# Pricing and Hedging American-Style Options with Deep Learning

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## Abstract

**:**

## 1. Introduction

## 2. Calculating a Candidate Optimal Stopping Strategy

- (i)
- Simulate5 paths ${\left({x}_{n}^{k}\right)}_{n=0}^{N}$, $k=1,\dots ,K$, of the underlying process ${\left({X}_{n}\right)}_{n=0}^{N}$.
- (ii)
- Set ${s}_{N}^{k}\equiv N$ for all k.
- (iii)
- For $1\le n\le N-1$, approximate $\mathbb{E}\left[{G}_{{\tau}_{n+1}}\mid {X}_{n}\right]$ with ${c}^{{\theta}_{n}}\left({X}_{n}\right)$ by minimizing the sum$$\sum _{k=1}^{K}{\left(g({s}_{n+1}^{k},{x}_{{s}_{n+1}^{k}}^{k})-{c}^{\theta}\left({x}_{n}^{k}\right)\right)}^{2}\phantom{\rule{1.em}{0ex}}\mathrm{over}\theta .$$
- (iv)
- Set$${s}_{n}^{k}:=\left\{\begin{array}{cc}n\hfill & \mathrm{if}g(n,{x}_{n}^{k})\ge {c}^{{\theta}_{n}}\left({x}_{n}^{k}\right)\hfill \\ {s}_{n+1}^{k}\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$
- (v)
- Define ${\theta}_{0}:=\frac{1}{K}{\sum}_{k=1}^{K}g({s}_{1}^{k},{x}_{{s}_{1}^{k}}^{k})$, and set ${c}^{{\theta}_{0}}$ constantly equal to ${\theta}_{0}$.

- $I\ge 1$ denotes the depth and ${q}_{0},{q}_{1},\dots ,{q}_{I}$ the numbers of nodes in the different layers;
- ${a}_{1}^{\theta}:{\mathbb{R}}^{{q}_{0}}\to {\mathbb{R}}^{{q}_{1}},\dots ,{a}_{I}^{\theta}:{\mathbb{R}}^{{q}_{I-1}}\to {\mathbb{R}}^{{q}_{I}}$ are affine functions;
- For $j\in \mathbb{N}$, ${\phi}_{j}:{\mathbb{R}}^{j}\to {\mathbb{R}}^{j}$ is of the form ${\phi}_{j}({x}_{1},\dots ,{x}_{j})=(\phi \left({x}_{1}\right),\dots ,\phi \left({x}_{j}\right))$ for a given activation function $\phi :\mathbb{R}\to \mathbb{R}$.

## 3. Pricing

#### 3.1. Lower Bound

#### 3.2. Upper Bound, Point Estimate and Confidence Intervals

## 4. Hedging

#### 4.1. Hedging Until the First Possible Exercise Date

#### 4.2. Hedging Until the Exercise Time

## 5. Example

#### 5.1. Pricing Results

#### 5.2. Hedging Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1. | Meaning feedforward networks with a single hidden layer. |

2. | 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. |

3. | That is, ${X}_{n}$ is ${\mathcal{F}}_{n}$-measurable, and $\mathbb{E}[f\left({X}_{n+1}\right)\mid {\mathcal{F}}_{n}]=\mathbb{E}[f\left({X}_{n+1}\right)\mid {X}_{n}]$ for all $n\le N-1$ and every measurable function $f:{\mathbb{R}}^{d}\to \mathbb{R}$ such that $f\left({X}_{n+1}\right)$ is integrable. |

4. | 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 $x\in {\mathbb{R}}^{d}$ to construct good hedging strategies in Section 4. |

5. | As usual, we simulate the paths $\left({x}_{n}^{k}\right)$, $k=1,\dots ,K$, independently of each other. |

6. | Generated independently of ${\left({x}_{n}^{k}\right)}_{n=0}^{N}$, $k=1,\dots ,K$ |

7. | The use of nested simulation ensures that ${m}_{n}^{k}$ are unbiased estimates of ${M}_{n}^{\Theta}$, which is crucial for the validity of the upper bound. In particular, we do not directly approximate ${M}_{n}^{\Theta}$ with the estimated continuation value functions ${c}^{{\theta}_{n}}$. |

8. | Discounting is done with respect to the savings account. Then, the discounted value of money invested in the savings account stays constant. |

9. | That is, ${Y}_{m}$ is ${\mathcal{H}}_{m}$-measurable and $\mathbb{E}[f\left({Y}_{m+1}\right)\mid {\mathcal{H}}_{m}]=\mathbb{E}[f\left({Y}_{m+1}\right)\mid {Y}_{m}]$ for all $m\le NM-1$ and every measurable function $f:{\mathbb{R}}^{d}\to \mathbb{R}$ such that $f\left({Y}_{m+1}\right)$ is integrable. |

10. | |

11. | Independent of ${\left({y}_{m}^{k}\right)}_{m=0}^{M}$, $k=1,\dots ,{K}_{H}$. |

12. | 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. |

13. | 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). |

14. | That is, $\mathbb{E}\left[({W}_{t}^{i}-{W}_{s}^{i})({W}_{t}^{j}-{W}_{s}^{i})\right]={\rho}_{ij}(t-s)$ for all $i\ne j$ and $s<t$. |

15. | 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). |

16. | The hyperparamters ${\beta}_{1},{\beta}_{2},\epsilon $ were chosen as in Kingma and Ba (2015). The stepsize $\alpha $ was specified as ${10}^{-1}$, ${10}^{-2}$, ${10}^{-3}$ and ${10}^{-4}$ according to a deterministic schedule. |

**Table 1.**Price estimates for max-call options on 5 and 10 symmetric assets for parameter values of $r=5\%$, $\delta =10\%$, $\sigma =20\%$, $\rho =0$, $K=100$, $T=3$, $N=9$. ${t}_{L}$ is the number of seconds it took to train ${\tau}^{\Theta}$ and compute $\widehat{L}$. ${t}_{U}$ is the computation time for $\widehat{U}$ in seconds. 95% CI is the 95% confidence interval (6). The last column lists the 95% confidence intervals computed in Becker et al. (2019a).

d | ${\mathit{s}}_{0}$ | $\widehat{\mathit{L}}$ | ${\mathit{t}}_{\mathit{L}}$ | $\widehat{\mathit{U}}$ | ${\mathit{t}}_{\mathit{U}}$ | Point Est. | $95\%$ CI | DOS $95\%$ CI |
---|---|---|---|---|---|---|---|---|

5 | 90 | $16.644$ | 132 | $16.648$ | 8 | $16.646$ | $[16.628,16.664]$ | $[16.633,16.648]$ |

5 | 100 | $26.156$ | 134 | $26.152$ | 8 | $26.154$ | $[26.138,26.171]$ | $[26.138,26.174]$ |

5 | 110 | $36.780$ | 133 | $36.796$ | 8 | $36.788$ | $[36.758,36.818]$ | $[36.745,36.789]$ |

10 | 90 | $26.277$ | 136 | $26.283$ | 8 | $26.280$ | $[26.259,26.302]$ | $[26.189,26.289]$ |

10 | 100 | $38.355$ | 136 | $38.378$ | 7 | $38.367$ | $[38.335,38.399]$ | $[38.300,38.367]$ |

10 | 110 | $50.869$ | 135 | $50.932$ | 8 | $50.900$ | $[50.846,50.957]$ | $[50.834,50.937]$ |

**Table 2.**Average hedging errors and empirical hedging shortfalls for 5 and 10 underlying assets and different numbers M of rehedging times between consecutive exercise times ${t}_{n-1}$ and ${t}_{n}$. The values of the parameters r, $\delta $, $\sigma $, $\rho $, K, T and N were chosen as in Table 1. IHE is the intermediate average hedging error (8), IHS the intermediate hedging shortfall (9), HE the total average hedging error (10) and HS the total hedging shortfall (11). $\widehat{V}$ is our price estimate from Table 1. T1 is the computation time in seconds for training the hedging strategy from time 0 to ${t}_{1}=T/N$. T2 is the number of seconds it took to train the complete hedging strategy from time 0 to T.

$\mathit{d}$ | ${\mathit{s}}_{\mathbf{0}}$ | $\mathit{M}$ | IHE | IHS | IHS/$\widehat{\mathit{V}}$ | T1 | HE | HS | HS/$\widehat{\mathit{V}}$ | T2 |
---|---|---|---|---|---|---|---|---|---|---|

5 | 90 | 12 | $0.007$ | $0.190$ | $1.1\%$ | 102 | $-0.001$ | $0.676$ | $4.1\%$ | 379 |

5 | 90 | 24 | $0.007$ | $0.139$ | $0.8\%$ | 129 | $-0.002$ | $0.492$ | $3.0\%$ | 473 |

5 | 90 | 48 | $0.007$ | $0.104$ | $0.6\%$ | 234 | $-0.001$ | $0.367$ | $2.2\%$ | 839 |

5 | 90 | 96 | $0.007$ | $0.081$ | $0.5\%$ | 436 | $-0.001$ | $0.294$ | $1.8\%$ | $1546$ |

5 | 100 | 12 | $0.013$ | $0.228$ | $1.4\%$ | 102 | $0.006$ | $0.785$ | $4.7\%$ | 407 |

5 | 100 | 24 | $0.013$ | $0.163$ | $1.0\%$ | 131 | $0.006$ | $0.569$ | $3.4\%$ | 512 |

5 | 100 | 48 | $0.013$ | $0.118$ | $0.7\%$ | 252 | $0.007$ | $0.423$ | $2.5\%$ | 931 |

5 | 100 | 96 | $0.013$ | $0.089$ | $0.5\%$ | 470 | $0.006$ | $0.335$ | $2.0\%$ | $1668$ |

5 | 110 | 12 | $0.002$ | $0.268$ | $1.6\%$ | 102 | $-0.012$ | $0.881$ | $5.3\%$ | 380 |

5 | 110 | 24 | $0.002$ | $0.192$ | $1.2\%$ | 130 | $-0.012$ | $0.638$ | $3.8\%$ | 511 |

5 | 110 | 48 | $0.002$ | $0.139$ | $0.8\%$ | 262 | $-0.013$ | $0.474$ | $2.9\%$ | 950 |

5 | 110 | 96 | $0.002$ | $0.105$ | $0.6\%$ | 471 | $-0.010$ | $0.374$ | $2.3\%$ | $1673$ |

10 | 90 | 12 | $-0.015$ | $0.192$ | $0.7\%$ | 111 | $-0.010$ | $0.902$ | $3.4\%$ | 414 |

10 | 90 | 24 | $-0.014$ | $0.147$ | $0.6\%$ | 145 | $-0.011$ | $0.704$ | $2.7\%$ | 534 |

10 | 90 | 48 | $-0.015$ | $0.136$ | $0.5\%$ | 269 | $-0.011$ | $0.611$ | $2.3\%$ | 958 |

10 | 90 | 96 | $-0.015$ | $0.121$ | $0.5\%$ | 506 | $-0.012$ | $0.551$ | $2.1\%$ | $1792$ |

10 | 100 | 12 | $0.008$ | $0.230$ | $0.9\%$ | 111 | $0.015$ | $1.025$ | $3.9\%$ | 414 |

10 | 100 | 24 | $0.008$ | $0.176$ | $0.7\%$ | 152 | $0.014$ | $0.797$ | $3.0\%$ | 531 |

10 | 100 | 48 | $0.008$ | $0.150$ | $0.6\%$ | 271 | $0.016$ | $0.682$ | $2.6\%$ | 978 |

10 | 100 | 96 | $0.008$ | $0.132$ | $0.5\%$ | 512 | $0.014$ | $0.672$ | $2.6\%$ | $1803$ |

10 | 110 | 12 | $-0.029$ | $0.249$ | $1.0\%$ | 112 | $-0.026$ | $1.146$ | $4.4\%$ | 410 |

10 | 110 | 24 | $-0.029$ | $0.189$ | $0.7\%$ | 146 | $-0.027$ | $0.908$ | $3.5\%$ | 530 |

10 | 110 | 48 | $-0.029$ | $0.160$ | $0.6\%$ | 269 | $-0.026$ | $0.782$ | $3.0\%$ | 965 |

10 | 110 | 96 | $-0.029$ | $0.151$ | $0.6\%$ | 507 | $-0.024$ | $0.666$ | $2.5\%$ | $1777$ |

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## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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/jrfm13070158

**Chicago/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