ABCDEFGHIJKLMNOPQRSTUVWXYZ
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S0100
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T1
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r2.0000%t = T = 1
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u1.1S2uu121
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p1uu100
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A2uu
a2*S2+b2*e^rT
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t = 0
t = T/2 =
0.5
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S1u110
S2ud =S2du
100
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p1u?
p2ud = p2du
0
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S0100A1u
a1*S1 +b1*e^rt
A2ud = A2du
a2*S2+b2*e^rT
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p0?
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A0a0*S0 + b0
S2ud =S2du
100
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S1d90.90909091
p2ud = p2du
0
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p1d?
A2ud = A2du
a2*S2+b2*e^rT
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A1d
a1*S1 +b1*e^rt
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S2dd82.64
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p2dd0
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A2dd
a2*S2+b2*e^rT
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Step 1 find p1u
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*By self-financing:
a1*S2+b1*e^rT = a2*S2+b2*e^rT
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Construct a replicating portfolio A by long a1 units of stock and b1 units of $1 risk-free bond, so that A2 = p2
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#N/A1211.02020134a1=100
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#N/A1001.02020134b1=0
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=>a1=4.761904762
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b1=-466.761273
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By no arbitrage, A0 = c0
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p1u = a1*S1 + b1*e^rt =
52.35722202
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Step 2 find p1d
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Construct a replicating portfolio A by long a1 units of stock and b1 units of $1 risk-free bond, so that A2 = p2
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#N/A1001.02020134a1=0
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#N/A82.641.02020134b1=0.00
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=>a1=0
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b1=0
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By no arbitrage, A0 = c0
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p1u = a1*S1 + b1*e^rt =
0
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Step 3 find p0
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t = 0
t = T/2 =
0.5
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S1u110
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c1u52.35722202
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S0100A1u
a1*S1 +b1*e^rt
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c0?
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A0a0*S0 + b0
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S1d90.90909091
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c1d -
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A1d
a1*S1 +b1*e^rt
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*By self-financing:
a1*S2+b1*e^rT = a0*S2+b0*e^rT
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Construct a replicating portfolio A by long a1 units of stock and b1 units of $1 risk-free bond, so that A2 = c2
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#N/A1101.010050167a0=52.35722202
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#N/A90.909090911.010050167b0=0.00
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=>a0=2.742521154
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b0=-246.8393284
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By no arbitrage, A0 = p0
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p0 = a0*S0 + b0=
27.41278698
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Risk-neutral representation
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u1.1u1.1u1.1
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d0.9d0.9d0.9
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qu0.5288342085qu0.5288342085=>qu0.5288342085
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qd0.4711657915qd0.4711657915qd0.4711657915
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p1u52.35722202p1d0p027.41278698
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pu0.5288342085
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pd0.4711657915
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puu0.2796656201
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pud0.2491685884
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pdd0.221997203
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p027.41278698
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