Difference between replicating portfolio and option price

Q

Quant

Hello Quant Stack Exchange community,

I've been working on a discrete-time model for option pricing, where I calculate the replicating portfolio using the model and compare it with the real option prices dynamically. The equation I'm using to represent this is (similar to equation of Bakshi et al. 1997 - Empirical Performance of Alternative Option Pricing Models):

$$H_{t+1} = a_tS_{t+1} + C_t \cdot e^{(R \cdot \Delta t)} - P_{t+1} $$

Here, $H_{t+1}$ represents the error between the replicating portfolio and the option value. $a_t$ is hedge ratio, $S_{t+1}$ underlying price, $C_t$ is cash or bond, and $P_{t+1}$ is market option price. I have observed that $H_{t+1}$ can take both positive and negative values.

I'm curious about the implications of these positive and negative values of $H_{t+1}$ within the context of option pricing. Specifically, what does it signify when $H_{t+1}$ is positive, and conversely, when it's negative (from the view of option seller)?

Any insights or references to relevant literature would be greatly appreciated.

Thank you!
 

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