1. Chapter 13
Market-Making and
Delta-Hedging

2. 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

3. 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

4. Market-maker risk (cont.)

5. Market-maker risk (cont.)
Delta (D) and Gamma (G) as measures of exposure:
Suppose D is , 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 $ ($0.75 x )
The actual increase in the option’s 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

6. Delta-hedging
Market-maker sells one option, and buys D shares, implying a market value of x $40 – equal to $ needing to be borrowed.
Delta hedging for 2 days: (daily rebalancing and mark-to-market):
Day 0: Share price = $40, call price is $2.7804, and D =
Sell call written on 100 shares for $278.04, and buy shares.
Net investment: (58.24x$40) – $ = $
At 8%, overnight financing charge is
$ x (e0.08/365-1) = $ 0.45

7. Delta-hedging
Delta hedging on the next day: (daily rebalancing and marking-to-market):
Day 1: If share price = $40.5, call price is $3.0621,
and D =
Overnight profit/loss:
($ $40)x 58.24shares = $29.12 gain on stock position.
$ – $ = -$28.17 loss on written call option.
$29.12 – $28.17 – $0.45(overnight interest) = $0.50 profit.
New option D = Not delta-hedged anymore since we only have shares. We must thus buy new
shares, i.e additional shares costing 40.5x3.18 = $
Note: new market value that needs to be borrowed is
61.42 x $ $ = $

8. Delta-hedging On the last day: (daily rebalancing and mark-to-market):
Day 2: If share price = $39.25, call price is $2.3282
($ $40.50)x 61.42shares = -$76.78 loss on stock position.
$ – $ = $73.39 gain on written call option.
2, x (e0.08/365-1) = -$0.48 in overnight interest.
Overnight profit/loss: – $ $73.39 – $0.48 = – $3.87.

9. Delta-hedging (cont.)
Delta hedging for several days:

10. Delta-hedging (cont.) Delta hedging for several days: (cont.)
G: For large decreases in stock price D decreases, and the short call option position increases in value slower than the loss in stock value. For large increases in stock price D increases, and the option decreases in value faster than the gain in stock value. In both cases the net loss increases.
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.

11. Delta-hedging (cont.)

12. Mathematics of D-hedging (cont.)
D-G approximation:
Recall the under (over) estimation of the new option value using D alone when stock price moved up (down) by e.
(e = St+h – St)
Using the D-G approximation the accuracy can be improved a lot:
Example 13.1: S: $ $40.75, C: $ $3.2352,
and G:
Using D approximation:
C($40.75) = C($40) x = $3.2172
Using D-G approximation:
C($40.75) = C($40) x x x = $3.2355

13. Mathematics of D-hedging (cont.)
D-G approximation: (cont.)

14. Mathematics of D-hedging (cont.)
q: Accounting for time

15. Mathematics of D-hedging (cont.)
Market-maker’s profit when the stock price changes by e over an interval h:
Change in value
of stock
of option
Interest
expense
The
effect
of G
of q
Interest cost

16. Mathematics of D-hedging (cont.)
If s is annual, a one-standard-deviation move e over a period of length h is sSh. Therefore,

17. The Black-Scholes Analysis
Black-Scholes partial differential equation:
where G, D, and q are partial derivatives of the option price
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. The Black-Scholes Analysis (cont.)
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-neutral position by using options to hedge
Augment the portfolio 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. The Black-Scholes Analysis (cont.)
G-neutrality: Let’s G-hedge a 3-month 40-strike call with a 4-month 45-strike put, along with a certain number of shares of the underlying stock.
Recall that Gp = n1G1 + n2G2
And so, since we want Gp=0, we need n2 = n1 G1/G2
If the 40-strike option is option #1 and the 45-strike is option #2, we need to buy
n2 = GK=40, t=0.25/ GK=45, t=0.33=0.0651/0.0524= units of the 45-strike option for each unit (n1=1) of the 40-strike option.

20. The Black-Scholes Analysis (cont.)
Since the resulting D is equal to , we need to buy shares of stock in order to also be delta-neutral.

21. The Black-Scholes Analysis (cont.)

22. Market-making as insurance
Insurance companies have two ways of dealing with unexpectedly large loss claims:
Hold capital reserves
Diversify risk by buying reinsurance
Market-makers also have two analogous ways to deal with excessive losses:
Reinsure by trading out-of-the-money options
Hold capital to cushion against less-diversifiable risks
When risks are not fully diversifiable, holding capital is inevitable