Box Models III – Volatility
Box Model A is:
The number drawn from this box represents the value of the underlying at expiration.
Here is Model A-HV:
“HV” stands for “High Volatility” and “LV” stands for “Low Volatility”. I promised last time to compute the value of a put using these simple models, and demonstrate that puts also have higher values for underlyings with higher volatilities. Under Model A, a put 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 corresponding option value at expiration, we get:
The same calculation as before shows that the put option with strike 100 is worth $10 under Model A-HV, more than the $5 that the put option is worth under Model A. The higher volatility throws the underlying price further away from the current price of 100, and this benefits the put on the downside and does not hurt it any more on the upside, just the opposite effect of what we saw with calls.
Similarly, the value of a put under Model A-LV is like the number drawn from the box:
And the value of this put is $2.50, discounted again by the safe interest rate, less than the $5 (less discount) this put is worth under Model A.
Just as with calls, a higher volatility leads price to higher excursions (up or down) from the current price, and puts on underlyings with higher volatilities are worth more.