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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.

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    $\begingroup$ This question remains closed until the monetary reward is removed and the title changed. $\endgroup$
    – Attack68
    Nov 5, 2023 at 14:30
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    $\begingroup$ It is a question on an interesting and important topic, so I hope it can be reopened. $\endgroup$
    – nbbo2
    Nov 5, 2023 at 19:16
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    $\begingroup$ Do you know how to compute sticky delta? (Sticky strike is really just black Scholes delta). If you bump up and down, you adjust IV (because it's stuck to delta /moneyness). Once you get that, it's really just running examples with different vol surface shapes to see the impact. $\endgroup$
    – AKdemy
    Nov 6, 2023 at 13:37

3 Answers 3

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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 $$\frac{d C}{d S}=\frac{\partial BS}{\partial S} + \frac{\partial BS}{\partial \sigma}\times\frac{\frac{\partial^2 BS}{\partial S^2}\frac{\partial \sigma}{\partial \delta}}{1-\frac{\partial^2 BS}{\partial S \partial \sigma} \frac{\partial \sigma}{\partial \delta}}$$

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  • $\begingroup$ Where does the denominator $1-...$ come from when determining $\partial \sigma / \partial S$? $\endgroup$
    – Attack68
    Nov 8, 2023 at 20:25
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    $\begingroup$ Added the proof to the answer - hope that helps explain $\endgroup$
    – river_rat
    Nov 9, 2023 at 7:39
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    $\begingroup$ Thanks, taking the total derivative of $\partial \delta / \partial S$ instead of partial. Helpful. $\endgroup$
    – Attack68
    Nov 9, 2023 at 7:58
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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.

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  • $\begingroup$ if skew is downward sloping in a sticky delta regime (say 10v at ATM strike at \$100 and 9v at \$105), and the underlying moved up (say to \$105 which now has 10v as it is ATM), why would this result in a vega loss (in a sticky delta regime)? $\endgroup$
    – bng
    Dec 23, 2023 at 3:08
  • $\begingroup$ Rereading this response, I think this response may be ambiguously worded. Originally I thought this response meant to say "[in a sticky delta regime] you would make money but not as much as you'd think", but now I think it meant to say "you would make money [in a sticky strike regime] but not as much as you'd [would] in a sticky delta regime" $\endgroup$
    – bng
    Dec 26, 2023 at 16:51
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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:

  • Sticky strike is really just black Scholes delta computed with Finite Difference.
  • Sticky delta refers to adjusting IV when you bump up and down (because IV is stuck to delta /moneyness).

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:

  • a downward sloping IV (around P2) -> Sticky Delta will be lower
  • a flat IV (around P5) -> Sticky Delta will be identical
  • an upward sloping IV (around P8) -> Sticky Delta will be higher

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.

enter image description here

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:

enter image description here

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)))

enter image description here

As suggested at the begining:

  • Sticky Delta will be lower if the IV surface slopes downward
  • Sticky Delta will be identical if the IV surface is flat (around spot)
  • Sticky Delta will be higher if the IV surface slopes downward

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.

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