1. Implicit Volatility
Stefano Grazioli

2. Critical Thinking Easy meter
Extra time: homework assigned on Tue is due on Friday
Extra office hours 5:30 onwards

3. Your Opinion Matters Do you feel you are learning?
Is the class getting you to think on your own?
What do you like about it?
What can be improved?
Attitude towards the HT

4. Implicit Volatility
Goal seek and Solver

5. Volatility (sigma) is needed to compute delta
Volatility is a measure of the dispersion of the returns of a security over a unit of time
Implicit volatility is the volatility that a stock should have, given the observed market prices and assuming that BS is true.

6. Solving for sigma is too hard!
Price of a Call = S * N(d1) – Xe-rt * N(d2)
d1 = {ln(S/X) + (r + s 2/2)t}
st
d2 = d1 - st
S = current spot price,
X = option “strike” or “exercise” price,
t = time to option expiration (in years),
r = riskless rate of interest (per annum),
s = spot return volatility (per annum),
N(z) = probability that a standardized normal variable will be less than d. In Excel, this can be calculated using NORMSDIST(z). Delta for a Call = N(d1) Delta for a Put = N(d1) -1

7. Solving problems analytically
Preferred way to solve problems of the form “given that y=f(x) and that we know y, find x”
Example: I = C [(1+r)t -1]
If we know that I = $ and t=10 and C = $1000 what is r?
Analytical solution: First you solve the equation, i.e. r = (I/C+1)1/t – 1 then you plug in the numbers r = (218.99/1000+1)1/10 – 1 and finally you find out that r = 2%

8. Finding r iteratively Simplified example
Pick an arbitrary solution value (e.g., r = 1%) and compute the formula: I = 1000 [(1+r)10 -1]
Try another solution value (e.g., r=1.10%) and compute the formula
If the value of I obtained in step 2 is closer to the target value of I ($218.99) than the value of I obtained in step (1), keep increasing the value of r, else reduce it.
Keep on going until you find an r that generates a value of I close enough to the target I=$218.99

9. Iterative Solutions Easy to use BUT Often only approximations
Not guaranteed to find a solution
Might miss solutions in case there is more than one y x

10. Implicit Volatility
Demo

11. Excel Tools
Solver if you have more than one unknown “X”s – e.g., suppose that we did not know t and r
Goal “Y” can be a constant, or a min, or a max
Can do constraints on the Xs
Goal seek if you have one unknown “X” and one known “Y” – as in the example
Goal “Y” is a constant
No constraints (e.g., x > 0)

12. Getting feedback on your volatilities
Only once per each team
Volatilities = secret sauce

13. WINIT
What Is New In Technology?

14. Homework
Financial charts