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Anyone tried the Goldfarb manual pg 145 question 5?

Assume the stock price is 100, mu=10%, sigma=27%, Assume there are 256 trading days in a year, determine the stock price after 7 trading days. Risk-free rate is 5%. If we long 1mil call options and short 1mil call options K=90 and the number of puts to offset the net costs to zero is 183,497,713. Using B-S, we get the initial value as:

1) Call with K= 89 (Long) = 11.127

2) Call with K=90 (Short) = 10.135

3) Put with K=89 (Short) = 0.005

Now, is found that the 1% quantile stock price after 1 trading days is 96.17. It says in question 5 that, the values based on recognizing the new stock price and with options now having 6 days to maturity. The price per option new value is:

1) Call with K=89 (new value = 6.774)

2) Call with k=90 (new value = 5.830)

3) Put with K=89 (new value is 0.006)

I couldn't get this value using B-S. Anyone can help on this? The parameters I used would be r=0.05, T=6/256, S=96.17, sigma = 0.27

For example for 1)

I got d1 = 1.92, d2=1.88 and therefore I compute B-S for call which gives me a value of 7.31 instead of the value 6.774.

Thank you very much for that.

For SPDA product, I have some confusion on the statements they make as follows:

1) For SPDA (single premium deferred annuity) product, the crediting strategy offsets the tendency to lapse, thereby it does not exhibit negative convexity. (STATED IN THE MANUAL)

My question:

- Negative convexity is as rates increase, duration also increases. I thought that if there is an increase in market rates, the lapse would increase, duration would decrease. Normal cases, lapses make SPDA product to exhibit positive convexity so how does negative convexity comes into the picture?

2) Less responsive crediting strategy lead to sig. higher duration.

- My understanding is that lapse rates depend on the spread between market rates and the crediting rates. If there is an increase iin market rates, therefore with a less responsive crediting rate strategy ---> crediting rates would not move towards the market rates ---> there will increase in lapses ---> decrease in duration. CONTRADICT to the statement above.

I couldn't figure out the link between crediting strategy ---> lapse ---> duration.

Anyone can help? Thanks a lot for this :)

bagheera

04-21-2007, 04:24 AM

For SPDA product, I have some confusion on the statements they make as follows:

1) For SPDA (single premium deferred annuity) product, the crediting strategy offsets the tendency to lapse, thereby it does not exhibit negative convexity. (STATED IN THE MANUAL)

My question:

- Negative convexity is as rates increase, duration also increases. I thought that if there is an increase in market rates, the lapse would increase, duration would decrease. Normal cases, lapses make SPDA product to exhibit positive convexity so how does negative convexity comes into the picture?

2) Less responsive crediting strategy lead to sig. higher duration.

- My understanding is that lapse rates depend on the spread between market rates and the crediting rates. If there is an increase iin market rates, therefore with a less responsive crediting rate strategy ---> crediting rates would not move towards the market rates ---> there will increase in lapses ---> decrease in duration. CONTRADICT to the statement above.

I couldn't figure out the link between crediting strategy ---> lapse ---> duration.

Anyone can help? Thanks a lot for this :)

I agree with you on the first issue. For the second, if you assume for a moment that surrender charges deter lapses to a considerable extent, then a less responsive crediting strategy will indeed have a higher duration (relative to the more responsive crediting strategy case duration) because as market rates change the price of the policy will change (just like a fixed rate bond). On the other hand, a more responsive crediting strategy will tend to keep the price constant for changes in market rates (just like a floating rate bond), so duration will be lower than the earlier case.

But if surrender charges are not able to deter lapses sufficiently, the overall effect on the duration due to the responsiveness plus lapses appears to be unclear. If the crediting strategy is less responsive, the duration associated with it should decrease, all other things being equal, due to more lapses. And if the crediting strategy is more responsive then the duration associated with it would increase due to fewer lapses.

To sum up, there are two factors that affect duration: responsiveness and lapses. These two, both considered separately, decrease duration but they are somewhat negatively correlated as well. So when they come together it's either in the form 1. More responsive, fewer lapses 2. Less responsive, more lapses. Each of the "states" of these two factors in the two cases above exerts "pulls" in opposite directions on the duration. For example, if you consider case 1, "more responsive", in itself, should decrease duration but the resulting fewer lapses increase duration. I suppose it so turns out that case 2 eventually ends up having a higher duration i.e the effect of responsiveness on the duration appears to be stronger.

Sorry for this long-winded explanation. Hope I made some sense.

rsgoldfarb

04-23-2007, 12:16 AM

JY88 is correct. There was an error in my solution to this question.

The steps shown and the explanation are correct, but the option values after one day were calculated using a different stock price than the amount shown. At some point I changed the assumed volatility from 31% to 27% so that the solution would differ from the numbers in the reading and then mistakenly failed to update the stock price that was used in the option calculation.

The corrected values for the three option values are:

Call (K=89): 7.316

Call (K=90): 6.358

Put (K=89): 0.042

The final total change in value over 1 day, at the 99 percentile level, is -$6.7 million.

I apologize for the confusion this caused. The website contains the corrected solution attached to the Study Manual Updates file.

Richard Goldfarb

Thank you guys for helping.

Bagheera, let me see if I understand what you are trying to explain here.

Do you mean that;

1) a less responsive strategy - mean that as market rate changes price of the policy will change. In this case, if increase in market rates ---> decrease in price of policy ---> no lapses ---> increase in duration.

2) a more responsive crediting strategy - mean that as market rate changes price of policy will be constant. In this case, if increase in market rates ---> no change in price of policy ---> more lapses --> decrease in duration.

Ok then i see why less responsive crediting strategy would lead to higher duration.

Therefore with a more responsive crediting strategy, SPDA will not exhibit negative convexity.

Thank you for that. I think I get that :-p

bagheera

04-24-2007, 12:22 AM

Thank you guys for helping.

Bagheera, let me see if I understand what you are trying to explain here.

Do you mean that;

1) a less responsive strategy - mean that as market rate changes price of the policy will change. In this case, if increase in market rates ---> decrease in price of policy ---> no lapses ---> increase in duration.

2) a more responsive crediting strategy - mean that as market rate changes price of policy will be constant. In this case, if increase in market rates ---> no change in price of policy ---> more lapses --> decrease in duration.

Ok then i see why less responsive crediting strategy would lead to higher duration.

Therefore with a more responsive crediting strategy, SPDA will not exhibit negative convexity.

Thank you for that. I think I get that :-p

What I want to say is (quoted and edited from your post above) ,

1) a less responsive strategy- as market rates change price of the policy will change => non-zero duration (delta P > 0 and duration is 1/P*delta P/delta Y ). Also there will be increase in lapses (since policy is less responsive with respect to crediting rate changes) leading to a reduced Macaulay (hence modified) duration too. But decrease is not significant since first effect is stronger. So net duration is higher compared to second case below.

2) a more responsive crediting strategy - means that as market rate changes price of policy will be constant. In this case, duration is zero. Resulting fewer lapses (due to responsiveness) lead to increase in duration, but effect is small. So net duration is lower than in case 1.

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