The overnight profit formula from a textbook (possibly Derivative Markets by McDonald) is the following:
where Delta is the delta of the option, S is the stock price, t is a specific point in time, h is a small movement in time, V is the value of the option, r is the continuous risk-free rate.
Assuming the market maker sold a put and delta hedged by shorting the stock.
My question is that if this is applied in a multi-period fashion, are we assuming that we are losing interest on the option we sold?
For example: at time
t, we sold a put for
7 dollars and shorted
50 worth of stock to delta hedge. At time
t+h, the put is worth
15 dollars and we sold additionally 40, making it
90 to delta hedge. Our profit from the last term of the equation is (a positive number since delta of a put is negative).
Then in the next period, the interest would be
To me, this is counter-intuitive, because if we bought a call for 5 dollars and delta hedged at time t, and if the call price shot up 10 dollars at t+h, it would mean we're earning interest on the 15 dollars from time t+h to t+h+j, for some j in the future, when we only paid for the call at time t for 5 dollars.
A somewhat related question that may be too simple to start a new question is that is there a way to arrive at the profit/loss number just by looking at the portfolio value alone? For example, say MM sold a put and shorted stocks to delta hedge--the portfolio consists of a shorted put, shorted stock, and cash that is presumably invested in risk-free bonds. From time to time, the value of the stock and put changes, and so does the portfolio. Will it be possible to construct a portfolio that consists of the shorted put, shorted stock, risk-free bonds, and the interest earned, so that the profit/loss can be ascertained from the closing value of the portfolio value?