Pricing and Hedging American-Style Options with Deep Learning
2. Calculating a Candidate Optimal Stopping Strategy
- Simulate5 paths , , of the underlying process .
- Set for all k.
- For , approximate with by minimizing the sum
- Define , and set constantly equal to .
- denotes the depth and the numbers of nodes in the different layers;
- are affine functions;
- For , is of the form for a given activation function .
3.1. Lower Bound
3.2. Upper Bound, Point Estimate and Confidence Intervals
4.1. Hedging Until the First Possible Exercise Date
4.2. Hedging Until the Exercise Time
5.1. Pricing Results
5.2. Hedging Results
Conflicts of Interest
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Meaning feedforward networks with a single hidden layer.
This covers Bermudan options as well as American options that can only be exercised at a given time each day. Continuously exercisable options must be approximated by discretizing time.
That is, is -measurable, and for all and every measurable function such that is integrable.
The main difference between this algorithm and the one of Longstaff and Schwartz (2001) is the use of neural networks instead of linear combinations of basis functions. In addition, the sum in (3) is over all simulated paths, whereas in Longstaff and Schwartz (2001), only in-the-money paths are considered to save computational effort. While it is enough to use in-the-money paths to determine a candidate optimal stopping rule, we need accurate approximate continuation values for all to construct good hedging strategies in Section 4.
As usual, we simulate the paths , , independently of each other.
Generated independently of ,
The use of nested simulation ensures that are unbiased estimates of , which is crucial for the validity of the upper bound. In particular, we do not directly approximate with the estimated continuation value functions .
Discounting is done with respect to the savings account. Then, the discounted value of money invested in the savings account stays constant.
That is, is -measurable and for all and every measurable function such that is integrable.
Independent of , .
The computations were performed on a NVIDIA GeForce RTX 2080 Ti GPU. The underlying system was an AMD Ryzen 9 3950X CPU with 64 GB DDR4 memory running Tensorflow 2.1 on Ubuntu 19.10.
Bermudan max-call options are a benchmark example in the literature on numerical methods for high-dimensional American-style options; see, e.g., Longstaff and Schwartz (2001); Rogers (2002); García (2003); Broadie and Glasserman (2004); Haugh and Kogan (2004); Broadie and Cao (2008); Berridge and Schumacher (2008); Jain and Oosterlee (2015); Becker et al. (2019a, 2019b).
That is, for all and .
Simulation based methods work for any price dynamics that can efficiently be simulated. Prices of max-call options on underlying assets with different price dynamics were calculated in Broadie and Cao (2008) and Becker et al. (2019a).
The hyperparamters were chosen as in Kingma and Ba (2015). The stepsize was specified as , , and according to a deterministic schedule.
|d||Point Est.||CI||DOS CI|
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Becker, S.; Cheridito, P.; Jentzen, A. Pricing and Hedging American-Style Options with Deep Learning. J. Risk Financial Manag. 2020, 13, 158. https://doi.org/10.3390/jrfm13070158
Becker S, Cheridito P, Jentzen A. Pricing and Hedging American-Style Options with Deep Learning. Journal of Risk and Financial Management. 2020; 13(7):158. https://doi.org/10.3390/jrfm13070158Chicago/Turabian Style
Becker, Sebastian, Patrick Cheridito, and Arnulf Jentzen. 2020. "Pricing and Hedging American-Style Options with Deep Learning" Journal of Risk and Financial Management 13, no. 7: 158. https://doi.org/10.3390/jrfm13070158