price of call option, C 1 is the value of call option computed by the Black-Scholes model, and C 2 is the value of call option computed by eorem 3. It is obvious that E(|C 2 − C|) < E(|C 1 − C|). In the following, we will support the statement by the hypothesis test (i.e., ttest function of MATLAB R2018b).
In fact, for the hypothesis test, e t−statistics equals −5.9311 with degree of freedom 61, and its p value is 7.6261 × 10 −8 . us, we accept H 1 , i.e., E(|C 2 − C|) < E(|C 1 − C|). at is, the prices of call options computed by eorem 3 are far nearer to their actual prices than those computed by the Black-Scholes model.

Comparison of Pricing Effect on Put Options.
ere are 62 put options on September 4, 2018, and the current price of underlying asset S 0 � 2.55. Based on the estimated parameters μ � 2.557%, σ � 0.2005, r � 1.768%, and λ � 0.013306 for put options, the put options of Shanghai 50 ETF on September 4, 2018, are priced by our pricing formula in eorem 2 and Black-Scholes model, respectively; see Table 4 for detailed data, where T is the expiration time of put option, K is the strike price of put option, S 0 is the initial price of underlying asset, τ is the left expiration time of put option, d 1 , d 2 , and d 3 are the parameters in eorem 3, P 1 is the value of put option computed by the Black-Scholes model, P 2 is the value of put option computed by eorem 3, and P is the actual closing price of put option.
According to simple computing, the expectation and variance of absolute errors follow as E P 1 − P � 0.00836, Var P 1 − P � 0.0056946, where P is the actual closing price of put option, P 1 is the value of put option computed by the Black-Scholes model, and P 2 is the value of put option computed by eorem 3. It is obvious that E(|P 2 − P|) < E(|P 1 − P|). In the following, we will support the statement by the hypothesis test (i.e., ttest function of MATLAB R2018b).
In fact, for the hypothesis test, e t−statistics equals −4.7567 with degree of freedom 61, and its p value is 6.2304 × 10 −6 . us, we accept H 1 , i.e., E(|P 2 − P|) < E(|P 1 − P|). at is, the prices of put options computed by eorem 3 are far nearer to their actual prices than those computed by the Black-Scholes model. erefore, our pricing formulas in eorem 2 and eorem 3 have less absolute errors than those of the Black-Scholes model for both call options and put options.
at is, the fitting effect of our pricing formulas is superior to that of the Black-Scholes model.

Empirical Analysis of S&P 500 Options
In the section, we will present empirical analysis on S&P 500 options and use the data of April 1 and 2, 2019 to compare the fitting effect of our pricing model and the Black-Scholes pricing model. All used data come from the Chicago Board Options Exchange, which includes actual option price C or P, trading date t, exercise date T, strike price K, and current price of underlying asset S 0 . We consider the out-of-money put and call options, which are more liquid and actively traded than in-the-money options. And observations with trading volume below average, prices less than $0.5 or left expiration time less than 10 days or longer than 360 days are discarded. e annualized historical volatility σ � 0.1174 based on closing prices of the underlying asset over the last month. According to the Board of Governors of the Federal Reserve System (https://www.federalreserve.gov/releases/ h15/data.htm), the annualized risk-free interest rate is 2.43% on April 1 and 2.42% on April 2, 2019.

Parameter Estimation.
We consider the daily logarithmic returns of S&P 500 index closing prices from January 2015 to December 2018. e augmented Dickey-Fuller test shows the Dickey-Fuller statistic equals −10.283 with lag order 10 and p value is 0.01, which indicates that the time series of returns is stationary. Furthermore, the autocorrelogram shows the coefficients of autocorrelation mostly fall within double standard deviations (see Figure 1). us, we accept that the time series of daily logarithmic returns of S&P 500 index closing prices are stationary and independent. We take the annual average return rate based on 252   19. According to simple computation, we obtain μ � 4.932% and the left expiration time τ � (T − t)/365 (years), where t � 2019/04/01, and then we work out d 1 , d 2 , and d 3 by eorem 2; see Table 5 for detailed data, where C is the actual closing price of call option, τ is the left expiration time of call option, K is the strike price of call option, and d 1 , d 2 , and d 3 are the parameters in eorem 2. Furthermore, we obtain the estimated value of the risk-compensation factor λ � −0.0246533 by the least square method with R 2 � 0.9532.
ere are 62 put options on April 1, 2019, and the current price of underlying asset S 0 � 2867. 19. Similarly, we obtain μ � 4.932% and the left expiration time τ � (T − t)/365 (years), where t � 2019/04/01, and then we work out d 1 , d 2 , and d 3 by eorem 3; see Table 6 for detailed data, where P is the actual closing price of put option, τ is the left expiration time of put option, K is the strike price of put option, and d 1 , d 2 , and d 3 are the parameters in eorem 3. Furthermore, we obtain the estimated value of the riskcompensation factor λ � 0.0269767 by the least square method with R 2 � 0.9561.

Comparison of Pricing Effect.
In the section, we will compare the fitting effect of our pricing formulas with the Black-Scholes model.

Comparison of Pricing Effect on Call Options.
ere are 25 call options on April 2, 2019, and the current price of underlying asset S 0 � 2867.24. Based on the estimated parameters μ � 4.932%, σ � 0.1174, r � 2.42%, and λ � −0.0246533 for call options, the call options of S&P 500 on April 2, 2019, are priced by our obtained pricing formula in eorem 1 and the Black-Scholes model, respectively; see Table 7 for detailed data, where K is the strike price of call option, τ is the left expiration time of call option, d 1 , d 2 , and d 3 are the parameters in eorem 3, C is the actual closing price of call option, C 1 is the value of call option computed by the Black-Scholes model, and C 2 is the value of call option computed by eorem 1.
According to simple computing, the expectation and variance of absolute errors follow as E C 1 − C � 6.902386, Var C 1 − C � 6.810772, where C is the actual closing price of call option, C 1 is the value of call option computed by the Black-Scholes model, and C 2 is the value of call option computed by eorem 2. It is obvious that E(|C 2 − C|) < E(|C 1 − C|). In the following, we will support the statement by the hypothesis test (i.e., test function of MATLAB R2018b).
In fact, for the hypothesis test, the t−statistics equals −2.1465 with degree of freedom 24, and its p value is 0.01854. us, we accept H 1 , i.e., E(|C 2 − C|) < E(|C 1 − C|). at is, the prices of call options computed by eorem 2 are far nearer to their actual prices than those computed by the Black-Scholes model.

Comparison of Pricing Effect on Put Options.
ere are 39 put options on April 2, 2019, and the current price of underlying asset S 0 � 2867.24. Based on the estimated parameters μ � 4.932%, σ � 0.1174, r � 2.42%, and λ � 0.0269767 for put options, the put options of S&P 500 on April 2, 2019, are priced by our obtained pricing formula in eorem 3 and the Black-Scholes model, respectively; see Table 8 for detailed data, where K is the strike price of put option, τ is the left expiration time of put option, d 1 , d 2 , and d 3 are the parameters in eorem 3, P is the actual closing price of put option, P 1 is the value of put option computed by the Black-Scholes model, and P 2 is the value of put option computed by eorem 3.
According to simple computing, the expectation and variance of absolute errors follow as where P is the actual closing price of put option, P 1 is the value of put option computed by the Black-Scholes model, and P 2 is the value of put option computed by eorem 3. It is obvious that E(|P 2 − P|) < E(|P 1 − P|). In the following, we will support the statement by the hypothesis test (i.e., ttest function of MATLAB R2018b).
In fact, for the hypothesis test,       e t−statistics equals −4.4919 with degree of freedom 38, and its p value is 0.00019. us, we accept H 1 , i.e., E(|P 2 − P|) < E(|P 1 − P|). at is, the prices of put options computed by eorem 3 are far nearer to their actual prices than those computed by the Black-Scholes model. erefore, our pricing formulas in eorems 2 and 3 have less absolute errors than those of the Black-Scholes model for both call options and put options. at is, the fitting effect of our pricing formulas is superior to that of the Black-Scholes model.

Conclusion
In this study, we obtain the pricing formula of European options, including European call option and European put option, in a risk-aversive market. Corollaries of our obtained results improve the Black-Scholes model owning to its much weaker conditions. It follows from our obtained results that European option value depends on the drift coefficient μ of its underlying security, which does not display in the Black-Scholes model only because μ � r in a risk-neutral financial market according to the no-arbitrage opportunity principle. Empirical analyses show that the fitting effect of our pricing formulas is superior to that of the Black-Scholes model.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.