Abstract
In order to meet a variety of demands, modern financial institutions issue many exotic options besides the vanilla options we have introduced in Chaps. 2 and 3. An exotic option is an option that is not a vanilla put or call. It usually is traded between companies and banks and not quoted on an exchange. In this case, we usually say that it is traded in the over-the-counter market.
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Notes
- 1.
Actually, \(G_{1}\left ({S}^{{\prime}},T; S,t,B_{l}\right )d{S}^{{\prime}}\) is the probability of the price at time T being in \(\left [{S}^{{\prime}},{S}^{{\prime}} + d{S}^{{\prime}}\right ]\) with the lowest price during the time period \(\left [t,T\right ]\) being greater than B l . Let us explain this fact. Consider all the paths of the price during the time period \(\left [t,T\right ]\) that start from S at time t. For any path that hits the lower barrier, the contribution to the option value is 0 because the option dies. Only those paths that never hit the lower barrier have contribution to the option value. A path that never hits the lower barrier S = B l during the time period \(\left [t,T\right ]\) is a path whose lowest price during the time period \(\left [t,T\right ]\) is greater than B l . From the expression for V (S, t), we see that the value of a down-and-out option is equal to the discounting factor times an integral of the product of the payoff function and \(G_{1}\left ({S}^{{\prime}},T; S,t,B_{l}\right )\) on [B l , ∞). Consequently, \(G_{1}\left ({S}^{{\prime}},T; S,t,B_{l}\right )d{S}^{{\prime}}\) actually is the probability of the price at time T being in \(\left [{S}^{{\prime}},{S}^{{\prime}} + d{S}^{{\prime}}\right ]\) with the lowest price during the time period \(\left [t,T\right ]\) being greater than B l . This fact will be used when we derive closed-form solutions for lookback options in Sect. 4.4.3.
- 2.
- 3.
A similar problem is given as a part of Problem 16 for readers to prove.
- 4.
\(S_{n}(t_{i}^{+})\) can also be expressed as \(S_{n}(t_{i}^{+}) =\min (S_{n-1}(t_{i}^{-}),\max (S,S_{n}(t_{i}^{-})))\).
- 5.
If Q − 1 = Q T, which is equivalent to QQ T = I or Q T Q = I, then Q is called an orthogonal matrix.
- 6.
This function is referred to as Green’s function for the n-dimensional heat equation.
- 7.
The value of this function has to be obtained by numerical methods. Here we write down the approximate expression derived in [26] by Drezner and Wesolovsky with the coefficients based on the abscissas and weight factors for Gaussian integration of ∫ − 1 1 f(x)dx on p. 916 of the handbook [1] by Abramowity and Stegum (editors). When | ρ | < 0. 7, it is approximated by
$$\displaystyle\begin{array}{rcl} & & \rho \sum _{i=1}^{5}\left \{W_{ i}\mathrm{{e}}^{\left [X_{i}\rho x_{1}x_{2}-\frac{1} {2} (x_{1}^{2}+x_{2}^{2})\right ]\Big/\left [1-{(X_{i}\rho )}^{2}\right ]}\Big/\sqrt{1 - {(X_{ i}\rho )}^{2}}\right \} + N(x_{1})N(x_{2}); {}\\ \end{array}$$when | ρ | ≥ 0. 7, it is approximated by
$$\displaystyle\begin{array}{rcl} & & N_{2}(x_{1},x_{2}; \mbox{ sgn }(\rho )) -\mbox{ sgn }(\rho )a\mathrm{{e}}^{-x_{1}x_{2}^{{\prime}}/2}\Bigg\{\frac{1} {6\pi }\left (3 - c{b}^{2} + c{a}^{2}\right )\mathrm{{e}}^{-{b}^{2}/(2{a}^{2})} {}\\ & & - \frac{1} {3\sqrt{2\pi }} \frac{b} {a}\left (3 - c{b}^{2}\right )N(-b/a) +\sum _{ i=1}^{5}W_{ i}\mathrm{{e}}^{- \frac{{b}^{2}} {2y_{i}^{2}} }\left [{ \mathrm{{e}}^{-x_{1}x_{2}^{{\prime}}/(1+\sqrt{1-y_{i }^{2}})} \over \mathrm{{e}}^{-x_{1}x_{2}^{{\prime}}/2}\sqrt{1 - y_{ i}^{2}}} - 1 - cy_{i}^{2}\right ]\Bigg\}, {}\\ \end{array}$$where
$$\displaystyle\begin{array}{rcl} & & W_{1} = 0.0188540425,\quad W_{2} = 0.0380880594,\quad W_{3} = 0.0452707394, {}\\ & & W_{4} = 0.0380880594,\quad W_{5} = 0.0188540425, {}\\ & & X_{1} = 0.0469100770,\,\quad X_{2} = 0.2307653449,\,\quad X_{3} = 0.5000000000, {}\\ & & X_{4} = 0.7692346551,\,\quad X_{5} = 0.9530899230, {}\\ & & \mbox{ sgn}(\rho ) = \left \{\begin{array}{ll} 1,\quad &\mbox{ if}\quad \rho \geq 0,\\ - 1,\quad &\mbox{ if} \quad \rho <0, \end{array} \right. {}\\ & & N_{2}(x_{1},x_{2}; 1) = N(\min (x_{1},x_{2})),\quad N_{2}(x_{1},x_{2}; -1) =\max (0,N(x_{1}) - N(-x_{2})), {}\\ & & x_{2}^{{\prime}} = \mbox{ sgn}(\rho )x_{ 2},\quad a = \sqrt{1 {-\rho }^{2}},\quad b = \vert x_{1} - x_{2}^{{\prime}}\vert,\quad c = (4 - x_{1}x_{2}^{{\prime}})/8, {}\\ & & y_{i} = aX_{i},\quad i = 1, 2, 3, 4, 5. {}\\ \end{array}$$The authors claim that for any x 1, x 2, − 1 ≤ ρ ≤ 1, its maximum error is 2 ×10 − 7. However in the FORTRAN program given in that paper there are two typos. In the book [43] by Hull, another approximate expression given in [25] by Drezner is also shown. (In order to know how to get its coefficients of expression, see the paper [74] by Steen, Byrne, and Gelbard.) Its accuracy is four decimal places and, averagely speaking, it needs more computational time than the expression shown here.
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Zhu, Yl., Wu, X., Chern, IL., Sun, Zz. (2013). Exotic Options. In: Derivative Securities and Difference Methods. Springer Finance. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7306-0_4
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