Delta of a Call Option in a Local Volatility Model

Summary

The document poses a question about deriving the delta of a standard call option when the underlying follows a local volatility process, with volatility specified as a function of both price and time. It asks how the option value responds to changes in the current underlying price under that model.

It also asks whether the implied volatility associated with the local volatility model depends on the underlying price when the option is represented using a Black–Scholes price. No derivation or answer is included, so the document provides no method, numerical evidence, or guidance on computing the sensitivity. Its useful content is limited to identifying the relationship between local volatility, option delta, and state-dependent implied volatility as questions for analysis.

Key ideas

• The question concerns call option delta under a price- and time-dependent local volatility process.
• It asks whether the Black–Scholes implied volatility induced by local volatility varies with the underlying price.
• The document provides no derivation, answer, or evidence for calculating delta.

Tags

# How to calculate Delta of an option in the Local Volatility model?


# How to calculate Delta of an option in the Local Volatility model?












Let $dS/S = \sigma(S,t) dW$.

Let the local volatility be known, i.e, we know the formula $\sigma(S,t)$.

How do I derive $\Delta$ of a regular call in this model?

Let BS be the Black Scholes price, and let $\sigma_{imp}$ be the implied volatility induced by above local volatility. Then, does $\sigma_{imp}$ depend on $S$ as well?

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