Option Pricing

Abstract

Call and put options provide a standard tool for hedging exposure to foreseeable risk. The pricing of options is inherently coupled to the price of the underlying asset, hence a model for pricing options must couple innately to the model for the underlying asset price. This chapter details option pricing based upon a so-called double subordination price model for the asset. Subordination models offer the ability to include more of the stylized facts of asset prices, increasing the accuracy of option prices. This chapter details the application of a double subordinated model to capture the mean, variance, skewness, and kurtosis, as well as intrinsic time features of the return process for one of the optimized domestic REIT portfolios.

Appendices

Appendix 1

Although (12.13) implies that the model (12.10) has eight parameters, μ, γ, ρ, σ, μ_U, λ_U, μ_T, λ_T, only six are identifiable. Using the relationship (12.15) between the CF and the MGF, from (12.13) ${K}_{X_1}(w)$ can be written as

$$ {K}_{X_1}(w)=\mu w+{c}_K\left(1-\sqrt{g(w)}\right), $$

$$ g(w)=1-2{c}_g\left(1-\sqrt{h(w)}\right)-2{d}_{\gamma }w, $$

$$ h(w)=1-2{d}_{\rho }w-{d}_{\sigma }{w}^2, $$

(12.61)

where c_K = λ_U/μ_U, ${c}_g={\mu_U}^2{\lambda}_{\mathcal{T}}/\left({\lambda}_U{\mu}_{\mathcal{T}}\right)$, ${d}_{\gamma }=\gamma\ {\mu_U}^2/{\lambda}_U,\kern0.5em {d}_{\rho }=\rho\ {\mu_{\mathcal{T}}}^2/{\lambda}_{\mathcal{T}}$, and ${d}_{\sigma }={\sigma}^2\ {\mu_{\mathcal{T}}}^2/{\lambda}_{\mathcal{T}}$. From (12.61) we have the following derivatives with respect to w:

$$ {h}^{\prime }(w)=-2{d}_{\rho }-2{d}_{\sigma }w, $$

$$ {h}^{\prime \prime }(w)=-2{d}_{\sigma }, $$

$$ {h}^{(p)}(w)=0,\kern0.5em p=3,4,\dots; $$

(12.62)

$$ {g}^{\prime }(w)={c}_g\left[{h}^{-1/2}{h}^{\prime}\right]-2{d}_{\gamma }, $$

$$ {g}^{\prime \prime }(w)={c}_g\left[-\frac{1}{2}{h}^{-3/2}{\left({h}^{\prime}\right)}^2+{h}^{-1/2}{h}^{\prime \prime}\right], $$

$$ {g}^{\prime \prime \prime }(w)={c}_g\left[\frac{3}{4}{h}^{-5/2}{\left({h}^{\prime}\right)}^3-\frac{3}{2}{h}^{-3/2}{h}^{\prime }{h}^{\prime \prime }+{h}^{-1/2}{h}^{\prime \prime \prime}\right], $$

$$ {g}^{(4)}(w)={c}_g\left[-\frac{15}{8}{h}^{-7/2}{\left({h}^{\prime}\right)}^4+\frac{9}{2}{h}^{-5/2}{\left({h}^{\prime}\right)}^2{h}^{\prime \prime }-\frac{1}{2}{h}^{-3/2}\left[3{\left({h}^{\prime \prime}\right)}^2+4{h}^{\prime }{h}^{\prime \prime \prime}\right]+{h}^{-1/2}{h}^{(4)}\right]; $$

(12.63)

and

$$ {K_{X_1}}^{\prime }(w)=\mu -\frac{c_K}{2}\left[{g}^{-1/2}{g}^{\prime}\right], $$

$$ {K_{X_1}}^{\prime \prime }(w)=-\frac{c_K}{2}\left[-\frac{1}{2}{g}^{-3/2}{\left({g}^{\prime}\right)}^2+{g}^{-1/2}{g}^{\prime \prime}\right], $$

$$ {K_{X_1}}^{\prime \prime \prime }(w)=-\frac{c_K}{2}\left[\frac{3}{4}{g}^{-5/2}{\left({g}^{\prime}\right)}^3-\frac{3}{2}{g}^{-3/2}{g}^{\prime }{g}^{\prime \prime }+{g}^{-1/2}{g}^{\prime \prime \prime}\right], $$

$$ {K_{X_1}}^{(4)}(w)=-\frac{c_K}{2}\left[-\frac{15}{8}{g}^{-7/2}{\left({g}^{\prime}\right)}^4+\frac{9}{2}{g}^{-5/2}{\left({g}^{\prime}\right)}^2{g}^{\prime \prime }-\frac{1}{2}{g}^{-3/2}\left[3{\left({g}^{\prime \prime}\right)}^2+4{g}^{\prime }{g}^{\prime \prime \prime}\right]+{g}^{-1/2}{g}^{(4)}\right]. $$

(12.64)

Noting that h(0) = g(0) = 1, the dependence of the derivatives in (12.62) through (12.64), when evaluated at w = 0, can be summarized as follows:

$$ {h}^{\prime }(0)={h}^{\prime}\left({d}_{\rho}\right), $$

$$ {h}^{\prime \prime }(0)={h}^{\prime \prime}\left({d}_{\sigma}\right), $$

$$ {h}^{(p)}(w)=0,\kern0.5em p=3,4,\dots; $$

(12.65)

$$ {g}^{\prime }(0)={g}^{\prime}\left({c}_g,{h}^{\prime }(0),{d}_{\gamma}\right)={g}^{\prime}\left({c}_g,{d}_{\rho },{d}_{\gamma}\right), $$

$$ {g}^{(p)}(0)={g}^{(p)}\left({c}_g,{h}^{\prime }(0),{h}^{\prime \prime }(0)\right)={g}^{(p)}\left({c}_g,{d}_{\rho },{d}_{\sigma}\right),p=2,3,\dots; $$

(12.66)

and

$$ {K_{X_1}}^{\prime }(0)={K_{X_1}}^{\prime}\left(\mu, {c}_K,{g}^{\prime }(0)\right)={K_{X_1}}^{\prime}\left(\mu, {c}_K,{c}_g,{d}_{\rho },{d}_{\gamma}\right), $$

$$ {K_{X_1}}^{(p)}(0)={K_{X_1}}^{(p)}\left({c}_K,{g}^{\prime }(0),{g}^{\prime \prime }(0),\dots, {g}^{(p)}(0)\right)={K_{X_1}}^{(p)}\left({c}_K,{c}_g,{d}_{\rho },{d}_{\gamma },{d}_{\sigma}\right), $$

$$ p=2,3,\dots . $$

(12.67)

Thus, all derivatives of ${\left.{K}_{X_1}(w)\right|}_{w=0}$, or equivalently all moments of the process X₁, depend only on the six identifiable parameters μ, c_K,  c_g, d_ρ, d_γ, d_σ.

Appendix 2

Using the definitions,

$$ \overline{a}=1+a,\kern0.5em {d}_{\lambda }=\frac{\lambda_{\mathcal{T}}}{\lambda_U},\kern0.5em {d}_{\rho }=\frac{\varrho }{\lambda_{\mathcal{T}}},\kern0.5em {d}_{\sigma }=\frac{\sigma^2}{\lambda_{\mathcal{T}}}, $$

the terms in (12.49) are

$$ {c}_1=1-2{d}_{\lambda }+2{d}_{\lambda }{\left({b}_1^2+{b}_2^2\right)}^{1/4}\cos \left(\frac{\theta_b}{2}\right),\kern0.5em {c}_2=2{d}_{\lambda }{\left({b}_1^2+{b}_2^2\right)}^{1/4}\sin \left(\frac{\theta_b}{2}\right), $$

$$ {\theta}_c=\mathrm{atan}\left(\frac{c_2}{c_1}\right)\in \left(\left.-\frac{\pi }{2},\frac{\pi }{2}\right]\right., $$

where

$$ {\displaystyle \begin{array}{ll}{b}_1=1-2{d}_{\rho}\overline{a}-{d}_{\upsigma}\left({\overline{a}}^2-{v}^2\right),& \kern0.5em {b}_2=-2\left({d}_{\rho }+\overline{a}{d}_{\upsigma}\right)v,\\ {}{\theta}_b=\mathrm{atan}\left(\frac{b_2}{b_1}\right)\in \left(\operatorname{}-\frac{\pi }{2},\frac{\pi }{2}\right]\operatorname{}.\end{array}} $$

For sufficiently large values of v, we have the approximations:

$$ {b}_1\approx {d}_{\sigma }{v}^2,\kern0.5em {b}_2=-2\left({d}_{\rho }+\overline{a}{d}_{\sigma}\right)v,\kern0.5em {\left({b}_1^2+{b}_2^2\right)}^{1/4}\approx {d}_{\sigma}^{1/2}v, $$

$$ \cos \left(\frac{\theta_b}{2}\right)\approx 1,\kern0.75em \sin \left(\frac{\theta_b}{2}\right)\approx -\frac{d_{\rho }+\overline{a}{d}_{\sigma }}{d_{\sigma }v}, $$

$$ {c}_1\approx 2{d}_{\lambda}\sqrt{d_{\sigma }}v,\kern0.5em {c}_2\approx -2{d}_{\lambda}\frac{d_{\rho }+\overline{a}{d}_{\sigma }}{\sqrt{d_{\sigma }}},\kern0.5em {\left({c}_1^2+{c}_2^2\right)}^{1/4}\approx \sqrt{c_1}, $$

$$ \cos \left(\frac{\theta_c}{2}\right)\approx 1. $$

(12.68)