What is the effect of a short stock position on the value of an option?

Question

I'm reading the book "Option Volatility and Pricing" by Natenberg and there is a paragraph I'm having trouble understanding.

Chapter 7 (Risk Measurement 1), Page 98, Paragraph 2:

The value of a stock option will also depend on whether a trader has a long or short stock position. If a trader’s option position also includes a short stock position, he is effectively reducing the interest rate by the borrowing costs required to sell the stock short (see the section “Short Sales” in Chapter 2). This will reduce the forward price, thereby lowering the value of calls and raising the value of puts. As a consequence, the trader who is carrying a short stock position ought to be willing to sell calls at a lower price or buy puts at a higher price. If the trader either sells calls or buys puts, he will hedge by purchasing stock, which will offset his short stock position.

The fact that option values depend on whether the trader hedges with long stock or short stock presents a complication that most traders would prefer to avoid. This leads to a useful rule for stock option traders:

Whenever possible a trader should avoid a short stock position.

My Doubts are:

1. What exactly is the meaning of "If a trader’s option position also includes a short stock position"? Why or under what conditions would you say an option position "includes" a short stock position?

Is he talking about delta hedging? Is it implicitly assumed that option traders will delta hedge their portfolios? I'm skeptical because up until this point in the book no mention is made of delta hedging.

2. Why would the valuation of an option depend on the existence of short stocks in the portfolio as mentioned in the statement: "As a consequence, the trader who is carrying a short stock position ought to be willing to sell calls at a lower price or buy puts at a higher price"

I believe that valuation of a fincancial instrument would be independent of what you already hold or your cost to finance the purchase.

Again it seems, as explained in the next sentence, that either selling calls or buying puts will implicitly involve a long stock position for the purpose of delta hedging. Hence an existing short stock position combined with either selling calls or buying puts, and hence going long stock in effect neutralises the short position and hence saves us from the borrowing costs.

Is this a fair analysis? Again it boils down to the question of whether delta hedging is ubiquitous enough for it to be implicitly assumed.

3. The last line says "Whenever possible a trader should avoid a short stock position."

I understand this is obvious just looking at the borrowing cost. Is that it? Are there subtleties that I'm missing here? Can anyone expand on why else would a trader want to avoid a short stock position?

Answer 1

It isn't clear to me what Natenberg is saying in the excerpt because his writing style isn't the most user friendly. Is there additional context for this quote such as in a chapter about arbitrage strategies? Some random thoughts without a complete answer for you...

If you look at conversions and reversals, the value of a put is connected by the pricing variables. If you assume that the stock is at the strike price and do some gross simplification then:

Put + Carry Cost = Call + Dividend

-which means that call premium will by higher than put premium by the amount of the carry cost and put premium will be higher than call premium if there's a dividend.

I think that Natenberg is alluding to a Reversal where one is short the stock and short the put. The borrow rate for the short stock decreases the credit received from the proceeds of the short sale. That in turn, decreases 'value' of the put/call pricing spread to the trader (not the pricing of options). I can't suss out the full meaning of this since I only understand the basics of arbitrage.

Per your comments:

"If the trader either sells calls or buys puts, he will hedge by purchasing stock, which will offset his short stock position."

I don't think the overall discussion of carry cost here has anything to do with delta neutral hedging. That's a function of obviously, delta, not option pricing. And yet he sneaks a line in here implying adjusting delta (selling calls or buying puts adds negative delta so some stock is bought to get delta back in line).

An option position that includes a short stock position is a hedged position and it has nothing to do with the aforementioned arbitrage.

Valuation of options has nothing to do with the existence of other legs in one's position. The quote "As a consequence, the trader who is carrying a short stock position ought to be willing to sell calls at a lower price or buy puts at a higher price" has to do with the value of a trader's position not the pricing mechanism.

"Whenever possible a trader should avoid a short stock position."

This is nonsensical. Your short a stock when you want to be short, heeding the borrow cost and the ex-dividend dates. Options have a wider B/A spread, time premium and a delta issue. If the put is 100 delta then the B/A is usually very wide. If ATM (50 delta) then you need twice as many puts (narrower B/A spread). Either way, it's a poor duplicate of the short of the underlying. Apart from the borrow cost (which is almost negligible in liquid large caps) and the ex-div date, the only advantage the put has is limited risk in the amount of the premium paid.

To repeat, Natenberg's writing style isn't the most user friendly and it often leads to stretching one's imagination to grasp what he is alluding to.

Answer 2

What exactly is the meaning of "If a trader’s option position also includes a short stock position"?

It means that they have either sold shares short, or they have a position that somehow mimics selling shares short.

Why or under what conditions would you say an option position "includes" a short stock position?

Depending on the context, any position that is negatively correlated with the stock price can be considered "short", but here, given that he is distinguishing between short on the stock, selling calls, and buying puts (all of which are "short" in this general sense), he probably is using it more restrictively.

Why would the valuation of an option depend on the existence of short stocks in the portfolio as mentioned in the statement: "As a consequence, the trader who is carrying a short stock position ought to be willing to sell calls at a lower price or buy puts at a higher price"

If one is doing a risk-sensitive valuation, then reducing risk increases value. But selling short, selling calls, and buying puts are all in the same direction, so that doesn't apply.

Again it seems, as explained in the next sentence, that either selling calls or buying puts will implicitly involve a long stock position for the purpose of delta hedging.

Do you mean "Selling calls or buying puts would be paired with a long stock position"? If you buy long and sell short, you're back where you started minus transaction costs.