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Oke guys, I've been trying and trying and I just can't figure it out:
We have a portfolio z of the stock and an American put. z acts as numeraire. If we express an asset f (call/put/forward etc.) in multiples of Z (i.e. f/z) then there is a probability measure on the binomial tree that makes f/z into a martingale. Show the previous statement.
I tried the following but I think I'm totally on the wrong side:
W = So + P, where p is the put so also a 'f'.
X = f/z
df = (r+λσ)fdt + σfdWt
dz = dS + dP = [(r+λσ)fdt + σfdWt + μSdt + σSdWt]
dX = d(f.z^-1)
= F.Z^-1[(r+λσ)dt + σ(f)dWt] - F.Z^-1[(r+λσ)dt + σ(f)dWt + μdt + σ(s)dWt]
= F.Z^-1[(r+λσ)dt + σ(f)dWt - (r+λσ)dt - σ(f)dWt - μdt - σ(s)dWt]
= F.Z^-1[-μdt - σ(s)dWt]
Setting λ = μ/σ(s) and from Girs. Theorem we get: dWt* = dWt - λdt
We get: F.Z^-1[-μdt - σ(s)[(dWt - μ/σ(s)dt]] = (f/z)*-σ(s)dWt
This is sooooo wrong! Any suggestions?
We have a portfolio z of the stock and an American put. z acts as numeraire. If we express an asset f (call/put/forward etc.) in multiples of Z (i.e. f/z) then there is a probability measure on the binomial tree that makes f/z into a martingale. Show the previous statement.
I tried the following but I think I'm totally on the wrong side:
W = So + P, where p is the put so also a 'f'.
X = f/z
df = (r+λσ)fdt + σfdWt
dz = dS + dP = [(r+λσ)fdt + σfdWt + μSdt + σSdWt]
dX = d(f.z^-1)
= F.Z^-1[(r+λσ)dt + σ(f)dWt] - F.Z^-1[(r+λσ)dt + σ(f)dWt + μdt + σ(s)dWt]
= F.Z^-1[(r+λσ)dt + σ(f)dWt - (r+λσ)dt - σ(f)dWt - μdt - σ(s)dWt]
= F.Z^-1[-μdt - σ(s)dWt]
Setting λ = μ/σ(s) and from Girs. Theorem we get: dWt* = dWt - λdt
We get: F.Z^-1[-μdt - σ(s)[(dWt - μ/σ(s)dt]] = (f/z)*-σ(s)dWt
This is sooooo wrong! Any suggestions?
