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Greeks and Option Hedges

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Greeks and Option Hedges

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    1. 12-1 Greeks and Option Hedges

    2. 12-2 Option Greeks To replicate an option your synthetic option should change in value just like the real option when the parameters change. What happens to option price when one input changes? Delta (D): change in option price when stock price increases by $1 Gamma (G): change in delta when option price increases by $1 Vega: change in option price when volatility increases by 1% Theta (q): change in option price when time to maturity decreases by 1 day Rho (r): change in option price when interest rate increases by 1% Greek measures for portfolios The Greek measure of a portfolio is weighted average of Greeks of individual portfolio components

    3. 12-3 Illustration of delta and gamma

    4. 12-4 Delta Hedging Problems: 1-3, 7-10 (use Excel), 17

    5. 12-5 Who delta hedges? Portfolio insurers Market makers Anyone who wants to replicate options

    6. 12-6 What Do Market Makers Do? Provide immediacy by standing ready to sell to buyers (at ask price) and to buy from sellers (at bid price) Generate inventory as needed by short-selling Profit by charging the bid-ask spread

    7. 12-7 What Do Market Makers Do? (contd) Their position is determined by the order flow from customers In contrast, proprietary trading relies on an investment strategy to make a profit

    8. 12-8 Market-Maker Risk Market makers attempt to hedge in order to avoid the risk from their arbitrary positions due to customer orders Option positions can be hedged using delta-hedging Delta-hedged positions should expect to earn risk-free return if updated continuously.

    9. 12-9 Position on a written call option Is ? positive or negative?

    10. 12-10 Market-Maker Risk (contd) Delta (D) and Gamma (G) as measures of exposure Suppose D is 0.5824, when S = $40 (Table 13.1 and Figure 13.1) A $0.75 increase in stock price would be expected to increase option value by $0.4368 ($0.75 x 0.5824) The actual increase in the options value is higher: $0.4548 This is because D increases as stock price increases. Using the smaller D at the lower stock price understates the the actual change Similarly, using the original D overstates the the change in the option value as a response to a stock price decline Using G in addition to D improves the approximation of the option value change

    11. 12-11 Delta-Hedging Market-maker sells one option, and buys D shares to hedge option risk Delta hedging for 2 days: (daily rebalancing and mark-to-market): Day 0: Share price = $40, call price is $2.7804, and D = 0.5824 Sell call written on 100 shares for $278.04, and buy 58.24 shares. Net investment: (58.24x$40) $278.04 = $2051.56 At 8%, overnight financing charge is $0.45 [$2051.56x(e-0.08/365-1)] Day 1: If share price = $40.5, call price is $3.0621, and D = 0.6142 Overnight profit/loss: $29.12 $28.17 $0.45 = $0.50 Buy 3.18 additional shares for $128.79 to rebalance Day 2: If share price = $39.25, call price is $2.3282 Overnight profit/loss: $76.78 + $73.39 $0.48 = $3.87

    12. 12-12 Delta-Hedging (contd) Delta hedging for several days

    13. 12-13 Delta-Hedging (contd) Delta hedging for several days (cont.) G: For large increases in stock price D increases, and the option decreases in value faster than the gain in stock value. This G- risk can be serious, and is often hedged especially if delta can not be updated near-continuously q : If a day passes with no change in the stock price, the option becomes cheaper. Since the option position is short, this time decay increases the profits of the market-maker. Interest cost: In creating the hedge, the market-maker purchases the stock with borrowed funds. The carrying cost of the stock position decreases the profits of the market-maker.

    14. 12-14 Mathematics of ?-Hedging D-G approximation (contd)

    15. 12-15 Mathematics of ?-Hedging (cont.) D-G approximation Can estimate new option value using D alone when stock price moved up (down) by e. (e = St+h St) Using the D-G the accuracy can be improved a lot Example 13.1: S: $40 $40.75, C: $2.7804 $3.2352, G: 0.0652 Using D approximation C($40.75) = C($40) + 0.75 x 0.5824 = $3.2172 Using D-G approximation C($40.75) = C($40) + 0.75 x 0.5824 + 0.5 x 0.752 x 0.0652 = $3.2355

    16. 12-16 Mathematics of ?-Hedging (contd) q: Accounting for time

    17. 12-17 The Black-Scholes Analysis Black-Scholes partial differential equation where G, D, and q are partial derivatives of the option price computed at t Under the following assumptions: underlying asset and the option do not pay dividends interest rate and volatility are constant the stock moves one standard deviation over a small time interval The equation is valid only when early exercise is not optimal (American options problematic)

    18. 12-18 The Black-Scholes Analysis (contd) Advantage of frequent re-hedging Varhourly = 1/24 x Vardaily By hedging hourly instead of daily total return variance is reduced by a factor of 24 The more frequent hedger benefits from diversification over time Three ways for protecting against extreme price moves Adopt a G- position by using options to hedge Augment the portfolio by by buying deep-out-of-the-money puts and calls Use static option replication according to put-call parity to form a G and D-neutral hedge

    19. 12-19 Portfolio insurance

    20. 12-20 Delta-hedged portfolio positions Strategy: Replicate delta of put option continuously through time Began being widely used in the 1980s, devised largely by Leland & Rubinstein It is blamed in part for -22% crash of October 1987 Still used widely today (with safeguards).

    21. 12-21 Two basic methods of portfolio insurance Buy put options Expensive Not enough liquidity Create your own put options Need to match put greeks for a given put at $K

    22. 12-22 Creating synthetic put options Sell stock (futures) and invest proceeds at risk-free rate Sell quantity sufficient to match the ? of the put option As the value of the portfolio increases, the ? of the put becomes less negative decrease short position in stock (futures) --- buy stock (futures) As the value of the portfolio decreases, the ? of the put becomes more negative increase short position in stock (futures) --- sell stock (futures) These adjustment are potentially destabilizing especially if everyone is doing it together...

    23. 12-23 portfolio insurance example A fund manager has a portfolio that mirrors the image of the SP500, and is worth $100 million. The value of the SP500 is 1452.63, and the portfolio manager would would like to buy insurance against a reduction of more than 5% in value over the next 6 months. The risk-free rate is 7%, the dividend yields on the portfolio and the SP500 is 3.5%, and the standard deviation of the index is 30% per annum. Questions: how much would it cost to simply buy european put options? What would be the outcome if SP500 ends up at 1200? How could the firm hedge the position dynamically with synthetic put option? What would be the outcome if SP500 ends of at 1200?

    24. 12-24 Portfolio insurance example pg 2 Using Put Options the manager is concerned about a 5% drop in value, therefore the manager should use a put option on the index with X = .95(1452.63) = 1380 the value of the put option at X=1380 : the number of contracts to be purchased N = (1)(100,000,000/145,263) = 688 put options the cost of insurance = 688(75.17)100 = 5,171,696 at expiration ST = 1200 (value fell by 1200/1452.63 - 1= -17.39% ) portfolio value = 10,000,000(1-.1739) = 82,610,000 option payoff = (1380-1200)*100*688 = 12,384,000 total payoff = $94,994,000 which is essentially a loss of 5% (rounding error)

    25. 12-25 Portfolio insurance example pg 3 Using synthetic options what is the delta of the put option with X=1380? Put delta = e-qt(N(d1) - 1) = e-.07(.5)(.6665-1) = -0.3220 what to do? Short 32.2% of portfolio and invest at risk-free rate update periodically illustrate a bi-weekly update with index ending at 1200. 26 weeks in six-month period, or 13 bi-weekly periods shares of index initially held = 100,000,000/14542.63=68,840 shares of index to be shorted = .322(68,840) = 22,166 Note, we could short on the run sell on the run

    26. 12-26 Portfolio insurance example pg 4 The dynamic portfolio insurance process Note the put delta approaches -1.0, implying full hedge at maturity Payoff at expiration if we sold on the run we have $94,943,229 in cash at expiration

    27. 12-27 Portfolio insurance example pg 5 Why may we not end up with $95,000,000 ? 1. Transaction Costs in example we are short of $95 million w/ zero t-costs their inclusion would drop us even lower than $94.9 million result 2. The price decline may not involve smooth changes an extreme case would cost us another (roughly) $4.6 million: 3. Continuous adjustment if we update continuously (every second) we would end up at $95m no matter how volatile the price descent (w/o t-costs).

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