# How does a Delta Hedged portfolio yield the Risk-free?

Here I'm considering the simple case of a dealer writing call options on a stock and hedging the short position with a "textbook" Delta Hedge, i.e. goes long on $N_c \times Delta$ stocks (where $N_c$ is the number of written calls and $Delta$ is the Delta of the options).

The value of my portfolio is given by: $$V_t = N_c \cdot Delta_t \cdot S_t - N_c \cdot c_t = N_c ( Delta_t \cdot S_t - c_t )$$

Assuming that nothing changes (stock price, delta, etc.) and pretending that the hedge is really effective, i.e. the portfolio is really risk-free and there is no Gamma effect or anything else that could disrupt the hedge, then this portfolio should earn the risk-free rate... Why?

I understand the arbitrage pricing principle that a hedged/riskless amount of money invested should earn the risk-free rate, but I cannot see the economic explanation.

What are the components growing at the risk-free? What is the arbitrage strategy that would force the value of a Delta hedged portfolio to drift at the risk-free rate?

Here some more thoughts on this "dilemma":

• I know that $Delta_t$ and $c_t$ actually decrease with time till they go to zero at expiry of the option ceteris paribus (please, correct me if I'm wrong)... but still I don't see how this would explain any risk-free yield.

• Very often, I see the argument saying that the proceeds from the short position on the calls is invested at the risk-free rate. But this cannot be the explanation, can it? Aren't my proceeds expensed in the purchase of the stocks? And second, the posit is that the whole delta hedged portfolio yields the risk-free and not just the short call proceeds.

• Here I found a nice explanation by Alex C: Why/How does a hedged portfolio make profits? saying:

An Investment Bank earns a profit by selling you an option at a slightly higher price than the theoretical price, or buying it back from you at a slightly lower price. They call this "earning a spread". Then they hedge the option, so as not to make any [further] gains or losses on it (other than the risk free rate).

Another way they could earn a profit is if they have a more accurate estimate of volatility than other people have. But that is not easy to do consistently.

However this is just telling me how option writers charge me more to make a profit. It is not telling me what is the arbitrage opportunity that compels the value of that portfolio to drift at the risk-free rate.

• Where in the formula for the call price (and the Delta) do I see that there is a drift that would "magically" make a delta hedged portfolio drift at the risk-free rate?
• You will not find what you are looking for. The requirement that the portfolio earns the risk free rate is something we are imposing in order to calculate the option price.
– dm63
May 10 '18 at 13:09
• $Delta_t$ and $c_t$ do not (necessarily) go to zero; they are random variables. In particular, if the option expires in the money, each will have positive value (and $Delta_t$ will go to one).
– Drew
May 12 '18 at 2:44

I'll put here the answer provided in a comment by @dm63 (thanks by the way):

The requirement that the portfolio earns the risk free rate is something we are imposing in order to calculate the option price.

I disagree with the view that it’s only a definition. There has to be a physical reality underlying the claim. One cannot just decree that the portfolio grows at the risk-free rate.

Note the portfolio grows at the risk-free rate, it doesn’t necessarily mean you always earn it, you may well be paying it.

In your particular example, the dealer has received the Call premium $C$ and has borrowed $\frac{\partial{C}}{\partial{S}}-C$ to fund its delta hedge.

That term is the value of the portfolio, and since it is money borrowed (by model assumption at the risk-free rate), it costs the risk-free rate to its holder.

• In every reference I've found on the topic, authors/textbooks explicitly show the portfolio yielding the risk-free rate: a positive profit, not a cost. This value is then used as benchmark to see, at $t+1$, how much is the portfolio is off. As you said, the strategy involves borrowing funds and selling stocks each time the portfolio is rebalanced... but that is always shown as a "separate topic". At each period, a new benchmark would be calculated for the next period as $V_{t+1} = V_t \cdot e^{rt}$ May 10 '18 at 19:40
• Right but nothing preclude $V_t$ from being negative. It still "grows" at the risk-free rate, or "yields" the risk-free rate, only negatively i.e. it's a cost to the holder.
– Ivan
May 10 '18 at 20:17
• I agree with you... but textbooks don't 😅I think academics are just repeating a mantra without inquisitive spirit. I swear I couldn't find a paper or a person showing an arbitrage strategy proving that $V_{t+1} = V_t \cdot e^{rt}$ must hold. That's why I got the feeling that, so far, the best explanation is the one given by @dm63... his explanation probably comes from practical experience and also somehow matches the idea that $V_t \cdot e^{rt}$ is "just" a desirable outcome (represented as a benchmark) and not an inevitable result of arbitrage pricing rules May 10 '18 at 20:53