Delta Hedging The Greeks

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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

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 up by $1.60 when a stock goes up 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 =

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

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?

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

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

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

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 = - 125,000 i.e., we need 125,000 short calls.

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 + 1,000,000 = $0 We have a Delta neutral portfolio (yay!)

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 x * 1/|Delta-1| short put x Long stock X * 1/Delta short call or x * 1/|Delta-1| long put

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

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 = - 111,111 so, we need to buy back 13,889 calls (=125,000-111,111) to maintain delta neutrality.

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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 More recommendations