P1. T3. Hull, Chapter 10: Properties of Stock Options (Study notes)

[LMFRM]

Hi David,
On Hull, Chapter 10: Properties of Stock Options in p108, it is mentioned that
"Volatility : Greater volatility increases the value of both a call and a put option", but in p111 that 'In general, we can say that for American put, the early exercise becomes more attractive as Volatility decrease". I would say when the volatility increase.
Could you explain it ?
Thks,

 

[Alex_1]

Hi @LMFRM , I think we have to look at two aspects of option pricing simultaneously: the evolution of the risk-free rate and of the volatility (I just looked through the notes and also throught some very helpful posts in this forum, so the below is just a short summary with some additions from my side):

If due to an interest rate increase the interest amount which you earn on the strike price (X) is greater than what Hull calls "insurance element lost" it is optimal to exercise an American put early.
If the volatility decreases, the insurance character of the option becomes less valuable rendering the option itself less valuable (you are less able to hedge against the respective risk), so it is more profitable for you as an option holder to exercise the American put early (you want to exercise the put before it becomes even less valuable).

Let me know if the explanation helps.

Thanks.

 

[David Harper CFA FRM]

And just to expand on @Alex_1 's helpful insight (as I find this difficult too), below is Hull's Exhibit 10.6 (8th edition)/Exhibit 11.6 (new 9th edition). The nonlinear line plots put the typical put option value (vertical axis) against stock price (horizontal axis). For an American put, we can think of the put value as a combination of intrinsic value (i.e. ,K - S) + time value (TV, which increases with volatility) + early exercise (EE) value.

I like this loose relationship: P = p + EE = (intrinsic + TV) + EE
... because it shows that higher volatility can increase put value (P) even as the value of EE decreases. That is, nothing necessarily wrong with:
Higher volatility --> ↑P = [no impact on intrinsic value] + ↑[time value] + ↓[early exercise]

Hull's point with the plot (below) is that for an American put, when the stock drops to point (A), where S < K such that the put has sufficient intrinsic value, the total value of the put has converged to intrinsic value of the put. At lower stock prices, both the time value and the early exercise components are squeezing to zero as the intrinsic value overwhelms the total option value. Per Hull's arrows, higher volatility pulls the non-linear price line up, which effectively moves (A) to the left: higher volatility renders the time value more valuable but the early exercise feature less valuable; as curved price line shifts up, gap between curve and straight (intrinsic value) expands but (A) moves to the left (as A shifts to the left, there is less benefit to early exercise).

In summary, chart-wise: the 45-degree dotted-line anchored at (K) is fixed regardless. Higher volatility increases the time value of the option (represented by gap between 45 degree dotted line and nonlinear price line) which increases the total option value (shifts up), however it also effectively shifts the vertical (A) line to the left (requiring an increasingly deep in the money put to validate an immediate exercise)

The extreme converse illustrates maybe better: imagine an ITM American put option, strike at $20 with stock price at $8. This option has $12 of intrinsic value and can be immediately exercised for $12 payoff. There are two difference between a put and a call:

1. Hull's point that the put has limited further upside, unlike a call; the stock can only go to zero. This skews (constrains) the insurance element
2. But also the expected drift of the stock is positive. Imagine the volatility decreases down to zero. Then the time value is zero (or worse): with an upward drift in stock price, our $12 intrinsic value is not only shrinking, but it's shrinking into future value. So, at zero volatility, while our overall put value has decreased to its lowest (ceteris paribus), the early exercise feature value is has increased to its maximum because we can't afford to wait (zero volatility has eliminated the insurance character"). I hope that's interesting!

 

[LMFRM]

Thks Alex and David, it is more clear for me.