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I've plotted the charts of Volga of Vanilla Call/Put using finite difference method, and found they are the same, and an asymmetrical shape of observed for both. Any intuitive way to explain the behaviour, or why ITM call/OTM put has higher Volga?

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  • $\begingroup$ These charts look correct. ericbenhamou.net/documents/Encyclo/volga.pdf The Vega of a Put and a Call are the same (due to put call parity), so the volga (the derivative of vega with respect to $\sigma$) of Put and Call are also the same. Also, it is true that it goes to zero when $S=K$. But your main question, about the asymmetric peaks, I don't know the answer. $\endgroup$
    – Alex C
    Aug 29 '19 at 0:06
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The shape you observe is really only due to spot being higher for ITM calls & OTM puts.

The plots are definitely correct. You can quickly check by using standard closed form volga, which is $$vega*\frac{d1*d2}{\sigma}$$.

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Changing risk free rate and dividends just shifts the whole graph slightly left or right respectively.

Volga is $$vega*\frac{d1*d2}{\sigma}$$ hence $$ S * e^{-d*t}*n(d1) * \sqrt \tau*\frac{d1*d2}{\sigma}$$ with $d_1= \frac{\ln S- \ln K +(r-d+\frac{1}{2}\sigma^2)\tau }{\sigma \sqrt{\tau}}$ and $d_2= d_1 - \sigma \sqrt{\tau}$.

As $n(d1)>0$ whenever $\sigma \sqrt{\tau}>0$, it can only be explained with two things, spot $S>0$, or the term $\frac{d1*d2}{\sigma}$. The latter is a bit tricky, and could be similar to what happens with delta at a first glance. The $d1*d2$ part may look daunting, but PCs allow you to deal with this quickly. The below uses Julia.

function d1d2Vol(S,K,t,rf,d,σ) d1 = ( log(S/K) + (rf - d + 1/2*σ^2)*t ) / (σ*sqrt(t))
  d2 = d1 - σ*sqrt(t)
  vega_c = S * exp(-d*t)*n(d1) * sqrt(t)*0.01
  volga = vega_c*((d1*d2)/σ)
  return d1*d2/σ, vega_c, volga
end

spotRange = 10:10:300
K         = 150
rf        = 0 
d         = 0
σ         = 0.2
t         =1
df = DataFrame(d1d2Vol.(spotRange,K,t,rf,d,σ))
rename!(df,[:d1d2v,:vega,:volga] )

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This looks like $(d1*d2)/σ$ actually has the opposite effect of what is observed. Plotting it shows this indeed.

plot(spotRange,[df.d1d2v], label= L"\frac{d1*d2}{\sigma}",size=(800,650), legendfontsize=20, xlabel = "Spot", ylabel= L"\frac{d1*d2}{\sigma}")

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So the only thing left to explain the difference (that also must offset the opposite effect of $(d1*d2)/σ$) is the spot price. If you take a closer look at Black Scholes, this also makes intuitive sense. The standard formula as shown here is providing a result in currency.

function BSM(S,K,t,rf,d,σ)
  d1 = ( log(S/K) + (rf - d + 1/2*σ^2)*t ) / (σ*sqrt(t))
  d2 = d1 - σ*sqrt(t)
  c  = exp(-d*t)S*N(d1) - exp(-rf*t)*K*N(d2)
  vega = S * exp(-d*t)*n(d1) * sqrt(t)*0.01 
  volga = vega*((d1*d2)/σ)
  return c, vega, volga
end

Now, pricing two identical options with $S=90$, $K=100$, $t=1$ year, zero rates and dividends and the only difference being vol of $\sigma=0.2$ and $\sigma=0.21$ you can see how adding vega is close to the new value of the option after increasing vol.

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However, that is dependent on the actual value of spot (and strike). Assume you look at Berkshire which is currently around USD 421,350.

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Vega still works, but the value is vastly different to before. "Issue" here is that the BS price is in cash, not percent of underlying. If you transform it into a comparable percentage price, you see that both options ae priced the same way (which they should as it is the same inputs and same moneyness).

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You can easily transform one volga into the other by dividing by spot of the volga, and multiple by other spot value.

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A final sanity check is to exclude S in the volga formula and plot the same value again. Indeed, as the size of $(d1*d2)/σ$ suggests, the highest point is now actually on the left hand side.

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