Although I have only discussed the hedging of delta risks thus far, as can be seen in Table 7.13, the vega risks inherent in the put option are, in fact, the largest risks vis-a-vis the delta, gamma, and rho risks. Despite this, there is no necessity for the hedger to hedge the vega risks. In fact, it is not uncommon for some practitioners to take on these risks unhedged and only run a deltahedging program.^{[1]}

Given the above backdrop, should the hedger decide to hedge the vega risks, he or she is now faced with the task of trying the extend the deltahedging analysis to incorporate delta-vega hedging attributes. To do this, one has to incorporate the following mechanics:

■ Introduce option assets that can be used to manage the vega risks.^{[2]}

■ Be able to simulate implied volatilities (and/or produce a real-world scenario of implied volatilities).

■ Introduce bid-offer spreads on implied volatilities.

Given the above caveats, to illustrate the implementation of the vegahedging program and keep the illustration manageable, I will assume that:

■ The hedge asset is a nine-year, at-the-money put option.

■ Every time the net portfolio vega risks are neutralized, the hedger sells all the existing assets at market price and transacts into the then-prevailing 9-year at-the-money put option.^{[3]}

■ Liquidating options are done at mid-market volatility levels, while the purchase (sales) of new at-the-money options are done at higher (lower) implied-volatility levels.

■ The monitoring and neutralizing of the net portfolio vega/delta risks would last until the first year of hedging is completed. Once this one- year mark is reached, the hedger will enter into a 9-year put option that perfectly replicates the option sold (hence this asset may not be an at- the-money option).

Spot-implied volatilities across all maturities and strikes are assumed to be constant (i.e., the implied volatility surface is trivially parallel to the strike and term axis).^{[4]}

■ Term structure of spot-implied volatilities is assumed to move as an entity^{[5]} and is simulated using the simple lognormal pdf.^{[6]} More precisely, I will assume that 1nσt+u is normally distributed with mean 1nσt – 0.02u and variance 0.04" – where σt+u represents the term structure of spot volatilities at time t + u.

One can now use the above assumptions and implement a vega-hedging program (similar to what was done to a delta-hedging program) and obtain the results in Table 7.17.^{[7]}

Table 7.17 shows the impact of the correlation between the volatility and stock-price jumps and net portfolio volatility-trading limits. From the table, two important observations can be made:

TABLE 7.17 Hedge P8cL in a 2 Percent Bid-Offer Spread in Both Volatility and Stock Price Movements and $0

Volatility

Trading

Limit

Perfect Positive Correlation

No Correlation

Perfect Negative Correlation

Average

Stdev

Average

Stdev

Average

Stdev

$-

$4,477.20

$508.90

$4,397.81

$1,113.30

$4,367.66

$1,364.96

$5.00

$328.83

$158.41

$1,891.63

$900.02

$1,945.53

$1,234.27

$10.00

$127.32

$66.94

$1,076.97

$742.27

$1,208.11

$1,040.31

$15.00

$76.68

$41.35

$669.61

$588.34

$703.59

$926.68

$20.00

$56.45

$25.92

$451.75

$455.59

$499.35

$751.46

$25.00

$48.13

$19.43

$317.78

$362.52

$389.95

$604.90

$30.00

$43.04

$17.99

$231.49

$297.91

$310.49

$495.19

$35.00

$38.67

$17.76

$171.77

$239.34

$252.40

$407.82

$40.00

$34.19

$16.97

$132.74

$197.57

$205.75

$338.47

$45.00

$29.43

$12.42

$105.07

$164.08

$171.08

$289.13

$50.00

$28.27

$11.75

$84.01

$135.93

$139.90

$240.09

$55.00

$27.37

$11.03

$68.27

$111.97

$116.33

$204.33

$60.00

$26.48

$10.41

$55.88

$93.07

$96.59

$171.13

$65.00

$25.71

$9.81

$46.64

$77.21

$80.69

$147.30

$70.00

$25.10

$8.97

$39.39

$62.11

$67.40

$123.29

1. Correlations between stock-price and volatility-rate jumps do matter, in that the more negative the correlation, the higher the hedge costs. While all the information is computed when the net portfolio delta-trading limit is 0, having a nonzero limit would definitely have an impact on the results

2. In the hedging strategy discussed, I used a 9-year option that I rolled over each vega rebalancing moment. In practice nothing stops me from using options of other maturities (e.g., a 3-year put or 1-year call option – assuming that such options are available in abundance). In such an instance, the reader should realize that while it is theoretically possible to neutralize the vega of a 10-put option using short-dated options, volatilities trade like yield curves.^{[8]} As a consequence, in practice, as discussed earlier, it is customary to have three net portfolio vega-trading limits (short term, medium term, and long term) – where the short term would possibly contain option maturities that are three years and lesser, the long term would possibly contain option maturities that are seven years and longer. Doing this allows the hedger to more prudently quantify and manage vega risks.

[1] One rationale for this behavior is due to the mindset that the 10-year historical volatility is a better measure for the 10-year realized volatility. Since the 10-year implied volatility tends to exceed the 10-year historical volatility by a big margin (which sometimes can be as much as 15 percent – when the 10-year historical volatility is as low as 17 percent) and the historical volatility is quite stable (in that the distribution of 10-year historical volatility tends to have a small standard deviation), practitioners think of this risk as negligible. As a consequence of this, deltas are computed using these historical volatilities. It is important for the reader to understand that with this thinking, the hedger cannot expect the mark-to-market value of his/her 10-year put option to be consistent with what the capital markets would indicate, because the capital markets' pricing of these options is driven off implied volatilities. (See Chapter 6 and footnote 30).

[2] Since there is a large variety of options that can be used (e.g., at the-money, in-the- money, and out-of-the-money options with maturities of one, two, or six months, and so on), the hedger is faced with the daunting task of deciding how best to select the right option to manage this risk and to neutralize the vega risks, the hedger first needs to identify the option's maturity and strike before deciding on the quantity of such options needed. See also footnote 15. Furthermore, the consequence of using options as hedge assets and managing the vega risks to historical volatility (though this can be done) would be a nightmare of trying to keep track of the cash flow going in and out of the hedging program. To understand this better, consider the instance when the 10-year option (in Table 7.13) is underwritten when the implied and historical volatilities are 35 and 20 percent, respectively. A 35 percent (20 percent) implied (historical) volatility gives a premium of about $560 ($166) and a vega of about 2770 (2267) for this 10-year option and charges $560 for the sale of the option. Suppose now that 1 year has passed and the 9-year hedge option has the same implied (historical) volatilities, giving it a respective premium of $578 ($179) and a vega of about 2790 (2330). If the hedger uses the historical volatility to manage the hedging program, then he/she would be needing 0.97, 9-year, at-the-money put options leading to a cost of $174 – which unfortunately would not be the actual amount paid out as the market would demand a premium of $562, making it difficult (but not impossible) to keep track of the cash flow going in and out of the hedging program. This problem does not occur if the hedger runs the entire hedging program using implied volatilities.

[3] This is a departure from what I did for the delta-hedging program where it was sufficient for me to top up or scale down the deltas by buying (selling) additional (redundant) shares.

[4] As a consequence of this, forward volatilities would trivially also be the same as the spot volatilities.

[5] See comment following Table 7.10 on the bucketing of vegas.

[6] This is an example of a simple volatility generator. One can easily use something that is more complex like the Heston's model or any other volatility generator to customize the generation of implied volatilities.

[7] To do this, I had to first simulate the volatility of the stock using the lognormal assumption for future spot-volatility movements, use that volatility to simulate stock prices, which are used to compute the vega of the 10-year put as it decays through time. At each time of rebalancing, I had to compute the amount of 9-year at-the- money options that I had to buy so that the vegas are neutralized. Once this amount is determined, I had to compute the net deltas of the portfolio (comprising the sold put and the newly acquired 9-year put) and then check to see what the portfolio delta is and if I should neutralize it (in which case I would need to transact in stocks). I continued this process until the end of the first year when I unwound all my positions to transact in a 9-year option with a strike price of $40 (the same level as the original 10-year put option that was originally sold). In this way, all the Greeks will be completely neutralized until the maturity of the sold option.

[8] Like the yield curves where different maturities have different characteristics (i.e., the short end of the yield curve does not trade like the long end of the yield curve), the volatility term structures are similar. As a consequence, as in the yield curve, the trader needs to understand the dynamics between short-, medium-, and long-term volatility movements and how one may (or may not) affect the other.

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