In Reading 29, section 4.1 which is about Dynamic Hedging, the curriculum keeps saying that a Delta Hedged portfolio should earn the risk-free rate… How???
I’ve gone through plenty of threads here but no convincing answer…
In a Delta Hedged portfolio, if stocks go up by 1USD, the calls will go down by 1USD × Delta. Assuming an astounding perfect hedge (no gamma effect, change in price, mis-pricing of option, etc.) the value of my portfolio should not change, i.e. change in value = 0, and not a risk-free.
So a Delta Hedged portfolio is given by
V = – N × c + N × Delta × S
where N = number of written call options; c = call option premium, S = stock price
In this model, I don’t see what could possibly earn the risk-free rate! It’s not a swap exchanging a risky return for the risk-free, it’s not a forward locking the return… it’s a scheme where ∆S × Delta should offset ∆c.
True that, ignoring Gamma effects and other things, Delta decreases with time till it goes to zero at expiry… but so does the call price ceteris paribus. Now, I’m not sure if Delta and the call price are supposed to decrease at the same pace, but I guess so. Anyways, having factors decreasing with time date still does not explain what’s the source of the risk-free.
Are the proceeds from the short position invested at the risk-free? If so, then just (N × c) should grow at the risk free… not V. Anyways those proceeds are not invested at the risk-free… they are used in the purchase of the stock (as the sign in the model above shows).
I understand the arbitrage pricing principle that a hedged/riskless amount of money invested should earn the risk-free rate… but in this specific case I cannot see the economic explanation. Please help!