Lately, I've read a basic portfolio insurance strategy which is to construct a protective put option portfolio using black-scholes pricing model.

Assuming a put option is available, and the stock's current price is S=$56, X=$50, vol=0.3, risk free rate = 8%, T=1, then after 1 year, you will lose no more than $6 on the share. We know that this protection isn't come for free, it cost p=$2.38 (using Black-scholes formula), which means that you need $58.38 for one share.

Suppose we are going to invest $1000 for the share and option, then after 1 year, the whole value of our initial investment should never dip below $856.5. (=1000/58.38*50)

So here come's the question, assume that there's no put available, and we want to construct a put using B-S model, according to the formula

P = -S(t)N(-d1) + N(-d2)Xe^-r(1-t)

buying a put is equivalent to investing N(-d2)Xe^-r(1-t) in a risk-free bond that matures at time 1 and investing -S(t)N(-d1) in the stock. (short position in stock and long position in bond)So total investment, protective put = S(t) + P(t) = S(t)N(d1)+Xe^r(1-t)N(-d2), therefore, it give the proportion invested in stock w0= S(t)N(d1)/[S(t)N(d1)+Xe^r(1-t)N(-d2)], bonds 1-w0.

According to the deduced formula above, If the analysis and strategy are correct, we can adjust the proportion of stocks and bonds week by week (52weeks), then after 1 year, the portfolio value shouldn't dip below $856.5 as calculated above.

But after I did the simulation, it didn't go as the I thought it would be. I generated 100 stock paths in a year, and each of them with the year end price below the X=$50, as we want to see the worse situation. I found that among the 100 final portfolio values, some of them are actually below $856.5, which means the protection is not sure.

Then I tried to find how low it can go down, so I further made another 100 simulation with the final portfolio values lower than the 856.5, and I found that the average -1.5% rate from 856.5 with standard deviation 5%. In such case, this strategy does protect our investment, but we can't find our maximum loss and where our protection level is. Assume that my simulation and programming are correct (I tested for hurdreds of times), then my question is why is that and where the problem is, is any missing assumption?

Thanks for your answer.

All the best