Pricing European options under a diffusion model with psychological barriers and leverage effect

ABSTRACT

This paper derives closed-form formulae for European options under a diffusion model when there exist psychological barriers and leverage effect in underlying dynamics. The state space of the proposed model is divided into different subregions by psychological barriers, and in each subregion, the model behaves like a constant elasticity of variance process. Within this framework, we derive both Laplace transform- and spectral expansion-based analytical solutions, which allow fast and accurate calculation of option prices. Numerical results are presented to explore the nontrivial properties of our model.

KEYWORDS:

• European options
• psychological barrier
• leverage effect
• constant elasticity of variance process

JEL CLASSIFICATION:

• G13

Disclosure statement

No potential conflict of interest was reported by the author(s).

ORCID

Shiyu Song  http://orcid.org/0000-0001-9141-9525

Notes

1 In the classical diffusion theory (see, e.g. Itô and McKean 1974), the speed density $\mathfrak{m}(\cdot )$ and the scale density $\mathfrak{s}(\cdot )$ are two fundamental elements to construct the infinitesimal generator of a regular diffusion. In the current model we have $\mathfrak{s}(x)={\mathrm{e}}^{-{\int }_{S^{\ast }}^x\frac{2r}{{\sigma }^2(y)y}\mathrm{d}y},\mathfrak{m}(x)=\frac{2}{{\sigma }^2(x)x^2\mathfrak{s}(x)}.$

2 $\mathfrak{S}$ may be equal to $({\mathbb{e}}_1,y]$, $[{\mathbb{e}}_1,y]$, $[y,{\mathbb{e}}_2)$ or $[y,{\mathbb{e}}_2]$ depending on the hitting direction and whether the boundary point ${\mathbb{e}}_1$ or ${\mathbb{e}}_2$ is regular reflecting. For example, in the up hitting case, if ${\mathbb{e}}_1$ is (resp. is not) a regular reflecting boundary, then $\mathfrak{S}$ represents $[{\mathbb{e}}_1,y]$ (resp. $({\mathbb{e}}_1,y]$).

3 We use the notation $f^{-}(x)=\underset{ϵ\downarrow 0}{lim}\frac{f(x)-f(x-ϵ)}{{\int }_{x-ϵ}^x{\mathfrak{s}}^X(z)\mathrm{d}z},$ where ${\mathfrak{s}}^X(\cdot )$ denotes the scale density.

4 Throughout the paper the notations without superscripts are related to the process $\{S_t,t\ge 0\}$ unless otherwise specified.

5 We do not differentiate the series (14(14) $P(S,T)=K{\mathrm{e}}^{-rT}Z(S)+\sum _{n=1}^{\mathrm{\infty }}({\iota }_n^{(2)}-K{\iota }_n^{(1)}){\mathrm{e}}^{-(r+{\lambda }_n)T}{\phi }_n(S),$(14) ) with respect to S term by term to get the explicit delta because the verification of the uniform convergence of the ‘pseudo-density’ requires to estimate the large-n asymptotics for eigenvalues which is hard to achieve under the current model. See the discussion about this point for the hitting time densities of CIR and OU diffusions in Linetsky (2004c).

6 We use Mathematica 10 running on a PC with the Intel Core i7-6700 CPU and 8GB of RAM for all calculations in this paper.

7 Following one of the reviewers' suggestion, to further test the performance of the Laplace transform approach, we also consider the put prices when ${\beta }_1={\beta }_2=-0.5$. We find that, when S takes values from 100 to 120, the Laplace transform approach can produce the same results as those obtained by using the spectral expansion approach; however, when S takes 80 and 90, the former approach does not work. By careful observation, we find that when S<100, the term $B(-m_1+\frac{1}{2},{\kappa }_1,m_1;z_1(S))$ appears in the expression of ${\mathrm{\Lambda }}_{2,\theta }(S)$ (see Lemma 3.1 and Proposition 3.4), where $-m_1+\frac{1}{2}:=h=0$ and $m_1:=l=\frac{1}{2}$ under our parameter settings. Thus, we have $h-l+\frac{1}{2}=0$, which implies that the representation of $B(h,k,l;x)$ in Lemma 3.1 is invalid in such a case since $\mathrm{\Gamma }(h-l+\frac{1}{2})$ does not exist. So we think this is the reason why the Laplace transform approach fails in the case when ${\beta }_1={\beta }_2=-0.5$ and S<100. To further verify this point, we consider several other cases like when ${\beta }_1={\beta }_2=-\frac{1}{3}$ and $-\frac{1}{5}$ (for both cases $h-l+\frac{1}{2}\ne 0$), and the results indicate that the Laplace transform approach is still applicable for these cases (for small T we need enhance the precision of internal calculations appropriately in the Wynn-Rho algorithm; the Mathematica code for the algorithm is available at http://www.pe.tamu.edu/valko/public_html/Nil/). In summary, it seems that the Laplace transform approach based on the Wynn-Rho algorithm works well for most cases except for those with particular choices of parameter values like ${\beta }_1={\beta }_2=-0.5$ which make the expressions of Laplace transforms not computable.

8 The expression of the option's gamma is not provided in the paper, but it can be easily obtained by differentiating (19(19) $\mathrm{\Delta }(S)={\mathrm{e}}^{-rT}[K{\mathcal{L}}^{-1}\{{\mathrm{\Lambda }}_{1,\theta }^{\prime }(S)\}(T)+{\mathcal{L}}^{-1}\{{\mathrm{\Lambda }}_{2,\theta }^{\prime }(S)\}(T)],$(19) ) with respect to S. What's more, when there are no psychological barriers (or equivalently, when the model (1(1) $\frac{\mathrm{d}{S}_t}{S_t}=r\mathrm{d}t+\sigma (S_t)\mathrm{d}{B}_t,S_0=S>0,$(1) ) becomes an ordinary CEV process), the closed-form solutions for the delta and gamma can be found in Larguinho, Dias, and Braumann (2013). Based on those results, one may compare the option's deltas and gammas with and without psychological barriers. Obviously, in the latter case, the option's gamma is a continuous function with respect to the initial underlying price.

9 One definition of $\{L_t(x),t\ge 0\}$ is given by the following limit form (see Protter 2004, 225): $L_t(x)=\underset{ϵ\to 0}{lim}\frac{1}{2ϵ}{\int }_0^t{\mathbb{1}}_{\{|S_s-x|<ϵ\}}\mathrm{d}⟨S{⟩}_s,$ where $\{⟨S{⟩}_t,t\ge 0\}$ denotes the quadratic variation process of $\{S_t,t\ge 0\}$.

Additional information

Funding

We acknowledge financial support from the National Natural Science Foundation of China (11801407, 71701145, 71790594, 11631004, 71532001), and MOE (Ministry of Education in China) Project of Humanities and Social Sciences (17YJC790146).