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© Stefano Grazioli - Ask for permission for using/quoting: grazioli@virginia.edu

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Lab on Friday Office hours added Tue & Th 3-6pm (and beyond) Easy meter

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Questions? Team formation Simplified IPs on Beta for testing

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© Stefano Grazioli - Ask for permission for using/quoting: grazioli@virginia.edu

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Objective: obtain the right type and quantity of securities to counterbalance the movements of a security that we own. Delta Neutral Portfolio

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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). O 2 – O 1 U 2 – U 1 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

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I own 100,000 IBM 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?

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We 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

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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

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

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gain/loss from options = - gain/loss from stocks N options * (O 2 -O 1 ) = - N stocks * (U 2 -U 1 ) N options = - N stocks * (U 2 -U 1 )/(O 2 -O 1 ) N options = - N stocks * 1/Delta call N options = - 100,000 * 1/0.8 N options = - 125,000 i.e., we need 125,000 short calls.

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Suppose 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

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Delta of a Call Option = N(d1) Delta of a Put Option = N(d1) -1 d1 = {ln(S/X) + (r + 2 /2) t} t

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1 Short callDelta long stock 1 Long callDelta short stock 1 Short put|Delta-1| short stock 1 Long put|Delta-1| long stock 1 Short stock1/Delta long call or 1/|Delta-1| short put 1 Long stock1/Delta short call or 1/|Delta-1| long put If your position is......this is what you need

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There is a catch. Delta changes with time....

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Delta changes with S, r, and t. Since they all change in time, the hedge needs to be periodically readjusted – a practice called rebalancing (r, 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 = - N options * 0.9 N options = - 111,111 so, we need to buy 13,889 calls (=125,000-111,111) to maintain delta neutrality.

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When/how to rebalance Balancing a whole portfolio Other types of hedging

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© Stefano Grazioli - Ask for permission for using/quoting: grazioli@virginia.edu

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Give yourself plenty of time Test the numbers!

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