Return to Answer

In the Black-Scholes-Merton model, with model option price $V$ as a function of underlying price $S_t$, strike price $X$, continuously compounded risk-free rate $r$, continuously compounded dividend yield $y$, time-to-maturity (in year fractions) $\tau$ and implied volatility $\sigma$, our $\Delta$ is defined as

$$ \Delta\equiv \frac{\partial V}{\partial S_t}=e^{-y\tau}\mathrm{N}\left(d_1\right) $$ with $$ d_1\equiv \frac{\ln S- \ln X +(r-y+\frac{1}{2}\sigma^2)\tau }{\sigma \sqrt{\tau}} $$

Let $B\equiv Xe^{-r\tau}$ the discounted strike and $\tilde{S}\equiv Se^{-y\tau}$ the 'yield-discounted' spot price, then

$$ \begin{align} \frac{\partial \Delta}{\partial \sigma}&=e^{-y\tau}\mathrm{n}\left(d_1\right)\left(\frac{\partial d_1}{\partial \sigma}\right)\\ &=e^{-y\tau}\mathrm{n}\left(d_1\right)\left(\frac{1}{2}\sqrt{\tau}-\frac{\ln \tilde{S} - \ln B }{\sigma^2\sqrt{\tau}}\right) \end{align} $$

As $\mathrm{n}\left(d_1\right)> 0$ whenever $\sigma\sqrt{\tau}>0$, the sign of the change in $\Delta$ as a function of $\sigma$ depends on whether

$$ \frac{1}{2}\sigma^2\tau \lessgtr\ln \tilde{S} - \ln B $$ i.e. whehther the (logarithmic) moneyness is within 1/2 of the term variance. HTH?

In the Black-Scholes-Merton model, with model option price $V$ as a function of underlying price $S_t$, strike price $X$, continuously compounded risk-free rate $r$, continuously compounded dividend yield $y$, time-to-maturity (in year fractions) $\tau$ and implied volatility $\sigma$, our $\Delta$ is defined as

$$ \Delta\equiv \frac{\partial V}{\partial S_t}=e^{-y\tau}\mathrm{N}\left(d_1\right) $$ with $$ d_1\equiv \frac{\ln S- \ln X +(r-y+\frac{1}{2}\sigma^2)\tau }{\sigma \sqrt{\tau}} $$

Let $B\equiv Xe^{-r\tau}$ the discounted strike and $\tilde{S}\equiv Se^{-y\tau}$ the 'yield-discounted' spot price, then

$$ \begin{align} \frac{\partial \Delta}{\partial \sigma}&=e^{-y\tau}\mathrm{n}\left(d_1\right)\left(\frac{\partial d_1}{\partial \sigma}\right)\\ &=e^{-y\tau}\mathrm{n}\left(d_1\right)\left(\frac{1}{2}\sqrt{\tau}-\frac{\ln \tilde{S} - \ln B }{\sigma^2\sqrt{\tau}}\right) \end{align} $$

As $\mathrm{n}\left(d_1\right)> 0$ whenever $\sigma\sqrt{\tau}>0$, the sign of the change in $\Delta$ as a function of $\sigma$ depends on whether

$$ \frac{1}{2}\sigma^2\tau \lessgtr\ln \tilde{S} - \ln B $$ i.e. whether the (logarithmic) moneyness is within 1/2 of the term variance. HTH?

In the Black-Scholes-Merton model, with model option price $V$ as a function of underlying price $S_t$, strike price $X$, continuously compounded risk-free rate $r$, continuously compounded dividend yield $y$, time-to-maturity (in year fractions) $\tau$ and implied volatility $\sigma$, our $\Delta$ is defined as

$$ \Delta\equiv \frac{\partial V}{\partial S_t}=e^{-y\tau}\mathrm{N}\left(d_1\right) $$ with $$ d_1\equiv \frac{\ln S- \ln X +(r-y+\frac{1}{2}\sigma^2)\tau }{\sigma \sqrt{\tau}} $$

Let $B\equiv Xe^{-r\tau}$ the discounted strike and $\tilde{S}\equiv Se^{-y\tau}$ the 'yield-discounted' spot price, then

$$ \begin{align} \frac{\partial \Delta}{\partial \sigma}&=e^{-y\tau}\mathrm{n}\left(d_1\right)\left(\frac{\partial d_1}{\partial \sigma}\right)\\ &=e^{-y\tau}\mathrm{n}\left(d_1\right)\left(\frac{1}{2}\sqrt{\tau}-\frac{\ln \tilde{S} - \ln B }{\sigma^2\sqrt{\tau}}\right) \end{align} $$

Clearly, this expression can be positive / negative depending on whether

$$ \frac{1}{2}\sigma^2\tau \lessgtr\ln \tilde{S} - \ln B $$ i.e. whehther the (logarithmic) moneyness is within 1/2 of the term variance. HTH?

In the Black-Scholes-Merton model, with model option price $V$ as a function of underlying price $S_t$, strike price $X$, continuously compounded risk-free rate $r$, continuously compounded dividend yield $y$, time-to-maturity (in year fractions) $\tau$ and implied volatility $\sigma$, our $\Delta$ is defined as

$$ \Delta\equiv \frac{\partial V}{\partial S_t}=e^{-y\tau}\mathrm{N}\left(d_1\right) $$ with $$ d_1\equiv \frac{\ln S- \ln X +(r-y+\frac{1}{2}\sigma^2)\tau }{\sigma \sqrt{\tau}} $$

Let $B\equiv Xe^{-r\tau}$ the discounted strike and $\tilde{S}\equiv Se^{-y\tau}$ the 'yield-discounted' spot price, then

$$ \begin{align} \frac{\partial \Delta}{\partial \sigma}&=e^{-y\tau}\mathrm{n}\left(d_1\right)\left(\frac{\partial d_1}{\partial \sigma}\right)\\ &=e^{-y\tau}\mathrm{n}\left(d_1\right)\left(\frac{1}{2}\sqrt{\tau}-\frac{\ln \tilde{S} - \ln B }{\sigma^2\sqrt{\tau}}\right) \end{align} $$

As $\mathrm{n}\left(d_1\right)> 0$ whenever $\sigma\sqrt{\tau}>0$, the sign of the change in $\Delta$ as a function of $\sigma$ depends on whether

$$ \frac{1}{2}\sigma^2\tau \lessgtr\ln \tilde{S} - \ln B $$ i.e. whehther the (logarithmic) moneyness is within 1/2 of the term variance. HTH?

In the Black-Scholes-Merton model, with model option price $V$ as a function of underlying price $S_t$, strike price $X$, risk-free rate of return $r$, time-to-maturity (in year fractions) $\tau$ and implied volatility $\sigma$, our $\Delta$ is defined as

$$ \Delta\equiv \frac{\partial V}{\partial S_t}=\mathrm{N}\left(d_1\right) $$ with $$ d_1\equiv \frac{\ln S- \ln X +(r+\frac{1}{2}\sigma^2)\tau }{\sigma \sqrt{\tau}} $$

Let $B\equiv Xe^{-r\tau}$ the discounted strike, then

$$ \begin{align} \frac{\partial \Delta}{\partial \sigma}&=\mathrm{n}\left(d_1\right)\left(\frac{\partial d_1}{\partial \sigma}\right)\\ &=\mathrm{n}\left(d_1\right)\left(\frac{1}{2}\sqrt{\tau}-\frac{\ln S - \ln B }{\sigma^2\sqrt{\tau}}\right) \end{align} $$

Clearly, this expression can be positive / negative depending on whether

$$ \frac{1}{2}\sigma^2\tau \lessgtr\ln S - \ln B $$ i.e. whehther the (logarithmic) moneyness is within 1/2 of the term variance. HTH?

In the Black-Scholes-Merton model, with model option price $V$ as a function of underlying price $S_t$, strike price $X$, continuously compounded risk-free rate $r$, continuously compounded dividend yield $y$, time-to-maturity (in year fractions) $\tau$ and implied volatility $\sigma$, our $\Delta$ is defined as

$$ \Delta\equiv \frac{\partial V}{\partial S_t}=e^{-y\tau}\mathrm{N}\left(d_1\right) $$ with $$ d_1\equiv \frac{\ln S- \ln X +(r-y+\frac{1}{2}\sigma^2)\tau }{\sigma \sqrt{\tau}} $$

Let $B\equiv Xe^{-r\tau}$ the discounted strike and $\tilde{S}\equiv Se^{-y\tau}$ the 'yield-discounted' spot price, then

$$ \begin{align} \frac{\partial \Delta}{\partial \sigma}&=e^{-y\tau}\mathrm{n}\left(d_1\right)\left(\frac{\partial d_1}{\partial \sigma}\right)\\ &=e^{-y\tau}\mathrm{n}\left(d_1\right)\left(\frac{1}{2}\sqrt{\tau}-\frac{\ln \tilde{S} - \ln B }{\sigma^2\sqrt{\tau}}\right) \end{align} $$

Clearly, this expression can be positive / negative depending on whether

$$ \frac{1}{2}\sigma^2\tau \lessgtr\ln \tilde{S} - \ln B $$ i.e. whehther the (logarithmic) moneyness is within 1/2 of the term variance. HTH?