l="//html.scirp.org/file/6-1490474x204.png" /> potential given by (73) can still be a good starting point for obtaining an approximate value for the potential.
6. Numerical Results and Option Price Simulations
In order to test this method and to solve the simulation problem for the option price solution of the non equilibrium Black-Scholes model, the behaviour of an European call option is simulated, using the 90-days futures of the e-mini S&P 500 from September 1998 to June 2007. The contract is set having the same underlying asset, opening and expiring dates than the S&P 500 futures. The option strike price is stablished as the underlying price at the opening date of the contract, assuming the market is going to be flat, in such a way that the option price is
where will be the empirical simulated option market price at i-day, is the e-mini S&P 500 future price and K is the option strike price. As it is well known E-mini S&P 500 options are priced in index points up to two decimals. One E-mini S&P 500 option can be exercised into one E-mini S&P 500 futures contract and since each contract has a multiplier of $50, the option price must also be multiplied by $50 to get a corresponding dollar value and every one point of change in the price of the option or the underlying futures for that matter is worth $50 per contract.
The e-mini S&P 500 futures contracts used to simulate the option are specified in Table 1.
The results are shown in the case of the first contract (e-mini S&P 500 from 12/ 03/1998 to 10/06/1998). Figure 2 shows the mispricing in (68) between the simulated option price and the Black-Scholes price. For this calculation, the standard deviation of the underlying returns from the previous 90 days is estimated and the three-months USA Treasury rate r at the initial day of the contract is taken as the risk- free rate. The estimated numerical values in fact are and.
Now Equation (72) can be solved via Newton-Raphson to obtain the empirical function daily for this contract as it can be seen in Figure 3. Then a continuous potential model for this function is proposed of the form and a non-linear Levenberg-Marquardt regression is performed in order to fit parameters a, b and c. The estimated parameter values are, and and Figure 3 shows the results.
At this point, the time-dependent potential can be obtained by using Equation (73)
Table 1. E-mini S&P 500 contracts.
Figure 2. Mispricing.
Figure 3. Empirical (continuous line) and estimated (dashed line).
as shown in Figure 4.
Now by replacing the continuous potential in the interacting Black- Scholes Equation (13) and integrating it by means of the Crank-Nicholson method, the interacting solution for the option price of a call option can be derived, as shown in Figure 5.
Clearly, the calibration of the potential allows to fit a more exact price than that of the traditional Black-Scholes model without considering arbitrage. The behavior of the interacting versus the usual Black-Scholes models can be tested for option pricing in terms of the performance measure discussed before. The computed values of the are: 14,980.76 for the Black-Scholes model and 1705.44 for the interacting Black-Scholes model, which difference is clearly visible in Figure 5.
When the calibrated model is used with its respective potential for simulating the rest of the contracts considered in series of Table 1, similar results are found, that in all the cases defeat Black-Scholes predictions as showed in Figure 6.
7. Conclusions and Further Research
In this work, the arbitrage effects for a non-equilibrium quantum Black-Scholes model of option pricing are calibrated. This calibration procedure rests heavily on the semi- classical approximation of the interacting Black-Scholes model, which permits to con- struct an equation for the interaction potential, from which the arbitrage bubble and the interaction potential can be estimated. By using this estimated potential, the price trajectory of a real call option can be simulated for several contracts of the S&P index, which allow to take into account any market imperfection and price desaligments. Even though a semi-classical approximation for the solution of the interacting Schrödinger equation is used, the results are extremely good in predicting the real option price and its trajectory for every contract simulated.
Since in real life, market imperfections always happen, almost on a regular basis,
Figure 4. Interacting potential.
Figure 5. Simulated option price P (continuous line), Black-Scholes model price B-S (dashed line) and interacting Black-Scholes model price CPV (dotted line) for the e-mini S&P 500 contract from 12/03/1998 to 10/06/1998.
Figure 6. (a) (b) (c) (d) (e) (f): Simulated option price P (continuous line), Black-Scholes model price B-S (dashed line) and interacting Black-Scholes model price CPV (dotted line) for e-mini S&P 500 contracts in Table 1.
hence arbitrage processes form part of the normal operation of the stock exchange, and logically mispricing is always going to exist. If this mispricing could be calibrated using the potential of the interacting Black-Scholes, even in a small part, it is expected that those results are always going to outperform the traditional Black-Scholes formulation. In this context, this model and its calibration procedure could be used very easily to simulate in a more exact fashion option pricing of any underlying asset.
Future research could be directed to capture different potential patterns for different underlying assets and different market situations. Even in this case, the potential is short-lived and circumstantial, for example in the case of bubbles, rebounds, crises or critical information (for example, when Bernanke talked!), it is possible to use this methodology to capture the potential of the contract in a similar situation and to simulate the new contract. Alternatively, if the situation is normal and no special conditions are foreseen, a good practice would be to use the immediately preceding contract in order to calibrate the potential and therefore the quantum model; considering the reasons given above, in almost all the cases, it is expected that this model will defeat the traditional Black-Scholes model.
Conflicts of Interest
The authors declare no conflicts of interest.
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