# William K. Bertram's sharpe formula checking

I have some issues to verify by simulation the formulas in the paper of William K. Bertram "Analytic solutions for optimal statistical arbitrage trading". first, the reversion parameter alpha=180 in the paper seems too high as i guess the OU will diverge after a few steps. I am simulating X as:

def simulate_OU(alpha,nu,x0,T):

X=np.zeros(T)
X[0]=x0

N=np.random.normal(0,1,size=T)

delta=1

for i in range(1,T):
X[i]=X[i-1]-alpha*X[i-1]*delta+nu*np.sqrt(delta)*N[i]

return X


if i take x0=0, alpha=180, nu=0.1538 and T=100 The process diverges. Alpha should normally be <1 Could someone show me what i am doing wrong here? Thanks in advance

• My guess (although I did not find this in the paper) is that things are being done in units of years, and therefore if you want to simulate daily you need $delta=\frac{1}{365}$. Then alpha*delta will be less than half. – Alex C Oct 4 '19 at 20:19