衍生品证券DerivativesChapter13.ppt

1,Derivative Securities SAIF Dr. Ming Guo,Chapter 13 Market-Making and Delta-Hedging,2,Chapter Outline,What do market-makers do? Market-makers option risk in the absence of hedging Dynamic delta-hedging Delta-Gamma approximation, theta accounting for time, understanding market-makers profit Market-making as insurance,3,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 Their position is determined by the order flow from customers In contrast, proprietary trading relies on an investment strategy to make a profit,4,Risk of Market-Maker,The market-maker has arbitrary position generated by fulfilling customer orders (reactive) Option positions can be hedged using delta-hedging: to hedge by taking an offsetting positions in shares () Because of the cost difference between options and shares, the market-maker must invest capital (B) to maintain the position The hedged position earns the risk-free rate Delta-hedge is a key to understand option pricing,5,The Risk of Unhedged Position,Suppose a market-maker sells a call option The risk is that the share price will rise,6,The Risk of Unhedged Position(contd),We can measure the profit by marking-to-market the position: what would be the gain or loss if we liquidate the position today? Suppose that the market-maker does not hedge the written option. If the stock price rises to $40.75, the option price is $3.2176. Market-makers profit is $0.4372. If the stock price does not change, the market-maker will profit because of time decay: option value is smaller one day closer to expiration.,7,Overnight Profit for Unhedged Short Call,8,Delta (D) as a Measure of Exposure,Delta (D) describes the price sensitivity of the option D is 0.5824 when S = $40 A $0.75 increase in stock price would be expected to increase option value by $0.4368 ($0.75 0.5824) The actual increase in option price (according to the option pricing model) is higher: $0.4372 Delta only approximately measures the change of option value in response to change in stock price,9,Delta-Hedging,Market-maker sells one option, and buys D shares 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: (58.24 $40) $278.04 = $2051.56 At 8%, overnight financing charge is $2051.56 (e0.08/365 1) = $0.45,10,Delta-Hedging (contd),Day 1: If share price = $40.5, call price is $3.0621, and D = 0.6142 Overnight profit: 58.24 ($40.50 $40) 100 ($3.0621 $2.7804) $0.45 = $29.12 $28.17 $0.45 = $0.50 Dynamic rebalancing Additional shares to buy: 100 (0.6142 0.5824) = 3.18 Additional investment: 3.18 $40.50 = $128.79,11,Delta-Hedging (contd),Mark-to-market profit The net cash flow generated by always borrowing to fully fund the position Not entirely self-financing (with $0.50 leftover can be pocketed) Day 2: If share price = $39.25, call price = $2.3282 Interest = $2181.30 (e0.08/365 1) = $0.48 Overnight profit: 61.42 ($39.25 $40.50) 100 ($2.32823.0621) $0.48 = $76.78 + $73.39 $0.48 = $3.87,12,Delta-Hedging (contd),Daily Profit calculation over 5 days for a market-maker who delta-hedges.,13,Source of Hedging P&L,Gamma (G): gamma is positive for call option. For large decreases in stock price D decreases, and the option decreases in value slower than the loss in stock value For large increases in stock price D increases, and the option increases in value faster than the gain in stock value In both cases the net loss increases.,14,Source of Hedging P&L (contd),Theta (q): theta is negative for most options. 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.,15,Hedging Profit and Loss,16,One SD Movement of Stock Price,The market-maker approximately breaks even for one-standard-deviation move in stock ,17,The Mathematics of Delta-Hedging,Delta, gamma, and theta all play a role in determining the profit on a delta-hedged position. How to use delta, gamma, and theta to approximate the change in option price? Delta-Gamma approximation Theta: accounting for time How to use the mathematical approximation to understand the market-makers profit?,18,Gamma (G) as a Measure of Exposure,Gamma measures the change in delta when stock price changes: to obtain an average delta during stock price changes Using G in addition to D improves the precision of the option value change For a call, as stock price increases, delta also increases Using the smaller delta at the lower stock price understates the actual option value change Similarly, delta overstates the change in option value when stock price declines,19,Delta-Gamma Approximation,D-G approximation If G is approximately constant, the average delta is simply the average of (St) and (St+h).,20,Delta-Gamma Approximation,D-G approximation The Gamma correction is always positive 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 0.5824 = $3.2172 Using D-G approximation: C($40.75) = $3.2172 + 0.5 0.752 0.0652 = $3.2355,21,Delta-Gamma Approximation (contd),22,Theta: Accounting for Time,Greeks are provided in table 13.1.,23,Market-Makers Profit,Market-makers profit when the stock price changes by e over an interval h:,(13.7),24,Market-Makers Profit (contd),Sources of market-makers profit Time decay benefits the market-maker Interest and gamma work against the market-maker It is the magnitude, not the direction of the stock price move that determines the profit If s is annual, e is one-standard-deviation move over a period of length, e2 = s2S2h,25,Market-Makers Profit (contd),26,Market-Makers Profit (contd),The profit of a delta-hedged market-maker The losses from $1-move (-1.970) is substantially larger than gains from no move (1.283) Small moves are more probable than large moves For normally distributed stock price Stock price moves greater than 1s occur about 1/3 of the time The market-maker expects to make small profits about 2/3 of the time, and large losses 1/3 of the time On average, the market-maker will break even,27,Delta-Hedging in Practice,Delta-hedging does not eliminate risk Unexpected dividend payment Negative gamma induces risks from large price moves Three ways to protect against extreme price moves To adopt a Gamma-neutral position by using options To augment the portfolio by buying deep-out-of-the-money puts and calls To use static option replication according to put-call parity to form a G and D-neutral hedge,28,Gamma-Neutrality,29,Gamma-Neutrality (contd),30,Gamma-Neutrality (contd),The