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Say I am a market maker.

The ask (me selling it) formula is pretty common in textbooks etc by no arbitrage: $$F = S \cdot \text{exp}(r-d)$$

where $r$ is interest rate and $d$ -- dividend.

Again, by no arbitrage, if I am buying it, i.e. quoting a bid, would it be: $$F = S \cdot \text{exp}(d-r)$$

?

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  • $\begingroup$ Say $d=0$ and $r=0.1$, what would happen? $\endgroup$ – Bob Jansen Mar 11 at 9:01
  • $\begingroup$ @BobJansen We would have a spread between bid and ask, as expected. $\endgroup$ – A.L. Verminburger Mar 11 at 13:19
  • $\begingroup$ Seems suspiciously large though. What if $r<0$? $\endgroup$ – Bob Jansen Mar 11 at 17:55
  • $\begingroup$ @BobJansen No difference. But I worked from first principles. Turns out get same formula. Just high level seemed like it should be the reverse inside exponent (thinking in terms of rates, divs and repo) for reverse carry. Still strange that examples are always given as plain (ask) carry when deriving this and never the reverse. I guess any bid / ask is purely commission around this mid. $\endgroup$ – A.L. Verminburger Mar 11 at 18:50
  • $\begingroup$ Might be worthwhile to post your derivation as an answer. Personally, I don't remember examples being given as either ask or bid but as a mid price. Maybe, I never noticed the ambiguity. $\endgroup$ – Bob Jansen Mar 11 at 18:52
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The typical no arbitrage argument for carry (price change) comes from a bid perspective. I am selling you a forward (delivery of underlying) for some price $F$ in $\tau$ future term. I can go into the market and buy the underlying for $S$. To do that I need to borrow $S$.

Now there are two ways to go about borrowing and I am not sure which one happens in practice.

Let's say I just go and get an unsecured loan at a LIBOR rate of $r$ -- this is a financing expense. However I will get some benefit from dividend $d$ the underlying is paying. I want to be neutral at contract expiry so:

$$F-S \cdot \text{exp}\Big((r-d)\tau\Big) = 0$$ $$F = S \cdot \text{exp}\Big((r-d)\tau\Big)$$

Another way to look at borrowing would be from a secured perspective. I buy the underlying and immediately "lend it out" as collateral to counter-party C. Counter-party will transfer me the dividends $d$ and I will pay some secured rate $r'$ (presumably $< r$). End up with same expression, just different rate.

$$F = S \cdot \text{exp}\Big((r'-d)\tau\Big)$$

Because the rate would be lower I am assuming this is what actually happens in practice.

Now let's derive it from the reverse carry perspective.I am quoting a bid, so agree to buy underlying at future date for fixed price. Again, do not know how this happens in practice, so will assume can get an unsecured or secured loan of the underlying (not cash).

In an unsecured scenario I borrow the underlying stock from C, sell it immediately for $S$ and invest it in some unsecured money market account for $r$. I still have to pay C the dividend, so that's an expense. At maturity I will buy the asset for fixed price $F$ and return it to C. My portfolio at maturity $\tau$ should be hedged:

$$S \cdot \text{exp}\Big((r-d)\Big) - F = 0$$ $$F = S\cdot \text{exp}\Big((r-d)\tau\Big)$$

Another way would be that to borrow the underlying stock I need to post collateral in the form of cash and receive some reduced secured rate $r'$ -- but I guess it's nice to know that I get to keep the asset if C goes bust in return (can someone comment on which scenario actually happens in real life). I get the cash needed for collateral by immediately selling the borrowed underlying. Also pay dividend $d$ to C as previously. Get same equation, but with different (again presumably lower) rate:

$$F = S\cdot \text{exp}\Big((r'-d)\tau\Big)$$

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    $\begingroup$ The point is, your argument works exactly the same in reverse (ie from an offered perspective). The formula does not change. The only difference is that each variable in the formula needs to be taken at its bid or offer side depending what the hedge transaction is. Also, the rates are secured rates in practice. $\endgroup$ – dm63 Mar 13 at 15:44
  • $\begingroup$ @dm63 Thank you. Effectively is $r-r'=\text{repo rate}$ in my equations? $\endgroup$ – A.L. Verminburger Mar 14 at 6:33
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Here is the simple analogy

Price of the Food shop = Price of all the apples + Price of all the oranges + Price of all the equipment - Amount of Debt in the books.

Is the bid price the opposite?

*Ask the right question you will get the answer!

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