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I. Basic math.
II. Pricing and Hedging.
1. Basics of derivative pricing I.
2. Change of numeraire.
3. Basics of derivative pricing II.
A. Option pricing formula for an economy with stochastic riskless rate.
B. T-forward measure.
C. HJM.
4. Market model.
5. Currency Exchange.
6. Credit risk.
7. Incomplete markets.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Option pricing formula for an economy with stochastic riskless rate.


e use the notation

MATH (MMA numeraire)
The $\beta_{t}$ is the worth of one unit of reference currency invested in MMA at time $t=0$ . It is a traded asset suitable as a numeraire, see ( Suitable numeraire ).




According to the martingale result ( Risk neutral pricing ), price $V_{t}$ of a call stroke at $K$ written on the traded asset $S_{t}$ is given by MATH Let MATH , MATH . We rewrite the last formula for $t=0$ : MATH The expressions MATH and MATH may be regarded as parts of kernels for numeraire changes, see ( Change of numeraire definition ). Indeed, MATH where $\beta_{0}=1$ and the Prob $_{S}$ is the probability changed to the numeraire $S_{t}$ . Similarly, MATH where MATH is the price of riskless bond with maturity $T$ as observed at time $t$ and Prob $_{T}$ is the probability with respect to the numeraire MATH . We compute the probabilities as follows. MATH where the MATH is a martingale under Prob ${}_{S}$ . MATH where the MATH is a martingale under Prob ${}_{T}$ . We are unable to proceed further without some assumptions about the distribution. Under log-normal assumption, we compute both of the probabilities by setting MATH MATH where the variables $\xi$ are standard normal with respect to the corresponding measures and the volatility $\sigma$ is the same in both of the expressions as shown in the ( Change of measure recipe section ).





Notation. Index. Contents.


















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