▫️ 🧜🏾 🤳🏽 Opções matemática ou modelo Black-Scholes 🤚🏿 🤤 🦒

O interesse geral pelo modelo Black-Scholes (doravante - BS) se deve ao fato de que em algum momento seus autores revolucionaram a área de avaliação do valor justo de opções e outros instrumentos financeiros derivativos. Mais tarde, eles receberam o Prêmio Nobel por suas descobertas, e a fórmula analítica que derivaram tornou-se, talvez, a mais fundamental e conhecida no mundo das finanças.

O modelo BS não é menos interessante do ponto de vista da análise matemática e probabilística-teórica de baixo nível. O artigo discute em detalhes o processo de comprovar os princípios básicos e essenciais do modelo BS e também deduz uma fórmula analítica que é usada para avaliar o valor justo das opções.

Conceitos Básicos

Opção - um contrato pelo qual o comprador de uma opção recebe o direito , mas não a obrigação, de comprar ou vender um determinado ativo a um preço predeterminado, que é denominado preço de exercício ou preço de exercício.

Para fins de análise posterior, tal instrumento financeiro é mais precisamente representado como uma função que descreve os pagamentos de opções no momento do vencimento do contrato. Para uma compreensão mais simples e intuitiva, consideraremos uma opção do tipo Call, cuja função de pagamento é a seguinte.

onde é o preço do ativo subjacente, o preço do exercício.

Do ponto de vista prático, a função assume que o comprador da opção se beneficiará se o preço do ativo-objeto ultrapassar o preço de exercício x_s e coincidir com a diferença [x-x_s] . Caso contrário, o titular da opção receberá uma perda igual ao prêmio pago pela compra do contrato de opção.

, . , . , , ( ).

, C = max (x - x_s; 0) , , , , .

, , , . , .

, C = max (x - x_s; 0) x (t) . , x (t) , : $dx = xrdt + x \ sigma \ delta W$ *. , .

$dC = \ left (\ frac {\ parcial C} {\ parcial t} + xr \ frac {\ parcial C} {\ parcial x} + \ frac {\ sigma ^ 2} {2} \ frac {\ parcial ^ 2 C} {\ parcial x ^ 2} \ direita) dt + x \ sigma \ frac {\ parcial C} {\ parcial x} \ delta W \ qquad (1)$

, . (1) , . - .

$\ Pi = \ frac {\ parcial C} {\ parcial x} \ cdot x - C (x, t) \ qquad (2)$

, $\ Delta = \ frac {\ parcial C} {\ parcial x}$ - .

, , $\ Delta = const$ - : $d \ Pi = \ Delta \ cdot dx - dC$ . , , *, (1) . :

$d \Pi = \Delta(xrdt + x\sigma \delta W) - \left [ \left(\frac{\partial C}{\partial t} + xr\frac{\partial C}{\partial x} + \frac{\sigma^2}{2} \frac{\partial ^2 C}{\partial x^2}\right)dt + x\sigma\frac{\partial C}{\partial x} \delta W \right ] \qquad (3)$

, $\Delta = \frac{\partial C}{\partial x}$ , $x\sigma\frac{\partial C}{\partial x} \delta W$ :

$d \Pi = - \left [\frac{\partial C}{\partial t} + \frac{\sigma^2}{2} \frac{\partial ^2 C}{\partial x^2}\right]dt \qquad (4)$

, . $\tau$ , $\tau = T-t$ , . , , $\tau$ , . , : $\frac{\partial C}{\partial t } = - \frac{\partial C}{\partial \tau }$ .

B,S - : $d \Pi = \Pi rdt$ , . (4) , $\Pi$ (2) .

$\frac{\partial C}{\partial \tau }dt - \frac{\sigma^2}{2} \frac{\partial ^2 C}{\partial x^2}dt =\left( \frac{\partial C}{\partial x} \cdot x - C(x, t) \right ) rdt$

, :

$\frac{\partial C}{\partial \tau } + rC = \frac{\sigma^2 x^2}{2} \frac{\partial ^2 C}{\partial x^2} + rx\frac{\partial C}{\partial x } \qquad (5)$

, . , , , .

. $y = \ln x$ . x^2 , .

, , x^2 , :

$\frac{\partial C}{\partial \tau } + rC = \frac{\sigma^2}{2} \frac{\partial ^2 C}{\partial y^2} + R\frac{\partial C}{\partial y } \qquad (6)$

$R = r - \frac{\sigma^2}{2}$

, $y = \ln x$

$\frac{\partial C}{\partial x} = \frac{\partial C}{\partial y} \cdot \frac{dy}{dx} = \frac{\partial C}{\partial y} \cdot \frac{1}{x}$

, $y = \ln x$

$\frac{\partial ^2 C}{\partial x^2} =\left ( \frac{\partial C}{\partial y} \cdot \frac{dy}{dx}\right )_x '= (\frac{\partial C}{\partial y})_x '\cdot \frac{dy}{dx} + \frac{\partial C}{\partial y} \cdot (\frac{dy}{dx})_x' = \left( (\frac{\partial C}{\partial y})_y' \cdot y'\right ) \cdot y'+ \frac{\partial C}{\partial y} \cdot y'' =$ $= \left( \frac{\partial^2 C}{\partial y^2} \frac{1}{x}\right ) \frac{1}{x} - \frac{\partial C}{\partial y} \frac{1}{x^2} = \frac{1}{x^2}\left ( \frac{\partial^2 C}{\partial y^2} - \frac{\partial C}{\partial y}\right )$

$\frac{\partial C}{\partial \tau } + rC = \frac{\sigma^2 x^2}{2} \cdot \frac{1}{x^2}\left ( \frac{\partial^2 C}{\partial y^2} - \frac{\partial C}{\partial y}\right ) + rx \cdot \frac{1}{x} \frac{\partial C}{\partial y} \Leftrightarrow \frac{\partial C}{\partial \tau } + rC = \frac{\sigma^2}{2} \left( \frac{\partial ^2 C}{\partial y^2} -\frac{\partial C}{\partial y } \right )+ r\frac{\partial C}{\partial y }$

$\frac{\partial C}{\partial \tau } + rC = \frac{\sigma^2}{2} \left( \frac{\partial ^2 C}{\partial y^2} -\frac{\partial C}{\partial y } \right )+ r\frac{\partial C}{\partial y } \Rightarrow_1 \frac{\partial C}{\partial \tau } + rC = \frac{\sigma^2}{2} \frac{\partial ^2 C}{\partial y^2} - \frac{\sigma^2}{2} \frac{\partial C}{\partial y } + r\frac{\partial C}{\partial y } \Rightarrow_2$ $\frac{\partial C}{\partial \tau } + rC = \frac{\sigma^2}{2} \frac{\partial ^2 C}{\partial y^2} + (r - \frac{\sigma^2}{2}) \frac{\partial C}{\partial y } \Rightarrow_3 \frac{\partial C}{\partial \tau } + rC = \frac{\sigma^2}{2} \frac{\partial ^2 C}{\partial y^2} + R\frac{\partial C}{\partial y }$

$R = r - \frac{\sigma^2}{2}$

, . : $C(e^y, \tau) = e^{\alpha y + \beta \tau} \cdot U(y, \tau)$ , $\alpha$ $\beta$ , , :

$\frac{\partial U}{\partial \tau } = \frac{\sigma^2}{2} \frac{\partial ^2 U}{\partial y^2} \qquad (7)$

$Ue^{\alpha y + \beta \tau}$

$\left (U e^{\alpha y + \beta \tau} \right )_\tau ' + rUe^{\alpha y + \beta \tau} = \frac{\sigma^2}{2} \left (U e^{\alpha y + \beta \tau} \right )_{yy} '' + R\left (U e^{\alpha y + \beta \tau} \right )_y ' \qquad (*)$

$\tau$

$\left (U e^{\alpha y + \beta \tau} \right )_\tau ' = \frac{\partial U}{\partial \tau} \cdot e^{\alpha y + \beta \tau} + U \cdot \left ( e^{\alpha y + \beta \tau} \right )_\tau ' = \frac{\partial U}{\partial \tau} \cdot e^{\alpha y + \beta \tau} + \beta U \cdot e^{\alpha y + \beta \tau}$

$\left (U e^{\alpha y + \beta \tau} \right )_y ' = \frac{\partial U}{\partial y} \cdot e^{\alpha y + \beta \tau} + U \cdot \left ( e^{\alpha y + \beta \tau} \right )_y ' = \frac{\partial U}{\partial y} \cdot e^{\alpha y + \beta \tau} + \alpha U \cdot e^{\alpha y + \beta \tau}$

$\left( { \frac{\partial U}{\partial y}} \cdot e^{\alpha y + \beta \tau} + \alpha U \cdot e^{\alpha y + \beta \tau} \right)_y '= \left( { \frac{\partial U}{\partial y}} \cdot e^{\alpha y + \beta \tau} \right)_y' + \left (\alpha U \cdot e^{\alpha y + \beta \tau} \right )_y' =$ $\left( { \frac{\partial^2 U}{\partial y^2}}e^{\alpha y + \beta \tau} + \alpha \frac{\partial U}{\partial y} e^{\alpha y + \beta \tau} \right) + \left ( \alpha \frac{\partial U}{\partial y} e^{\alpha y + \beta \tau} + \alpha^2 U e^{\alpha y + \beta \tau}\right ) =$ $e^{\alpha y + \beta \tau} \left( \frac{\partial^2 U}{\partial y^2} + 2\alpha \frac{\partial U}{\partial y} + \alpha^2 U \right)$

$e^{\alpha y + \beta \tau}$

$\frac{\partial U}{\partial \tau} + \beta U + rU = \frac{\sigma^2}{2}\left( \frac{\partial^2 U}{\partial y^2} + 2\alpha \frac{\partial U}{\partial y} + \alpha^2 U \right) + R\left ( \frac{\partial U}{\partial y} + aU \right )$

$\alpha = -\frac{R}{\sigma^2}$ , $\beta = -(r + \frac{R^2}{2 \sigma^2})$ , :

$\frac{\partial U}{\partial \tau} - (r + \frac{1}{2}\frac{R^2}{\sigma^2}) U + rU = \frac{\sigma^2}{2}\left( \frac{\partial^2 U}{\partial y^2} - \frac {2R}{\sigma^2} \frac{\partial U}{\partial y} + \frac{R^2}{\sigma^4} U \right) + R\left ( \frac{\partial U}{\partial y} - \frac{R}{ \sigma^2}U \right )$

$\frac{\partial U}{\partial \tau } = \frac{\sigma^2}{2} \frac{\partial ^2 U}{\partial y^2}$

(7) :

$P(y, \tau, y_0) = \frac{1}{\sigma \sqrt{2 \pi \tau }} \cdot \exp(-\frac{(y-y_0)^2}{2\sigma^2 \tau}) \qquad (8)$

$(P)_\tau '$ , $(P)_{yy} ''$ (7) .

$e^{*} = \exp(-\frac{(y-y_0)^2}{2\sigma^2 \tau})$

$\tau$ :

$\frac{\partial P}{\partial \tau} = \left (\frac{1}{\sigma \sqrt{2 \pi \tau }} \right )_ \tau ' \cdot e^{*} + \frac{1}{\sigma \sqrt{2 \pi \tau }} \cdot \left (\exp(-\frac{(y-y_0)^2}{2\sigma^2 \tau}) \right )_\tau ' =$ $- \frac{1}{2\sigma \tau \sqrt{2\pi \tau}} \cdot e^{*} + \frac{1}{\sigma \sqrt{2\pi \tau}}\cdot e^{*} \cdot \frac{(y-y_0)^2}{2\sigma^2 \tau^2} = e^{*} \left (\frac{(y-y_0)^2}{2 \sigma^3 \tau^2 \sqrt{2\pi \tau}} - \frac{1}{2\sigma \tau \sqrt{2\pi \tau}} \right )$

$\frac{\partial P}{\partial y} = \frac{1}{\sigma \sqrt{2\pi \tau}} \cdot (e^{*})_y' = - \frac{1}{\sigma \sqrt{2\pi \tau}} \cdot e^{*} \cdot \frac{(y-y_0)}{ \sigma^2 \tau} \Rightarrow$ $\frac{\partial^2 P}{\partial y^2} = \left (- \frac{1}{\sigma \sqrt{2\pi \tau}} \cdot e^{*} \cdot \frac{(y-y_0)}{\sigma^2 \tau} \right )_y' = \left (- \frac{1}{\sigma^3 \tau \sqrt{2\pi \tau}} \cdot \left ( e^{*} \cdot (y-y_0) \right ) \right )_y' =$ $- \frac{1}{\sigma^3 \tau \sqrt{2\pi \tau}} \cdot \left[\left( e^{*} \cdot (-\frac{(y-y_0)}{\sigma^2 \tau}) \cdot (y-y_0) \right ) + e^{*} \cdot 1\right] = e^{*} \cdot \left (\frac{(y-y_0)^2}{\sigma^5 \tau^2 \sqrt{2\pi \tau}} - \frac{1}{\sigma^3 \tau \sqrt{2\pi \tau}} \right )$

(7) e^* . :

$\frac{(y-y_0)^2}{2 \sigma^3 \tau^2 \sqrt{2\pi \tau}} - \frac{1}{2\sigma \tau \sqrt{2\pi \tau}} = \frac{\sigma^2}{2}\left (\frac{(y-y_0)^2}{\sigma^5 \tau^2 \sqrt{2\pi \tau}} - \frac{1}{\sigma^3 \tau \sqrt{2\pi \tau}} \right )$

, u(s) :

$\int_ {-\infty}^{+\infty} u(s) P(y, \tau, s)ds,$

$\tau$ , (7) , - . , (7) :

$U(y, \tau) = \int_{-\infty}^{+\infty} u(s) P(y, \tau, s)ds =\frac{1}{\sigma \sqrt{2 \pi \tau }} \int_{-\infty}^{+\infty} u(s) \cdot \exp \left(-\frac{(y-s)^2}{2\sigma^2 \tau}\right)ds \qquad (9)$

(9) u(s) . , u(y) = U(y;0) . - , $\tau \mapsto 0$ - $\delta (y-s)$ :

$U(y; 0) = \int _{-\infty}^{+\infty}u(s) \delta (y-s)ds = u(y)$

: f(x) [a;b] , g(x) , $c \in[a,b]$ , :

$\int_{a}^{b} f(x)g(x) dx = f(c) \int_{a}^{b}g(x)dx$

$\varepsilon > 0$ .

$\tau \mapsto 0$ "" $\int_{-\infty}^{y- \varepsilon }, \int_{y-\varepsilon }^{+\infty}$ :

$U(y, \tau) \approx \int_{y- \varepsilon}^{y+ \varepsilon} u(s) P(y, \tau, s)ds$

$d \in [y-\varepsilon; y+\varepsilon ]$ ,

$\int_{y- \varepsilon}^{y+ \varepsilon} u(s) P(y, \tau, s)ds = u(d) \int_{y- \varepsilon}^{y+ \varepsilon} P(y, \tau, s)ds.$

, $\lim_{\tau \mapsto 0} \int_{y- \varepsilon}^{y+ \varepsilon} P(y, \tau, s)ds = 1$ , $u(d) \int_{y- \varepsilon}^{y+ \varepsilon} P(y, \tau, s)ds \approx u(d)$ . , $\varepsilon >0$ , $\tau \mapsto 0$ , $d \mapsto y$ . :

$u(y) = U(y;0) = e^{-\alpha y}\cdot C(e^{y}; 0) = e^{-\alpha y} \cdot \max (e^y -x_s;0)$

, $\max (e^y -x_s;0)=0$ $y < \ln x_s$ , (9) :

$U(y, \tau) = \int_{\ln x_s}^{+\infty} (e^s - x_s)\frac{e^{-\alpha s}}{\sigma \sqrt{2 \pi \tau}} \exp\left(-\frac{(y-s)^2}{2 \sigma^2 \tau}\right)ds \qquad (10)$

(10) $U(y;\tau)$ , , $C(e^y, \tau) = e^{\alpha y + \beta \tau} \cdot U(y, \tau)$ . , $U(y, \tau)$ $C(e^y, \tau)$ .

(10) .

, -:

$C = x_0F \left [ \frac{\ln(xe^{r\tau} / x_s)}{\sigma \sqrt{\tau}} + \frac{\sigma \sqrt{\tau}}{2} \right ] - x_se^{-r\tau}F \left [ \frac{\ln(xe^{r\tau} / x_s)}{\sigma \sqrt{\tau}} - \frac{\sigma \sqrt{\tau}}{2} \right ]$

, , $\sigma -$ .

, :

$z = \frac{(s-y)}{ \sigma \sqrt{\tau}}; \qquad s = z\sigma \sqrt{\tau} + y \qquad$ $z(\ln x_s) = \frac{\ln x_s - y}{\sigma \sqrt{\tau}} =: \gamma \text{ - }$ $dz = \left (\frac{(s-y)}{2 \sigma \sqrt{\tau}} \right )' ds \Rightarrow dz = \frac{ds}{\sigma \sqrt{\tau}} \Rightarrow ds = \sigma \sqrt{\tau}dz$

$U(y, \tau) =\int_{\gamma }^{+\infty}\left (e^{(1-\alpha)(\sqrt{\tau} \sigma z + y)} - x_s e^{-\alpha(y + \sqrt{\tau}\sigma z)} \right ) \frac{e^{-\frac{z^2}{2}}}{\sigma \sqrt{2 \pi \tau}} \sqrt{\tau }\sigma dz =$ $\frac{1}{\sqrt{2 \pi }} \int_{\gamma }^{+\infty}\left (e^{(1-\alpha)(\sqrt{\tau} \sigma z + y) -\frac{z^2}{2} } - x_s e^{-\alpha(y + \sqrt{\tau}\sigma z)-\frac{z^2}{2}} \right ) dz$

$U(y, \tau) =\frac{1}{\sqrt{2 \pi }}\left [ \int_{\gamma }^{+\infty}e^{(1-\alpha)(\sqrt{\tau} \sigma z + y)-\frac{z^2}{2}} dz -\int_{\gamma }^{+\infty} x_s e^{-\alpha(y + \sqrt{\tau}\sigma z)-\frac{z^2}{2}} dz \right ]$

, .

, , , , $e^{-\frac{v^2}{2}}$ , .

, $-\infty$ , . , . :

$\ frac {1} {\ sqrt {2 \ pi}} \ int _ {\ gamma} ^ {+ \ infty} e ^ {- \ frac {z ^ 2} {2}} dz = F (- \ gamma) \ qquad (*)$

, , . , , , . :

, $d_1 = - \ left (\ gamma - \ sigma \ sqrt {\ tau} (1- \ alpha) \ right)$ , $d_2 = - \ left (\ gamma + \ alpha \ sigma \ sqrt {\ tau} \ right)$ , *.

, , . , :

$\ gamma = \ frac {\ ln x_s - y} {\ sigma \ sqrt {\ tau}}; \ qquad \ alpha = - \ frac {R} {\ sigma ^ 2}; \ qquad R = r - \ frac {\ sigma ^ 2} {2}; \ qquad y = \ ln x.$

$d_1 = \ frac {\ ln (xe ^ {r \ tau} / x_s)} {\ sigma \ sqrt {\ tau}} + \ frac {\ sigma \ sqrt {\ tau}} {2}; \ qquad d_2 = \ frac {\ ln (xe ^ {r \ tau} / x_s)} {\ sigma \ sqrt {\ tau}} - \ frac {\ sigma \ sqrt {\ tau}} {2}.$

, . $C (e ^ y, \ tau)$ , , $C (e ^ y, \ tau) = e ^ {\ alpha y + \ beta \ tau} \ cdot U (y, \ tau)$ , , ** $e ^ {\ alpha y + \ beta \ tau}$ .

. , $e ^ {\ alpha y + \ beta \ tau} \ cdot e ^ {y (1- \ alpha) + \ frac {1} {2} \ sigma ^ 2 \ tau (1- \ alpha) ^ 2}$ , $e ^ {\ alpha y + \ beta \ tau} \ cdot e ^ {{\ alpha y + a ^ 2 \ sigma ^ 2 \ tau / 2}}$ $e ^ {- r \ tau}$ . , :

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Opções matemática ou modelo Black-Scholes

Conceitos Básicos

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