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Borrow, dividends and what happens at expiry

OPT · Chapter 512 min readAsked at Optiver, IMC, Akuna, Wolverine

Assumes Static arbitrage: the constraints every quote must respect.

After this lesson you should be able to

  • Put borrow cost and dividends into put–call parity correctly.
  • Decide when to exercise an American option early.
  • Manage pin risk and understand the mechanics of assignment.

The clean theory assumes you can short freely, that dividends are known, and that expiry is an instant. None of that is quite true, and the gaps are where option desks make and lose money. These are the details that separate someone who has priced options from someone who has read about them.

Equation 5.1

Parity, with the real-world terms

The spot term is discounted by the dividend yield qq — and a stock borrow cost enters in exactly the same place, because paying to borrow is economically identical to the stock paying a dividend you do not receive.

C−P=Se−qT−Ke−rTC - P = S e^{-qT} - K e^{-rT}
qq
Dividend yield, or dividend yield plus borrow cost for a hard-to-borrow name.
Se−qTSe^{-qT}
What a share is worth to someone who will not collect the dividends.

Proposition 5.2

Hard-to-borrow names

When a stock is expensive to borrow, the synthetic short built from options is cheaper than the real one, so puts trade rich to calls relative to naive parity. Reading the parity relationship backwards recovers the market’s implied borrow rate — which is often the only place that rate is visible at all.

Holds when

  • A borrow rate of tens of per cent shows up as a parity gap that looks like an arbitrage and is not.
  • The implied borrow can move violently, and a position that was flat becomes a bet on it.
  • Any apparent parity violation should prompt "what is the borrow?" before anything else.

Proposition 5.3

When to exercise early

Never exercise an American call on a non-dividend-paying stock: you would throw away the remaining time value and pay the strike early. With a dividend, exercising just before the ex-date can be right if the dividend exceeds the time value you give up. American puts are different — exercising early to receive the strike and start earning interest on it can be rational at any time, and deep in-the-money puts on low-volatility names frequently should be.

Holds when

  • Call: compare the dividend against the remaining time value of the corresponding put.
  • Put: compare the interest earned on the strike against the time value given up.
  • The American premium over European is largest for deep in-the-money puts and for calls facing large dividends.

Example 5.4

You hold a $50\$50 American call on a stock at $60\$60, going ex a $2\$2 dividend tomorrow. The $50\$50 put trades at $0.30\$0.30 and rates are negligible. Exercise?

Show the worked solutionHide the worked solution

Worked solution

  1. Formula
    exercise if D>P+K(1−e−rT)\text{exercise if } D > P + K(1 - e^{-rT})
  2. Substitute
    D=2.00,P=0.30,rT≈0D = 2.00, \quad P = 0.30, \quad rT \approx 0
  3. Solve
    2.00>0.30+02.00 > 0.30 + 0
  4. Answer
    Yes — exercise before the ex-date\text{Yes — exercise before the ex-date}

Sanity check. Exercising captures a $2\$2 dividend and gives up $0.30\$0.30 of insurance against the stock falling below $50\$50. The put price *is* the time value you forfeit, which is why parity makes the comparison exact rather than approximate.

Definition 5.5

Pin risk

Pin risk — At expiry with the stock sitting on a strike, you do not know whether your short options will be assigned. Guess wrong in either direction and you carry an unhedged stock position over the weekend. It is a genuinely binary exposure, it is not compensated, and it arrives when you have the least time to react.

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