Appendix B: Proof of Theorem 3.2
Proof of Equation (3.2.4) and (3.2.5) in Theorem 3.2
By applying the martingale pricing method, the market value of the second-type guarantees at time t, 0≤ t ≤ T0 ≤ T1 ≤ . . . ≤ TN, is derived as follows:
Substituting GTN(II) as shown in (3.2.2) into (B.1), we know
(B.1) = EQt
Therefore, ψt(n)(II) can be derived below.
ψt(n)(II) = EQt
According to the definition of MT(n)
N(II), we know
‧
By “The Law of Iterated Conditional Expectation ” in Duffie (1988), (B.12) and (B.13) can be obtained as follows.We solve (B-1) as follows.
(B-1) = EQT
where ETηi+1(·) denotes the expectation under the forward martingale measure QTi+1 defined by the Radon-Nikod´ym derivative dQdQTi+1 = P (Ti+1,Tβi+1)/P (Ti,Ti+1)
Ti+1/βTi .
‧
We then solve (B-1a) and (B-1b), respectively.
(B-1a) = ETTi+1i
Iα is an indicator function with
Ti are determined below.
P (Ti, Ti) We define each variable at time t as follows.
Ci(t) = P (t, Ti)/P (t, Ti+1), (B.25)
Ei(t) = St/P (t, Ti+1). (B.26)
By employing Proposition 2.3 and Itˆo’s Lemma and substituting ¯σP(t,·) defined in (2.2.7) for σP(t,·), the dynamics of (B.25) and (B.26) under the forward measure QTi+1 can be obtained as given below. Under the forward measure QTi+1, the random variables defined in (B.25) and (B.26) are martingales, and their dynamics can be written as follows.
dCi(t)
‧
Solving the stochastic differential equations (B.28), we obtain:
Ei(Ti+1)
where PTri+1(·) denotes the probability under the forward martingale measure QTi+1.
By inserting (B.30) into PTri+1(·) , the probability can be obtained after rearrangement as follows:
PTri+1
Using (B.30), we know ETTi+1i where PRri(·) denotes the probability under the martingale measure Ri which is defined by the Radon-Nikod´ym derivative
‧
From the Radon-Nikod´ym derivative dRi
dQTi+1 , we know that
dWtTi+1 = dWtRi+ γEi (t)dt (B.39) Under the measure Ri, we obtain (B.40) by substituing (B.39) into (B.30)
STi+1
By inserting (B.40) into PRri(·) , the probability can be obtained after rearrangement as follows:
PRri
By combing (B.22), (B.31), (B.32), (B.37) with (B.41), (B-1a) can be obtained below.
(B-1a) = P (Ti, Ti)
P (Ti, Ti+1)N (di3)− P (Ti, Ti)
P (Ti, Ti+1)N (− di3). (B.42) From (B.30), we obtain
(B-1b) = ETTi+1
And we obtain (B-1) as shown in (B.46).
(B-1) = P (Ti, Ti+1)ETTi+1i {CTi+1} = 2N(di3). (B.46) Besides, (B.47) can be obtained by using (A.36).
EQt Inserting (B.13) and (B.47) into (B.8), we derive the result as follows.
ψt(n)(II) = EQt
Therefore, the proof of Theorem 3.2 is completed.
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