Put-Call Parity with Regime-Dependent Rates and Dividends

Summary

The document asks how to apply put-call parity when interest rates and dividend yields vary across states of a Markov regime-switching model. It describes a setting in which each regime may also use a different underlying price model, and transitions are governed by a continuous-time Markov chain. In the constant-rate case, the usual adjustment to put prices uses the present value of the underlying less the present value of the strike.

The text raises the problem but supplies no derivation, formula, numerical example, or answer. In a regime-switching setting, the relevant discounting and expected dividend treatment generally depend on the path of regimes over the option’s life, so a state-by-state substitution into the constant-parameter formula is not justified without further assumptions. The document is useful as a statement of a derivatives-pricing question, but it does not establish a solution or explain how to implement one.

Key ideas

• The question concerns put-call parity when rates and dividend yields change with a Markov regime.
• Each regime may have its own parameters and even a different underlying price model.
• The constant-parameter parity adjustment cannot simply be applied independently by regime without additional justification.
• The document provides no answer, derivation, or numerical evidence.

Tags

# Put-call parity under a regime-switching model


# Put-call parity under a regime-switching model












I need some help. I'm given $J$ different regimes, each one characterized by its own parameters $(r_i, \delta_i,\sigma_i,...)$ with $i\in \mathcal{J}= \{1,2,...,J\}$ ($r$ = risk-free interest rate, $\delta$ = continuous dividend yield). For instance, we can have regime $1$ under Black-Scholes model with parameters $(r_1,\delta_1,\sigma_1)$ and then regime $2$ under Variance-Gamma model with parameters $(r_2,\delta_2,\sigma_2,\nu_2,\theta_2)$. Let $\alpha_{t\{t\in[0,T]\}}$ be the Markov chain taking values $i\in \mathcal{J}$ and $Q := \{q_{ij},1 \leq i,j \leq J\} \in R^{J×J}$ its associated intensity matrix (also known as generator), such that the matrix of transition probabilities is defined as $P(\Delta t)=exp(Q \Delta t)$. In this framework, I need to price call options in Matlab exploiting the put-call parity, however if $r$ and $\delta$ were constant for all the regimes then, given that $Prices$ is the vector containing the put prices for each regime, I could write \begin{align} & for \ \ j=1:J \\ & \ \ \ \ \ \ Prices(j)=Prices(j)+S0*exp(-\delta*T)-K*exp(-r*T); \\ & end \end{align} where $S0$ is the spot price, $K$ is the strike price and $T$ the maturity. How can I generalize the put-call parity with $r_i$ and $\delta_i$ depending on state $i$? Any help will be highly appreciated.

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