Stochastic Discounting and Convexity of Cash-Account Call Prices

Summary

The document compares the price of a European call written on a one-dollar cash account when the discount factor is stochastic with the price under a deterministic discount factor having the same expectation. It expresses the random-rate call payoff using the stochastic discount factor and applies the convexity of the positive-part payoff. Jensen’s inequality then gives a lower bound: the stochastic-discount-factor call price is at least as large as the corresponding price computed using the expected discount factor.

This establishes a price comparison for the specified cash-account payoff and matching expected discount factor. The passage introduces the question of how stochastic interest rates affect implied volatility for equity calls, but the displayed argument alone does not settle that broader comparison. It explicitly notes independence between rates and stock price for a preceding observation, while the derivation shown is not a general result for equity options with arbitrary rate-stock dependence. No numerical example or empirical evidence is supplied, and the excerpt does not spell out conditions for converting the price inequality into an implied-volatility ordering.

Key ideas

• The cash-account call payoff can be written as the positive part of one minus strike times the stochastic discount factor.
• Convexity of the positive-part function and Jensen’s inequality yield a lower bound using the expected discount factor.
• The bound compares cash-account call prices under stochastic and deterministic discounting with the same expected discount factor.
• The displayed argument does not by itself establish a general implied-volatility ordering for equity calls.

Tags

# Implied Equity Volatility under Stochastic Interest Rate


# 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.

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