# Dynamic Hedging Managing Vanilla and Exotic Options - Nassim T.

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time and possible asset prices.

• There are two kind of topography reports: strike (or static) topography and gamma (or dynamic) topography.

• Topography has considerable advantages as it allows for transcending the Greeks as shown in Table 14.3.

Bucketing and Topography 233

Table 14.3 Old Days Strike Topography

March June

Strike P Ñ P Ñ

65 26

70 -20

75 -20

80 21

85 5 1

90 11 2

95 -5 - 19 11 -2

100 -72 - 71

105 2 70

110 1 23 11

115 -9

120

125

Strike Topography (or Static Topography)

¦ Static Topography is a two-dimensional map of the position displaying the distribution of the exposure iniace value horizontally (across expirations) and vertically (across strikes).

When option trading was still at a primitive state and option traders were dealing with only one or, at the most, two expirations, they used to maintain their open positions on a card provided by the clearing house so they could rapidly see what they had in their inventory. The clearing houses handed out the cards without charge. Table 14.3 depicts a sample card.

The trader could thus examine the position beyond the deltas and gammas and trade his position instead of trading his Greeks. Such a method, indeed, has an admirable tutorial value as it forces the trader to le^rn the intricacies of option trading without having recourse to pseudo-mathematical methods like the Greek reports. Instead of waking up at night with cold sweats and shouting the abstract, "I do not enjoy my being short 22.23 gammas and 71 weighted vegas," the trader would phrase his worries about the position in more precise terms such as, "I am short so many of the 105 calls that I need serious protection."

Modern book runners, however, do not have such luxury. As there are no set expiration dates and no set strikes, the card in Table 14.2 would need a few thousand lines and columns to reflect the topography of a large over-the-counter portfolio. In addition, it does not allow for any option structure beyond the vanilla.

234 Measuring Option Risks

So over-the-counter books need a design that allows traders to bundle the exposures in time and space points and examine their concentration risks. The strike topography is generally designed as shown in Table 14.4.

The over-the-counter strike topography displays the face value exposure per bidimensional bucket. The report shows horizontally (strike-wise) the net of the face value exposures between midpoints on the grid and vertically (time periods) the net of face value between points. So the 1W/100 bucket displays the net of all the trades between two days and one week and every option struck between 99 and 101.

The strike topography should deliberately exclude most nonvanilla options, which can be remedied in a separate report. American options, in addition should be handled with care, because their nominal expiration is not exactly the expected one. Only soft American instruments should be included on the matrix. A more thorough treatment of American options corresponds to shifting their nominal maturity into the "omega" or real time to expiration.

An application of the topography method is the method of squares, explained in Chapter 9.

Adding Correlated Instruments. It is possible to incorporate more than one similar commodity by summing them up, that is assuming 100% correlation. Such a method aims only at examining the strike concentration or the origin of gammas and vegas and can thus simplify without deluding the trader about his position. For example, if one thinks that it would be necessary to examine the risks of the French franc and the German mark in one report for topography purposes only, then all the USD-FRF strikes would be converted into USD-DEM. The real improvement, however,

Table 14.4 Over-the-Counter Strike Topography

Spot 80- 85 90 93 96 98 100 102 104 107 110 115 120 +

Id -83 86 -12 -33 -41 -10 -9 90 97 16 -49 56 15

2d 47 9 18 68 -15 -73 20 54 13 -54 -67 -24 -4

lw -33 66 -18 -25 45 66 -35 50 -71 18 27 23 -58

2 w -38 34 12 44 55 54 -53 -41 47 64 -28 -9 37

lm 35 -17 -55 34 3 52 43 7 -8 -15 30 -27 13

2m 12 45 2 -25 33 38 -20 15 5 -21 1 -26 -34

3m -14 -27 21 13 -28 -5 22 -6 -35 13 -24 39 6

6m -9 -11 1 23 20 -28 28 6 -11 -29 29 -15 -18

9m -2 -14 -1 8 -6 -6 0 -17 5 1 3 1 3

iy 6 -7 -6 2 1 -3 -2 5 9 1 10 9 5

2Ó 2 5 -2 4 -1 2 4 -1 -2 5 -2 -4 3

çó -4 3 1 0 -5 -2 0 -2 -3 -5 3 -4 5

5y+ -1 4 0 1 -5 -4 0 4 -2 -4 -5 -4 -2

Bucketing and Topography 235

would come from the hardly practicable three-dimensional map that allows for the slopes of the correlation.

Scaled Strike Topography. Since the difference between strikes that are close to the money is more meaningful for short-term options than for back-month structures, it is necessary to scale. For all intents and purposes the "gap" between the 100 and the 101 strikes is insignificant for a 5-3'ear option and very annoying for an overnight structure. The method as shown in Table 14.5 used to account for the risk difference is the scaling method that, in place of strikes, examines standard deviations. At 15.7 volatility, a 1 standard deviation difference between strikes is 1 percentage point, that is, between 100 and 101. In one year, it is the square root of 252 (252 days of business), namely 15.7. So the equivalent to an overnight gap of 1 point is the one-year gap of 15.7 points. Hence, assuming the spot traded at 100 (and no drift), the report would put in the same column the 101 overnight strike and the 115.7 one-year strike.

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