Options 4: Hedging Options

This notebook illustrates how a position in the underlying asset can be used to hedge an option (delta hedging).

Load Packages and Extra Functions



In [1]:

    
using Dates, Distributions

include("jlFiles/printmat.jl")
include("jlFiles/printTable.jl")









    Out[1]:





printTable2



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using Plots

#pyplot(size=(600,400))
gr(size=(480,320))
default(fmt = :svg)

A First-Order Approximation of the Option Price Change

"Delta hedging" is based on the idea that we can approximate the change in the option price by

$C_{t+h}-C_{t}\approx \Delta_t \left( S_{t+h}-S_{t}\right)$,

where $\Delta_t$ is the derivative of the call option price wrt. the underlying asset price.

In the Black-Scholes model, the Delta of a call option is

$\Delta=\frac{\partial C}{\partial S}=e^{-\delta m}\Phi\left( d_{1}\right),$

where $d_1$ is the usual term in Black-Scholes.

Similarly, the Delta of a put option is

$\frac{\partial P}{\partial S}=e^{-\delta m}[\Phi\left( d_{1}\right)-1]$.

The file included in the next cell contains the functions Φ() and OptionBlackSPs() from the chapter on the Black-Scholes model.

The subsequent cell defines a function for the $\Delta$ of the Black-Scholes model.



In [3]:

    
include("jlFiles/OptionsCalculations.jl")









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OptionBlackSPs



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"""
Calculate the Black-Scholes delta
"""
function OptionDelta(S,K,m,y,σ,δ=0,PutIt=false)
    d1 = ( log(S/K) + (y-δ+0.5*σ^2)*m ) / (σ*sqrt(m))
    d2 = d1 - σ*sqrt(m)
    if PutIt
        Δ = exp(-δ*m)*(Φ(d1)-1)
    else
        Δ = exp(-δ*m)*Φ(d1)
    end
    return Δ
end









    Out[4]:





OptionDelta



In [5]:

    
(S,K,m,y,σ) = (42,42,0.5,0.05,0.2)

Δc = OptionDelta(S,K,m,y,σ)               #call
Δp = OptionDelta(S,K,m,y,σ,0,true)        #put

printlnPs("Δ:")
printTable([Δc Δp (Δc-Δp)],["call","put","difference"],[" "],width=12)









    



        Δ:
         call         put  difference
        0.598      -0.402       1.000



In [6]:

    
S_range    = 30:60          #different spot prices
Δc_S_range = OptionDelta.(S_range,K,m,y,σ)

plot( S_range,Δc_S_range,
      linecolor = :red,
      legend = false,
      title = "Black-Scholes call option delta",
      xlabel = "current asset price",
      ylabel = "option price" )









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Hedging an Option

The example below shows how a delta hedging works for a European call option when the price of the underlying changes (from 42 on day 0 to 43 on day 1). For simplicity, we assume that the Black-Scholes model is a good description of how the option price is set.



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(S₀,S₁,K,m,y,σ) = (42,43,42,0.5,0.05,0.2)

C₀ = OptionBlackSPs(S₀,K,m,y,σ)   #option price at S₀
Δ₀ = OptionDelta(S₀,K,m,y,σ)      #Delta at S₀
M₀ = C₀ - Δ₀*S₀                    #on money market account

C₁ = OptionBlackSPs(S₁,K,m-1/252,y,σ)   #option price at S₁ (it's one day later) 
dC = C₁ - C₀                      #change of option value
dV = Δ₀*(S₁-S₀) - (C₁-C₀)           #change of hedge portfolio value

xy = [S₀,Δ₀,C₀,M₀,S₁,C₁,dC,dV]
printTable(xy,[" "],["S₀","Δ₀","C₀","M₀","S₁","C₁","dC","dV"])

printblue("\nV changes much less in value than the option (abs(dV) < abs(dC)): 
the hedge helps")









    



            
S₀    42.000
Δ₀     0.598
C₀     2.893
M₀   -22.212
S₁    43.000
C₁     3.509
dC     0.616
dV    -0.018


V changes much less in value than the option (abs(dV) < abs(dC)): 
the hedge helps



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