Dr William Wu Chengyuan is ranked among the Top 5 Kaggle competitors based in Singapore.
On Kaggle, competitors are ranked globally based on their performance in machine learning competitions. Rankings are determined by medal placements, leaderboard positions, and overall competition points.
⚡ Kaggle competitions typically involve:
- End-to-end model development
- Feature engineering
- Cross-validation strategies
- Ensemble methods
- Efficient experimentation leveraging state-of-the-art algorithms
👤 Profile Highlights:
- Rank: Kaggle Master 🎖️
- Category: Competitions 💻
- National Ranking: Top 5 in Singapore 🇸🇬
- Peak Global Ranking: 93rd out of 200,000+ competitors (Top 0.1%) 🌍
- Focus Areas: Large Language Models (LLMs) 🤖, Machine Learning 📈, and Deep Learning 🧠
🏆 Kaggle Gold Medals:
- Kaggle Gold Medal (2nd/1692, Top 0.2% in LLMs – You Can’t Please Them All)
- Kaggle Gold Medal (4th/1427, Top 0.3% in ARC Prize 2024)
- Kaggle Gold Medal (8th/2437, Top 0.4% in Google Jigsaw – Agile Community Rules Classification)
Dr William Wu works in artificial intelligence and machine learning, with practical experience in building predictive models and optimizing performance under real-world constraints.
🔗 LinkedIn: https://www.linkedin.com/in/wu-chengyuan/
🔗 Kaggle: https://www.kaggle.com/williamwu88
and invests in the stock and money market account so that the portfolio value 
at each time ![t\in [0,T]](https://s0.wp.com/latex.php?latex=t%5Cin+%5B0%2CT%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
is equal to 
. This is equivalent to the condition that 
for all 
.
for all 
and 
.
and 
calculated previously in ![\displaystyle\begin{aligned}&\Delta(t)(\alpha-r)S(t)\, dt+\Delta(t)\sigma S(t)\, dW(t)\\&=\left[-rc(t,S(t))+c_t(t,S(t))+\alpha S(t)c_x(t,S(t))+\frac{1}{2}\sigma^2 S^2(t)c_{xx}(t,S(t))\right]\, dt\\&\quad+\sigma S(t)c_x(t,S(t))\, dW(t)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%26%5CDelta%28t%29%28%5Calpha-r%29S%28t%29%5C%2C+dt%2B%5CDelta%28t%29%5Csigma+S%28t%29%5C%2C+dW%28t%29%5C%5C%26%3D%5Cleft%5B-rc%28t%2CS%28t%29%29%2Bc_t%28t%2CS%28t%29%29%2B%5Calpha+S%28t%29c_x%28t%2CS%28t%29%29%2B%5Cfrac%7B1%7D%7B2%7D%5Csigma%5E2+S%5E2%28t%29c_%7Bxx%7D%28t%2CS%28t%29%29%5Cright%5D%5C%2C+dt%5C%5C%26%5Cquad%2B%5Csigma+S%28t%29c_x%28t%2CS%28t%29%29%5C%2C+dW%28t%29%5Cend%7Baligned%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
terms, we get 
. This is known as the delta-hedging rule.
terms, we have
, then we have derived the Black-Scholes-Merton partial differential equation:
for all 
.
, where 
is the strike price of the call option.
denote the value of a European call option at time ![\displaystyle\begin{aligned}dc(t,S(t))&=c_t(t,S(t))\, dt+c_x(t,S(t))\, dS(t)+\frac{1}{2}c_{xx}(t,S(t))\, dS(t)\, dS(t)\\&=c_t(t,S(t))\, dt+c_x(t,S(t))(\alpha S(t)\, dt+\sigma S(t)\, dW(t))+\frac{1}{2}c_{xx}(t,S(t))\sigma^2 S^2(t)\, dt\\&=\left[ c_t(t,S(t))+\alpha S(t) c_x(t,S(t))+\frac{1}{2}\sigma^2 S^2(t)c_{xx}(t,S(t))\right]\, dt\\&\quad+\sigma S(t)c_x(t,S(t))\, dW(t)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7Ddc%28t%2CS%28t%29%29%26%3Dc_t%28t%2CS%28t%29%29%5C%2C+dt%2Bc_x%28t%2CS%28t%29%29%5C%2C+dS%28t%29%2B%5Cfrac%7B1%7D%7B2%7Dc_%7Bxx%7D%28t%2CS%28t%29%29%5C%2C+dS%28t%29%5C%2C+dS%28t%29%5C%5C%26%3Dc_t%28t%2CS%28t%29%29%5C%2C+dt%2Bc_x%28t%2CS%28t%29%29%28%5Calpha+S%28t%29%5C%2C+dt%2B%5Csigma+S%28t%29%5C%2C+dW%28t%29%29%2B%5Cfrac%7B1%7D%7B2%7Dc_%7Bxx%7D%28t%2CS%28t%29%29%5Csigma%5E2+S%5E2%28t%29%5C%2C+dt%5C%5C%26%3D%5Cleft%5B+c_t%28t%2CS%28t%29%29%2B%5Calpha+S%28t%29+c_x%28t%2CS%28t%29%29%2B%5Cfrac%7B1%7D%7B2%7D%5Csigma%5E2+S%5E2%28t%29c_%7Bxx%7D%28t%2CS%28t%29%29%5Cright%5D%5C%2C+dt%5C%5C%26%5Cquad%2B%5Csigma+S%28t%29c_x%28t%2CS%28t%29%29%5C%2C+dW%28t%29%5Cend%7Baligned%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
, as well as the identities 
, 
, and 
.
. Let 
. According to the Ito-Doeblin formula, we have![\displaystyle\begin{aligned}d(e^{-rt}c(t,S(t)))&=df(t,c(t,S(t)))\\&=f_t(t,c(t,S(t)))\, dt+f_x(t,c(t,S(t)))\, dc(t,S(t))\\&\quad+\frac{1}{2}f_{xx}(t,c(t,S(t)))\, dc(t,S(t))\, dc(t,S(t))\\&=-re^{-rt}c(t,S(t))\,dt+e^{-rt}\, dc(t,S(t))\\&=e^{-rt}[-rc(t,S(t))+c_t(t,S(t))+\alpha S(t)c_x(t,S(t))\\&\quad+\frac{1}{2}\sigma^2S^2(t)c_{xx}(t,S(t))]\, dt+e^{-rt}\sigma S(t)c_x(t,S(t))\, dW(t).\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7Dd%28e%5E%7B-rt%7Dc%28t%2CS%28t%29%29%29%26%3Ddf%28t%2Cc%28t%2CS%28t%29%29%29%5C%5C%26%3Df_t%28t%2Cc%28t%2CS%28t%29%29%29%5C%2C+dt%2Bf_x%28t%2Cc%28t%2CS%28t%29%29%29%5C%2C+dc%28t%2CS%28t%29%29%5C%5C%26%5Cquad%2B%5Cfrac%7B1%7D%7B2%7Df_%7Bxx%7D%28t%2Cc%28t%2CS%28t%29%29%29%5C%2C+dc%28t%2CS%28t%29%29%5C%2C+dc%28t%2CS%28t%29%29%5C%5C%26%3D-re%5E%7B-rt%7Dc%28t%2CS%28t%29%29%5C%2Cdt%2Be%5E%7B-rt%7D%5C%2C+dc%28t%2CS%28t%29%29%5C%5C%26%3De%5E%7B-rt%7D%5B-rc%28t%2CS%28t%29%29%2Bc_t%28t%2CS%28t%29%29%2B%5Calpha+S%28t%29c_x%28t%2CS%28t%29%29%5C%5C%26%5Cquad%2B%5Cfrac%7B1%7D%7B2%7D%5Csigma%5E2S%5E2%28t%29c_%7Bxx%7D%28t%2CS%28t%29%29%5D%5C%2C+dt%2Be%5E%7B-rt%7D%5Csigma+S%28t%29c_x%28t%2CS%28t%29%29%5C%2C+dW%28t%29.%5Cend%7Baligned%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
, 
, and 
.
and in a stock modeled by geometric Brownian motion
: At time 
is invested in the money market account.
is due to two factors, the capital gain 
on the stock position and the interest earnings 
on the cash position.
: reflects an average underlying rate of return 
: reflects a risk premium 
for investing in the stock
: a volatility term proportional to the size of the stock investment
and the discounted portfolio value 
. We let ![\begin{aligned}d(e^{-rt}S(t))&=df(t,S(t))\\&=f_t(t,S(t))\, dt+f_x(t,S(t))\, dS(t)+\frac{1}{2}f_{xx}(t,S(t))\, dS(t)\, dS(t)\\&=-re^{-rt}S(t)\, dt+e^{-rt}\, dS(t)+0\\&=-re^{-rt}S(t)\, dt+e^{-rt}[\alpha S(t)\, dt+\sigma S(t)\, dW(t)]\\&=(\alpha-r)e^{-rt}S(t)\, dt+\sigma e^{-rt}S(t)\, dW(t)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7Dd%28e%5E%7B-rt%7DS%28t%29%29%26%3Ddf%28t%2CS%28t%29%29%5C%5C%26%3Df_t%28t%2CS%28t%29%29%5C%2C+dt%2Bf_x%28t%2CS%28t%29%29%5C%2C+dS%28t%29%2B%5Cfrac%7B1%7D%7B2%7Df_%7Bxx%7D%28t%2CS%28t%29%29%5C%2C+dS%28t%29%5C%2C+dS%28t%29%5C%5C%26%3D-re%5E%7B-rt%7DS%28t%29%5C%2C+dt%2Be%5E%7B-rt%7D%5C%2C+dS%28t%29%2B0%5C%5C%26%3D-re%5E%7B-rt%7DS%28t%29%5C%2C+dt%2Be%5E%7B-rt%7D%5B%5Calpha+S%28t%29%5C%2C+dt%2B%5Csigma+S%28t%29%5C%2C+dW%28t%29%5D%5C%5C%26%3D%28%5Calpha-r%29e%5E%7B-rt%7DS%28t%29%5C%2C+dt%2B%5Csigma+e%5E%7B-rt%7DS%28t%29%5C%2C+dW%28t%29%5Cend%7Baligned%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
![\begin{aligned}d(e^{-rt}X(t))&=df(t,X(t))\\&=f_t(t,X(t))\, dt+f_x(t,X(t))\, dX(t)+\frac{1}{2}f_{xx}(t,X(t))\, dX(t)\, dX(t)\\&=-re^{-rt}X(t)\, dt+e^{-rt}\, dX(t)+0\\&=-re^{-rt}X(t)\, dt+e^{-rt}[rX(t)\, dt+\Delta(t)(\alpha-r)S(t)\, dt+\Delta(t)\sigma S(t)\, dW(t)]\\&=\Delta(t)(\alpha-r)e^{-rt}S(t)\, dt+\Delta(t)\sigma e^{-rt}S(t)\, dW(t)\\&=\Delta(t)\, d(e^{-rt}S(t))\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7Dd%28e%5E%7B-rt%7DX%28t%29%29%26%3Ddf%28t%2CX%28t%29%29%5C%5C%26%3Df_t%28t%2CX%28t%29%29%5C%2C+dt%2Bf_x%28t%2CX%28t%29%29%5C%2C+dX%28t%29%2B%5Cfrac%7B1%7D%7B2%7Df_%7Bxx%7D%28t%2CX%28t%29%29%5C%2C+dX%28t%29%5C%2C+dX%28t%29%5C%5C%26%3D-re%5E%7B-rt%7DX%28t%29%5C%2C+dt%2Be%5E%7B-rt%7D%5C%2C+dX%28t%29%2B0%5C%5C%26%3D-re%5E%7B-rt%7DX%28t%29%5C%2C+dt%2Be%5E%7B-rt%7D%5BrX%28t%29%5C%2C+dt%2B%5CDelta%28t%29%28%5Calpha-r%29S%28t%29%5C%2C+dt%2B%5CDelta%28t%29%5Csigma+S%28t%29%5C%2C+dW%28t%29%5D%5C%5C%26%3D%5CDelta%28t%29%28%5Calpha-r%29e%5E%7B-rt%7DS%28t%29%5C%2C+dt%2B%5CDelta%28t%29%5Csigma+e%5E%7B-rt%7DS%28t%29%5C%2C+dW%28t%29%5C%5C%26%3D%5CDelta%28t%29%5C%2C+d%28e%5E%7B-rt%7DS%28t%29%29%5Cend%7Baligned%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
be a differentiable function and 
be a Brownian motion.
has nonzero quadratic variation.
to be a function of both 
.
be a function where the partial derivatives 
, 
, and 
are continuous, and let 
,
Chengyuan Wu 