Deriving SVI Put and Call Wing Slopes from Total Variance
Summary
The document clarifies how put and call slopes are defined in the SVI-JW volatility parametrization. The key distinction is that the wing slopes refer to total implied variance, rather than implied volatility or variance divided by time. Differentiating the SVI total-variance function with respect to log-moneyness gives limiting slopes in the far put and call wings, expressed using the slope parameter and correlation parameter.
The response then relates these limits to the JW quantities by scaling with the square root of at-the-money total implied variance. It gives the put measure as proportional to the negative of the put-wing slope and the call measure as proportional to the call-wing slope; both are expected to be positive under positive total variance and positive SVI slope. The discussion resolves a definitional mismatch in the question, but does not address calibration, arbitrage-free parameter constraints, or empirical fit of a volatility surface.
Key ideas
- SVI wing slopes are limits of the derivative of total implied variance with respect to log-moneyness.
- The far put and call slopes depend on the SVI slope and correlation parameters.
- The JW put and call quantities scale the wing slopes by inverse square root at-the-money total variance.
- The put quantity uses the opposite sign of the put-wing slope, while the call quantity follows its slope.
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Full text
# How are the call and put slopes in the SVI-JW parametrization derived?
# 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.
## Answer by Quantuple (score 3)
https://quant.stackexchange.com/a/58303
$(p_t,c_t)$ 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)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.