Abstract
This chapter closes the first part of the book which is on financial modeling and state price deflator construction. We give several examples of valuation of cash flows and basis financial instruments of the financial market, and we model and price defaultable bonds and derivatives of underlying financial instruments such as European put and call options. This provides the grounding for the valuation of insurance portfolios, in particular, if they include financial options such as minimal interest rate guarantees. Moreover, we define the Vasicek financial model which is going to be used as toy model in many subsequent examples.
Keywords
- State Price Deflator
- Basic Financial Instruments
- Minimum Interest Rate Guarantee
- Fundamental Theorem Of Asset Pricing (FTAP)
- Price Process
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Appendix: Proofs of Chap. 5
Appendix: Proofs of Chap. 5
In this section we prove the main statements of Chap. 5. For the proofs of Theorems 5.12 and 5.13 we refer to Appendices 11.2.2 and 11.2.3 because these proofs are of broader interest.
Proof of Proposition 5.5
The proof is similar to Example 5.2. The consistency condition w.r.t. φ implies the martingale requirement
The last expected value is the expectation of a log-normal distribution and gives
Henceforth, we obtain the requirement
which gives the -measurable drift term \(a_{t+1}^{(i)}= r_{t} - \frac{1}{2}(\sigma^{(i)} )^{2} + \lambda~r_{t} \sigma^{(i)} c^{(i)}\). □
Proof of Proposition 5.6
Using the affine term structure in the discrete time one-factor Vasicek model and the definition of the spot rate dynamics (3.15) we have
where we have used Theorem 3.5 in the last step. Henceforth, we obtain
Note that B(t,m)−βB(t+1,m)=1+λgB(t+1,m), see Theorem 3.2, which gives the claim. □
Proof of Proposition 5.8
Note, that we have
Using Propositions 5.5 and 5.6 we have under \(\mathbb{P}\)
Under the accounting condition \(A_{0}^{(i)}=P(0,m)\) and using the definition of K (i) (see (5.14)) we obtain
Henceforth, under the real world probability measure \(\mathbb{P}\), given , we obtain that \(\log A_{1}^{(i)}-\log P(1,m)\) has a Gaussian distribution with mean
and variance
But then the claim follows. □
Proof of Corollary 5.9
We have
Because \(l^{(i)}_{p}+1>0\), the last term determines the sign of the derivative \(\frac{\partial}{\partial p}~l^{(i)}_{p}\). Henceforth, we have for \(s_{1}^{(i)}>0\)
For the derivative w.r.t. the correlation parameter we obtain
for p<1/2 and λ≥0. □
Proof of Lemma 5.11
We have (A T −K)+−(K−A T )+=A T −K. This immediately implies
This proves the claim. □
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Wüthrich, M.V., Merz, M. (2013). Pricing of Financial Assets. In: Financial Modeling, Actuarial Valuation and Solvency in Insurance. Springer Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31392-9_5
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