Abstract
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 of its underlying asset, which does not display in the Black–Scholes model only because 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.
1. Introduction
The 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. The 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. Their approach assumes that the underlying assets evolve as dependent processes. The 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 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. The 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. That 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 risk-neutral 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. The empirical analyses show that the fitting effect of our pricing formulas is superior to that of the Black–Scholes model.
2. The Black–Scholes Formula
In this study, we will investigate European option pricing and compare our results with those of Black and Scholes [1]. Thus, 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. That is, the security price satisfies stochastic differential equation:where and are constant and .
Assumption 2. The short-term interest rate is known and is constant through time. That is, the risk-free bond price satisfies ordinary differential equation:where is constant.
Assumption 3. The security pays no dividends or other distributions.
Assumption 4. There are no transaction costs in buying or selling the security or the option.
Assumption 5. The security can be continuously transacted.
Assumption 6. The 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. There 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. There 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 aswhere is the initial price of underlying asset, is the strike price of option, is the short-term interest rate, is the diffusion coefficient of underlying asset, is the left expiration time of option, is the cumulative density function of standard normal distribution, and
3. 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 no-arbitrage 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 by . Then the value of European call option at expiration time with underlying asset and strike price follows ashere and in the sequel the operator . The value of European put option at expiration time with underlying asset and strike price 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 , where is the risk-compensation coefficient, is the risk measure, and is the risk size. After the seller has received the reasonable risk compensation, the seller takes risky asset as equivalent risk-free bond, so it follows from (6) that the value at time of European call option with underlying asset , strike price , and expiration time follows aswhere is the risk-free rate during . Analogically, it follows from (7) that the value of European put option with underlying asset , strike price , and expiration time 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 and the risk-compensation coefficient is , and assuming a European option with underlying asset , strike price , and expiration time , then its call-option value at time follows asand its put-option value at time follows aswhere is the risk-free rate during and .
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., for any risk variable .
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 and diffusion coefficient , the current price of underlying asset is , risk-free interest rate is through the time, the risk-compensation factor is , and the risk measure is standard deviation, then the value of European call option with strike price and left expiration time follows aswhere
The Proof of Theorem 2 refers to Appendix A.
If a financial market is risk-neutral, then investors treat expected return and deterministic return equally, so expectation yield equals risk-free yield , that is, . Otherwise, if , there will exist arbitrage opportunity. Furthermore, in a risk-neutral financial market, the risk-compensation factor equals zero, that is, . Thus, it yields from Theorem 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 and diffusion coefficient , the current price of underlying asset is , and risk-free yield is through the time, then the value of European call option with strike price and left expiration time follows aswhere
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 and diffusion coefficient , the current price of underlying asset is , risk-free yield is through the time, the risk-compensation factor is , and the risk measure is standard deviation, then the value of European put option with strike price and left expiration time follows aswhere
The Proof of Theorem 3 refers to Appendix B.
According to the analysis before Corollary 1, if a financial market is risk-neutral, then and . Furthermore, it yields from Theorem 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 and diffusion coefficient , the current price of underlying asset is , and risk-free yield is through the time, then the value of European put option with strike price and left expiration time follows aswhere
Remark 1. Although Corollaries 1 and 2 obtain the same values as those in Theorem 1 for European call option and European put option, Corollaries 1 and 2 need much weaker conditions than those of Theorem 1. That is, Corollaries 1 and 2 improve Theorem 1.
In fact, if Assumptions 1, 2, and 4 in Section 2 hold, Corollaries 1 and 2 hold. However, the conditions that Theorem 1 (i.e., the Black–Scholes model) holds are Assumptions 1 to 9 in Section 2.
4. 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 or , trading date , exercise date , strike price , and current price of underlying asset . In addition, the database also includes the historical volatility of Shanghai 50 ETF on September 3, 2018, and on September 4, 2018, and the 1-year deposit benchmark interest rate of Chinese Central Bank , 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.
4.1. Parameter Estimation
There are 48 call options on September 3, 2018, and the current price of underlying asset . We take the annual average return rate of the last month as the drift coefficient of the underlying asset . According to simple computation, we obtain and the left expiration time (years), where , and then we work out , , and by Theorem 2; see Table 1 for detailed data, where is the actual closing price of call option, is the expiration time of call option, is the strike price of call option, is the initial price of underlying asset, is the left expiration time of call option, and , and are the parameters in Theorem 2. Furthermore, we obtain the estimated value of the risk-compensation factor by the least square method with .
There are 60 put options on September 3, 2018, and the current price of underlying asset . Similarly, we obtain and the left expiration time (years), where , and then we work out , , and by Theorem 3; see Table 2 for detailed data, where is the actual closing price of put option, is the expiration time of put option, is the strike price of put option, is the initial price of underlying asset, is the left expiration time of put option, and , and are the parameters in Theorem 3. Furthermore, we obtain the estimated value of the risk-compensation factor by the least square method with .
In addition, we obtain the estimation of risk-free rate by minimizing the mean square error of the Black–Scholes model on September 3, 2018. Note that is a little higher than the 1-year deposit benchmark interest rate of Chinese Central Bank , so it is very reasonable that we take as the risk-free rate. Furthermore, it is far advantageous for the Black–Scholes model to improve its fitting effect.
4.2. Comparison of Pricing Effect
In the section, we will compare the fitting effect of our pricing formulas with the Black–Scholes model.
4.2.1. Comparison of Pricing Effect on Call Options
There are 62 call options on September 4, 2018, and the current price of underlying asset . Based on the estimated parameters , , , for call options, the call options of Shanghai 50 ETF on September 4, 2018, are priced by our obtained pricing formula in Theorem 3 and the Black–Scholes model, respectively; see Table 3 for detailed data, where is the expiration time of call option, is the strike price of call option, is the initial price of underlying asset, is the left expiration time of call option, , and are the parameters in Theorem 3, is the value of call option computed by the Black–Scholes model, is the value of call option computed by Theorem 3, and is the actual closing price of call option.
According to simple computing, the expectation and variance of absolute errors follow aswhere is the actual closing price of call option, is the value of call option computed by the Black–Scholes model, and is the value of call option computed by Theorem 3. It is obvious that . In the following, we will support the statement by the hypothesis test (i.e., t-test function of MATLAB R2018b).
In fact, for the hypothesis test,
The statistics equals with degree of freedom 61, and its value is . Thus, we accept , i.e., . That is, the prices of call options computed by Theorem 3 are far nearer to their actual prices than those computed by the Black–Scholes model.
4.2.2. Comparison of Pricing Effect on Put Options
There are 62 put options on September 4, 2018, and the current price of underlying asset . Based on the estimated parameters , , , and for put options, the put options of Shanghai 50 ETF on September 4, 2018, are priced by our pricing formula in Theorem 2 and Black–Scholes model, respectively; see Table 4 for detailed data, where is the expiration time of put option, is the strike price of put option, is the initial price of underlying asset, is the left expiration time of put option, , and are the parameters in Theorem 3, is the value of put option computed by the Black–Scholes model, is the value of put option computed by Theorem 3, and is the actual closing price of put option.
According to simple computing, the expectation and variance of absolute errors follow aswhere is the actual closing price of put option, is the value of put option computed by the Black–Scholes model, and is the value of put option computed by Theorem 3. It is obvious that . In the following, we will support the statement by the hypothesis test (i.e., t-test function of MATLAB R2018b).
In fact, for the hypothesis test,
The statistics equals with degree of freedom 61, and its value is . Thus, we accept , i.e., . That is, the prices of put options computed by Theorem 3 are far nearer to their actual prices than those computed by the Black–Scholes model.
Therefore, our pricing formulas in Theorem 2 and Theorem 3 have less absolute errors than those of the Black–Scholes model for both call options and put options. That is, the fitting effect of our pricing formulas is superior to that of the Black–Scholes model.
5. 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 . 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. The annualized historical volatility 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 on April 2, 2019.
5.1. Parameter Estimation
We consider the daily logarithmic returns of S&P 500 index closing prices from January 2015 to December 2018. The augmented Dickey–Fuller test shows the Dickey–Fuller statistic equals with lag order 10 and 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). Thus, 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 effective trading days as the drift coefficient of the underlying asset .
There are 41 call options on April 1, 2019, and the current price of underlying asset . According to simple computation, we obtain and the left expiration time (years), where , and then we work out , , and by Theorem 2; see Table 5 for detailed data, where is the actual closing price of call option, is the left expiration time of call option, is the strike price of call option, and , and are the parameters in Theorem 2. Furthermore, we obtain the estimated value of the risk-compensation factor by the least square method with .
There are 62 put options on April 1, 2019, and the current price of underlying asset . Similarly, we obtain and the left expiration time (years), where , and then we work out , , and by Theorem 3; see Table 6 for detailed data, where is the actual closing price of put option, is the left expiration time of put option, is the strike price of put option, and , and are the parameters in Theorem 3. Furthermore, we obtain the estimated value of the risk-compensation factor by the least square method with .
5.2. Comparison of Pricing Effect
In the section, we will compare the fitting effect of our pricing formulas with the Black–Scholes model.
5.2.1. Comparison of Pricing Effect on Call Options
There are 25 call options on April 2, 2019, and the current price of underlying asset . Based on the estimated parameters , , , and for call options, the call options of S&P 500 on April 2, 2019, are priced by our obtained pricing formula in Theorem 1 and the Black–Scholes model, respectively; see Table 7 for detailed data, where is the strike price of call option, is the left expiration time of call option, , and are the parameters in Theorem 3, is the actual closing price of call option, is the value of call option computed by the Black–Scholes model, and is the value of call option computed by Theorem 1.
According to simple computing, the expectation and variance of absolute errors follow aswhere is the actual closing price of call option, is the value of call option computed by the Black–Scholes model, and is the value of call option computed by Theorem 2. It is obvious that . 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 statistics equals with degree of freedom 24, and its value is 0.01854. Thus, we accept , i.e., . That is, the prices of call options computed by Theorem 2 are far nearer to their actual prices than those computed by the Black–Scholes model.
5.2.2. Comparison of Pricing Effect on Put Options
There are 39 put options on April 2, 2019, and the current price of underlying asset . Based on the estimated parameters , , , and for put options, the put options of S&P 500 on April 2, 2019, are priced by our obtained pricing formula in Theorem 3 and the Black–Scholes model, respectively; see Table 8 for detailed data, where is the strike price of put option, is the left expiration time of put option, , and are the parameters in Theorem 3, is the actual closing price of put option, is the value of put option computed by the Black–Scholes model, and is the value of put option computed by Theorem 3.
According to simple computing, the expectation and variance of absolute errors follow aswhere is the actual closing price of put option, is the value of put option computed by the Black–Scholes model, and is the value of put option computed by Theorem 3. It is obvious that . In the following, we will support the statement by the hypothesis test (i.e., t-test function of MATLAB R2018b).
In fact, for the hypothesis test,
The statistics equals with degree of freedom 38, and its value is 0.00019. Thus, we accept , i.e., . That is, the prices of put options computed by Theorem 3 are far nearer to their actual prices than those computed by the Black–Scholes model.
Therefore, our pricing formulas in Theorems 2 and 3 have less absolute errors than those of the Black–Scholes model for both call options and put options. That is, the fitting effect of our pricing formulas is superior to that of the Black–Scholes model.
6. 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 of its underlying security, which does not display in the Black–Scholes model only because 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.
Appendix
A. Proof of Theorem 2
In order to prove Theorem 2, we need the following lemma.
Lemma A.1. Assuming , then for any real number and positive real number , it follows that
Proof. If , denote the density function of by , and thenFor any real number and positive real number , it follows thatThe proof is complete.
Noting that the underlying asset follows a geometric Brownian motion with drift coefficient and diffusion coefficient and the current price of underlying asset is , it follows from It formula thatwhere is the standard Wiener process, soIt yields from Lemma A.1 and (A.5) thatIt follows from (A.6) to (A.8) thatFurthermore, it obtains from (A.9) and (A.10) thatThus, it yields from Proposition 1, (A.9), and (A.11) thatThe proof is complete.
B. Proof of Theorem 3
In order to prove Theorem 3, we first present another lemma as follows.
Lemma B.1. Assuming , then for any real number and positive real number , it follows that
Proof. If , denote the density function of by , and thenFor any real number and positive real number , it follows thatThe proof is complete.
Noting that the underlying asset follows a geometric Brownian motion with drift coefficient and diffusion coefficient and the current price of underlying asset is , it follows from It formula thatwhere is a standard Wiener process, soIt yields from Lemma B.1 and (B.5) thatIt follows from (B.6) to (B.8) thatFurthermore, it obtains from (B.9) and (B.10) thatThus, it yields from (B.9) and (B.11) thatThe proof is complete.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the National Natural Science Foundation of China (no. 11471120).