Using Implied Volatility for Delta Hedges and Realized Volatility for Paths

Summary

The document describes a Monte Carlo setup for studying the profit and loss of a long out-of-the-money call that is delta hedged over time. The option is purchased using the volatility assumed at inception, while simulated underlying price paths reflect the volatility expected to be realized. To calculate each hedge adjustment, the response says to keep using the purchased option's volatility in the Black–Scholes delta calculation.

The example contrasts a 10% purchased volatility with 15% realized volatility: the initial option value and each rehedging delta use the former, while the simulated stock path uses the latter. This separates pricing assumptions from the data-generating process and supports comparisons when purchased volatility is below, equal to, or above realized volatility. The discussion is brief and does not specify path dynamics, transaction costs, discrete rebalancing effects, or other model assumptions that affect simulated P&L.

Key ideas

• The simulation should distinguish the volatility used to price the option from volatility used to generate underlying paths.
• The initial option value and rehedging deltas use the purchased volatility assumption.
• The simulated underlying path uses the realized volatility assumption.
• P&L conclusions depend on path dynamics, hedge frequency, trading costs, and other model choices.

Tags

# How to simulate a delta hedged option strategy


# How to simulate a delta hedged option strategy












I'd like to do a montecarlo simulation of a $\Delta$ hedged strategy (long OTM call) to see how the PnL distributes on cases like:

- $\sigma_{bought} < \sigma_{realized}$

- $\sigma_{bought} > \sigma_{realized}$

- $\sigma_{bought} = \sigma_{realized}$.

For this, I calculate the purchased option price with $\sigma_{bought}$ at $t_0$ and then do a random walk for the underlying asset $S$ so I can modify the hedge amount on each step.

My problem is that in order to calculate the hedge amount variations on $t$, I do not only need a new $S_t$ but also a volatility as $\Delta$ depends on vol (at least with BSM formula: $N(d_1)$ where $d_1$ depends on vol).

Question: What volatility should I use on each step to calculate the new $\Delta$ of the option?

## Answer by jrolo (score 2)

https://quant.stackexchange.com/a/74696

Agreed with nbbo2. I did this exact thing a while back.

[So, for example If I want to compare 𝜎𝑏=10% with 𝜎𝑟=15%. That means I should calculate initial option value with 𝜎=10%, but then on each step: montecarlo simulation as well as 𝑑1(𝑓𝑜𝑟Δ) will use 𝜎=15%?]

Calculate initial option value with vol(b) = .10, and the hedging delta at each step with vol(b) = .10. But then use vol (r) = .15 to calculate the path of the stock.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.