Monte Carlo Simulation European Option Price

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Monte Carlo SimulationEuropean Option Price

• Jan Röman
• Group memberShanwei Huang
• An Gong
• Mesut Bora Sezen

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MMA 707 Analytical Finance I
1.Introduction 2.Simulation 3.Summary
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MMA 707 Analytical Finance I

• History of Monte Carlo Method
• Use and Main Fields of Monte Carlo
• Monte Carlo in Financial Engineering

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MMA 707 Analytical Finance I
'returns an array of n normally distributed
variables 'using box muller transformation Public
Function NRandVars(N As Single) As Variant
ReDim randArr(1 To N) As Variant Dim i, n2,
counter As Single n2 Application.Floor(N / 2,
1) Dim v1, v2, tmp, fac As Double counter
0 For i 1 To n2 Do v1 2 Rnd -
1 v2 2 Rnd - 1 tmp v1 v1 v2
v2 Loop Until tmp lt 1 fac Sqr(-2
Log(tmp) / tmp) counter counter 1
randArr(counter) v1 fac counter counter
1 randArr(counter) v2 fac Next i If
(N gt (n2 2)) Then Do v1 2 Rnd -
1 v2 2 Rnd - 1 tmp v1 v1 v2
v2 Loop Until tmp lt 1 fac Sqr(-2
Log(tmp) / tmp) counter counter 1
randArr(counter) v2 fac End If NRandVars
randArr End Function
Modules
'returns mean of an array Public Function
GetMean(x As Variant) As Double Dim i As
Single Dim tmpsum As Double tmpsum 0 Dim
N As Single N UBound(x, 1) - LBound(x, 1)
1 For i LBound(x, 1) To UBound(x, 1)
tmpsum tmpsum x(i, 1) Next i GetMean
tmpsum / N End Function
Public Function GetMaximum(x As Variant) As
Double Dim maxVal As Double maxVal -2 50
For i LBound(x, 1) To UBound(x, 1) If x(i,
1) gt maxVal Then maxVal x(i, 1) End
If Next i GetMaximum maxVal End Function
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MMA 707 Analytical Finance I
Mathematical formulas
We divide the maturity T into several
intervals,and this is the formula to calculate
the stock price .
The payoff function of the call and put option
is given by
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MMA 707 Analytical Finance I
Mathematical formulas
Then,we can extimate the premium of the
european option by using the following function.
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MMA 707 Analytical Finance I
Private Sub CommandButton1_Click() Dim nsim As
Single Application.Range("D20").Value ""
Application.Range("D21").Value ""
Application.Range("D24").Value "" S0
Application.Range("D11").Value 'underlying price
1 k Application.Range("D12").Value 'strike
T Application.Range("D15").Value 'maturity
sigma Application.Range("D13").Value
'volatility r Application.Range("D14").Value
'risk free rate nsteps Application.Range("D17"
).Value 'no of timesteps nsimulations
Application.Range("D18").Value ' no of mc
simulations Randomize dt T / nsteps
vsqrdt sigma dt 0.5 drift (r - sigma
2 / 2) dt ReDim callpayoffvec(1 To
nsimulations, 1 To 1) ReDim putpayoffvec(1 To
nsimulations, 1 To 1) Dim counter As Single
counter 1 Dim procounter As Single
probcounter 0 randvec NRandVars(nsteps
nsimulations) 'get the random number
For i 1 To nsimulations 'get the pay off by
"for" cycle st S0 For j 1 To nsteps
randvar randvec(counter) st st
Exp(drift vsqrdt randvar) Sheet2.Cells(j,
i).Value st counter counter 1 Next
j If st gt k Then probcounter probcounter
1 callpayoffvec(i, 1) Application.Max(st -
k, 0) putpayoffvec(i, 1) Application.Max(k -
st, 0) Next i MC_callprice Exp(-r T)
GetMean(callpayoffvec) 'get the option price
MC_putprice Exp(-r T) GetMean(putpayoffvec)
Application.Range("D20").Value MC_callprice
Application.Range("D21").Value MC_putprice
Application.Range("D24").Value (probcounter /
nsimulations) Dim ch As ChartObject
'plot chart Dim PRange As Range
Worksheets("sheet1").ChartObjects.Delete
Set PRange Application.Range("Sheet2!A1").Resize
(nsteps, nsimulations) Set ch
Worksheets("sheet1").ChartObjects.Add(280, 120,
480, 250) ch.Chart.ChartType xlLine
ch.Chart.HasLegend False ch.Chart.ChartWizar
d SourceWorksheets("sheet2").Range(PRange.Addres
s), _ CategoryTitle"simulation step",
ValueTitle"Avista value" End Sub
Main Program
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MMA 707 Analytical Finance I
Application
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MMA 707 Analytical Finance I

• Stochastic vs Deterministic
• Significance of Simulation
• Disadvantage

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Thank You!