# [SOLVED] Delta Hedging an Option over Time - Risk Free?

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!

Assuming a perfect hedge (again ignoring Gamma, time effect, volatility, etc.) I see an arbitrage opportunity if the portfolio didn’t earn the risk free… but is the answer to my question a matter of pricing??? But then, which component is earning the risk-free?

V = N × [Delta × S – c]

so to get V × ( 1 + rf × 1/T ) I must/should have that

N × [∆Delta × S – ∆c] = V × rf × 1/T

Is that the case??? Which means that as time passes, the call price should decrease by an amount that guarantees the risk-free ceteris paribus???

The proceeds from the short sale are invested in bonds (at assumed to be the risk-free rate). Vice versa if long an option the premium is financed by borrowing at the risk-free rate.

It follows the same principle as forward contracts, i.e. F = S * (1+r)n, if you own the underlying and short forward you earn the risk-free rate.

Thanks for replying, but I don’t think this is the answer. As I mentioned in my question:

The curriculum clearly states that the whole portfolio should grow at the risk-free rate. Plus, the value of the portfolio is given by:

V = Ns × S – Nc × c = Nc × Delta × S – N × c

which means that the proceeds from writing calls are quite naturally used to purchase the stocks, as needed for the Delta Hedge. If proceeds were invested at the risk-free, I should have a third component in the equation, shouldn’t I?!

And again (using continuous compounding): V should increase to V × et/365 and not just V + N × c × et/365

In the curriculum in Exhibit 22, I see mentioned “Value of Bonds Purchased”… but this is an odd naming to refer to the loan taken an repaid to adjust the number of stocks as Delta changes.

Whatever! I’ll crack this puzzle after the exam. Now I’d better just accept it as a dogma and focus on studying and practicing

Found a good answers here:

https://quant.stackexchange.com/questions/18336/why-how-does-a-hedged-portfolio-make-profits/18342#18342

Another explanation, more quantitative, here:

https://quant.stackexchange.com/questions/35707/a-simple-question-cost-of-delta-hedging-when-a-call-option-is-sold

but here the best answer is puzzling me… the best answer says that the P&L of a Delta Hedged portfolio is:

I think I’ve found the answer!!!

https://quant.stackexchange.com/questions/39718/how-does-a-delta-hedged-portfolio-yield-the-risk-free/39728#39728

Don’t forget that the delta of a call is convex. Even if at a point in time your portfolio is delta neutral, the moment the underlying price changes, yes the portfolio value will not change, but the delta will change and if you want to keep the portfolio delta-neutral, you will have to rebalance by purchasing or shorting further calls. The calls you buy or sell will need to be priced in such a way that the portfolio earns the risk-free rate, otherwise there would be arbitrage opportunities (that you are missing or someone is taking advantage of, if priced differently).