Implied Equity Volatility under Stochastic Interest Rate

I would like to draw some general conclusions for the effect of stochasticity of interest rate on the implied volatility of a European call of a stock. Below I show, trivially, the implied volatility of a European call on a cash account with stochastic interest rate is higher than one with the same expected discount factor if the interest rate is independent of the stock price.

The question is: what can we say in general terms about the relative magnitudes of the implied volatility when the interest rate is stochastic?

Consider a European call on a cash account of one dollar with a stochastic interest rate. Let the stochastic discount factor be $d$ \begin{align*} C(K) &:= E\big[d(d^{-1}-K)_+\big] = E\big[(1-Kd)_+\big] \\ &\ge \big(1-KE[d]\big)_+ = E[d]\bigg(\frac1{E[d]}-K\bigg)_+ =: C_0(K), \end{align*} $C_0(K)$ stands for the call price when the interest rate is not stochastic but with the same expected discount factor.

• The payoff does not look like that of a European option. Should it be $C(K) = E(d(S-K)^+)$? – Gordon Oct 25 '16 at 12:51
• @Gordon: You are absolutely right. I have corrected the statement of my question. It seems it is hard to say something general even when the interest rate is independent of the stock price. The stock price is not independent of the discount factor even if the interest rate is instantaneously independent of the stock price. – Hans Oct 25 '16 at 19:34
• The answer for this question should give you a clue for the relationship of the volatilities. – Gordon Oct 25 '16 at 19:50
• @Gordon: Nice analytical solution for the hybrid geometric Brownian stock price. The conclusion is what I expected. For other more complicated models, say Heston model with Hull-White, we can use the maximum principle of hyperbolic PDE to derive some inequality of the option price with respect to the volatility of the interest rate and the instantaneous correlation between equity and interest rate. I am wondering more about what can be said more generally than some specific diffusion models. – Hans Oct 25 '16 at 20:52
• For the general case, I have not yet considered. – Gordon Oct 25 '16 at 20:56