Financing Cash in a Multi-Period Delta-Hedged Option Portfolio
Summary
This discussion explains the financing term in the profit-and-loss formula for a delta-hedged option position. At each hedge adjustment, the stock position and option liability or asset leave a cash balance; a nonzero balance must be borrowed or invested, and its interest contributes to P&L in the following period. The answer applies this logic to a market maker short an option and holding a delta hedge, addressing why the cash account changes as the option and hedge values move.
It also describes the source of residual P&L: discrete rebalancing leaves exposure to the option’s nonlinear relationship between price and delta. In the idealized continuous-trading model, the hedge P&L is zero, while finite rebalancing intervals can produce deviations that generally shrink as rebalancing becomes more frequent. The exchange gives a conceptual explanation rather than a worked multi-period ledger, and the zero-P&L statement relies on ideal assumptions; real trading includes costs and other sources of error.
Key ideas
- The financing balance is determined by the current stock position minus the option value.
- A nonzero cash balance earns interest when invested or incurs interest when borrowed.
- Rebalancing changes both the hedge and the cash account over time.
- Discrete hedging leaves residual P&L because option delta varies nonlinearly with the underlying price.
- Continuous rebalancing would eliminate hedge P&L in the idealized model.
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# Profit and Loss on delta-hedged portfolio
# 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!
## Answer by Sanjay (score 1)
https://quant.stackexchange.com/a/44673
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 desiredShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.