Box Models III
Box Model A is:
The number drawn from this box represents the value of the underlying at expiration.
Let’s now play with Model A a little bit and try to see the effect of volatility. Here is Model A-HV:
“HV” stands for “High Volatility” and “LV” stands for “Low Volatility”. You can see that the A-HV model produces values of the underlying further away from its current price of 100, and the A-LV model produces values that are closer to the current price of 100, than the Model A. Under Model A, a call option with strike 100 is worth $5. How much is it worth under Model A (HV)? Again replacing each value of the underlying by the option value at expiration, we get:
The same calculation as before shows that the call option with strike 100 is worth $10, more than the $5 the call option is worth under Model A. You can see that the higher volatility throws the underlying price further away from the current price of 100 and that this benefits the call on the upside and does not hurt it any more on the downside.
Similarly, the value of a call under Model A-LV is like the number drawn from the box:
And the value of this call is $2.50, discounted again by the safe interest rate, less than the $5 this call is worth under Model A.
This little tinker toy example displays the essential reason that the value of a call option increases with the volatility of the underlying. What about puts? Try this yourself — I’ll do a put next time.