Expected Value of a Put option hedge (on a basket of assets)!

[Edwin]

I have liabilities that offer policyholders a guaranteed investment return at maturity. The payoff is simply max(Fund Value at maturity, Guaranteed Value) to the policyholder, for the liabilities, this is easily accessible in Prophet. The payoff for the company is max(Guaranteed Value - Fund Value at maturity, 0) (EMBEDDED PUT OPTION)!, relates to vanilla put with max(K - S_T, 0).

Now the underlying fund backing the liabilities is an Equity Fund, this equity fund is made up of assets similar to the JSE Top40 index. To hedge the Option, I therefore buy a Put Option on the JSE Top40 (BASEKET OPTION WITH 40 INDIVIDUAL STOCKS). Now I need to assess the effectiveness of the hedge by showing that in a years' time the Expected Value of the Option will be equal to the value of the Liabilities, though note that the underlying is a basket of Options!!!PLEASE KINDLY HELP

I work in the capital division of a life insurance company, and yes this is a real life situation that the CRO of our company wants the answer for!!

 

[manish.rex]

I have liabilities that offer policyholders a guaranteed investment return at maturity. The payoff is simply max(Fund Value at maturity, Guaranteed Value) to the policyholder, for the liabilities, this is easily accessible in Prophet. The payoff for the company is max(Guaranteed Value - Fund Value at maturity, 0) (EMBEDDED PUT OPTION)!, relates to vanilla put with max(K - S_T, 0).

Now the underlying fund backing the liabilities is an Equity Fund, this equity fund is made up of assets similar to the JSE Top40 index. To hedge the Option, I therefore buy a Put Option on the JSE Top40 (BASEKET OPTION WITH 40 INDIVIDUAL STOCKS). Now I need to assess the effectiveness of the hedge by showing that in a years' time the Expected Value of the Option will be equal to the value of the Liabilities, though note that the underlying is a basket of Options!!!PLEASE KINDLY HELP

I work in the capital division of a life insurance company, and yes this is a real life situation that the CRO of our company wants the answer for!!

The liability is the guarantee amount, and by purchasing the put option, barring basis risk, the exposure of guarantee amount seems to have been hedged.

As far as showing the hedge effectiveness with the way suggested by CRO (Now I need to assess the effectiveness of the hedge by showing that in a years' time the Expected Value of the Option will be equal to the value of the Liabilities) ,note that the expected option payoff has to be the difference between the fund value and guarantee amount one year from now, and not that the expected payoff from option will match the guarantee amount: if this was required, better buy a ZCB, which will compound into the guarantee amount.

It is the combination of the long position taken in fund + the option amount which is going to match the guarantee amount+ some upside payoff - the cost of purchasing the option(s). This is very well understood that can be shown as pay-offs under all scenarios of the fund value movement.

Two risks exsist which may prevent the above from ahppening:

1. Credit risk: if the options are bespoke/OTC, yes credit risk is very much there.The issuer of the option may refuse to pay on exercise.

2. Basis risk: how well the options on individual stocks move in retaion to the total fund value: does the sum of movement in options values off-set the movement in the equity fund value: this is important if the fund is replicating some index, as over time index composition may change from original, and it may not be possible to adjust the individual holdings of the options in the same way.

3. If the options are not on individaul stocks, but rather than on the basket of assets, then the misprising risk of such an option also exists (assumptions used in prising).

hope this helps

 

[Edwin]

The liability is the guarantee amount, and by purchasing the put option, barring basis risk, the exposure of guarantee amount seems to have been hedged.

As far as showing the hedge effectiveness with the way suggested by CRO (Now I need to assess the effectiveness of the hedge by showing that in a years' time the Expected Value of the Option will be equal to the value of the Liabilities) ,note that the expected option payoff has to be the difference between the fund value and guarantee amount one year from now, and not that the expected payoff from option will match the guarantee amount: if this was required, better buy a ZCB, which will compound into the guarantee amount.

It is the combination of the long position taken in fund + the option amount which is going to match the guarantee amount+ some upside payoff - the cost of purchasing the option(s). This is very well understood that can be shown as pay-offs under all scenarios of the fund value movement.

Two risks exsist which may prevent the above from ahppening:

1. Credit risk: if the options are bespoke/OTC, yes credit risk is very much there.The issuer of the option may refuse to pay on exercise.

2. Basis risk: how well the options on individual stocks move in retaion to the total fund value: does the sum of movement in options values off-set the movement in the equity fund value: this is important if the fund is replicating some index, as over time index composition may change from original, and it may not be possible to adjust the individual holdings of the options in the same way.

3. If the options are not on individaul stocks, but rather than on the basket of assets, then the misprising risk of such an option also exists (assumptions used in prising).

hope this helps

Hi Manish.rex, you don't know how much this helps....will appreciate more comments. I generally want to assume that the Index follows Geometric Brownian Motion and then use historical data to evaluate the expected value of the index?...please help....


P.S I am looking for a general quantitative banter or seasoned Acted forumnists' technobabble!

 

[Oxymoron]

Copulas and Monte Carlo?

1) Get the list of instruments which the put options are based upon. Compute their volatility and correlation (you could even do a dynamic correlation model and simulate correlation changes over time if that's needed).
2) Simulate the value of the instruments for a 1 year period (jointly using Copulas). Since the put options are based on risk neutral valuation, Mu = Rf.
3) Compute the put option value at the end of one year. (Strike - asset value at the end of year 1 for each asset ...and sum up)
4) See if this equals the value of liabilities.
5) I charge $10,000 if you've read this advice.

 

[Edwin]

Copulas and Monte Carlo?

1) Get the list of instruments which the put options are based upon. Compute their volatility and correlation (you could even do a dynamic correlation model and simulate correlation changes over time if that's needed).
2) Simulate the value of the instruments for a 1 year period (jointly using Copulas). Since the put options are based on risk neutral valuation, Mu = Rf.
3) Compute the put option value at the end of one year. (Strike - asset value at the end of year 1 for each asset ...and sum up)
4) See if this equals the value of liabilities.
5) I charge $10,000 if you've read this advice.

Lol....thanks for the technobabble....the simplest case we assume that the basket has a log-normal distribution and we then try to calculate the mean and variance of the basket distribution (GBM assumption see above). If you try this out, you'll see that the basket volatility is dependent on correlation - see HULL Technical note 28*. As correlation increases, so too does the basket vol....assuming GBM for the whole index > implying correlation is 1 > you will the maximum price...

....>the minimum value of the Index > a prudent answer?

 

[Oxymoron]

Lol....thanks for the technobabble....the simplest case we assume that the basket has a log-normal distribution and we then try to calculate the mean and variance of the basket distribution (GBM assumption see above). If you try this out, you'll see that the basket volatility is dependent on correlation - see HULL Technical note 28*. As correlation increases, so too does the basket vol....assuming GBM for the whole index > implying correlation is 1 > you will the maximum price...

....>the minimum value of the Index > a prudent answer?

It's interesting because correlations increase in bad economic climates where volatility is highest while decrease in good periods. That's where the dynamic correlation model comes in.

In all honesty though, you might want to do something you can interpret to your CRO.

 

[Edwin]

It's interesting because correlations increase in bad economic climates where volatility is highest while decrease in good periods. That's where the dynamic correlation model comes in.

In all honesty though, you might want to do something you can interpret to your CRO.

I think I'm getting close....maybe two more weeks! Will keep you posted