1. Delta Hedging
The Greeks

2. Critical Thinking Easy meter Extra office hours 5:30 onwards
I will be around fri and sat

3. 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

4. What is Delta? O2 – O1 U2 – U1 Delta =
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 up by $1.60 when a stock goes up 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 =

5. Delta is NOT constant 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

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
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 calls = - gain/loss from stocks
Ncalls * (O2-O1) = - Nstocks * (U2-U1)
Ncalls = - Nstocks * (U2-U1)/(O2-O1)
Ncalls = - Nstocks * 1/Deltacall
Ncalls = - 100,000 * 1/0.8
Ncalls = - 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?
Delta = Option price change / Underlier price change = 0.8
100,000 AAPL & -125,000 Calls on AAPL
so: Option price change = 0.8 * (-$10) = -$8
Change in Total Portfolio Value = 100,000 * (-$10) + (-125,000) * (-$8) =
= -1,000, ,000,000 = $0
We have a Delta neutral portfolio (yay!)

12. What Delta-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

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

14. Dynamic Delta Hedging Ncalls = - 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.
Ncalls = - 100,000 * 1/0.9
Ncalls = - 111, so, we need to buy back
13,889 calls (=125, ,111) to maintain delta neutrality.

15. What Is New In Technology?
WINIT
What Is New In Technology?

16. Demo
The Recommender

17. Algotrading: The Recommender
Start
Find the top Candidate Recommendation
and store it in Recommendation(i)
Pick the next recommendation for a family portfolio in Recommendations(i)
Automated execution?
Display all
Yes
Hedging today?
End No
Execute the top scored Recommendation No
Need to hedge this portfolio? A No
Recompute all metrics and Recommendations A
Generate and score Candidate Recommendations
No more
Display all
End
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