# Problem abt portfolio insurance using protective put

#### WolverineHK

##### Junior member
23 0
Dear all,

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?

All the best

#### Martinghoul

##### Senior member
2,690 276
I can't really tell you, as you're trying to debug too many potential issues all at once.

I would concentrate on a single thing at first, i.e. whether your put replication is working correctly. So in your simulation, are you able to confirm that at every node of your stock price path, your replicating portfolio taken by itself is, in fact, giving you the same return as the put?

#### WolverineHK

##### Junior member
23 0

I have checked very detaily for my program, it seems nothing wrong. Besides, how are you going to findthe put return for every node? Is it by executing the put before maturity at a node time Ti?

All the best,
Kenneth

#### Martinghoul

##### Senior member
2,690 276

I have checked very detaily for my program, it seems nothing wrong. Besides, how are you going to findthe put return for every node? Is it by executing the put before maturity at a node time Ti?

All the best,
Kenneth
The way I see it you have two potential places where you could have an issue and you need to isolate which one of the two is the culprit. One place it could be going wrong is the replication of the put. The other place is the calculation of the total payout in the final tree.

#### Hoggums

##### Senior member
2,176 878
I think you are over analysing the whole thing. Buying puts is a very simple strategy to hedge out potential losses at a price.

Calculating potential losses is actually very simple if you let the option expire. And from the purchase of the option you are instantly aware of your maximum loss.

E.g. let's say you have 1000x ABC shares purchased at 100p per share.

You buy a 1000x at the money (100p) put options on ABC for 12p per share costing you £120.

With this position the most you can lose is 12p x 1000 = £120.00 and for all share prices below 112p you will incur a loss - increasing to a maximum of £120 at 100p. For all prices above 112p you will make a profit. If you sell before option expiry then your losses will potentially be less - but never greater than £120.

Don't get too wound up in the black scholes mathematics, volatility changes daily which affects option prices, dividends change, interest rates change which makes your calculations nothing more than theory.

#### Martinghoul

##### Senior member
2,690 276
I thought the whole point is that there's no option mkt and you have to replicate the put payoff using the standard binomial tree? I agree it's very theoretical, but I thought that's the whole point.

#### WolverineHK

##### Junior member
23 0
Thanks Hoggum. Yes Martinghoul, you got me. Maybe I could send you my spreadsheet?

#### WolverineHK

##### Junior member
23 0
Moreover, how does ppl insure there portfolio return in real market when there's no option mkt?

#### Technically Fundamental

##### Senior member
2,810 178
Those formulas are a lot more detailed than the binomials I've been looking at but is it possible that the theoretical doesn't work for values of [email protected] < 50?

#### WolverineHK

##### Junior member
23 0
I don't think so, as the whole point is only if the [email protected] is below the strike price the option would be exercised, which means the whole strategy has its value when it's lower than 50.

#### Technically Fundamental

##### Senior member
2,810 178
Did you put that bit in?
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.

#### WolverineHK

##### Junior member
23 0
yeah, i put it in as principal cost

#### WolverineHK

##### Junior member
23 0
is it any possible way that I could post my spreadsheet here?

2,810 178