Boundary Conditions for an American Put Under Stochastic Volatility and Rates
Summary
The document derives a pricing PDE for an American put when the underlying price, volatility, and interest rate are stochastic and potentially correlated. It then asks how to specify boundaries beyond the terminal payoff and the large-underlying-price condition, including boundaries for zero and very high volatility and a zero interest rate.
The proposed conditions use reduced PDEs at zero volatility and zero rates, while the high-volatility limit sets the option value to zero. The source gives equations but no numerical experiments or validation, and it does not resolve every modeling issue: the early-exercise boundary is stated as a condition without a derivation, and the appropriateness of the limiting conditions depends on the chosen stochastic processes and numerical scheme. Readers should check the coefficients and assumptions against their own model before implementation.
Key ideas
- The pricing PDE includes drift, diffusion, cross-derivative, and discounting terms for the asset, volatility, and rate.
- At zero volatility, the document proposes a reduced PDE in the asset and interest-rate dimensions.
- At very high volatility, it states that the put value tends to zero.
- At a zero interest rate, it proposes a reduced PDE with volatility dynamics and a price-volatility cross term.
- Boundary conditions should be validated against the model assumptions and numerical implementation.
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Full text
# Boundary conditions of PDE from SV model with stochastic interest rate
# Boundary conditions of PDE from SV model with stochastic interest rate
The PDE for the American put option price $P(S,\sigma ,r,t)$ is \begin{align*} 0 =& P_t+P_SS(r-\delta)+P_\sigma a(\sigma)+P_r\alpha (r,t) \\ +& \frac{1}{2}P_{SS}S^2\sigma ^2 + \frac{1}{2}P_{\sigma \sigma}b^2(\sigma)+\frac{1}{2}P_{rr}\beta^2(r) \\ +& P_{S\sigma}\sigma Sb(\sigma)\rho _{12}+P_{Sr}\sigma S\beta(\sigma)\rho _{13}+P_{\sigma r}\beta(\sigma)b(\sigma)\rho _{23}-rP \end{align*} that was extracted from stochastic system \begin{align*} dS_t &= (r_t-\delta)S_tdt+\sigma _tS_tdW_t^{(1)} \\ d\sigma _t &=a(\sigma _t)dt+b(\sigma _t)dW^{(2)}_t\\ dr_t &= \alpha(r_t,t)dt+\beta (r_t)dW_t^{(3)} \end{align*} such that $$ dW^{(i)}_tdW^{(j)}_t=\rho_{ij}dt $$
I found this boundary conditions \begin{align} & P(\infty ,\sigma ,r,t)=0 \\ & P(S,\sigma ,r,T)=\max (K-S,0) \\ & P(\bar{S}(T-t),\sigma ,r\,,t\,)=\max (K-\bar{S}(T-t)\,,\,0\,) \\ & {{P}_{S}}(\bar{S}(T-t),\sigma ,r\,,t)=-1 \\ \end{align} Here, $\bar S(T-t)$ is the early exercise price, which depends on the option time-to-maturity $\tau =T-t$.
Now How can I find others boundary conditions ?
## Answer by user16651 (score 1)
https://quant.stackexchange.com/a/18519
When $\sigma=0$ , the boundary condition is little more complicated: \begin{align} P_t+(r-\delta)SP_S +\alpha P_r +\beta^2\frac{1}{2} P_{rr}-rP=0 \end{align} When $\sigma\rightarrow\infty$ , we have \begin{align} P(S,\infty,r,t)=0 \end{align} When $r=0$ , then \begin{align} P_t+aP_\sigma+\frac{1}{2}b^2P_{\sigma\sigma}+\sigma S b \rho_{12}P_{S\sigma}=0 \end{align}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.