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 Team submission possible from H17  me with team name, members (userids) and get the team # on collab.  Easy meter

 Questions?  Team formation / paper / opting out

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Profit & Loss Going long / short = flipping horizontally the payoff curve Profit & Loss Stock price short $10 long $10 price at which you bought it

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

Stock price Profit & Loss TCs always lower your payoff curveTC long - TC $10 Stock price short - TC Profit & Loss TC $10

Stock price Profit & Loss long - TC TCs always lower your payoff curve Stock price short - TC Profit & Loss TCTC $10$10

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 Combine different types of positions to obtain custom payoff curves.  Payoff curves are designed to achieve different objectives. Hedging is one of them.

1. Offsetting the position (not applicable) 2. One to one 3. One to many 4. Dynamic approaches 5. Synthetics (based on put/call parity) 6. Delta hedging (based on Black Scholes) 7. Delta + Gamma hedging (complex refinement)

Stock price Long position to hedge Total Payoff Profit & Loss Short position Perfect hedge, but guaranteed to lose money. Impossible do to when a position is illiquid (i.e., you cannot do it in the HT)

Stock price Profit & Losses Total Payoff long Stock short call strike Very popular - Neutral to moderately bullish

A short callGo long on the stock A long callGo short on the stock A short put... A long put... A short stock... A long stock... If our position is......this is what we (the system) should do

Stock price Profit & Losses short call long Stock Way out of the money – Inexpensive means to protect wealth from sharp downturns long put Total Payoff

Stock price Profit & Losses short call Total Payoff long on Stock Buy the stock if its price raises above strike, and sell it back if falls below. Yes, there is a catch....

 Typically useful for manually managing your portfolio  In the past: Most teams did Delta Hedging (next time) Some of the better teams did their own mix of Delta and Gamma hedging There is a dark horse…

Stock price Long position to hedge Total Payoff Profit & Loss Synthetic Short position Perfect hedge, but costly.

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

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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

 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 = / = +0.8  Example2: a put option is up by $0.5, when the stock is down by $1. Delta = 0.50 / = -0.5

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?

 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

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

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

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.

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, ,000,000 = $0 We have a Delta neutral portfolio

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

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

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

 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 ,000 = - N options * 0.9 N options = - 111,111 so, we need to buy 13,889 calls (=125, ,111) to maintain delta neutrality.

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

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 Give yourself plenty of time  Test the numbers!

 Teams!  APPL_COCTB  On Sunday reference code will be released. No more late hand-in  Easy meter