How are the call and put slopes in the SVI-JW parametrization derived?

In the SVI-JW parametrization, we have

$$w(k; a, b, \rho, m, \sigma) = a + b \left [ \rho(k-m) + \sqrt{(k-m)^{2} + \sigma^{2}} \right ]$$

Which gives us

\begin{align*} \sigma_{BS}(k) &= \frac{1}{\sqrt{t}}\sqrt{a + b \left [ \rho(k-m) + \sqrt{(k-m)^{2} + \sigma^{2}} \right ]} \\ \\ \\ \frac{\partial \sigma_{BS}}{\partial k} &= \frac{b\left [\rho + \frac{(k - m)}{\sqrt{(k-m)^{2} + v^2}}\right ]}{2\sqrt{t}\sqrt{a + b \left [ \rho(k-m) + \sqrt{(k-m)^{2} + \sigma^{2}} \right ]}} \end{align*}

We can evaluate ATM variance $$v_{t}$$ by setting $$k=0$$ in $$w(k; a, b, \rho, m, \sigma)$$, we can evaluate ATM skew $$\psi_{t}$$ by evaluating $$\frac{\partial \sigma_{BS}}{\partial k}|_{k=0}$$ and we can evaluate minimum implied variance $$\tilde{v}_{t}$$ by setting $$\frac{\partial \sigma_{BS}}{\partial k} = 0$$ and plugging $$k$$ into $$w(k)$$.

How can we find the put and call slopes $$p_{t}$$ and $$c_{t}$$? I assumed it would be the limit of the variance $$\frac{w(k)}{t}$$ when $$k \rightarrow \pm \infty$$ but this gives me $$\frac{b}{\sqrt{t}}(\rho \pm 1)$$ which does not match Gatheral's results in his original paper.

Link to original paper Arbitrage-free SVI volatility surfaces by Jim Gatheral, Antoine Jacquier here.

$$(p_t,c_t)$$ are respectively related to the put/call slopes of the total implied variance, not variance $$w(k,t)=\sigma^2(k,t) t$$
Under SVI $$w(k) = a + b \left(\rho(k-m) + \sqrt{(k-m)^2 + \sigma^2} \right)$$ such that $$\frac{\partial w}{\partial k}(k) = b \left( \rho + \frac{k-m}{\sqrt{(k-m)^2+\sigma^2}} \right)$$ and $$\lim_{k \to \pm \infty} \frac{\partial w}{\partial k}(k) = b \left( \rho \pm 1 \right)$$ (see also here end of p.5)
Now, remembering should you define: $$p_t := \frac{1}{\sqrt{w_t}} b (1-\rho)$$ $$c_t := \frac{1}{\sqrt{w_t}} b (1+\rho)$$ with $$w_t$$ the ATMF total implied variance ($$w_t = v_t t$$ in the JW space) then you have that indeed:
• $$p_t$$ is proportional to the opposite of the put slope of total implied variance and is expected to be positive (because $$w_t$$ and $$b$$ are positive)
• $$c_t$$ is proportional to the call slope of total implied variance and is expected to be positive (because $$w_t$$ and $$b$$ are positive)
• However the original paper has definitions for ($c_{t}$, $p_{t}$) of $\frac{b}{\sqrt{w_{t}}} (1 \pm \rho)$ where $w_{t} = w(0, t)$. Where is the $\frac{1}{\sqrt{w_{t}}}$ term coming from? And why is the sign different for $p_{t}$?
• Hi, I've edited my answer to make it clearer. The term $1/\sqrt{w_t}$ is basically a scaling term. And indeed, $p_t \propto -\text{put slope}$ and $c_t \propto +\text{call slope}$. Sep 28, 2020 at 12:29