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The stock exchanges provide the data of option prices using theoretical formulations such as Black-Scholes formula. The dataset necessary for training an artificial neural network (ANN) to address option pricing models, specifically the Black-Scholes model, has already been produced using the same formulas. However, the literature claims that ANNs can capture more realistic and broadly applicable market assumptions, including non-linearity and stochastic volatility.

I wonder how can we substantiate, through mathematical reasoning, the claim that ANNs are more effective than exact and numerical solutions? OR

On what basis should I justify ANNs can potentially capture and generalize complex relationships between the variables in a way that analytical or numerical methods may not?

In a paper entitled "A neural network model for estimating option prices", the author claims 'Approximately for half of the cases that they examined, mean squared error for the neural network is smaller than that of Black-Scholes, which implies the good performance of ANN relative to Black and Scholes'.

I know how to calculate the MSE error of ANN predictions from Black-Scholes formula. What does the phrase 'smaller than that of Black-Scholes mean'? How is the MSE error of the BS formula calculated? Relative to what it is calculating? From where that true data we get? How does that true data generate?

Can I say the error comes due to continuous evolution of the Black–Scholes models over time, such as regime switching, Jump diffusion, variable volatility, stochastic volatility, fractal dynamics, etc.?

Thanks for your valuable response in advance.

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    $\begingroup$ I might be missing something, but in case it's price quoted, you have no error with black Scholes because it perfectly defines the implied vol needed to get the market price.if it's vol quoted, you don't observe market prices and it's black Scholes that is used to translate quotes into prices. Hence, I don't think there is any error, unless the author compares the results to Black Scholes prices from non sensible inputs like historical vol. $\endgroup$
    – AKdemy
    Dec 14, 2023 at 8:47
  • $\begingroup$ @AKdemy I got your point that there wont be any error if we are using implied vol. But my doubt still remains as I am desperate to know how to compare ANN results with numerical solutions if Black-Scholes models lacks analytical solutions (as in the case of variable coefficients). $\endgroup$
    – Messi Lio
    Dec 14, 2023 at 9:33
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    $\begingroup$ Not sure what you mean to be honest? Even if American, you can just solve the PDE, there is no need for a closed form solution. To be honest, I haven't read the paper because it's behind a paywall and also quite dated. $\endgroup$
    – AKdemy
    Dec 14, 2023 at 9:43
  • $\begingroup$ @AKdemy We use some market data (which is already calculated by BS formula) to train ANN and claim ANN is efficient than BS formula or efficient than Numerical solutions of BSE. Why it is ? How it is sensible? This is my basic doubt. Hope you got my doubt. Thank you in advance. $\endgroup$
    – Messi Lio
    Dec 14, 2023 at 9:52
  • $\begingroup$ @AKdemy In this paper, I have seen ''The option price thus obtained (as per Black & Scholes formula) are compared with the closing price of the option traded at NSE'' in the section In section 5.2. (Training of ANN ). My basic doubt here is : How can we compare NSE price and BS formula as NSE itself uses BS formula to calculate option price?. Thank you. $\endgroup$
    – Messi Lio
    Dec 14, 2023 at 13:33

1 Answer 1

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The paper that you last linked in a comment can be accessed free of charge and if the original paper is using something similar it's a tautology that the ANN model is better.

The author describes that he uses (section 4.1)

  • the closing spot price but
  • for option prices he uses all exercise prices available on a given day (no matter the time)
  • a fixed risk free interest rate (obtained from discount rates of Treasury bills),
  • a fixed volatility, obtained as the 60 day standard deviation of historical returns
  • no dividends
  • only discrete days (not exact time to expiry)

After that, the author computes the value of nifty call options from these inputs with Black Scholes and compares the results with market quoted prices.

Subsequently, the author assigns weights to all inputs (section 4.3) and runs the model. The claim is that this is better than Black Scholes, which is in my humble opinion complete bogus for at least the following reasons:

  • you would need the spot price at the time of the option quote
  • nifty pays dividends (I think, didn't check in detail) but they are completely neglected
  • using a fixed historical IV is completely mispricing options as shown for example in this answer
  • risk free interest rates are not constant across all expiries
  • exact time to expiry is important, especially for options with shorter maturity (the link showing how using historical vol misprices options also uses exact time to expiry, visible on the code where T+m is computed)

To sum up, by assigning weights to each input the author simple corrects the inputs into Black Scholes for all mis-specifications the author has.

The procedure is completely ignoring how Black Scholes is used in the market and I am honestly surprised you can actually write papers like that.

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  • $\begingroup$ I got your point regarding the paper. However, I am really exited to know from you that how stock exchanges evaluate option prices (quote, true out put)? thank you. $\endgroup$
    – Messi Lio
    Dec 14, 2023 at 17:30

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