Looking at price profiles such as Figures 7.1-7.6 is useful for getting an overall feel for how the value of a portfolio changes with stock price and time. FIowever, it is usually necessary to have much more detailed breakdown of all the risks a portfolio is exposed to and these risks need to be expressed in dollar terms. To illustrate one way in which the risks can be examined we return to combination #3 example given above. In order to make the numbers more meaningful we multiply all the position sizes by 100. The example portfolio thus comprises a short position in 200 three-month $95 strike puts, long 100 three-month $105 strike puts and long 100 six-month $115 strike calls. For simplicity we assume that all options were purchased or sold at the appropriate fair value using a volatility of 15% when the stock price was $100 and that interest rates are zero. Initially it is necessary to define three shift parameters: a volatility shift, a time shift and a stock price shift.
These parameters will set the risk definitions associated with changes in the relevant variable.
Volatility Shift
A major source of risk to an options portfolio is the change in the overall market price of volatility. In the example portfolio we are iissuming that the portfolio is set up when all options are trading
(7.2)
at prices derived by using an input volatility of 15%. What if this suddenly increases by a given amount, say 1% to 16%? The given amount is the volatility shift used to define such volatility (or vega) risk and throughout the rest of this book we will set this to 1% although it is possible to use any figure. Columns headed (+ vol) will refer to the changes caused by increasing the volatility by 1%.
Time Shift
The time shift parameter in the illustrated example is set to 1 day (1/365th of a year) but of course can be set to a longer time period. This means that when reading the risk table, the columns headed (+ time) will refer to the changes caused by the passage of one day.
Stock Price Shift
Whether long or short volatility, the portfolio manager will need to have an idea as to how rapidly the delta of the portfolio is changing with respect to changes in the underlying stock price, i.e. the position gamma. Rather than give an instantaneous rate of change it is useful to have a measure that gives the difference in position delta when the underlying moves a specific amount—the stock price shift. In this example we set the stock price shift to the very small value of $0.10.
The two main concerns are (i) how does the value of the portfolio change and (ii) how does the stock exposure of the portfolio change? Table 7.3 gives the risk breakdown of the example portfolio and is divided into two sections. The left-hand side gives the profit and loss in dollars and the right-hand side gives information about the total market exposure in stock units. We first discuss the elements of the row corresponding to the stock price of $100.
• Profit and Loss = 0. The total portfolio value at $100 is $45,572. The portfolio was constructed with the stock price at $100, so the prices will not have changed and therefore the net profit at this level is zero.
• Profit and Loss (+ vol) = -$144. This figure of -$144 represents the change in the overall profit if volatility were to increase by the amount defined by the volatility shift parameter. So, other things being equal, if all the volatilities increase by 1%, the portfolio will drop in value by $144. Similarly, other things being equal, if all the volatilities decrease by 1%, the portfolio would increase in value by $144.
Table 7.3 Risk statistics: - 200 X (3-month $95 puts) + 100 X (3-month $105 puts) + 100 X (6-month
$115 calls). Vol shift = 1%, time shift = 1 day, stock price shift = $0.10, portfolio value at $100 =
$45,572
Profit and loss($) Equivalent stock position (stock units)
Stock Profit Profit Profit ESP ESP ESP Gamma
price (+ vol) (+ time) (+ vol) (+ time)
76 -135,620 -36 +3 +9,970 -20 +2 -2
SO -96,027 -231 +20 +9,977 -90 +8 -10
84 -58,231 -848 +72 +8,970 -222 +19 -33
88 -25.923 -1,904 +164 +6,945 -273 +24 -69
92 -4,396 -2,656 +235 +3,673 -54 +7 -90
96 +3,302 -2,089 +203 +300 +341 -23 -72
100 0 -144 +65 1,611 +582 -42 -21
104 -6,707 +2,132 -95 -1,409 +508 -35 +29
108 -9,186 +3,682 -196 +343 +256 -15 +55
112 -3,260 +4.215 -221 +2,636 +25 +1 +57
116 +11,649 +4,017 -198 +4,755 107 +9 +48
120 +34,182 +3,463 -160 +6,432 -160 +10 +36
124 +62,511 +2,788 -122 +7,664 -172 +9 +26
• Profit and Loss (+ time) = +$65. This is the change in the portfolio value caused by one day passing. So, other things being equal, the portfolio will increase in value by $65 by the next day.
These sums are small compared to the overall portfolio value, because at this stock price the positive and negative effects of the volatility changes and time decay on the prices of the individual components almost completely cancel out. A breakdown of the effects of these changes on the individual components is shown in Table 7.4.
• ESP = -1,611 stock units. The equivalent stock position (ESP) gives the net exposure of the position. At $100 this is —1,611 and so the position would feel as if the portfolio were short of 1,611 units of stock. A volatility player not interested in being exposed to the direction of the underlying could of course remove this risk by buying 1.611 units of stock.
• ESP( +- vol) = +582 stock units. This represents the change in the stock exposure caused by a general increase in volatility. Other things being equal, it the volatility increases by 1%. the exposure increases by 582 units >md so the resultant exposure would now be =
—1,611 + 582 = —1,029 units. If the volatility decreased by 1% the exposure would decrease by •~'<S2 units resulting in an exposure of — 1,611 — 582 = —2,193 units.
• ESP(+ time) = -42 stock units. This represents the change in the exposure with the passage of time. At this stock price level, the exposure of this portfolio will decrease by 42 stock units to -1,611 - 42 = -1653 units the next day.
• Gamma = -21 stock units. This represents the level of rehedging activity necessary to stay delta neutral or the rate at which the stock exposure is altering. At a stock price near $100 this portfolio would require adjustments of 21 units per $0.10 move The negative sign indicates that at this level the portfolio is short volatility and so small increases (decreases) in stock price would require buying (selling) 21 units per $0.10 move.
In summary then, at $100 the portfolio will suffer a small loss if volatility increases and gain in value with the passage of time. At $100 the position is slightly short of the market and slightly short of volatility. We now consider the effects on the same portfolio of the (slightly unrealistic) situation of the stock price suddenly increasing to $116.
• Profit and Loss = +11,649. The portfolio will have increased in value by $11,649.
• Profit and Loss (+ vol) = +$4,017. This is saying that if the volatility increases by 1%
there will be an additional profit of $4,017 and so the total profit would now be that due to the price move plus that due to the volatility move or 11,649 + 4,017 = $15,666. If, however, the volatility falls by 1% the profit would be reduced by 4,017 = to $7,632. If the volatility were to alter by more than 1%, the effect on the overall profit would be significant. At this stock level the vega risk to the portfolio is much larger than at $100.
• Profit and Loss (+ time) - -$198. The time decay at this stock price level is now -$198 per day. These losses would be subtracted from the corresponding profit of $11,649.
• ESP = +4,755 stock units. At this level the portfolio has an exposure of long 4,755 stock units and this will increase with increasing stock price. In the limit, at very high stock prices, this figure will approach a maximum of 10,000 and is all due to the long 100 call options.
• ESP(+ vol) = -107 stock units. Other things being equal, the exposure will decrease by 107 units if the volatility increases by 1%. So at a volatility of 16% the net stock exposure would be = 4,755 - 107 = 4,648 units.
• ESP(+ time) = +9 stock units. Time passing will increase the stock exposure by 9 units a day.
• Gamma = +48 stock units. Up at this level, the gamma is positive and equal to 48 stock units per $0.10 price move. The move from $100 to $116 has resulted in the portfolio changing from one of short volatility to one of
long volatility. At $100 the negative curvature of the 200 short puts dominates the portfolio, whereas at $116 the dual effects of the positive curvature of the long $105 put and the long $115 call dominate.
Software that produces risk tables like that given in Table 7.3 allow an options manager to predict, with a fair degree of confidence, what will happen to his portfolio at any stock price and if market conditions change. More importantly, the output allows a volatility player to see, before it happens, how his hedging activity will alter under a variety of different scenarios. In addition to giving the overall portfolio characteristics, it is sometimes instructive to look in more detail at the contribution of each of the individual components to the total. Since the total portfolio characteristics are simply the sum of the individual characteristics the examination of such breakdowns is straightforward. The software produces subsets of individual characteristics like those in Table 7.4 showing the contributions to the total vega and theta effects on the profit.
Table 7.4 Breakdown of profit changes at stock price = $100
Vega risk (volatility increase by 1%) Theta risk (time decrease by one day)
Option Position One option Portfolio One option Portfolio
(a) (b) size (c) - (d) size (e)
-(a) X (b) (a) X (d)
3-month
($95) put -200 +15.22 -3,044 -1.27 +254
3-month
($105) put +100 +16.37 +1,637 -1.37 -137
6-month
($115) call +100 +12.63 +1,263 -0.52 -52
Total $144 +$65