1. Financial Strategies
Stefano Grazioli

2. Critical Thinking
Easy meter

3. Delta Hedging
The Greeks

4. Delta Neutral Portfolio
Delta Hedging
Objective: determine what is the right type and quantity of securities to counterbalance the movements of a security that we own.
Delta Neutral Portfolio

5. What is Delta? Delta = O2 – O1 U2 – U1
Delta is a parameter. Roughly, it is the change in an option price when the underlying stock price changes by a unit (e.g., one dollar).
O2 – O1
U2 – U1
Example1: we observe that a call option price goes down by $1.60 when a stock goes down by $2. Delta = / = +0.8
Example2: a put option is up by $0.5, when the stock is down by $1. Delta = 0.50 / = -0.5
Delta =

6. Balancing a Position in the HT
I own 100,000 AAPL stocks.
I am bearish - I think that the Stock price may go down.
What kind and how many options do I need, in order to counter-balance possible price changes and preserve my portfolio value?

7. Delta Hedging Example
We want to hedge 100,000 long AAPL stocks that we found in our IPs.
First, we need to find a security that counterbalances that behavior
Stock price
long Stock
Current Price

8. Hedging a Long Stock long call short call short put long put
Stock price
long Stock
Current Price
Profit & Loss
Profit & Loss
long call
Stock price
short call
Stock price
strike
strike
Profit & Loss
Profit & Loss
long put
short put
Stock price
Stock price
strike
strike

9. Short calls have the right behavior (also long puts)
- How many short calls?
long Stock
short call
Strike
Stock price
Current Price

10. How many short calls are needed to make our position price-neutral?
gain/loss from options = - gain/loss from stocks
Noptions * (O2-O1) = - Nstocks * (U2-U1)
Noptions = - Nstocks * (U2-U1)/(O2-O1)
Noptions = - Nstocks * 1/Deltacall
Noptions = - 100,000 * 1/0.8
Noptions = - 125, i.e., we need 125,000 short calls.

11. Numeric Check
Suppose that the APPL stock price decreases by $10. What happens to my portfolio?
by assumption:
Option price change / Underlier price change = 0.8
so: Option price change = 0.8 * (-$10) = -$8
Change in Portfolio value = 100,000 * (-$10) + (-125,000) * (-$8) =
= -1,000, ,000,000 = $0
We have a Delta neutral portfolio (yay!)

12. Computing Delta (homework)
Delta of a Call Option = N(d1)
Delta of a Put Option = N(d1) -1
d1 = {ln(S/X) + (r + s 2/2) t}
s t
N() is the standard normal cumulative distribution function and it is provided in Excel

13. What Hedges What If your position is... ...this is what you need
x Short call
x * Delta long stock
x Long call
x * Delta short stock
x Short put
x * |Delta-1| short stock
x Long put
x * |Delta-1| long stock
x Short stock
x * 1/Delta long call or n 1/|Delta-1| short put
x Long stock
X * 1/Delta short call or n 1/|Delta-1| long put

14. Need for Recalibration
There is a catch.
Delta changes with time....

15. Dynamic Delta Hedging Noptions = - 111,111 so, we need to buy back
Delta changes with S, r, s and t. Since they all change in time, the hedge needs to be periodically readjusted – a practice called rebalancing (r, s are fixed in the HT).
Example: Yesterday we wanted to hedge 100,000 long stock and so we shorted 125,000 calls. But today the delta is 0.9.
100,000 = - Noptions * 0.9
Noptions = - 111, so, we need to buy back
13,889 calls (=125, ,111) to maintain delta
neutrality.

16. WINIT
What Is New In Technology?

17. Strategy: Offset the Position with a Synthetic Security
Profit & Loss
Perfect hedge, but costly.
Synthetic Short position
Long position to hedge
Stock price
Total Payoff

18. Put-Call Parity
For European Ps and Cs that have the same strike K, and expire by the same time t:
P + S = C + K e-rt
We can solve for S, P, or C, effectively synthesizing a security with a combination of the other two and some interest-earning cash.
Example: S = C + Ke-rt – P and
- S = - C - Ke-rt + P