RIT H3: Delta hedging a call option

Transcription

1 RIT H3: Delta hedging a call option Trading strategies that reduce risk. Overview: We re working on the equity derivatives desk at an investment bank, and We write (sell) a call option on SAC stock. Large profit, but also high risk. To keep the profit we need to hedge. To hedge, buy SAC shares. How many shares? Price of SAC changes our risk changes. Buy/sell to adjust our hedge. 2018, Joel Hasbrouck, All rights reserved 2 1

2 Put and call options: a review Background Your Foundations of Finance text (Investments or Essentials of Investments; Bodie, Kane, and Marcus). The NASDAQ Options Trading Guide at , Joel Hasbrouck, All rights reserved 3 Terminology An American call gives the owner the right to buy the underlying stock (S) at exercise/strike price X up to and including maturity date T. To actually buy using the call, the owner exercises the option. Notify broker Pay X to the seller of the call. Receive the stock from the seller of the call. A European option can only be exercised at maturity. A put gives the holder the right to sell the underlying 2018, Joel Hasbrouck, All rights reserved 4 2

3 Options are in zero net supply. An option comes into existence when it is traded. The person who sells a call has written the call, and is short the call. The number of longs equals the number of shorts equals the open interest 2018, Joel Hasbrouck, All rights reserved 5 The intrinsic value of an American call with exercise price X. If the current stock price is S, the gain from immediate exercise is Intrinsic value = Max S X, 0 CallCall option withwith X X 40= Max(S X, 0) S Max S X,0 2018, Joel Hasbrouck, All rights reserved 6 3

4 Payoff = intrinsic value at expiration. S T : the price of the underlying (stock) at expiration time T. If we re long (we own the call) payoff = Max S T X, 0 If we re short (we ve sold/written the call), payoff = Max S T X, 0 Payoff profit 2018, Joel Hasbrouck, All rights reserved 7 Example: Right now (time 0) MSFT is trading at S 0 = $110 Alan thinks MSFT will. He buys a one-month call option with X = 100. Beth thinks MSFT will (or stay the same). She sells the call to Alan. Alan pays Beth the market price of the call $15. Alan is long the call; Beth is short; the open interest increases by one. At maturity 2018, Joel Hasbrouck, All rights reserved 8 4

5 If S T = $120, Alan exercises the call. He pays Beth $100; she gives him a share of MSFT which he sells for $120. Alan s payoff is 20 = Max ,0 Alan s profit is = 5. Where does Beth get the share? She buys it in the market, paying $120. Beth s payoff is = 20 = Max ,0 Beth s profit is = 5 (a loss) 2018, Joel Hasbrouck, All rights reserved 9 If S T = 99, Alan doesn t exercise. His payoff is zero. His profit is 15. Beth s payoff is zero. She keeps the 15 (her profit). If S T = 106 Alan: payoff =, profit = Beth: payoff =, profit = FAQ How did we know that Beth had to go out and buy MSFT to deliver? Maybe she already owned it. The market price of the call in this example is C = 15. Where did this come from? 2018, Joel Hasbrouck, All rights reserved 10 5

6 C, the value of the call Call with X 40 Call option with X = C 30 Max(S X, 0) S Max S X,0 Call Value The call value is computed from the Black-Scholes equation. The vertical difference between the two lines is the time value of the call. Example: If X = 40, S = 48 and C = 15, then The intrinsic value is Max 48 40,0 = 8 The time value is 15 8 = , Joel Hasbrouck, All rights reserved 11 The inputs to the Black-Scholes formula S the current stock price ($ per share) X the exercise price of the call option ($ per share) T time to maturity (years) r the risk-free interest rate (annual) A 4% rate would be entered as r = 0.04 σ the volatility of the underlying. The standard deviation of the stock s annual return. Example: Over time the S&P index average annual return is about 10%, with a standard deviation of about 20% Entered as σ = 0.20 σ is forecast over the life of the call. S, X, T, and r are observable; σ is an estimate. 2018, Joel Hasbrouck, All rights reserved 12 6

7 The Black-Scholes equation for C, the value of a call C = ตS N d 1 ถX Current Exercise stock price price e r T N(d 2 ) Present value factor r is the (risk-free) interest rate for borrowing and lending. T is the time remaining to maturity. d 1, d 2, and N( ) are given on the next slide. This variant of the equation is correct for a European call on a non-dividend paying stock. 2018, Joel Hasbrouck, All rights reserved 13 d 1 = ln S X + r+σ2 2 T σ T d 2 = d 1 σ T N(d) is the cumulative distribution for the standard normal density evaluated at d. N d is given by the Excel function NORM.S.DIST(d,TRUE) 2018, Joel Hasbrouck, All rights reserved 14 7

8 Spreadsheet: RIT - Excel Support - H3 v02.xlsm Resources Class notes (after midterm) Values set for the SAC call in the RIT H3 case. Contains RTD calls to update calculations while H3 is running. Back to the H3 story We re on the equity derivatives desk. A customer wants to buy a call option on SAC. They d prefer to buy an exchange-traded option. No such option exists: the customer comes to us. Acting as dealer, we sell to the customer. We first analyze pricing and hedging. (Also see the H3 case brief, posted to NYU Classes.) 2018, Joel Hasbrouck, All rights reserved 16 8

9 Pricing the call X = 50, r = 0, σ = 0.15 ( 15% per year ), T = 20 trading days. Annualize using trading days: T = 20 = C = $0.843/share We need to factor in a profit for ourselves so we quote a price of $1.41 to the customer. Check that $1.41 is the Black-Scholes value when σ = 0.25 (25%). How to justify this to the customer? The customer can do the calculations: they know the mark-up. We have a profit of about $1.410 $0.843 = $ , Joel Hasbrouck, All rights reserved 17 At start of run I manually changed the stock price to 50. With σ = 15%, C = $

10 Size The customer wants to buy 200 call (contracts). The standard contract size is 100 shares. In total, we are selling options on 20,000 shares. All prices are quoted on a per-share basis, but when the customer buys, they pay us $ = $28,200. Our mark-to-market profit is 20,000 $0.567 = $11, , Joel Hasbrouck, All rights reserved 19 Hedging Immediately after selling to the customer, we book a mark-tomarket profit. But we are short share calls. If at expiration, S T = $49, the customer won t exercise. We keep $28,200 as a profit. If S T = $60, the customer exercises. We buy 20,000 shares at $60, sell to the customer at $50. Our profit is $28,200 $200,000 = $171,800 (a loss). We want to hedge this uncertainty. 2018, Joel Hasbrouck, All rights reserved 20 10

11 Can we focus on the worst case scenario? At the start of the case, S = 50. If S T > 50, the customer exercises. They buy 20,000 from us at $50. Shouldn t we lock in a $50 purchase price for SAC by buying all the shares we might need right now? 2018, Joel Hasbrouck, All rights reserved 21 Hedging intuition: At a short time ( one day ) prior maturity If we think that S T > $50, we want to hold 20,000 shares (which we ll give to the customer). Reasonable if the current stock price is S $75. The call is deep in the money. If we think that S T < $50, we ll want to hold zero shares. Reasonable if S $20. The call is deep out of the money 2018, Joel Hasbrouck, All rights reserved 22 11

12 If we think that S T < 50 and S T > 50 are equally likely We ll either want 0 or 20,000 shares: let s buy 10,000. We might have this view if S $50. The call is at the money. Then, if the stock price climbs above $50, we buy more. If the stock price drops, we d sell off some. Can we make this more precise? 2018, Joel Hasbrouck, All rights reserved 23 Delta The stock price is currently at S 50. If the stock price went up by $1, how much would the value of the call change? or down by $1. This is measured by the delta of the call: Δ = dc C or ds S 2018, Joel Hasbrouck, All rights reserved 24 12

13 SAC Call Black-Scholes value of the call Δ Delta = slope = Check this using the spreadsheet. To hedge, we need to be long sh of stock SAC Stock 2018, Joel Hasbrouck, All rights reserved 25 The recipe Sell share call options at $1.41 Receive cash of $28,200. Buy = 10,169 shares of stock Pay with borrowed money Are we really hedged? What happens if S changes by ± $0.01? Using the spreadsheet, If S = 50.01, C = , Δ = If S = 49.99, C = , Δ = , Joel Hasbrouck, All rights reserved 26 13

14 Mark to market balance sheets S 0 Assets Liabilities Cash received (20,000 $1.41) 28,200 16,857 Call ~20,000 $0.843 Stock (10,169sh $50) 508, ,429 Loan / charge to capital 11,343 Net worth Cash received 28,200 16,756 Call ~20,000 $0.838 Stock (10,169sh $49.99) 508, ,429 Loan / charge to capital 11,343 Net worth Cash received 28,200 16,959 Call ~20,000 $0.848 Stock (10,169sh $50.01) 508, ,429 Loan / charge to capital 11,343 Net worth 27 Call value and delta over time (days to expiration) Call Value Stock Price T 20 T 5 T , Joel Hasbrouck, All rights reserved 28 14

15 Delta Stock Price T 20 T 5 T , Joel Hasbrouck, All rights reserved 29 H3 When we sell the customer the calls, we book a profit. Revenue from sale of calls less mark-to-market value. We hedge by going long (buying) the stock. We keep adjusting the hedge through maturity (20 days) At maturity, we settle with the customer. 2018, Joel Hasbrouck, All rights reserved 30 15

16 Settlement at maturity If at maturity S T < X (the exercise price) The call is out of the money. The intrinsic value of the call is 0. If we hedge correctly, we won t own any shares. If S T > X The call is in the money. The intrinsic value of the call is S T X per share If we hedge correctly, we ll own one share per call (a total of = 20,000 shares) 2018, Joel Hasbrouck, All rights reserved 31 Settlement when S T > X = 50 Settlement by delivery The customer pays us $50 = $1,000,000 We deliver = 20,000 shares of SAC Settlement in cash We sell our 20,000 shares for 20,000 S T We pay the customer S T X 2018, Joel Hasbrouck, All rights reserved 32 16

17 H3: How to play. The spreadsheet is set up to run without modification. But you must load the RTD library first (see instructions for ALGO1). Start the RIT client, then start Excel and load the H3 spreadsheet. The spreadsheet has calculations on the left and a plot of your open delta on the right. How close is your SAC position to where it should be? To activate the plot, right after the market starts, click on START REAL-TIME CHART The data and chart should update in real time. 2018, Joel Hasbrouck, All rights reserved , Joel Hasbrouck, All rights reserved 34 17

18 And at the end of the run 2018, Joel Hasbrouck, All rights reserved 35 Download performance reports Price (left scale, red line) At this resolution, the ask and bid aren t distinguishable. Position (right scale, black line) 36 18

19 $10 per trade. Adj Profit = 11,340 Avg absolute portfolio Δ commissions = 11,340 1,118 1,580 = 8, Suggestions Configure your desktop so that you can see The Excel chart of your open delta. Two RIT client order entry panels (buy and sell), or the book trader panel configured for one-click orders. Objectives Accurate hedging low average portfolio Δ Low trading costs (commissions) 2018, Joel Hasbrouck, All rights reserved 38 19

20 Practical complications Adjusting the hedge requires us to trade in the direction of the market. We sell when S falls, buy when S rises. Do our trades push the price against us? Are there other traders who are also trying to delta-hedge? Not an issue in the H3 case: the market for SAC is very liquid no matter what traders do. What if there is a significant news announcement when the market is closed? 2018, Joel Hasbrouck, All rights reserved 39 Hedging and jumps Delta hedging works best when successive price movements are small. Slow accumulations of lowintensity information Example SPY, April 15, 2011 Delta hedging does not work well when prices move due to large information shocks. 2018, Joel Hasbrouck, All rights reserved 40 20

21 Selling a call on ACOR, 14 April , Joel Hasbrouck, All rights reserved 41 At 10AM, a customer wants a one-year ACOR call with X = $20. T = 1; assume σ = 0.5, r = 0. At 10AM, the stock price is S $21. From Black-Scholes C = $2.976 and Δ = N d 1 = We sell ten 100-share call options to a customer at $6 We hedge by buying 623 shares of ACOR. At about 13:20 the stock price goes to $27. At S = $27, C = $7.576 and Δ = , Joel Hasbrouck, All rights reserved 42 21

22 Mark-to-market positions Time Assets Liabilities 10:00 Cash received from sale of calls 6,000 2,976 Calls, mark-to-mkt $ Stock (623 $21) 13,083 13,083 Loan / charge to capital 3,024 Net worth 14:00 Cash 6,000 7,576 Calls $ Stock (623 $27) 16,821 13,083 Charge to capital 2,162 Net worth (a loss of 862) 2018, Joel Hasbrouck, All rights reserved 43 22