Modeling Stock Borrow Costs in Binomial Option Valuation
Summary
The document considers how hard-to-borrow fees affect option values and why they should not be treated as a simple change to the risk-free rate or dividend yield. Dividends influence the stock’s expected value through its ex-dividend behavior, while a borrow fee applies to the stock value being borrowed. In a delta-hedged replication, the number of shares held changes with the option’s delta, so the fee depends on the stock position rather than the bond financing amount.
A one-period binomial replication illustrates the mechanism for a call. The hedge holds delta shares, which earn the stock-lending rate, and borrows or lends cash to match the option payoff in both up and down states. Including lending income increases the required cash borrowing in the setup and therefore reduces the call’s value. This is a conceptual single-step explanation; it does not provide a full multi-period model or address calibration and market-specific borrow constraints.
Key ideas
- A stock borrow fee applies to the value of the stock position, not directly to the financing bond amount.
- Dividend yield and stock borrow cost represent different cash-flow effects in option models.
- A binomial replication can show how lending income on delta shares affects the cash balance.
- In the presented call setup, stock-lending income raises the financing requirement and lowers call value.
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# Option Valuation With Hard To Borrow Rates
# Option Valuation With Hard To Borrow Rates
How would you include -in a simple way- high borrow rates, say 10%.
Intuitively, for PUTs I'd set r as r - borrow_rate, to include the negative carry of the borrow. So If I'm selling puts, value would increase to compensate the fact that keeping the short stock (the delta hedge) is costly.
Calls should be worth less as I'm receiving the short rate interest for holding the delta hedge.
Is it plausible this? Else, you could add the fee on the q. Which seems to me more easy as the borrow pnl is dependent on the underlying process and not the Bond Price (strike). Do you think this would apply to indices like a weighted average borrow rate added to the q dividend process?.
Best,
## Answer by rmacey (score 0)
https://quant.stackexchange.com/a/83763
Putting the borrow fees on q is problematic. Dividends impact the value of the stock in that stock values are thought to fall as they pay dividends. This is captured in S e^{-qT}. Putting them in the borrowing rate is also problematic. The borrowing rate is applied to the amount that is borrowed in that a long call is a dynamically levered position. The fee to borrow the stock applies to the stock price, a different amount.
Recall that both the Black-Scholes and Binomial model for calls are based on replicating the payoff of the call using a dynamic allocation to a levered stock position (the number of shares of stock is given by the delta). A nice way to see the impact of the borrowing fee is to consider a binomial model with a single step (move).
A single period binomial model where the stock is being lent and earning a rate can be expressed as:
delta * Su – r * B + l * delta * S = Cu
delta * Sd – r * B + l * delta * S = Cd
where S can go up to Su or down to Sd. The delta of the move will be (Cu - Cd) / (Su - Sd) where the Cu and Cd are the values of the call at expiration which are max(Su - K, 0) and max(Sd - K, 0). Let's say (l) is the rate to lend the stock. Over the period, owning delta shares of the stock will be delta * S * l. The borrowing is r*B. Solving for B gives (delta * Su + l * delta * S – Cu) / r.
The value of the call is delta * S - B. The l * delta * S increases the value of B and thus reduces the value of the call.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.