call price with random interest rate.pdf
TRANSCRIPT
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8/11/2019 Call Price with Random Interest Rate.pdf
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Call Price with Random Interest Rate
Mauricio Bedoya
September 2014
To understand this blog, we must know
1. Forward price ofS(t).
2. Radon-Nikodym Derivative Theorem.
3. Change of Measure.
Normally the blogs that we have developed, take 2 or 3 pages maximum. This one will takemore than 3 pages, but I will guide you step by step during the whole procedure.
Forward price ofS(t).
Mathematically, the Forward price ofS(t) is
F[S(t), T] =S(t) er(Tt)
= S(t)
B(t, T)
(1)
with B(t, T) =er(Tt).
Because the Forward price is defined positive, the elements that characterize its behaviourmust be defined positive too. We can assure that B(t, T) is always positive. At the same time,we can use the Geometric Brownian Motion under a risk neutral world, to characterize S(t),and guarantee non negativity. With this in mind, we can estimate dF[S(t), T] from equation 1(using Ito Quotient Rule)
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dF[S(t), T]
F[S(t), T] =
dS(t)
S(t)dB(t, T)
B(t, T) + (
dB(t, T)
B(t, T))2
dt2 = 0 dS(t)
S(t)dB(t, T)
B(t, T)
dt2 = 0 and dt dw= 0dF[S(t), T]
F[S(t), T] =
S(t)(r dt + (s) dw(t))
S(t)r dt B(t, T)
B(t, T)
dF[S(t), T]
F[S(t), T] =(s) dw(t)
t0
dF[S(u), T]
F[S(u), T] Lebesgue Integral=
t0
(s) dw(u)
Ito IntegralLn[
F[S(t), T]
F[S(0), T]=(s) w(t)
1
2 2 t
F[S(t), T] =F[S(0), T] e(s)w(t) 122t t [t,T]
(2)
Equation 2 result is Martingala (easy to prove) and characterize the evolution ofF[S(t), T].
Radon-Nikodym Derivative Theorem.
In this section, I will NOT restate the Theorem (search it in google or Wikipedia). In English,how do we get the Radon-Nikodym Derivative expression
ew(t)122t (3)
Assume that we have w(t) N[0, 1]; and define a new variable Y(t) = w(t)+ . Operating, itseasy to verify that Y(t) N[, 1]. Now, lets estimate the quotient
dYdW
likelihood
dY
dW =
12e
(w(t))2
2
12e
w2(t)2
=e122+w(t)
(4)
Change of Measure
To characterize the change of measure, we must select a numeraire (N(t)). Next, we can say
that
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EN[I{...}] = 1N(0)
E[D(t) N(T) I{...}] (5)
were
E[D(t) N(T)
N(0) |f(t)] = Radon Nikodym Derivative (6)
with D(t) =ert. If we define N(t) = S(t), and implement equation 6, we will get equation 3.
Now, lets put the all in practices while we estimate the price of the European Call Option withrandom interest rates. The discounted option pay-off (Risk Neutral World) is
C(0)=
E[D(T)(S(T)K) I{S(T)K}]
=E[D(T) S(T) I{S(T)K}]KE[D(T) I{S(T)K}]=S(0)
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S(0)E[D(T)S(T)I{S(T)K}]KB(0, T) 1B(0, T) E[D(T)B(T, T)I{S(T)K}]
=S(0)ES(0)[I{S(T)K}]K B(0, T)EB(0,t)[I{S(T)K}](7)
In equation 7 we have two measures (numeraires): one is B(0,T) and the other one is S(0).In equation 1, we use B(t,T) as numeraire. However, we havent use S(0) as numeraire. Fromequation 1, we know that
F[S(T), T] =S(T)
Using S(t) as numeraire in equation 1 we get the relation
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F[S(t), T]=er(Tt)
S(t)(8)
Applying Ito Quotient Rule to the previous equation we get
d( 1F[S(t),T]
)
1F[S(t),T]
= dB(t, T)
B(t, T)
dS(t)
S(t)+ (
dS(t)
S(t))2
dS(t)
S(t)dB(t, T)
B(t, T)
=r dt(r dt + (s) dw(t)) + 2(s) dt0
=(s)((s) dt dw(t)) dws(t)
=(s) dws(t)
(9)
Under Ws(t), 1F[S(t),T]
is Martingala. To prove this, just integrate the previous equation in the
interval [0,T] and take expectation. Using equation 9 in 7, we get
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C(0)=S(0)ES(0)[I{F[ST,T]K}]K B(0, T)EB(0,t)[I{F[ST,T]K}]=S(0)ES(0)[I{F[S(0),T]e(s)w(T) 122TK}
change measure here
]K B(0, T)EB(0,t)[I{F[S(0),T]e(s)w(T)
12
2TK}]
=S(0)ES(0)[I{F[S(0),T]e 122T(s)ws(T)K}]K B(0, T)EB(0,t)[I{F[S(0),T]e(s)w(T)12 2TK}]
=S(0)ES(0)[I{ 122T(s)ws(T)Ln( KF[S(0),T] )}]K B(0, T)EB(0,t)[I{(s)w(T) 122TLn( KF[S(0),T]}]=S(0)ES(0)[I
{ws(T)Ln(
F[S(0),T]
K )+12
2T(s)
}]K B(0, T)EB(0,t)[I
{w(T)Ln(
F[S(0),T]
K ) 12
2T(s)
}]
=S(0)ES(0)[I{Z(T)
Ln(F[S(0),T]
K )+12
2T(s)
T
}]K B(0, T)EB(0,t)[I
{Z(T)Ln(
F[S(0),T]
K )12
2T(s)
T
}]
=S(0) N(d1)K B(0, T) N(2)
(10)
with N characterizing the CDF of an Standard Normal Distribution.
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