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I have a question of which I am uncertain on how to answer, that is:

Assume the Black and Scholes differential equation for option pricing with constant risk free rate, $ r $ and constant volatility $ \sigma^2 $. For a new type of q-option, the defined payout at $ t = T $ is

$$ G_q(S,T) = (S-E)^2 \theta(S-E). $$

At time $ t $ the price of the underlying asset is found to be $ S_1 $. You consider to buy a q-option with $ T-t = 30 $ days at a strike price $ E = 1.2S_1 $. Without explicitly calculating the price of the option $ G_q(S,t) $ for an arbitrary time $ t $, would you expect that the price of this q-option underestimates or overestimates the market price for this specific option? That is, would you expect to pay more or less for the considered q-option as compared to the theoretical price?

Is the answer to this question just that the theoretical price neither over- nor underestimates the market price? That is, per definition they should be the same?

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  • $\begingroup$ The market does not believe that $r$ and $\sigma^2$ are constant, the theoretician in this question assumes that they are. The prices they set will be different. $\endgroup$
    – nbbo2
    Mar 14, 2020 at 9:45
  • $\begingroup$ @noob2 Given what you are saying, in general, would this mean that the theoretical price is lower or higher than actual price? $\endgroup$
    – Jon
    Mar 14, 2020 at 11:20
  • $\begingroup$ First find out (in your lecture notes or textbook) how for a vanilla option the BS price and the market price differ (hint: it depends whether $S>E$ or $S<E$). Then from this fact draw a conclusion about the situation for this strange option called a q-option. $\endgroup$
    – nbbo2
    Mar 14, 2020 at 11:56

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