Approaches to Pricing Zero-Rebate Barrier Options Under Heston
Summary
The document asks how to approximate a down-and-out call price when the underlying follows the Heston stochastic-volatility model. It states the option payoff with a barrier condition, contrasts the problem with a Black–Scholes closed-form expression, and supplies a Fourier integral for a Heston hitting probability in the special case of zero correlation and zero drift. A Riccati equation and related terms define part of that expression.
This provides a useful starting point for researchers considering barrier pricing: barrier survival depends on the path, and a hitting-probability result may inform an approximation. However, the text is a question rather than a proposed pricing method or validated result. Its supplied probability is restricted to special parameter assumptions and does not directly provide the requested option price under general Heston dynamics. No numerical evidence, error analysis, or comparison of approximation methods is presented.
Key ideas
- A down-and-out option’s value depends on whether the underlying crosses its barrier during the option’s life.
- The document contrasts a Black–Scholes formula with the absence of a stated closed form under Heston dynamics.
- It gives a Fourier representation for a hitting probability under zero correlation and zero drift.
- The hitting-probability result is a restricted starting point rather than a complete general pricing approximation.
- The document does not report numerical validation or compare candidate methods.
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# Zero-rebate barrier option pricing under the Heston model
# Zero-rebate barrier option pricing under the Heston model
I'm trying to derive an approximation for the zero-rebate barrier option under the Heston model: $$dS_t=\mu S_tdt+\sqrt{v_t}S_tdW^S_t$$ $$dv_t=\kappa(\bar{v}-v_t)dt+\eta\sqrt{v_t}dW^v_t,\quad d\langle W^S,W^v\rangle_t=\rho dt$$ The payoff of a down-and-out option is: $$\mathcal{C}_T=(S_T-k)\mathbb{I}_{\{S_T\geq K\}}\mathbb{I}_{\{(\max_{0\leq t\leq T}S_t=:)m_T\geq H\}}$$
Under the Black-Scholes dynamics we have the closed-form solution: $$\mathcal{C}_t=S_te^{r\tau}\left(\Phi\left(\frac{\ln(S_t/K)+\nu\tau}{\sigma\sqrt{\tau}}\right)-\left(\frac{H}{S_t}\right)^{1+2r/\sigma^2}\Phi\left(\frac{\ln(H^2/(S_tK))+\nu\tau}{\sigma\sqrt{\tau}}\right)\right) - K\left(\Phi\left(\frac{\ln(S_t/K)+(\nu-\sigma^2)\tau}{\sigma\sqrt{\tau}}\right)-\left(\frac{H}{S_t}\right)^{-1+2r/\sigma^2}\Phi\left(\frac{\ln(H^2/(S_tK))+(\nu-\sigma^2)\tau}{\sigma\sqrt{\tau}}\right)\right)$$ where $\nu=r+\frac{\sigma^2}{2}$ Obviously, under the Heston dynamics we don't have a closed-form solution. However, I would like to know whether there exists approximations allowing to price the down-and-out call option in a similar fashion.
As a starting point, I can use the result of the hitting probability of Heston for $\rho$ and $\mu$ equal to 0, i.e. the probability that during the time interval $[0,t]$ the process $S_t$ was positive, with $x$ and $y$ are initial values, solved via the Fourier transform in $x$: $$\frac{2}{\pi}\int^\infty_0 \frac{\sin (\omega x)}{\omega}\bigg(\frac{\Delta(\omega)e^{-m_-(\omega)\kappa t}}{m_+(\omega)+m_-(\omega)e^{-\Delta(\omega)\kappa t}}\bigg)^{\frac{2\theta\kappa}{\xi^2}}e^{-\frac{2y\kappa}{\xi^2}B(\omega,\kappa t)}d\omega$$ where $B(\omega,\tau)$ is a solution of the Riccati equation $$\frac{\partial B}{\partial \tau}(\omega,\tau)=-B(\omega,\tau)-B(\omega,\tau)^2+\frac{\xi^2 \omega^2}{4\kappa^2}, B(\omega,0)=0;$$ $$\Delta(\omega)=\sqrt{1+(\xi\omega/\kappa)^2}, m_{\pm}(\omega)=\frac{\Delta(\omega)\pm 1}{2}$$
Any hints in this direction as much as completely different solutions are much appreciated.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.