I’ve seen some previous posts around this subject, but none have definitively addressed it.
Can someone please explain why a delta hedging strategy should earn the risk free rate, both from a mathematical perspective and from a practical one?
My understanding is that all things equal (i.e. if the stock price does not change) the delta hedged portfolio will earn the risk free rate due to the decrease in delta (which i am assuming is because of time decay, is that correct?)
Also, if the stock price does move (increase for example) delta will move just enough (i.e increase from the stock price increase and decrease from the passage of time) to make it so that the portfolio is earning the risk free rate. Am i correct in my reasoning here?
I think that if one were to prove this mathematically, it would be a byproduct of the black scholes, given that the options delta is one of the terms of the BS.
If someone could confirm/refute my reasoning and provide a good mathematical explanation, that would be much appreciated.
I don’t think it’s because the movement is just enough to ensure you earn the Risk free rate, if you are fully hedged you are eliminating the said risk exposure and therefore earn the risk free rate. If you were to earn nothing the hedging strategy would not be worth implementing. Same principle applies in currency hedging for example when you eliminate foreign exposure by hedging the currency exposure AND the equity exposure.
Well thanks for taking the time to reply, but stating that eliminating risk exposure should earn you the risk free rate is as generic as they come. I was looking for a more mechanical i.e. mathematical proof. I am sure that this is an inherent property embedded within any option pricing model i just haven’t figured out how to prove it yet. Any other thoughts?
The risk free rate (and time) is the only unhedged variable in your pricing model. If your portfolio is delta neutral, then you earn the risk free rate.
If you hold 100 stocks for $100, and short 100 calls at a strike price of $1, and the price now is $1 at a delta of one, you’re premium is invested at the risk free rate, and the call payoff and the stock pay-off hedge one another (until the stock moves down, then you have to sell some shares).
Investing your premium in the risk free rate ensures that only your premium will earn the risk free rate, what about the rest of the portfolio?
Investing your premium and/or proceeds from the sale of the underlying asset in the risk free rate ensures that your portfolio CONTINUES to earn the risk free rate.
The fact that a delta hedged position earns the risk free rate has nothing to do with investing your premium at the risk free rate.
What i’ve been able to infer thus far is this, given a delta hedged portfolio and assuming no change in the underlying asset (until the expiration of the derivative), then that derivative’s (option) payoff is known, and therefore its price is the PV of its payoff at the risk free rate. which means that the portfolio will end up earning the risk free rate.
Generic explanations like (delta hedging makes your portfolio delta neutral and then you earn the risk free rate) are not helpful in this context, they add no value.
Although it isn’t easy to type the math here. But if you know how to derive the BSM equation you’ll find out why the portfolio earns the risk free rate. The reason the portfolio earns the rf rate is the fundamental reason behind how the equation came up. The portfolio consisting of ‘delta’ shares and short option. Doing the math (you can check it up on the net) you eliminate the delta times change in share price terms and the randomness in the equation. This removal of uncertainty says that the portfolio should earn the risk free rate over time. Sorry for being wordy on it. The math is too time taking here. Will do it for you when free (since I’m going this out from my mobile). Before that ensure you know a bit of stochastic calculus.
The delta of a portfolio should be zero, the delta of a stock is 1, the delta of the options should be -1, hence the hedge ratio that would achieve this by the multiplier of the number of contracts, or simply, changing the number of shares.
If you have a stocks worth $100 at a price of $1, and the delta is 0.5, then you can write call options for 200 shares, or buy put options for 200 shares. Let’s say the risk free rate is 10%. and the contracts expire in one year.
In the put example, if the stock goes to $9,999, what happens? You would have sold your shares as the price went up, and the delta of the put approached 0 (from -0.5 initially), so you did not really make a profit of $9,999 * 100, you only made $10, the risk free rate as you dynamically hedged the portfolio. So the final portfolio value was known beforehand, and it would have always ended up at $110, no matter what the stock price did.
If the price went to $0, then you would have bought 100 more shares gradually as the delta approached -1 for cumulative losses, or sold put options for cumulative profits, untill the exercise date where you would have either exercised all the put options for a $200 profit - premium paid - initial cost of shares - total value paid for the 100 extra shares at declining prices = $110, or $100 + realized put option selling profits - preimum paid - initial cost of shares = $110, any cash on hand would be invested at the risk free rate.
It would be difficult to write out the maths, you need a MCS to see if this is true. But that’s the theory anyway.
It may help to think in terms of the binomial option pricing model that uses risk neutral probabilities. A delta hedged portfolio (delta*Stock minus Call) will then earn the risk free rate since the risk neutral pricing equation holds for the derivative as well as its underlying
Mr. Smart I’m not sure who this asshat hadikadi is but if he’s not satisfied with this response he wont be satisfied with anything!! Let him be that guy who fails because he focuses too much on every little thing.
Mr. Smart’s explanation does not seem plausible, in my opinion. Continuing with his example of 100 shares long @ $1/share initial price, delta hedged with put options, lets assume that the delta hedging is carried out by adjusting the number of put options. If the stock increases to $10,000/share and the hedger’s total profit is indeed limited to $10 (and not $999,900), that implies a cumulative loss on his put options of $999,890. The long delta exposure is constant at 100 (100 shares long). The put option’s delta decreases (in absolute terms) as share price increases. Assuming a low delta of 0.001, the max number of put options that would be required to delta hedge = 100,000. That translates into an average premium of $9.9989 per $1 strike put ($999,890 put premium lost / 100,000 puts). Given that put premium for the $1 strike would reduce drastically in this scenario while the number of puts purchased at lower premiums increases drastically in order to offset the deteriorating delta as share price increases, the $9.99 AVERAGE premium implies a much higher than average premium for the $1 strike put to begin with. Such a high option premium that is significantly higher than 10x the beginning share price is not plausible.