Interpreting Replication Errors in Discrete-Time Option Hedging

# Difference between replicating portfolio and option price


# Difference between replicating portfolio and option price












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!

## Answer by KaiSqDist (score 0)

https://quant.stackexchange.com/a/78801

I did not read the paper, but looking at your description and from my understanding, it seems like a straightforward replication of delta hedging the option.

For example, if we wish to delta-hedge a short call position, one would go long on delta-times the amount of stock. In your case, the delta would be $a_t$ and the long stock position is financed with money $C_t$. However, as the stock price evolves to the next period $S_{t+1}$ and the call option price changes as well to $P_{t+1}$ (yes, confusing notation but bear with it), we happen to have a mismatch in pricing $H_{t+1}$:

\begin{equation} H_{t+1} = a_t S_{t+1} + C_te^{r\Delta t} - P_{t+1} \end{equation}

Therefore, from the perspective of an option seller (via the short option position), the pricing error $H_t$ is caused by the option gamma $\Gamma_t$, which is the sensitivity of the option delta $\Delta_t$ or $a_t$ to changes in the underlying stock price. Also, a negative (positive) $H_{t+1}$ simply means that the hedge position (consisting of the stock and the cash/bond) is worth more (less) than the call option $P_{t+1}$.

Last Comments: The problem you have encountered might be referencing $P_{t+1}$ for a long put position, so your question about asking in the POV of a option seller might not be a good question.