American put option with regime‐switching volatility (finite time horizon)—Variational inequality approach

We study the fair price of American put option with regime‐switching volatility. Assuming that volatility σ( t ) takes two different values σ 1 and σ 2 , applying Δ hedging technique we obtain a system of evolutionary variational inequalities, which possesses two free boundaries (optimal exercise boundaries). The following are the main results of this paper. Two free boundaries are monotonic and infinitely differentiable. The optimal exercise boundary of American put option with regime‐switching volatility in the bearish (or bullish) market is smaller (or higher) than the one of standard American put option. And the price of American put option with regime‐switching volatility in the bearish (or bullish) market is higher (or smaller) than the one of standard American put option. The solution of problem (1) is unique. These results are original in the option pricing with regime‐switching volatility, the proof is technical.