Hedging Option Books Using Neural-SDE Market Models

Abstract

We study the capability of arbitrage-free neural-SDE market models to yield effective strategies for hedging options. In particular, we derive sensitivity-based and minimum-variance-based hedging strategies using these models and examine their performance when applied to various option portfolios using real-world data. Through backtesting analysis over typical and stressed market periods, we show that neural-SDE market models achieve lower hedging errors than Black–Scholes delta and delta-vega hedging consistently over time, and are less sensitive to the tenor choice of hedging instruments. In addition, hedging using market models leads to similar performance to hedging using Heston models, while the former tends to be more robust during stressed market periods.

Keywords:

• Market models
• European options
• market simulators
• no-arbitrage
• neural-SDE
• hedging

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Nevertheless, different option pricing models may yield the same hedge ratios (and consequently hedging performance) if they are scale-invariant; see the in-depth analysis by Alexander and Nogueira (2007).

2 There are attempts to estimate some parameters using time series of the underlying and calibrating the remaining parameters to option prices (for example, Bates 2000; Broadie, Chernov, and Johannes 2007). However, this approach remains inconsistent, as it still relies on daily recalibration of the model, which leads to time-varying calibrated parameters, whereas in theory the model parameters are assumed constant over time.

3 We use liquidly traded European style vanilla options as hedging instruments. Though we only consider call options here, by put-call parity, put options can also be used.

4 The agent could invest the residual cash from the hedged portfolio (1(1) $\mathrm{\Pi }(S,{\xi }_1,\dots ,{\xi }_d)=V(S,{\xi }_1,\dots ,{\xi }_d)-X^{S}S-\sum _{i=1}^d{X}^{C_i}{C}_i(S,{\xi }_1,\dots ,{\xi }_d),$(1) ) in a risk-free asset. If, for example, the risk-free asset has non-zero interest yield $r_t$, then the hedging error in (2(2) ${\mathcal{E}}_t\left(\mathrm{\Delta }t\right)={\mathrm{\Pi }}_{t+\mathrm{\Delta }t}-{\mathrm{\Pi }}_t.$(2) ) would be revised to ${\mathcal{E}}_t\left(\mathrm{\Delta }t\right)={\mathrm{\Pi }}_t{e}^{{\int }_t^{t+\mathrm{\Delta }t}{r}_s\mathrm{d}\mathrm{s}}-{\mathrm{\Pi }}_{t+\mathrm{\Delta }t}$ instead.

5 We have assumed that the drift and diffusion coefficients are independent of S, in the sense that the underlying asset's log-return and volatility surfaces (represented by ξ) are unaffected by its price level. Nevertheless, it is possible, and simple, to add S as an additional argument and train the model with neural nets (as is done in Cohen, Reisinger, and Wang 2021).

6 We assume independent Brownian motions that drive the randomness in S and ξ in the main text of Cohen, Reisinger, and Wang (2022) for simplicity, but build a model for S and ξ jointly with a full ${\mathbb{R}}^{(d+1)\times (d+1)}$ covariance matrix, in Appendix B.2 of Cohen, Reisinger, and Wang (2022). Since we would like to see how MV-based delta hedging behaves differently from its sensitivity-based counterpart using market models, we need a market model that captures the correlation between S and ξ, according to Remark 3.2. Hence, here we allow a fully specified diffusion matrix (by neural networks) when building a market model.

7 However, these assumptions will not generally hold for neural-SDE market models. It remains to be investigated empirically how the violation of these assumptions impacts hedging performance using neural-SDE market models.

8 A down-and-out call option is a vanilla call option if its barrier has not been hit by its expiry date. However it becomes worthless if its barrier is hit at any time before it expires.

9 Nevertheless, the static hedging relation is usually derived with certain model assumptions, so that the replicating portfolios may only approximate the option's payoff. The impact of the approximation errors on hedging performance needs to be examined with empirical tests.

10 These prices are calculated from implied volatilities that are interpolated using a methodology based on a kernel smoothing algorithm, according to the OptionMetrics' IvyDB Europe reference manual (OptionMetrics 2020).

11 In practice, some of these portfolios are constructed to trade views on volatility directions or skews, hence hedging them may contradict the intended purpose of making profits when market moves as expected. By including these portfolios in the hedging analysis, we are more motivated by examining the performance of hedging different risk profiles than giving practical implications.

12 These portfolios are constructed from options with specified time-to-expiries and moneynesses (deltas), which are in general impossible to match perfectly with traded options. Therefore, we find the options that have closest time-to-expiries and moneynesses to the desired specifications when constructing portfolios.

13 This does not indicate that arbitrage was present in the market on these dates, but rather that our two-factor model fails to represent prices accurately, and so the best fitting factor values do not represent the prices in an arbitrage-free way. This was not observed in our training data (which includes the 2007-8 financial crisis, see Figure A8).

Additional information

Funding

This publication is based on work supported by the EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling (EP/L015803/1) in collaboration with CME Group. Samuel Cohen and Christoph Reisinger acknowledge the support of the Oxford–Man Institute for Quantitative Finance, and Samuel Cohen also acknowledges the support of the Alan Turing Institute under the Engineering and Physical Sciences Research Council grant EP/N510129/1.