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Imagine there exists a typical negative skew for some underlying I want to price a variance swap on. Critically, let’s say we are in a sticky strike world (the vols of each strike will not change with a move in spot).

Let’s say S=100 today and we price up a variance swap using the 1/K^2 replication formula and we get a fair strike of 20% (in vol terms), for example.

Tomorrow, S=102 and ATM vol has decreased because we have rode the skew curve down, but each strike has the same vol as yesterday. Now I need to remark the fair strike of the variance swap. Will it be the same as yesterday?

Initially, I thought yes if we strictly use the 1/K^2 methodology, but none of the individual strikes have changed, then the fair strike should be no different. However, intuitively, it seems strange that even though ATM vols remark lower, the fair strike wouldn’t change either. So I am not sure what is actually correct. Any help would be great - from an intern trying to learn :)

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    $\begingroup$ Sure, the vols stick to the strikes, but since the spot has moved the option prices will all change and so, unless by a miraculous conincidence, the varstrike will change too. $\endgroup$
    – Frido
    Commented Jun 18, 2023 at 19:30

2 Answers 2

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The fair strike computed via replication equals the integral of weighted prices of out-of-the-money options over all strikes. As you wrote correctly, these weights are being inversely proportional to squared strikes, an application of the Black Scholes closed-form formula for gamma, which ensures results in constant dollar gamma.

In your example, the OTM options changed with the change in ATMS. Since your prices for OTM options (and ATM) are different now for every weight, you should end up with a different variance swap strike.

In what follows, I'll rely heavily on the excellent answers given to this question. The par rate (or variance swap strike) is defined as (copied from @Quantuple in the linked question):

$$ K_{var} = \frac{2}{B(0,T)T} \left[ \int_0^{F(0,T)} \frac{P(K,T)}{K^2} dK + \int_{F(0,T)}^\infty \frac{C(K,T)}{K^2} dK \right] $$ where $T$ is the contract's maturity date, $B(0,T)$ the discount factor, $P(K,T)$ and $C(K,T)$ European option prices with strike $K$ and maturity $T$ and $F(0,T)$ the forward price. There are two documents from JP Morgan Variance Swaps and Just what you need to know about Variance Swaps that contain the formula and details, with the latter being more concise. Personally, I recommend reading Towards a Theory of Volatility Trading by Peter Carr et al.

I'll express the variance strike as $\sqrt{K_{var}}\times 100$. Moreover, I assume dividends and interest rates are zero to skip discounting and to simplify the formula somewhat because ATMS = ATMF. The Julia equivalent code of @will's answer, made interactive and suitable for your question, looks like this:

 # load packages 
using Distributions, Plots ,DataFrames, Interact,Plots, PlotThemes, QuadGK 
N(x) = cdf(Normal(0,1),x)

# generic put call pricer (cp = call_put_flag, 1 = call -1 = put)
function BSM(S,K,t,rf,d,σ, cp)
    d1 = ( log(S/K) + (rf - d + 1/2*σ^2)*t ) / (σ*sqrt(t))
    d2 = d1 - σ*sqrt(t)
    opt  = cp*exp(-d*t)S*N(cp*d1) - cp*exp(-rf*t)*K*N(cp*d2)
  return opt
end

# compute vol surface 
function vols(k, vol_atm, convexity, skew, s)
    v = 0.5*convexity*k^2 + (skew - convexity*s)*k + vol_atm + 0.5*convexity*s^2 - skew*s
    return v
end

# create interactive chart 
@manipulate for ATM=0.0:0.01:0.5, Skew=-0.001:0.0001:0.001, Convexity=-0.0001:0.00005:0.0005, s1 = 10:1:90, s2 = 12:1:92  ;
    atm_vol,convexity, skew, ks, T, r,d = ATM, Convexity, Skew, 10:1;80, 1, 0,0
    IVs = [vols(k, atm_vol, convexity, skew, S1) for k in ks]
    inputs = ks .=> IVs 
    res1 = [BSM.(s1,k,T,r,d,vols(k, atm_vol, convexity, skew, s1), k < s1 ? -1 : 1).* k^-2 for k in ks]
    res2 = [BSM.(s2,k,T,r,d,vols(k, atm_vol, convexity, skew, s1), k < s2 ? -1 : 1).* k^-2  for k in ks]
    
    # compute fair VS strike as the integral of OTM calls and put options 
    vs_strike1 = round(100*sqrt(2*(quadgk(k -> BSM.(s1,k,T,r,d,vols(k, atm_vol, convexity, skew, s1), -1)* k^-2, 0, s1)[1] + quadgk(k -> BSM.(s1,k,T,r,d,vols(k, atm_vol, convexity, skew, s1), 1)* k^-2, s1,150)[1])) , digits = dig)
    vs_strike2 = round(100*sqrt(2*(quadgk(k -> BSM.(s2,k,T,r,d,vols(k, atm_vol, convexity, skew, s1), -1)* k^-2, 0, s2)[1] + quadgk(k -> BSM.(s2,k,T,r,d,vols(k, atm_vol, convexity, skew, s1), 1)* k^-2, s2,150)[1])) , digits = dig)
    p2 = plot([k for (k,v) in inputs], [v for (k,v) in inputs], label = "Vol Surface", xlabel = "Strikes", ylabel = "IV")
    vline!([s1], label = "ATMS = $(s1) with IV = $(round(vols(s1, atm_vol, convexity, skew, s1), digits =4))")
    vline!([s2], legendposition = :topright, label = "ATMS = $(s2) with IV = $(round(vols(s2, atm_vol, convexity, skew, s1), digits = 4))")
    p1 = plot(ks, res1, size=(950,700), left_margin=3Plots.mm, label =  "ATM Spot = $(s1) => Fair VS Strike = $(vs_strike1)", title = "Weighted (1/K^2) OTM Put and Call Options")
    plot!(ks, res2, label = "ATM Spot = $(s2) => Fair VS Strike = $(vs_strike2)", xlabel = "Strikes")
    vline!([s1], label = "ATMS = $(s1) with IV = $(round(vols(s1, atm_vol, convexity, skew, s1), digits =4))")
    vline!([s2], legendposition = :bottomleft, label = "ATMS = $(s2) with IV = $(round(vols(s2, atm_vol, convexity, skew, s1), digits = 4))")
    plot(p1,p2, layout = (2,1) )
    end

Explanation:

  • BSM is standard Black Scholes Merton, with the last input defining call (1) and put (-1)
  • The code uses a simple functional form to create a vol surface that exhibits skew and curvature
  • s1 refers to the original ATMS and ATM vol as defined by the surface logik refers to this spot value
  • The area under the weigthed prices is the fair VS strike, which is computed using package called QuadGK, which applies a numerical integration scheme using an adaptive Gauss-Kronrod integration technique: the integral in each interval is estimated using a Kronrod rule (2*order+1 points) and the error is estimated using an embedded Gauss rule (order points). The interval with the largest error is then subdivided into two intervals and the process is repeated until the desired error tolerance is achieved.

Result:

Keeping the vol surface, while moving along the skew will lead to different OTM option prices and different VS strikes as can be seen in the GIF below.

enter image description here

Because the quality of the GIF is not great (size limit of imgur), I also include a screenshot:

enter image description here

You can find a more detailed chart that highlights the area under the OTM options here. The chart can also be used to show that skew is less important compared to convexity because the fair VS strike becomes larger relative to the ATM level.

Side remark, due to practical difficulties in replicating the actual log payout across strikes, the market for equity index varswaps usually trades at a basis to the replicating portfolio.

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Variance Swap fair strike through spot moves

Hi Saul5813,

I've put together the variance swap fair strike as spot moves, under a linear skew model where v(k) = 0.3 - 0.1*(k/s0 - 1). For the purpose of looking at sticky strike, I fixed the vols beforehand with s0 = 100.

I've annotated the Derman approximation where he notes the variance swap strike as being ~= σ_atmf + sqrt(1 + 3T * skew^2). Where Derman has skew as the slope between the 90% and 100% strikes.

Locally you can see it holds up nicely, and I suspect as T -> 0, the approximation will work better.

I also marked on the ATMF forward vol with s0 = 100 of 30%. You can see that the swap strike moves to the original ATMF strike when s ~= $105.

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