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In the previous section, we looked at the parallel and total gamma profiles of the hedging portfolios for different parametrizations. We recall that the total gamma of the MR portfolio is negative and almost twice as big in magnitude compared with other portfolios. Although this implies that the MR model is worse than the

others in terms of the gamma profiles, it is not the full story. Large negative gamma will only be a problem if it is not compensated by theta which is the Greek with respect to the time decay of an option. In this section, we analyze the gamma-theta balance for the hedging portfolio of the Bermudan. See Chapter 25, Volume 3 of Andersen and Piterbarg [2010] for a detailed discussion on this topic. Generally speaking, suppose we want to evaluate the change in value of the hedging portfolio as time advances forward by 1 trading day given a certain size of perturbation for all co-initial swap rates. By Taylor expansion, we have that

V_tport_+h ≈V_tport+∂V port t ∂t h+∆(t)+ 1 2 > A(t), (3.26)

whereh equals 1 trading day, > is the row vector of perturbation for all co-initial swap rates,∆(t) is the time-tdelta vector and A(t) is the time-tgamma matrix.

Note that the term ∂V

port

t

∂t is referred to as the theta and it can be calculated numerically. Since we assume that the proportions of hedging instruments do not change over the time period [t, t +h], the change in the portfolio’s value purely comes from the change in the Bermudan and hedging instruments’ values. As we delta hedge the Bermudan at timet, the vector∆(t) is a zero vector and hence∆(t)

will not contribute to the change in the portfolio’s value. It is then clear that the theta and the gamma of the portfolio should provide a good balance so thatV_tport_+h

will not differ too much fromV_tport, i.e. we would want that ∂V

port t ∂t h+ 1 2 >_{A(t) is}

as close to zero as possible (gamma-theta balance). Ideally, investors would like the sum ∂V port t ∂t h+ 1 2

>_{A(t) to be positive to ensure that the portfolio is not losing its}

value.

In practice, it is not unusual to observe that all co-initial swap rates will move up (or down) together so we will fix the perturbation to be of the same sign and size for all rates. The term>A(t), thus, has a close link to the total gamma as displayed below.

Bermudan vanilla swaptions co-initial swaps portfolio

MR 192 -369 -7 -184

HW 196 -286 2 -88

α= 0.05 199 -292 -8 -101

α= 0.3 196 -299 -7 -110

α= 5 195 -282 -6 -93

One step cov 196 -282 -6 -92

Table 3.22: Contribution of the gamma >A(t) to the change in values of the portfolios as all co-initial swap rates move up (or down) by 1 bp, i.e. > =

We now display the theta of the portfolio including the Bermudan and its hedging instruments in the following tables. It is clear from the results that we get the scenario: long theta (positive), short gamma (negative) for the HW, one step and weighted covariance portfolios. However, for the MR portfolio we obtain the scenario: short theta and short gamma. This means that only the HW, one step and weighted covariance portfolios tend to have the gamma-theta balance. In a practical sense, the gamma-theta balance has a direct implication. A long theta-short gamma position makes money in calm markets or small in magnitude equivalently, but loses money in volatile markets or big in magnitude. A short theta-long gamma position will do the opposite. The case of a short theta-short gamma position as implied by the MR portfolio will lose money in all market scenarios.

Bermudan vanilla swaptions co-initial swaps portfolio

MR -1806 1774 0 -32

HW -1119 1412 0 293

α= 0.05 -1162 1403 0 241

α= 0.3 -1132 1447 0 315

α= 5 -1098 1392 0 294

One step cov -1091 1391 0 300

Table 3.23: The change in values of the portfolios as time advances by 1 trading day (theta).

We now look further into why the MR portfolio has negative theta while others have the opposite. It is seen from the tables that the theta contribution from vanilla swaptions of the MR portfolio is positive and even bigger in size than those of other portfolios. The main factor that leads to negative theta of the MR portfolio is that the Bermudan’s theta following this parametrization is a lot more negative compared with others. While the Bermudan of the MR model loses 1806 after each trading day, the equivalent figures for other models is only roughly 1100. The reason for this is also clear. In order to compute the theta by finite difference, one will have to move time forward (i.e. time to expiry will decrease equivalently) and keep all other market data (implied volatilities and discount bonds) the same since theta is the partial derivative with respect to time. For example, we calculate the theta of the European swaption with today’s value Ve_Ti,j

0 by finite difference as e V_Ti,j 0+h−Ve i,j T₀

h with σei,j and y

i,j

0 both kept fixed. See also Chapter 22, Andersen and

Piterbarg [2010] for different ways considered by practitioners to calculate the theta. However, we note that computing the European swaption’s value with time to maturityTi−hand implied volatilityeσi,j is equivalent to the one with time to expiry Ti and implied volatility eσi,j

q

Ti−h

fore, computing the theta for the Bermudan and all European swaptions by finite difference can be done equivalently by adjusting all the involved implied volatilities in the swaption matrix: _eσi,j →eσi,j

q

Ti−h

Ti for alli, jand keep the discount bonds and time to expiry the same. This implies that the shorter the expiry is, the more the corresponding implied volatilities decrease. This adjustment of implied volatilities will not affect the correlation structure of the MR model but it will change that of the HW model as well as the one step and weighted covariance models immediately (similar to the vega analysis). Specifically, the correlation of any pair of co-terminal swap rates at their setting dates tends to decrease, e.g. this can be seen easily from the parametrization of the HW process. As a result, the Bermudan price produced by the HW, one step and weighted covariance models will decrease less than the MR model as time advances forward. This is again another interesting feature that differentiates the parametrizations by time and by expiry.

We now display the full result for the portfolios of payer Bermudan swaptions with other strikes in the following tables. We only show the results for the one step covariance case as similar conclusions are drawn for the weighted covariance model. Overall, we observe that one always gets a better gamma-theta balance for the HW and one step covariance models compared with the MR model. Note that when the strike is low (3% and 4%), we have the theta-gamma balance for the MR model but the theta is relatively small indicating that it might not be sufficient to offset the gamma risk when the market is volatile. For the large strike case (7%), a long gamma-long theta position is observed for the HW and one step covariance model indicating that the portfolios’ values tend to increase regardless of market movements.

Strike Model Component 3% 4% 5% 6% 7% MR Bermudan -23 207 192 118 73 vanilla swaptions -53 -568 -369 -168 -79 co-initial swaps 49 14 -7 -8 -7 portfolio -27 -347 -184 -58 -13 HW Bermudan -25 211 196 117 72 vanilla swaptions -30 -413 -286 -135 -65 co-initial swaps 50 21 2 -1 0 portfolio -5 -181 -88 -19 7 One step cov Bermudan -29 217 196 116 71

vanilla swaptions -23 -413 -282 -130 -60 co-initial swaps 51 17 -6 -9 -7

portfolio -1 -179 -92 -23 4

Table 3.24: Contribution of the gamma >A(t) to the change in values of the portfolios of payer Bermudans with different strikes as all co-initial swap rates move up (or down) by 1 bp, i.e. >= (±0.0001, . . . ,±0.0001).

Strike Model Component 3% 4% 5% 6% 7% MR Bermudan -243 -2684 -1806 -866 -445 vanilla swaptions 250 2685 1774 846 428 co-initial swaps 0 0 0 0 0 portfolio 7 1 -32 -20 -17 HW Bermudan -83 -1301 -1119 -631 -350 vanilla swaptions 142 1969 1412 716 373 co-initial swaps 0 0 0 0 0 portfolio 59 668 293 85 23 One step cov Bermudan -62 -1262 -1091 -621 -346

vanilla swaptions 107 1969 1391 695 357 co-initial swaps 0 0 0 0 0

portfolio 45 707 300 74 11

Table 3.25: The change in values of the portfolios of payer Bermudan swaptions with different strikes as time advances by 1 trading day (theta).

As estimated from the data, it is observed that one standard deviation move per day for swap rates is roughly±3 bp. We display the movement of the portfolios with respect to this scenario in the table below. Given this level of movement of the

co-initial swap rates, we can see that the values of the HW and one step covariance portfolios always increase whilst that of the MR portfolio decreases with certainty.

Strike Model Component 3% 4% 5% 6% 7% MR Bermudan -278 -2374 -1517 -690 -335 vanilla swaptions 171 1833 1220 594 309 co-initial swaps 74 20 -10 -12 -10 portfolio -33 -521 -307 -108 -36 HW Bermudan -121 -984 -825 -456 -243 vanilla swaptions 97 1350 984 513 276 co-initial swaps 76 31 2 -1 0 portfolio 52 397 161 56 33 One step cov Bermudan -105 -936 -798 -448 -239

vanilla swaptions 73 1350 969 500 266 co-initial swaps 76 25 -9 -13 -11 portfolio 44 439 162 39 16

Table 3.26: The gamma-theta balance (∂V

port

t ∂t h+

1

2

>_{A(t)) of the portfolios of payer}

Bermudan swaptions with different strikes as co-initial swap rates move 3 bp after 1 trading day, i.e. h= 1 trading day and> = (±0.0003, . . . ,±0.0003).