Abstract
In this Chapter, we will discuss that in order to find the value of an American style derivative, what kind of mathematical problems needs to be solved. When we have such discussions, we mainly take American options as examples. However the methods can be used for other American style derivatives.
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Notes
- 1.
If \(V -{ \partial V \over \partial S} S < 0\), the seller indeed borrows \(-\left (V -{ \partial V \over \partial S} S\right )\), the money needed to buy \({ \partial V \over \partial S}\) shares, from somewhere.
- 2.
If \(S_{l} = -\infty \), then the first “ ≤ ” needs to be changed into “ < ,” and if S u = ∞, then the second “ ≤ ” needs to be changed into “ < .” In what follows, the same notation is used.
- 3.
In this book we call this problem and the like a free-boundary problem. An LC problem usually involves free boundaries. Thus it is not strange that some people call an LC problem a free-boundary problem.
- 4.
This result can also be obtained from direct calculation, which is left for readers as Problem 20.
- 5.
Here we assume that the value of the dividends depends on S, just like what we did in Sect. 2.2.2.
References
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Zhu, Yl., Wu, X., Chern, IL., Sun, Zz. (2013). American Style Derivatives. In: Derivative Securities and Difference Methods. Springer Finance. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7306-0_3
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DOI: https://doi.org/10.1007/978-1-4614-7306-0_3
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