Higher Order Derivatives Analytics
Case Study
A simple mid-sized portfolio should be able to achieve 800k of cost reductions - How is so much left on the table?
Correct hedging the first time reduces overall costs as the need to unwind / reverse incorrect hedges is minimised, and traditional second order risk is bad at estimating this risk.
How bad? Take a simple book that only has one trade - long a 2y2y swaption straddle
(For net 1 unit of gamma) the gamma exposure looks like the below table:
If the 2y point moved +4bps and the 4y point moved +2bps, then the 2y2y fwd move should be zero (discounting assumptions etc left aside for simplicity).
Risk change and hence PL should both be zero but this is not the case for traditional gamma, showing a negative gamma PL for a long option, leading to 2 problems:
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Incorrect delta bucketing - correcting these "phantom" deltas reduces incorrect hedging
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Incorrect PL - incorrect risk bucketing leads to PLAT failures and higher capital costs
This is not due to higher order risk. Below, we highlight 5 benign scenarios and the delta changes and gamma PNLs calculated under both traditional gamma (TG) and PLATSON:
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Parallel shift (TG is accurate as per the implicit assumption)
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Bear Flatten
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Bear Steepen
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Bull Flatten
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Bull Steepen
The delta section (in green) shows that for non-parallel moves, traditional gamma achieves correct Net Delta but incorrect Gross Delta due to incorrect bucketing. This is the root cause of the non-linear PL errors.
Gross Delta Bucketing Error (%) is defined as ABS ( Gross Error / Gross delta )
On the above scenarios where 2s4s flattens/steepens <1bp, PLATSON gamma gives improved gross delta predicts of 25-100%.
The simple trade was modelled over the last 1 year of USD rate moves, during which 2y and 4y swap were correlated at 99% and a (generous) 50% tolerance for Gross Delta Bucketing Error has been designated. Only on the days when the error exceeds this is a charge of 0.05 basis points applied to the incorrect delta (to account for brokerage, operations, bid/offer, clearing fees, compression fees etc)
For the simple case of 1bn USD 2y2y ATM straddle, 66k USD is bled - on a daily basis this is unlikely to be picked up but across a complex portfolio, can add up to significant haemorrhaging over the year.
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PLATSON accuracy even at 5% tolerance is north of 95% with minimal incorrect hedging costs compared to TG accuracy at <1%
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For the previous year, TG incorrect bucketing at 50% accuracy Tolerance level can cost an extra 66k for 1bn 2y2y ATM Straddle (1/3 Normal Vega).
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A tolerance of 100% implies that the bucketed gamma risk from traditional gamma gives zero insight into the curve structure of the resulting gamma. (Hedging to TG model, once real risk is known all the hedges put in place would need corrected by as much as would have been hedged initially. Gamma would be as well represented by aggregating all the gamma risk into one (or other) of the affected buckets
- If the portfolio is vega and gamma hedged, inaccuracies from traditional gamma increase (this also occurs for PLATSON but to a much smaller extent ) - 75% Gamma and 95% Vega hedging the savings rise to 103k. (Delta hedging is negligible to the non-linear PL)
A sample portfolio of trades that might be found in a reasonable mid-sized portfolio as per below is taken and Delta, Vega and Gamma hedges were added as per the risk indicated (0.5y Vega risk appears from the gamma hedging). All trades are run through the same scenario (2017 data, so fairly benign volatility).
The results below (in kUSD) for this delta (100%), vega (95%) and gamma (75%) hedged portfolio show potential savings of around 800k USD per annum, simply from having inaccurately bucketed delta. This allows for a fairly large margin of error for traditional gamma.
