# Profit and Loss on delta-hedged portfolio

The overnight profit formula from a textbook (possibly Derivative Markets by McDonald) is the following:

$$\Delta _{t}(S_{t+h}-S_{t})-(V_{t+h}-V_{t})-(e^{rh}-1)(\Delta_{t}S_{t}-V_{t}),$$

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).

$$-(e^{rh}-1)(-50-7)$$

Then in the next period, the interest would be

$$-(e^{rh}-1)(-90-15)$$

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?

Thanks!

• @AlexC But say, we sold a put option, we also need to short the stock, so we're not borrowing any money but we're getting money from the proceed from the stock, and investing it in RF instruments? If this is the case, the option value at some time in the future should have no effect on our interest earned? – user101998 Mar 19 '19 at 15:26
• Sorry, I was thinking of a call instead of a put. – Alex C Mar 19 '19 at 15:29
• @AlexC No worries, but I think a call and a put should be analogous, so the value of the option sold at time t shouldn't affect the interest earned (in my mind)... – user101998 Mar 19 '19 at 15:30
• hmm.. What is te meaning of the word MM? – Sanjay Mar 19 '19 at 15:46
• @Sanjay Market Maker. I'll make the edit. – user101998 Mar 19 '19 at 15:53

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?

Yes! you do pay/receive interest if you borrow/store money.

In a delta hedge portfolio at any time $$t$$ you should have $$\Delta_t$$ amount of the stock.

$$\Delta_{t}S_{t}-V_{t} = \text{Value of postion in stock - value of position in option}$$

If $$\Delta_{t}S_{t}-V_{t} \neq 0$$ you need to borrow/store money and pay/receive interest. When you change your position from time to tome the then your money account changes as well. There is no Fokus/Pokus in that.

And Yes, it is possible to make/lose money in a Delta hedged portfolio mainly (but not solely) because of the non-linearity of $$\Delta(S_t)$$ in $$S_t$$. In theory the PnL (Profit-Loss) is zero if you trade continuously which is not possible in reality. When you time between rebalancing periods ($$h$$ in your notation) becomes smaller and you rebalance more often then PnL becomes closer to zero as desired

• But it seems inconceivable that I'd be earning money on the put I sold a while back, when I'm not getting any further cash flow from it? – user101998 Mar 19 '19 at 15:35
• You sold a put? What did you sell it for? Money, right? That money earns interest. – user3296 Mar 19 '19 at 22:40
• @user3296 Sure, but if the value of the put went up, why am I earning interest on the additional value that I didn't have? – user101998 Mar 20 '19 at 14:35