The “Greek” Δ - Derivation & Graphs.

Define BS formulae and associated functions.

d1[S_,K_,r_,delta_,sigma_,T_,t_]:=(Log[S/K]+(r-delta+sigma^2/2)(T-t))/(sigma*Sqrt[T-t]);

d2[S_,K_,r_,delta_,sigma_,T_,t_]:=(Log[S/K]+(r-delta-sigma^2/2)(T-t))/(sigma*Sqrt[T-t]);

CNorm[d_]:=CDF[NormalDistribution[0,1],d];

VCall[S_,K_,r_,delta_,sigma_,T_,t_]:=S*Exp[-delta*(T-t)]*CNorm[d1[S,K,r,delta,sigma,T,t]]-K*Exp[-r*(T-t)]*CNorm[d2[S,K,r,delta,sigma,T,t]];

VPut[S_,K_,r_,delta_,sigma_,T_,t_]:=K*Exp[-r*(T-t)]*CNorm[-d2[S,K,r,delta,sigma,T,t]]-S*Exp[-delta*(T-t)]*CNorm[-d1[S,K,r,delta,sigma,T,t]];

The “Greek” Δ = ∂ V/∂ (Subscript[S, t]):

Compute and simplify the derivative for Δ:

K=.;
D[VCall[S,K,r,delta,sigma,T,t],S]

FullSimplify[D[VPut[S,K,r,delta,sigma,T,t],S]]

FullSimplify[D[VPut[S,K,r,delta,sigma,T,t],S]]

Define Function Representation For ∆:

CDelta[S_,K_,r_,delta_,sigma_,T_,t_]:=(1/2) (E^(delta (t-T))) (1+Erf[(((-delta+r+((sigma^2)/2)) (-t+T)+Log[(S/K)])/((Sqrt[2]) sigma (Sqrt[-t+T])))]);

PDelta[S_,K_,r_,delta_,sigma_,T_,t_]:=-(1/2) (E^(delta (t-T))) Erfc[(((-delta+r+((sigma^2)/2)) (-t+T)+Log[(S/K)])/((Sqrt[2]) sigma (Sqrt[-t+T])))];

Limiting Behavior of Δ:

S → ∞ Limits

Limit[CDelta[S,K,r,delta,sigma,T,t],S->Infinity,Direction->-1,Assumptions->{r>0&&sigma>0&&delta>0&&K>0&&T>t>0}]

Limit[PDelta[S,K,r,delta,sigma,T,t],S->Infinity,Direction->-1,Assumptions->{r>0&&sigma>0&&delta>0&&K>0&&T>t>0}]

S → 0 Limits

Limit[CDelta[S,K,r,delta,sigma,T,t],S->0,Direction->-1,Assumptions->{r>0&&sigma>0&&delta>0&&K>0&&T>t>0}]

Limit[PDelta[S,K,r,delta,sigma,T,t],S->0,Direction->-1,Assumptions->{r>0&&sigma>0&&delta>0&&K>0&&T>t>0}]

Plot (Subscript[Δ, call]) For Various Strike Prices

Plot[{CDelta[S,100,.05,.02,.1,1,0],CDelta[S,90,.05,.02,.1,1,0],CDelta[S,110,.05,.02,.1,1,0],CDelta[S,120,.05,.02,.1,1,0],CDelta[S,80,.05,.02,.1,1,0]},{S,60,130}, AxesLabel->{"(Subscript[S, t])","(Subscript[Δ, call])"},PlotLabel->"(Subscript[Δ, call]) for call w/ various strike prices K when \nr = 5%, δ = 2%, σ = 10%, T=  1, and t = 0", PlotStyle->{{Red,Thick},{Blue,Thick},{Green,Thick},{Green,Thick,Dashed},{Blue,Thick,Dashed}}, PlotLegends->LineLegend[{Directive[Blue,Dashed,Thick],Directive[Blue,Thick],Directive[Red,Thick],Directive[Green,Thick],Directive[Green,Dashed]},{"K=80","K=90","K=100","K=110","K=120"},LegendFunction->"Frame"],GridLines->Automatic]

Plot (Subscript[Δ, put]) For Various Strike Prices

Plot[{PDelta[S,100,.05,.02,.1,1,0],PDelta[S,90,.05,.02,.1,1,0],PDelta[S,110,.05,.02,.1,1,0],PDelta[S,120,.05,.02,.1,1,0],PDelta[S,80,.05,.02,.1,1,0]},{S,60,130}, AxesLabel->{"(Subscript[S, t])","(Subscript[Δ, put])"},PlotLabel->"(Subscript[Δ, put]) for put w/ various strike prices K when \nr = 5%, δ = 2%, σ = 10%, T=  1, and t = 0", PlotStyle->{{Red,Thick},{Blue,Thick},{Green,Thick},{Green,Thick,Dashed},{Blue,Thick,Dashed}}, PlotLegends->LineLegend[{Directive[Blue,Dashed,Thick],Directive[Blue,Thick],Directive[Red,Thick],Directive[Green,Thick],Directive[Green,Dashed]},{"K=80","K=90","K=100","K=110","K=120"},LegendFunction->"Frame"],GridLines->Automatic]

Plot (Subscript[Δ, call]) For Various Expiration Times

Plot[{CDelta[S,100,.05,.02,.1,5,0],CDelta[S,100,.05,.02,.1,2,0],CDelta[S,100,.05,.02,.1,1,0],CDelta[S,100,.05,.02,.1,1/2,0],CDelta[S,100,.05,.02,.1,1/100,0]},{S,60,130}, AxesLabel->{"(Subscript[S, t])","(Subscript[Δ, call])"},PlotLabel->"(Subscript[Δ, call]) for call w/ various times T when \nK=100, r = 5%, δ = 2%, σ = 10%, and t = 0", PlotStyle->{{Red,Thick},{Blue,Thick},{Green,Thick},{Green,Thick,Dashed},{Blue,Thick,Dashed}}, PlotLegends->LineLegend[{Directive[Red,Thick],Directive[Blue,Thick],Directive[Green,Thick],Directive[Green,Thick,Dashed],Directive[Blue,Thick,Dashed]},{"T=5","T=2","T=1","T=1/2","T=1/100"},LegendFunction->"Frame"],GridLines->Automatic]

Plot (Subscript[Δ, put]) For Various Expiration Times

Plot[{PDelta[S,100,.05,.02,.1,5,0],PDelta[S,100,.05,.02,.1,2,0],PDelta[S,100,.05,.02,.1,1,0],PDelta[S,100,.05,.02,.1,1/2,0],PDelta[S,100,.05,.02,.1,1/100,0]},{S,60,130}, AxesLabel->{"(Subscript[S, t])","(Subscript[Δ, put])"},PlotLabel->"(Subscript[Δ, put]) for put w/ various times T when \nK=100, r = 5%, δ = 2%, σ = 10%, and t = 0", PlotStyle->{{Red,Thick},{Blue,Thick},{Green,Thick},{Green,Thick,Dashed},{Blue,Thick,Dashed}}, PlotLegends->LineLegend[{Directive[Red,Thick],Directive[Blue,Thick],Directive[Green,Thick],Directive[Green,Thick,Dashed],Directive[Blue,Thick,Dashed]},{"T=5","T=2","T=1","T=1/2","T=1/100"},LegendFunction->"Frame"],GridLines->Automatic]

Plot (Subscript[Δ, call]) For Various Volatilities

Plot[{CDelta[S,100,.05,.02,.75,1,0],CDelta[S,100,.05,.02,.5,1,0],CDelta[S,100,.05,.02,.1,1,0],CDelta[S,100,.05,.02,.05,1,0],CDelta[S,100,.05,.02,.01,1,0]},{S,60,130}, AxesLabel->{"(Subscript[S, t])","(Subscript[Δ, call])"},PlotLabel->"(Subscript[Δ, call]) for call w/ various volatilities σ when  \nK = 100, r = 5%, δ = 2%, T=  1, and t = 0", PlotStyle->{{Red,Thick},{Blue,Thick},{Green,Thick},{Green,Thick,Dashed},{Blue,Thick,Dashed}}, PlotLegends->LineLegend[{Directive[Red,Thick],Directive[Blue,Thick],Directive[Green,Thick],Directive[Green,Thick,Dashed],Directive[Blue,Thick,Dashed]},{"σ=0.75","σ=0.5","σ=0.1","σ=0.05","σ=0.01"},LegendFunction->"Frame"],GridLines->Automatic]

Plot (Subscript[Δ, put]) For Various Volatilities

Plot[{PDelta[S,100,.05,.02,.75,1,0],PDelta[S,100,.05,.02,.5,1,0],PDelta[S,100,.05,.02,.1,1,0],PDelta[S,100,.05,.02,.05,1,0],PDelta[S,100,.05,.02,.01,1,0]},{S,60,130}, AxesLabel->{"(Subscript[S, t])","(Subscript[Δ, put])"},PlotLabel->"(Subscript[Δ, put]) for put w/ various volatilities σ when  \nK = 100, r = 5%, δ = 2%, T=  1, and t = 0", PlotStyle->{{Red,Thick},{Blue,Thick},{Green,Thick},{Green,Thick,Dashed},{Blue,Thick,Dashed}}, PlotLegends->LineLegend[{Directive[Red,Thick],Directive[Blue,Thick],Directive[Green,Thick],Directive[Green,Thick,Dashed],Directive[Blue,Thick,Dashed]},{"σ=0.75","σ=0.5","σ=0.1","σ=0.05","σ=0.01"},LegendFunction->"Frame"],GridLines->Automatic]

Plot (Subscript[Δ, call]) For Various Dividend Reinvestment Rates δ

Plot[{CDelta[S,100,.05,.001,.25,1,0],CDelta[S,100,.05,.01,.25,1,0],CDelta[S,100,.05,.1,.25,1,0],CDelta[S,100,.05,.2,.25,1,0],CDelta[S,100,.05,.4,.25,1,0]},{S,60,130}, AxesLabel->{"(Subscript[S, t])","(Subscript[Δ, call])"},PlotLabel->"(Subscript[Δ, call]) for call w/ various dividend rates δ when  \nS = 100, r = 5%, σ = 25%, T=  1, and t = 0", PlotStyle->{{Red,Thick},{Blue,Thick},{Green,Thick},{Green,Thick,Dashed},{Blue,Thick,Dashed}}, PlotLegends->LineLegend[{Directive[Red,Thick],Directive[Blue,Thick],Directive[Green,Thick],Directive[Green,Thick,Dashed],Directive[Blue,Thick,Dashed]},{"δ=0.001","δ=0.01","δ=0.1","δ=0.2","δ=0.4"},LegendFunction->"Frame"],GridLines->Automatic]

Plot (Subscript[Δ, put]) For Various Dividend Reinvestment Rates δ

Plot[{PDelta[S,100,.05,.001,.25,1,0],PDelta[S,100,.05,.01,.25,1,0],PDelta[S,100,.05,.1,.25,1,0],PDelta[S,100,.05,.2,.25,1,0],PDelta[S,100,.05,.4,.25,1,0]},{S,60,130}, AxesLabel->{"(Subscript[S, t])","(Subscript[Δ, put])"},PlotLabel->"(Subscript[Δ, put]) for put w/ various dividend rates δ when  \nS = 100, r = 5%, σ = 25%, T=  1, and t = 0", PlotStyle->{{Red,Thick},{Blue,Thick},{Green,Thick},{Green,Thick,Dashed},{Blue,Thick,Dashed}}, PlotLegends->LineLegend[{Directive[Red,Thick],Directive[Blue,Thick],Directive[Green,Thick],Directive[Green,Thick,Dashed],Directive[Blue,Thick,Dashed]},{"δ=0.001","δ=0.01","δ=0.1","δ=0.2","δ=0.4"},LegendFunction->"Frame"],GridLines->Automatic]

Surface Plot of (Subscript[Δ, call]) For Varying Stock and Strike Prices

DeltaSurfacePlot = Plot3D[CDelta[S,K,.05,.02,.75,1,0],{S,50,130},{K,50,120}, AxesLabel->{"(Subscript[S, t])","K", "(Subscript[Δ, call])"},PlotLabel->"(Subscript[Δ, call])(S,K) when r = 5%, δ = 2%, σ = 0.75, T=  1, and t = 0", ColorFunction->Function[{x,y,z},Hue[0.65*(1-z)]]]

Surface Plot of (Subscript[Δ, call]) For Varying Volatility and Strike Prices

Plot3D[CDelta[100,K,.05,.02,sigma,1,0],{sigma,0,1},{K,70,120}, PlotPoints->45,AxesLabel->{"σ","K", "(Subscript[Δ, call])"},PlotLabel->"(Subscript[Δ, call])(σ, K) when S = 100, r = 5%, δ = 2%, T=  1, and t = 0", ColorFunction->Function[{x,y,z},Hue[0.65*(1-z)]]]

Contour Plot of (Subscript[Δ, call]) For Varying Volatility and Strike Prices

Quiet[ContourPlot[CDelta[100,K,.05,.02,sigma,1,0],{sigma,0,1},{K,70,120}, Contours->45,AxesLabel->{"σ","K"},PlotLabel->"(Subscript[Δ, call])(σ, K) when S = 100, r = 5%, δ = 2%, T=  1, and t = 0", PlotLegends->Automatic, ColorFunction->"TemperatureMap"]]

Surface Plot of (Subscript[Δ, call]) For Varying Volatility and Strike Prices - Close Up

Quiet[Plot3D[CDelta[100,K,.05,.02,sigma,1,0],{sigma,0,.01},{K,101,105}, PlotPoints->45,AxesLabel->{"σ","K", "(Subscript[Δ, call])"},PlotLabel->"(Subscript[Δ, call])(σ, K) when S = 100, r = 5%, δ = 2%, T=  1, and t = 0.", ColorFunction->Function[{x,y,z},Hue[0.65*(1-z)]],PlotRange->All]]

Contour Plot of Call Δ For Varying Volatility and Strike Prices - Close Up

Quiet[ContourPlot[CDelta[100,K,.05,.02,sigma,1,0],{sigma,0,.01},{K,101,105}, Contours->45,AxesLabel->{"σ","K"},PlotLabel->"(Subscript[Δ, call])(σ,K) when S = 100, r = 5%, δ = 2%, T=  1, and t = 0", PlotLegends->Automatic, ColorFunction->"TemperatureMap"]]