Options Part III – Delta Hedging

Please see related posts

Options Trading Part I – NFW Edition

Options Trading Part II – The Currency of Options Trading


For the next little bit I’ll use a specific made-up example, of a fictional pet insurance company[1] with ticker symbol PAWS.[2]

Let’s say PAWS shares trade at $100 per share, and I am planning to sell 1,000 3-month puts struck at $90 for $2.70. I bank $2,700 (1,000 shares x $2.70), and I give the put buyer the option to sell me 1,000 shares at $90 a piece at any time in the next 3 months. [For a definition of what a put is, I recommend starting here.  and then read here.]

As a retail investor, I’m hoping the shares stay roughly where they are, and certainly above $90 per share for the next 3 months. If they drop below $90/share, I am forced to buy them at a price above the market.

Without getting into heavy math, we can see intuitively why an option on a stock would be more valuable, or cost more, for a more volatile stock. If you have a 3 month option on a stock, and the stock moves only slightly during those three months, the owner of the option will have no opportunity to profit. The more dramatically the stock moves during those three months, the more the owner of the option may profit.

Perhaps not intuitive at first glance, however, is the idea that an options trader (the pro, not you and me) cares almost exclusively about the volatility of the stock – the frequency and magnitude of price changes during the time period of his option – and not about the price direction of a stock, up or down.

An options trader trades “volatility” for a living rather than stocks, and the value of all calls and puts in an option trader’s portfolio fluctuates with the rise and fall of volatility, rather than the price of stocks.

In practice, that means that if the trader owns an option, he hopes (or can be said to have ‘bet’) that the stock becomes more volatile. Conversely, if he has sold an option, he hopes (or can be said to have ‘bet’) that the stock becomes less volatile over the time horizon of the option.

In practice the professional options trader – or “vol trader” as he or she may be known – has a portfolio which is net long or net short volatility. And just importantly, in most cases the vol trader will seek to be “flat” or neutral with respect to the underlying stock, or stock market, or whichever market he or she[3] trades.

Hedging market exposure to the stock in order to isolate volatility exposure

Do you want to go deeper down the options trading rabbit hole with me? Why not? Let’s hum a few more bars of the volatility tune to learn about what options traders actually do for a living.

When an options trader buys my 1,000 puts on PAWS struck at $90 per share he typically will want to leave his portfolio exposed to volatility, but not exposed to the underlying stock. After all, he’s a ‘Vol trader,’ with a  view to the historical, present, and future value of volatility, but typically without any particular responsibility for a view on the historical, present and future price of the underlying stock. But initially, at least, he’s a little bit exposed to the price of the stock.

At the moment he begins to own my puts – with the right to sell me some 1,000 PAWS shares at $90 – he becomes ‘long’ volatility but slightly ‘short’ some notional amount of PAWS shares.


The notional ‘short’ PAWS shares needs some explanation. You see, he’s slightly short PAWS shares despite the fact that he hasn’t sold any yet.

Even though he hasn’t sold me any, there is some non-zero probability that he will end up selling me PAWS shares 3 months from now, so he has a contingent future short exposure to the stock, the contingency being that PAWS shares drop below $90 per share.

This positive probability of selling shares in the next 3 months makes the vol trader somewhat exposed to the direction of the market. And, generally speaking, a vol trader doesn’t want to be exposed to the direction of the market.

Notional market exposure and “the Delta”

Let’s assume the vol trader knows – and in fact he would know based on the measure of the historical volatility of PAWS shares – that there’s a 20% probability that PAWS stock goes below $90 in the next 3 months. That makes the options trader 20% “short” 1,000 shares of PAWS, on a probability-weighted basis.

In the options trading world this notional market exposure is known as the ‘Delta.’

The delta is used in practice to calculate how the options trader can hedge his market exposure to PAWS shares, which he doesn’t want. With a 20% short position on 1,000 shares, the right thing for the options trader to do is to purchase 200 shares of PAWS (20% of 1,000) at the market price of $100 per share.

This purchase – assuming he’s calculated the delta correctly – leaves him ‘market neutral’ with respect to the future price of PAWS but ‘long’ volatility with respect to future fluctuations – in either direction – in PAWS stock.

He’s long puts on 1,000 shares, and he’s also long enough shares to cover – on a probability-weighted basis – the expect amount of shares he may sell.

Now he’s good. And ready.

Trading the delta

The interesting part for an options trader begins as soon as he’s isolated his exposed to volatility only, so next I’ll describe good scenarios for the options trader.

Let’s assume PAWS drops the next day to $90 per share.

My fictional pet insurance company

For the next part I’m mostly going to ignore the option seller’s situation (my situation) however because – as noted earlier – no retail investor should be doing this.[4]

Our options trader, who is long the 1,000 puts and long 200 PAWS shares as a hedge, now has a great opportunity based on the market’s dramatic move downward. The delta of a $90 put with the market at $90 per share will be roughly 50%, meaning the trader is now 30% under-exposed to the underlying stock.

The delta changes, remember, because it reflects the probability-weighted exposure for an options trader to the stock market price over the remaining three months. Once the stock has dropped to $90, we can assume that there’s roughly a 50-50 chance that these puts will be exercised – meaning a 50-50 chance the trader will sell 1,000 shares to the put seller at $90, three months from now.

Our vol trader can, and should, purchase an additional 300 shares of PAWS to remain ‘market neutral’ to PAWS shares.

Once he buys 300 shares to add to his original 200 shares, he owns 500 shares total, and he owns 1,000 puts on PAWs with a Delta of 50. Once again, he’s good and ready.

He’s long volatility, but neutral to PAWS, exactly how a ‘vol’ trader should be.

The next day, PAWS rockets back upwards to a price of $100 per share.

Then what happens?

The original put seller (still me, I guess?) lets out a big sigh of relief that his puts are back ‘out of the money.’[5]

Interestingly, however, our options trader is also made happy by the quick move. He couldn’t care less that the puts he owns might expire unexercised, because he cares instead that the volatility of the stock has spiked.

Why is volatility so good for him?

The delta of the PAWS shares at $100 shifts back in this example to something close to 20%, leaving the trader ‘long’ PAWS shares by about 300 shares. The options trader, in order to shift back to ‘neutral’ on the PAWS stock, gets to sell 300 shares at $100.

This part is kind of cool, if you’re an options trader long ‘vol’ on PAWS.

In the course of two trading sessions, our options trades has bought 300 shares at $90 and sold 300 shares at $100, pocketing the riskless difference of $3,000. [300 shares x $10 price move.]

An options trader who is ‘long’ volatility will always have the happy circumstance of buying low and selling high in the course of ‘delta-hedging’ his exposure to the underlying stock.[6]

If PAWS shares go up again, to $110, his delta shrinks further and he will sell some of his original 200-share delta hedge at an even higher price. If PAWS shares drop, he will buy low at the new low share price to hedge his delta. All the while remaining ‘market neutral.’

The more the shares move over the course of the next three months the more the vol trader delta hedges profitably. Wash, Rinse, Repeat.

On the other hand if PAWS never moves over the three month period in our example, our options trader loses the money he spent on the premium.

That, in super-simplified form, is how options trading works.

The retail investor speculating in options rarely delta hedges or even understands how to calculate volatility, putting him at an extraordinary disadvantage with this type of speculating.

You can still get lucky with a leveraged long or short position, and everybody knows it is better to be lucky than good.

But again, I would only wish this type of retail speculating on my worst enemy.


Please see related posts:

Options Trading Part I – NFW Edition

Options Trading Part II – The Currency of Options Trading



[1] Here’s how I imagine pet insurance working: You pay $10/month to PAWS, and in return little Fifi gets medical costs covered up to a certain amount, plus some lump sum ($10K?) to compensate you in case Fifi goes missing or gets hit by a truck. I’m making this business up but I’m certain many dog owners would be willing to buy this type of insurance.

[2] I learned after I wrote this that there is an actual penny stock with ticker symbol PAWS and I’d recommend getting involved with that penny stock even less than I would recommend selling puts. Run away!

[3] Apologies in advance, I’m going to go all gender-specific in my pronouns for the rest of the post so I don’t have to keep adding “or she” to every clause. This is just to say that I’m sorry about this and I hope to make it up to you some day.

[4] But understand that the option seller (me in this example) begins to wet his pants because losses start quickly from here.

[5] Importantly, he has time to step away from his day-trading desk to change his pants.

[6] Of course this cuts both ways – an options trader who is ‘short’ volatility will be in the uncomfortable delta-hedging position of buying high and selling low if the underlying stock makes volatile moves over the life of the option.

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