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We consider the Black-Scholes model with time-dependent volatility $\sigma(t)$: $$ dS_{1}(t)=rS_{1}(t)dt+\sigma(t)S_{1}(t)dW(t) $$ The question: what constant $\hat{\sigma}$ one needs to apply such that for the process given by: $$ dS_{2}(t)=rS_{2}(t)dt+\hat{\sigma}S_{2}(t)dW(t) $$ the values of the European-style options for both models are the same?

My attempt is as follows: For European-style option models $S_{1}(t)$ and $S_{2}(t)$ at the maturity time $T$, the same price must be given. Therefore we can equate the model prices and integrate from time $0$ to $T$, which results in $$ rS(t)+\sigma(t)S(t)dW(t)=rS(t)+\hat{\sigma}S(t)dW(t) $$ or $$ \sigma(t)dW(t)=\hat{\sigma}dW(t) $$ Now, integrating from $0$ to $T$ gives for constant $\hat{\sigma}$ $$ \hat{\sigma}=\frac{\int^{T}_{0}\sigma(t)dW(t)}{\int^{T}_{0}dW(t)} $$

First of all, I do not know if this is the right way of solving the question, since according to the $It\bar{o}$ integral both the expectations of the numerator and denominator are equal to $0$, therefore this solution seems trivial or nonsense to me. Also, in this framework is this even a constant, since integrating w.r.t. Brownian motion does not produce a constant, it seems to me.

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    $\begingroup$ You’re right, what you wrote down is a random variable. The answer (which is analogous to how variance is additive when we sum up independent normals) is that $$\hat\sigma^2 = \frac{1}{T}\int_0^T \sigma^2(t)dt$$ $\endgroup$ – spaceisdarkgreen Mar 12 at 1:12
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My suggestion: Let's suppose that we want to price an European option. Let be $C^1$ the price of a vanilla option where the underlying asset is $S^1$. Then we will look for the implied volatility $\hat{\sigma}$ such that when we plug this parameter into the Black formula we recover $C^1$. This means: \begin{align*} C_{BS}(\hat{\sigma}) = C^1 \end{align*} where $C_{BS}(\sigma) = SN(d_1(\sigma)) - K\exp^{-r(T-t)}N(d_2(\sigma))$.

We can go even further by assuming the implied volatility depends only on the maturity of the option. In that case solving the Dupire formula in terms of implied volatility, we can back out the following: \begin{align*} \hat{\sigma}^2(T) = \frac{1}{T}\int_0^T\sigma^2(s)ds \end{align*}

Overall, the implied volatility is equal to the quadratic mean of the volality over the option life time.

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