Deep Learning in Asset Pricing

###### tags: `畢專` `paper` # Deep Learning in Asset Pricing [TOC] --- [paper](https://poseidon01.ssrn.com/delivery.php?ID=694020101096116126120024064117122121063069052046032018081088001105096025092073079125020120023035008127018120002126127122126086111073005045021069003001126074124084001007035067024097004083127005127026089024025082085067019064101102015112105087117100029072&EXT=pdf) --- ## Abstract >We use deep neural networks to estimate an asset pricing model for individual stock returns that takes advantage of the vast amount of conditioning information, while keeping a fully flexible form and accounting for time-variation. 我們使用深度神經網絡來估計單個股票收益的資產定價模型 deep neural network利用大量的條件信息，同時保持靈活的形式並說明時變。 >The key innovations are to use the fundamental no-arbitrage condition as criterion function, to construct the most informative test assets with an adversarial approach and to extract the states of the economy from many macroeconomic time series. 關鍵的創新是使用基礎無套利條件作為標準函數，用一種對抗的方法來建構信息最豐富的測試資產，並從許多宏觀經濟的時間序列中提取經濟狀況。 >Our asset pricing model outperforms out-of-sample all benchmark approaches in terms of Sharpe ratio, explained variation and pricing errors and identifies the key factors that drive asset prices. 夏普比率而言，我們的資產定價模型優於樣本外所有基準測試方法，並解釋了變化和定價錯誤，以及確定了會讓資產價格上漲的key factor。 >Our crucial innovation is the use of the no-arbitrage condition as part of the neural network algorithm. Significantly improves the risk premium signal and makes it possible to explain individual stock returns. 他們最重要的改革是使用無套利的條件in neural network algorithm, 並改善risk premium signal，使其能解釋單一的股票returns >No-arbitrage pricing theory provides a clear answer - expected returns differ because assets have different exposure to the stochastic discount factor (SDF) or pricing kernel. 從無套利理論可知每個asset有不同的exposure to SDF所以他們的預期報酬會不同 >The empirical quest in asset pricing for the last 40 years has been to estimate a stochastic discount factor that can explain the expected returns of all assets. 經驗上來說，**可以用SDF來解釋(知道)每個asset的預期報酬率**。 :::danger stochastic discount factor (SDF)隨機貼現因數: ::: * 目前遇到的四大挑戰 (1) the SDF could by construction depend on all available information, which means that the SDF is a function of a potentially very large set of variables. SDF可以通過構造依賴於所有可用信息，這意味著SDF是潛在的非常大的變量集的函數。 (2)the functional form of the SDF is unknown and likely complex. SDF的功能形式未知，可能很複雜。 (3) the SDF can have a complex dynamic structure and the risk exposure for individual assets can vary over time depending on economic conditions and changes in asset-specific attributes. SDF可以具有復雜的動態結構，並且由於經濟原因，單個資產的風險敞口會隨著時間的變化而變化，並且資產特定屬性也會發生變化。 (4) the risk premium(風險溢價) of individual stocks has a low signal-to-noise ratio, which complicates the estimation of an SDF that explains the expected returns of all stocks. 單個股票的風險溢價具有較低的信噪比，這使得對SDF的估算變得複雜，該SDF解釋了所有股票的預期收益。 :::info **風險溢價(risk premium):** 又稱風險貼水，指的是一個人在面對不同風險的高低、且清楚高風險高報酬、低風險低報酬的情況下，會因個人對風險的承受度影響其是否要冒風險獲得較高的報酬、或是只接受已經確定的所得，放棄冒風險可能得到的較高報酬，其確定的所得與較高的報酬之間的差。 * **財務學上把一個有風險的投資工具的報酬率與無風險報酬率的差額稱為“風險溢價”** * 以投資學的角度而言，風險溢價可以視為投資者對於投資高風險時，所要求的較高報酬。衡量風險時，通常的方法就是使用無風險利率 (Risk-free interest rate)，即政府公債之利率作為標的來與其他高風險的投資比較。高於無風險利率的報酬，這部份即稱為風險溢價高風險投資獲得高報酬，低風險就只有較低的報酬，**風險與風險溢價成正比關係**。 **訊號雜訊比(signal-to-noise ratio，縮寫為SNR或S/N):** ::: In this paper we estimate a general non-linear asset pricing model with deep neural networks for all U.S. equity data based on a substantial set of macroeconomic and firm-specific information. Our crucial innovation is the use of the no-arbitrage condition as part of the neural network algorithm. (1) What is the functional form of the SDF based on the information set? Popular models, for example the Fama-French ve-factor model, impose that the SDF depends linearly on a small number of characteristics. However, the linear model seems to be mis-specied and the factor zoo suggests that there are many more characteristics with pricing information. Our model allows for a general non-parametric form with a large number of characteristics. (2) What are the right test assets? The conventional approach is to calibrate and evaluate asset pricing models on a small number of prespeci ed test assets, for example the 25 size and book-to-market double-sorted portfolios of Fama and French (1992). However, an asset pricing model that can explain well those 25 portfolios does not need to capture the pricing information of other characteristic sorted portfolios or individual stock returns. Our approach constructs in a data-driven way the most informative test assets that are the hardest to explain and identify the parameters of the SDF. (3) What are the states of the economy? Exposure and compensation for risk should depend on the economic conditions. A simple 1 Electronic copy available at: https://ssrn.com/abstract=3350138 way to capture those would be for example NBER recession indicators. However, this is a very coarse set of information given the hundreds of macroeconomic time series with complex dynamics. Our model extracts a small number of state processes that are based on the complete dynamics of a large number of macroeconomic time series and are the most relevant for asset pricing. ::: danger Fama-French 3 factor model ::: ## Model ## Estimation ## Empirical Results for U.S. Equities ## Conclusion