Question

Review the derivation of the Black-Scholes formula (5.6.18). For this exercise, assume that our stock price at time u in the future is

\({{\bf{S}}_{\bf{0}}}{{\bf{e}}^{{\bf{\mu u + }}{{\bf{W}}_{\bf{u}}}}}\)where \({{\bf{W}}_{\bf{u}}}\) has the gamma distribution with parameters αu and β with β > 1. Let r be the risk-free interest rate.

a. Prove that \({{\bf{e}}^{{\bf{ - ru}}}}{\bf{E}}\left( {{{\bf{S}}_{\bf{u}}}} \right){\bf{ = }}{{\bf{S}}_{\bf{0}}}\,\,{\bf{if }}\,{\bf{and}}\,\,{\bf{only}}\,\,{\bf{if}}\,{\bf{\mu = r - \alpha log}}\left( {\frac{{\bf{\beta }}}{{{\bf{\beta - 1}}}}} \right)\)

b. Assume that \({\bf{\mu = r - \alpha log}}\left( {\frac{{\bf{\beta }}}{{{\bf{\beta - 1}}}}} \right)\). Let R be 1 minus the c.d.f. of the gamma distribution with parameters αu and 1. Prove that the risk-neutral price for the option to buy one share of the stock for the priceq at the time u is \(\begin{array}{l}{{\bf{S}}_{\bf{0}}}{\bf{R}}\left( {{\bf{c}}\left[ {{\bf{\beta - 1}}} \right]} \right){\bf{ - q}}{{\bf{e}}^{{\bf{ - ru}}}}{\bf{R}}\left( {{\bf{c\beta }}} \right)\,\,{\bf{where}}\\{\bf{c = log}}\left( {\frac{{\bf{q}}}{{{{\bf{S}}_{\bf{0}}}}}} \right){\bf{ + \alpha ulog}}\left( {\frac{{\bf{\beta }}}{{{\bf{\beta - 1}}}}} \right){\bf{ - ru}}\end{array}\)

c. Find the price for the option being considered when u = 1, q = \({{\bf{S}}_{\bf{0}}}\), r = 0.06, α = 1, and β = 10.

Short answer

a.\({e^{ - ru}}E\left( {{S_u}} \right) = {S_0}\,\,{\rm{if }}\,{\rm{and}}\,\,{\rm{only}}\,\,{\rm{if}}\,\mu {\rm{ = r - }}\alpha {\rm{log}}\left( {\frac{\beta }{{\beta - 1}}} \right)\)

b.one share of the stock for the price q at thew time u is

\(\begin{array}{l}{S_0}R\left( {c\left[ {\beta - 1} \right]} \right) - q{e^{ - ru}}R\left( {c\beta } \right)\,\,{\rm{where}}\\\\{\rm{c = log}}\left( {\frac{q}{{{S_0}}}} \right) + \alpha u\log \left( {\frac{\beta }{{\beta - 1}}} \right) - ru\end{array}\)

c.0.0454

Step-by-step solution

Step 1:Given information

stock price at time u in the future is

\({S_0}{e^{\mu u + {W_u}}}\) where \({W_u}\) has the gamma distribution with parameters αu and β with β > 1.Let r be the risk-free interest rate.

Calculation of Part(a)

Since \({W_w}\) follows the gamma distribution with parameters \(\alpha u\)and \(\beta > 1\) , the moment generating function for gamma distribution is as shown below :

\(\psi (t) = \frac{\beta }{{{{(\beta - t)}^{\alpha u}}}}\)

Therefore, by using the definition of the moment generating function the value for the mean is given by:

So, \(\exp \left( { - ru} \right)E\left( {{S_u}} \right) = {S_0}\) if and only if \(\exp \left[ {\left( {\mu - r} \right)u} \right]{\left( {\frac{\beta }{{\beta - 1}}} \right)^{\alpha u}} = 1\)

Solve above equation for \(\mu \) ,by taking logs on both the sides.

\(\begin{array}{l}\left[ {\left( {\mu - r} \right)u} \right] + \alpha u\log \left( {\frac{\beta }{{\beta - 1}}} \right) = 0\\\mu u - ru + \alpha u\log \left( {\frac{\beta }{{\beta - 1}}} \right) = 0\\\mu u = ru - \alpha u\log \left( {\frac{\beta }{{\beta - 1}}} \right)\\\mu = r - \alpha \log \left( {\frac{\beta }{{\beta - 1}}} \right)\end{array}\)

Hence it is proved that \({e^{ - ru}}E\left( {{S_u}} \right) = {S_0}\) , if and only if \(\mu = r - \alpha \log \left( {\frac{\beta }{{\beta - 1}}} \right)\)

Calculation of part (b)

Let ,
\(\begin{aligned}{}h\left( x \right) &= x - q\,{\rm{if}}\,x \ge q,\\ &= 0\,\,\,\,\,\,\,\,\,{\rm{if}}\,x < q\end{aligned}\)

The value of the option at time u is \(h\left( {{s_u}} \right)\) , the value for \({S_u}\) is greater than or equal to q if and only if\({W_u} \ge \log \left( {\frac{q}{{{S_0}}}} \right) - \mu u = c\).

Then the present value of the option is as shown below :

\(\begin{aligned}{}\exp \left( { - ru} \right)E\left[ {h\left( {{S_u}} \right)} \right)&= \exp \left( { - ru} \right)\int\limits_e^\infty {\left[ {{S_0}\,\exp \left( {\mu u + w} \right) - q} \right]} \frac{{{\beta ^{\alpha u}}}}{{\Gamma \left( {\alpha u} \right)}}{w^{\alpha u - 1}}\exp \left( { - \beta w} \right)dw\\ &= {S_0}\exp \left( {\left| {\mu - r} \right|u} \right)\frac{{{\beta ^{\alpha u}}}}{{\Gamma \left( {\alpha u} \right)}}\int\limits_c^\infty {{w^{^{\alpha u} - 1}}} \exp \left( { - w\left( {\beta - 1} \right)} \right)dw\\ &= - q\exp \left( { - ru} \right)\frac{{{\beta ^{\alpha u}}}}{{\Gamma \left( {\alpha u} \right)}}\int\limits_c^\infty {{w^{^{\alpha u} - 1}}} \exp \left( { - \beta w} \right)dw\\& = {S_0}\frac{{{{\left( {\beta - 1} \right)}^{\alpha u}}}}{{\Gamma \left( {\alpha u} \right)}}\int\limits_e^\infty {{w^{^{\alpha u - 1}}}} \exp \left( { - w\left( {\beta - 1} \right)} \right)dw - q\exp \left( { - ru} \right)R\left( {c\beta } \right)\\ &= {S_0}R\left( {c\left[ {\beta - 1} \right]} \right) - q\exp \left( { - ru} \right)R\left( {c\beta } \right)\end{aligned}\)

Hence it is proved that the risk neutral price for the option to buy one share of the stock for the price q at the time u is

\(\begin{array}{l}{S_0}R\left( {c\left[ {\beta - 1} \right]} \right) - q\exp \left( { - ru} \right)R\left( {c\beta } \right),{\rm{where}}\\{\rm{c = log}}\left( {\frac{q}{{{S_0}}}} \right) + \alpha u\log \left( {\frac{\beta }{{\beta - 1}}} \right) - ru\end{array}\)

Calculation of part (c)

. We know that \({\rm{c = log}}\left( {\frac{q}{{{S_0}}}} \right) + \alpha u\log \left( {\frac{\beta }{{\beta - 1}}} \right) - ru\)

Substituting u = 1, q = \({S_0}\) , r = 0.06, α = 1, and β = 10.

We get

\(\begin{aligned}{}c& = \log 1 + 1 \times 1 \times \log \left( {\frac{{10}}{{10 - 1}}} \right) - 0.06 \times 1\\& = 0 + \log \left( {\frac{{10}}{9}} \right) - 0.06\\ &= 0.0454\end{aligned}\)