American vs European Options

American and European option valuation: a whirlwind tour with Leisen Reimer

Below, we observe the difference between American and European Options making reference to Whaley (1986) who considers options on futures valuation. Check out pdf version. To speed up estimation we apply the Leisen Reimer (1996) binomial tree in lieu of Cox, Ross and Rubinstein. For a deep dive check out the last two video clips on this page. Immediately below, the VBA Code for Leisen Reimer is set up largely based on Espen Haug. We truncate the zero region of the tree and make use of dynamic memory. (The zero region in Leisen Reimer is quite subtantial). This further accelerates estimation. We estimate both American and European analogue options with varying stock prices. By setting dividend yield = the risk free rate we can deduce an estimation for options based on futures. The lower bounds for both European and American options are included in graphing and dividends and interest rates are dynamically adjusted.

Public Function LeisenReimerTrunc(AmeEurFlag As String, CallPutFlag As String, S As Double, X As Double, _

T As Double, r As Double, b As Double, v As Double, n As Integer) As Variant

Dim OptionValue() As Double

'Dim ReturnValue(3) As Double

Dim d1 As Double, d2 As Double

Dim hd1 As Double, hd2 As Double

Dim u As Double, d As Double, p As Double

Dim dt As Double, Df As Double

Dim i As Integer, j As Integer, z As Integer

n = Application.Odd(n)

ReDim OptionValue(0 To n)

If CallPutFlag = "c" Then

z = 1

ElseIf CallPutFlag = "p" Then

z = -1

End If

d1 = (Log(S / X) + (b + v ^ 2 / 2) * T) / (v * Sqr(T))

d2 = d1 - v * Sqr(T)

'// Using Preizer-Pratt inversion method 2

hd1 = 0.5 + Sgn(d1) * (0.25 - 0.25 * Exp(-(d1 / (n + 1 / 3 + 0.1 / (n + 1))) ^ 2 * (n + 1 / 6))) ^ 0.5

hd2 = 0.5 + Sgn(d2) * (0.25 - 0.25 * Exp(-(d2 / (n + 1 / 3 + 0.1 / (n + 1))) ^ 2 * (n + 1 / 6))) ^ 0.5

dt = T / n

p = hd2

u = Exp(b * dt) * hd1 / hd2

d = (Exp(b * dt) - p * u) / (1 - p)

Df = Exp(-r * dt)

For i = 0 To n

OptionValue(i) = Application.Max(0, z * (S * u ^ i * d ^ (n - i) - X))

For j = n - 1 To ((n + 1) / 2) Step -1

For i = (j - ((n - 1) / 2)) To j

If AmeEurFlag = "e" Then

OptionValue(i) = (p * OptionValue(i + 1) + (1 - p) * OptionValue(i)) * Df

ElseIf AmeEurFlag = "a" Then

OptionValue(i) = Application.Max((z * (S * u ^ i * d ^ (j - i) - X)), _

(p * OptionValue(i + 1) + (1 - p) * OptionValue(i)) * Df)

End If

ElseIf z = -1 Then

For j = n - 1 To ((n + 1) / 2) Step -1

For i = 0 To ((n - 1) / 2)

OptionValue((n + 1) / 2) = 0

If AmeEurFlag = "e" Then

OptionValue(i) = (p * OptionValue(i + 1) + (1 - p) * OptionValue(i)) * Df

ElseIf AmeEurFlag = "a" Then

OptionValue(i) = Application.Max((z * (S * u ^ i * d ^ (j - i) - X)), _

(p * OptionValue(i + 1) + (1 - p) * OptionValue(i)) * Df)

End If

For j = ((n + 1) / 2) - 1 To 0 Step -1

For i = 0 To j + 1

If AmeEurFlag = "e" Then

OptionValue(i) = (p * OptionValue(i + 1) + (1 - p) * OptionValue(i)) * Df

ElseIf AmeEurFlag = "a" Then

OptionValue(i) = Application.Max((z * (S * u ^ i * d ^ (j - i) - X)), _

(p * OptionValue(i + 1) + (1 - p) * OptionValue(i)) * Df)

End If

LeisenReimerTrunc = OptionValue(0)

End Function

In the video below, we set out the main factors that contribute to making American Futures Options more valuable than their European analogue.

We again emphasize below some of benefits of using Leisen Reimer (1996) when estimating American Options. We note that if set the cost of carry equal the zero then the binomial tree can be easily applied to the options on futures valuation.

Taking A Deeper Dive

Below we set two mini-lectures that examine differences in valuation when comparing American and European options. Some discussion of option properties and lower bonds are included.