I have read most of Shreve's Stochastic Calculus for Finance II. In it, the author prices various option types assuming an interest rate that is constant with respect to time.

We can expand this model using Hull-White to deal with stochastic interest rates. In this case, the interest rate is a stochastic process with respect to time.

In both of these scenarios, it appears to me that the "interest rate" is the short rate. Suppose you are hedging a short position in the call option. Suppose that the current value of the stock $S(t)$ at the current time $t$ requires a positive investment in the money market. A sudden change in $S$ from time $t$ to time $t+\epsilon$ could force the hedger to hold a negative investment in the money market. In this example, the hedger can flip from a positive investment to a negative investment in the money market in an arbitrarily small amount of time. That is why we are considering the short rate.

The Question

This all makes sense if the goal is a perfect hedge. However, yield curves are often not flat, and a hedger may decide that they will hold a positive position in they market account for a long period of time with high probability. This probability may be increased if the hedger is hedging options on multiple assets. In this case, the hedger may speculate by making a long term investment in the money market for the higher interest rate. Are there any good resources like books or research articles that talk about dealing with this situation?



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