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IT & FinanceStefano Grazioli
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© Stefano Grazioli - Ask for permission for using/quoting: [email protected]
Delta HedgingThe Greeks
Delta Hedging Objective: obtain the right
type and quantity of securities to counterbalance the movements of a security that we own.
DeltaNeutralPortfolio
What is 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: a call option price goes down by $1.60 when a stock goes down by $2.Delta = -1.60 / -2.00 = +0.8
Example2: a put option is up by $0.5, when the stock is down by $1. Delta = 0.50 / -1.00 = -0.5
Delta =
Balancing a Position
I own 100,000 IBM stocks.
I am bearish - I think that the Stock price
may go down.
What kind andhow many options do I need, in order to counter-balance possible price changes and preserve my portfolio value?
Delta Hedging ExampleWe want to hedge 100,000 long IBM stocks that we found in our IPs.
First, we need to find a security with the appropriate hedging behavior
Stock price
long Stock
Current Price
Hedging a Long Stock
Stock price
Profit & Loss long call
Stock price
short call
Profit & Loss
strike strike
Stock price
Profit & Loss
long put
Stock price
Profit & Loss
strike
short put
strike
Delta Hedging Example- Short calls have the right behavior (also long puts)
- How many short calls?
Stock price
short call
long Stock
Strike
Current Price
How many 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 = - 125,000 i.e., we need 125,000 short calls.
Numeric CheckSuppose that the IBM stock price decreases
by $10. What happens to my portfolio?
by assumption:
Option price change / Underlier price change = 0.8
so: Option price will change by 0.8 * (-$10) = -$8
Change in Portfolio value = 100,000 * (-$10) + (-125,000) * (-$8) =
= -1,000,000 + 1,000,000 = $0
We have a Delta neutral portfolio
Computing DeltaDelta 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
What Hedges What
1 Short call Delta long stock
1 Long call Delta short stock
1 Short put |Delta-1| short stock
1 Long put |Delta-1| long stock
1 Short stock 1/Delta long call or 1/|Delta-1| short put
1 Long stock 1/Delta short call or 1/|Delta-1| long put
If your position is... ...this is what you need
Need for Recalibration
There is a catch.Delta changes with time....
Dynamic Delta HedgingDelta 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 now the delta is 0.9.
100,000 = - Noptions * 0.9
Noptions = - 111,111 so, we need to buy 13,889 calls
(=125,000-111,111) to maintain delta neutrality.
Next Time When/how to rebalance Balancing a whole portfolio Other types of hedging
© Stefano Grazioli - Ask for permission for using/quoting: [email protected]
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© Stefano Grazioli - Ask for permission for using/quoting: [email protected]
HomeworkThe Spartan Trader
Suggestions Give yourself plenty of time Test the numbers!
