Abstract
In Part II, a whole array of very workable methods for the valuation and hedging of financial instruments was introduced. We now continue in Part III with the explicit valuation of the most important and common financial instruments.
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Notes
- 1.
Here, B A(R, t i, t i+1) is the discount factor based on a discrete, annual compounding interest rate R.
- 2.
A sequence x k, having the property that the quotient of two elements, one immediately following the other, is constant, x k+1∕x k = c for all k, is called a geometric sequence. Thus, for each of the elements of a geometric sequence, x k = x 1c k−1. The sum of the elements of a geometric sequence is called a geometricseries. The following formula holds for series of this type:
\(\sum \limits _{k=1}^{n}x_{k}=x_{1}\sum \limits _{k=1}^{n}c^{k-1}=x_{1}\frac {c^{n}-1}{c-1}\).
- 3.
Set k = i − m. The geometric series is then:
$$\displaystyle \begin{aligned} \sum_{i=m+1}^{i=n}b^{i-(m+1)} & =\sum_{k+m=m+1}^{k+m=n} b^{k+m-(m+1)}=\underset{\text{Geom. Series}}{\underbrace{\sum_{k=1} ^{n-m}b^{k-1}}}=\frac{b^{n-m}-1}{b-1}\\ & =b^{n-m-1}\frac{b(1-b^{m-n})}{b-1}=b^{n-m-1}\frac{b(1-b^{m-n})} {b(1-b^{-1})}. \end{aligned} $$ - 4.
Bonds cannot be interpreted as an underlying with a dividend yieldq when pricing options and forward contracts on bonds. The assumption of a dividend yield assumes payments relative to the underlying price, which is not a useful approximation for bonds. In addition, the accrued interest on the underlying is exchanged directly between the counterparties through the dirty price. In cash & carry arbitrage, only the “extra” cash flows which are not exchanged by the counterparties of the forward contract must be taken into consideration. These are just the coupon payments during the lifetime of the forward contract. These are received by the holder of the underlying, but not by the holder of the forward contract. Hence, the only consistent treatment of a bond is to interpret it as an underlying with dividend payments during the lifetime of the forward contract and to perform each calculation (for the spot price, the forward price and option strikes) with the dirty price (= clean price plus accrued interest).
- 5.
The actual price S(T) and not the originally agreed upon delivery price S(t 0, T)—where t 0 denotes the date the future contract was entered into—must be paid since the differences between the actual forward price of the underlying and the delivery price S(t 0, T) have already been settled on a daily basis because of the variation margin system.
- 6.
Small differences between similar, but not identical entities are frequently called basis.
- 7.
Here, the basis spread was assumed to be deterministic and time dependent. Though in reality, the basis will change over time, which could be simulated in a more complex model.
References
M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1972)
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Deutsch, HP., Beinker, M.W. (2019). Simple Interest Rate Products. In: Derivatives and Internal Models. Finance and Capital Markets Series. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-22899-6_15
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DOI: https://doi.org/10.1007/978-3-030-22899-6_15
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