I have been trying to get my head around these concepts but what I have found online has caused more confusion: specifically why a sticky delta model might lead to a higher delta or no. of contracts than a sticky strike model.
I get they are different models but not sure why sticky delta would be higher. I was looking at the S&P 500 and the delta being higher under sticky delta than sticky strike; I'm not sure why that would be.
By doing a bit of differential chasing you can get a closed form solution for the adapted aka sticky delta of a market call option. Recall that the price C of a market call option is given by the black-scholes equation and so $$C = BS(S, K, \sigma, T)$$ In the sticky-delta setting we know that the volatility $\sigma$ is functionally dependent on the black-sholes delta $$\sigma = f(\delta)$$ and $$\delta = \frac{\partial BS}{\partial S}(S,K,\sigma,T)$$
Taking differentials and solving shows that $$\frac{d C}{d S}=\frac{\partial BS}{\partial S} + \frac{\partial BS}{\partial \sigma}\times\frac{\frac{\partial^2 BS}{\partial S^2}\frac{\partial \sigma}{\partial \delta}}{1-\frac{\partial^2 BS}{\partial S \partial \sigma} \frac{\partial \sigma}{\partial \delta}}$$ So the impact to the normal sticky-strike delta is driven by the sign of the term $\frac{\partial \sigma}{\partial \delta}$ which is typically positive around the at-the-money forward level. Since around there gamma and vega are also positive, provided the steepness of the volatility curve in delta space is not too dramatic we will see that the adapted call delta is greater than the sticky-strike call delta.
(Edit - Adding proof)
$$\frac{dC}{dS} = \frac{\partial BS}{\partial S} + \frac{\partial BS}{\partial \sigma} \frac{\partial \sigma}{\partial \delta}\frac{d\delta}{dS}$$ $$\frac{d\delta}{dS} = \frac{\partial \delta}{\partial S} + \frac{\partial \delta}{\partial \sigma} \frac{\partial \sigma}{\partial \delta}\frac{d\delta}{dS} \implies \frac{d\delta}{dS}=\frac{\frac{\partial \delta}{\partial S}}{1-\frac{\partial \delta}{\partial \sigma}\frac{\partial \sigma}{\partial \delta}}$$ But $$\delta = \frac{\partial BS}{\partial S}(S,K,\sigma,T)$$ so $$\frac{d\delta}{dS}=\frac{\frac{\partial^2 BS}{\partial S^2}}{1-\frac{\partial^2 BS}{\partial S \partial \sigma}\frac{\partial \sigma}{\partial \delta}}$$ and thus $$\frac{d C}{d S}=\frac{\partial BS}{\partial S} + \frac{\partial BS}{\partial \sigma}\times\frac{\frac{\partial^2 BS}{\partial S^2}\frac{\partial \sigma}{\partial \delta}}{1-\frac{\partial^2 BS}{\partial S \partial \sigma} \frac{\partial \sigma}{\partial \delta}}$$
A simple example.
Suppose you have a call and spot rallies ($\Delta S >0$), say your skew is downward sloping, then you would make money but not as much as you'd think in a sticky delta regime.
This is because your positive delta pnl ($\delta_{BS} \Delta S$) is partially offset by the vega loss ($\nu \Delta \sigma_i $) as vol comes off ($\Delta \sigma_i < 0 $) for a fixed strike.
I always find charts intuitive. I'll look at calls, but it is straigthforward to extend it to puts as well.
As mentioned in a comment:
I use Julia to plot and compute the relevant values. The following code is used for plotting a dummy vol surface that has 3 different regions:
The graph on the left hand side (LHS) shows in the dashed green line a flat vol surface where the up and down shifts around spot have the same vol as an input. The orange dots on the RHS correspond to the orange dots on the LHS. The only difference are the two IVs at the very beginning and end, which makes the skew steeper.
using Plots, Distributions, DataFrames, PrettyTables
vola = [0.2474, 0.2471, 0.2470,0.2470,0.2470,0.2470, 0.2470, 0.2471, 0.2473]
vola_flat = [0.2471, 0.2471, 0.2471,0.2470,0.2470, 0.2470, 0.2471, 0.2471, 0.2471]
vola2 = [0.2484, 0.2471, 0.2470,0.2470,0.2470,0.2470,0.2470, 0.2471, 0.2483]
strike = 98.5:0.5:102.5
vol = Dict(zip(strike,vola))
vol_flat = Dict(zip(strike,vola_flat))
vol2 = Dict(zip(strike,vola2))
p1 = plot(strike, vola, ylims=(minimum(vola2)-0.0005,maximum(vola2)+0.0005),xlabel="Strike",
ylabel="IV", label = false, linewidth = 3)
plot!(strike, vola_flat, ylims=(minimum(vola2)-0.0005,maximum(vola2)+0.0005),xlabel="Strike",
ylabel="IV", label = false, linewidth = 2, linestyle = :dashdot, linecolor = :green)
[annotate!([strike[i]], [vola[i]+0.0001], "P$i") for i in 2:3:8]
[plot!([strike[i], strike[i]], [vola[i], 0], linestyle = :dashdot) for i in 2:3:8]
plot!(strike, vola, seriestype=:scatter, label = false, color = :orange)
p2 = plot(strike, vola2, ylims=(minimum(vola2)-0.0005,maximum(vola2)+0.0005),xlabel="Strike",
ylabel="IV", label = false, linewidth = 3)
plot!(strike, vola2, seriestype=:scatter, label = false,color = :green)
plot!(strike, vola, seriestype=:scatter, label = false, color = :orange)
[annotate!([strike[i]], [vola[i]+0.0001], "P$i") for i in 2:3:8]
[plot!([strike[i], strike[i]], [vola[i], 0], linestyle = :dashdot) for i in 2:2:8]
plot(p1,p2, left_margin = 2Plots.mm, legend = false)
plot!(size=(900,500))
Let's start by computing Black Scholes values and Delta first. The function looks like this (the cp_flag is allowing for calls (1) and puts (-1)):
function BSM(S,K,t,rf,d,σ, cp_flag)
d1 = ( log(S/K) + (rf - d + 1/2*σ^2)*t ) / (σ*sqrt(t))
d2 = d1 - σ*sqrt(t)
opt = cp_flag*exp(-d*t)S*N(cp_flag*d1) - exp(-rf*t)*cp_flag*K*N(cp_flag*d2)
return opt, exp(-d*t)*N(cp_flag*d1)
end
We evaluate this for different spot values, that correspond to the 3 different points,
df = DataFrame("Spot" => spot, "Call Constant Vol" => [BSM(spot,k,t,log(1+r*t)/t,log(1+d*t)/t,vol_flat[spot], 1)[1] for spot in spot],
"IV Constant" => vola_flat,
"Call BS Delta" => [round(BSM(spot,k,t,log(1+r*t)/t,log(1+d*t)/t,vol_flat[spot], 1)[2]*100, digits = 3) for spot in spot],
"Call Variable IV I" => [BSM(spot,k,t,log(1+r*t)/t,log(1+d*t)/t,vol[spot], 1)[1] for spot in spot],
"IV I" => vola,
"Call Variable IV II" => [BSM(spot,k,t,log(1+r*t)/t,log(1+d*t)/t,vol2[spot], 1)[1] for spot in spot],
"IV II" => vola2)
hl_1 = Highlighter((data,i,j) -> data[i,1] == 99.0, crayon"bg:dark_gray white bold")
hl_2 = Highlighter((data,i,j) -> data[i,1] == 100.5, crayon"bg:dark_gray white bold")
hl_3 = Highlighter((data,i,j) -> data[i,1] == 102.0, crayon"bg:dark_gray white bold")
PrettyTables.pretty_table(df, border_crayon = Crayons.crayon"blue",
header_crayon = Crayons.crayon"bold green",
formatters = ft_printf("%.2f" ),
highlighters = (hl_value(55.307),hl_value(57.643), hl_value(59.915), hl_1,hl_2, hl_3))
which yields the following table:
The Black lines indicate where we want to compute Delta. The Table already shows the corresponding Black Scholes Delta in orange color. It also computed all corresponding BSM values for the various spot values, either with the a flat vol (the shifted spots will use identical IVs) or actual vols according to the change in moneyness / delta the corresponds to the shift.
What is left now is to show what sticke strike and sticky delta refers to. As already mentioned, sticky strike just corresponds to BSM Delta. Finite Difference means that you bump up and down to get delta (see the link above for more details). In our case this looks like this:
df_delta = DataFrame( "Spot" => ["$(spot[i+1])" for i in 1:2:5],
"Sticky Strike" => [round((df[!, "Call Constant Vol"][i+2] - df[!, "Call Constant Vol"][i])/0.01, digits = 2) for i in 1:3:8],
"Sticky Delta I" => [round((df[!, "Call Variable IV I"][i+2] - df[!, "Call Variable IV I"][i])/0.01, digits = 2) for i in 1:3:8],
"Sticky Delta II" => [round((df[!, "Call Variable IV II"][i+2] - df[!, "Call Variable IV II"][i])/0.01, digits = 2) for i in 1:3:8])
PrettyTables.pretty_table(df_delta, border_crayon = Crayons.crayon"blue",
header_crayon = Crayons.crayon"bold green",
formatters = ft_printf("%.2f" ),
highlighters = (hl_value(55.31),hl_value(57.64), hl_value(59.91)))
As suggested at the begining:
The steeper, the more pronounced this effect. You can quickly check the flat surface case if you have Bloomberg. Load OVML (for FX) and compare Delta with sticky delta (both are displayed). Override IV manually and you will see both become identical.
Since the tables display the BSM values as well, it is also easy to find an explanation. Shifting spot down makes a call option worth more. Increasing IV as well. If the surface slopes downward, you have higher BSM values if you shift down, and lower if you shift up. Since the difference is what matters, you get smaller delta values. The opposite holds for upward sloping surfaces.
