Option Delta When Volatility Depends on Price or Delta

Summary

The document asks how to calculate a call option’s sensitivity to its underlying forward price when volatility is itself modeled as a function of that price or of the option’s delta. It defines the call price in terms of forward price, strike, volatility, maturity, and interest rate, then distinguishes the two proposed volatility dependencies. This raises a practical modeling issue: the usual partial derivative with respect to forward price holds volatility fixed, whereas a total sensitivity must account for how volatility changes as the forward price changes.

The document does not provide an answer, derivation, pricing model, or numerical example, so it offers no evidence for a particular formula. In the price-dependent case, a solution would need the volatility function and its derivative; in the delta-dependent case, delta and volatility may need to be solved together as an implicit relationship. The precise result also depends on the option pricing framework and assumptions. These limits make the text a useful statement of a derivatives-calculus question, rather than a complete method for computing a production hedge ratio.

Key ideas

• A call’s sensitivity can differ depending on whether volatility is held fixed or varies with the forward price.
• If volatility depends directly on forward price, its rate of change contributes to the total price sensitivity.
• If volatility depends on delta, the sensitivity relationship may be implicit and require a joint solution.
• The document poses the problem but supplies no derivation, model choice, or numerical answer.

Tags

# Delta of an option in two cases


# Delta of an option in two cases












Let C be the prime of a call in fi=unction of the price in term F, Strike K, volatilité $\sigma$ and maturity t: $C(F,K,\sigma,t,r) $ We assume that we know $\delta$

$\delta=\frac{\partial}{\partial F}C(F,K,\sigma,t,r)$

I would like to compute $\delta$ the variation of C in function of F in two cases :

1-If $\sigma=f(F,K)$

2-If $\sigma=g(\delta)$ Thanks everyone

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