Inv Management for Insurerers -- Chap 26 Joke?

[Eroboy]

There is a session talking about the problem with Black Scholes option valuation. I understand that Black Scholes is not perfect, but the example they demonstrate is just outrageous

They gave an example saying that Black Scholes fair value for a call is 5.25 and market price is 6 and manager shall not sell the call to make almost risk free profits because of the follow BS:

They say that the call worths 6.5(10*0.65) because ( initial stock worth 50 and with 65% worth 60 in the next period and 35% worth 40) based on manager's superior information on the stock return in the future.

That is just wrong!

First of all, they need to discount their option value (6.5)to reflect the time value of the money. They did that when they came up with the Black Scholes fair value.

Second of all, they need to find a discount rate for their option value(6.5). It is not risk free. Manager can have superior information but the stock can still go down. They can not discount at a risk free rate unless they form a risk free portfolio which is BS formula based on.

The return that they are talking about is nominal dollar return not risk adjusted return.

I have no clue how these kind of analysis can be published in a book edited by Fobozzi.

[Laurelinda]

Umm, I could be missing something here, but I think the sentence you're referring to says, "The expected value of the call is $6.5." This is the expected value at expiration, not the current fair value, so no discounting is necessary.

As for the rest, I'm not sure I follow you...isn't Exhibit 24 and the paragraph beneath it based on a riskless position constructed by buying one share and selling two calls? So I think the authors are assuming we've migrated into a risk-free world at that point.

[Eroboy]

Your are right 6.5 is the value at expiration at his expectation. I said it needs to be discounted to compare with today's market value 6 to see whether a short selling is worth or not even under his expectation of the stock distribution.

The return that I referred is the example the author used to illustration the idea (exhibit 24 to 25). That return is not risk adjusted return.

Besides, if I short a call at 6 and can replicate the payoff with 5.25 at today. Why shall I not short it unless I can not replicate it. You stock still have nominal 6% return plus a risk free payoff .75 at now without any additional up front cost.

[Will Durant]

I have no clue how these kind of analysis can be published in a book edited by Fobozzi.
Not sure what this last sentence means. I have never found Fabozzi particularly insightful or more accurate than other writers. His books are the worst on the syllabus as far as I'm concerned.

[Laurelinda]

Hey, I finally spent some time reading this chapter myself and I think I've figured out what's going on (granted the prose is unclear and Exhibit 25 has one error in it). I know Eroboy is probably long past this, but in case anyone else is back here in 4.3 with me...I thought I'd indulge myself in talking through the explanation I formulated tonight (comments welcome, ignoring my rambling is also welcome):

The point of the example is that selling a call that's overpriced based on BS risk-neutral pricing may not necessarily add to the expected return on your position in the underlying stock. In order to analyze real expected returns, you have to use the actuarial method of analyzing expected payoffs using realistic probabilities, which may not match the lognormal distribution assumption of BS.

The example on page 540 first establishes the BS risk-neutral price of a particular call maturing in one quarter with an at-the-money strike of $50, based on the assumption that the underlying stock price can only move up $10 or down $10 between now and maturity. I think this paragraph is pretty straightforward. No arbitrage argument: If you buy one share at $50 and sell two calls with strike $50, if the stock price moves up to $60, you will have ($60-2x$10) = $40, and if the stock moves down to $40, you will have ($40 - 2x$0) = $40. With a certain payoff of $40, this portfolio is riskless and can be discounted at the risk-free rate (assume 5% annual) for pricing. $40 / (1 + 5%/4) = $39.50, so the calls are worth ($50 - $39.50) / 2 = $5.25 each using BS risk-neutral valuation.

Now if you see a call priced higher than $5.25, say $6, you might assume that selling it is a good deal. In fact, it may hurt the expected return of your portfolio.

Imagine you purchase a share for $50 and know (realistically) that the probability of it increasing to $60 is 65%, and the probability of it decreasing to $40 is 35%. Your expected payoff is (60 x .65) + (40 x .35) = $53, and your expected return is (53/50)-1 = 6%.

Now imagine you also sell the overpriced $6 call.

Since you also receive the call premium when you purchase the stock, your initial investment is reduced to (50 - 6) = $44. Now if the stock increases, you have ($60 share - $10 call payoff) = $50 (note this is erroneously shown as $54 in Exhibit 25). If the stock decreases, you have ($40 share - $0 call payoff) = $40. Your expected payoff is (50 x .65) + (40 x .35) = $46.5, and your expected return is (46.5/44)-1 = 5.68%.

Why did the expected return take a hit? Because the expected realistic payoff of the call is (10 x .65) + (0 x .35) = $6.5. With a price of $6, the call has an expected return of (6.5/6)-1 = 8.33%. You've sold something that has a higher expected return, given your "knowledge" of the realistic return distribution, than your current position, and that has hurt your portfolio.

In order to enhance your return given the realistic distribution (which in this case is a grossly simplified and not very realistic distribution), the call needs to be priced high enough to have less than a 6% return, no matter what risk-neutral pricing says. (6.5 / 1.06 = $6.13)

I think this is what the example on pages 540-541 is trying to say...I hope I was coherent.

Good night everyone and have fun at the seminar! :toast:

[Eroboy]

Hey, I finally spent some time reading this chapter myself and I think I've figured out what's going on (granted the prose is unclear and Exhibit 25 has one error in it). I know Eroboy is probably long past this, but in case anyone else is back here in 4.3 with me...I thought I'd indulge myself in talking through the explanation I formulated tonight (comments welcome, ignoring my rambling is also welcome):

The point of the example is that selling a call that's overpriced based on BS risk-neutral pricing may not necessarily add to the expected return on your position in the underlying stock. In order to analyze real expected returns, you have to use the actuarial method of analyzing expected payoffs using realistic probabilities, which may not match the lognormal distribution assumption of BS.

The example on page 540 first establishes the BS risk-neutral price of a particular call maturing in one quarter with an at-the-money strike of $50, based on the assumption that the underlying stock price can only move up $10 or down $10 between now and maturity. I think this paragraph is pretty straightforward. No arbitrage argument: If you buy one share at $50 and sell two calls with strike $50, if the stock price moves up to $60, you will have ($60-2x$10) = $40, and if the stock moves down to $40, you will have ($40 - 2x$0) = $40. With a certain payoff of $40, this portfolio is riskless and can be discounted at the risk-free rate (assume 5% annual) for pricing. $40 / (1 + 5%/4) = $39.50, so the calls are worth ($50 - $39.50) / 2 = $5.25 each using BS risk-neutral valuation.

Now if you see a call priced higher than $5.25, say $6, you might assume that selling it is a good deal. In fact, it may hurt the expected return of your portfolio.

Imagine you purchase a share for $50 and know (realistically) that the probability of it increasing to $60 is 65%, and the probability of it decreasing to $40 is 35%. Your expected payoff is (60 x .65) + (40 x .35) = $53, and your expected return is (53/50)-1 = 6%.

Now imagine you also sell the overpriced $6 call.

Since you also receive the call premium when you purchase the stock, your initial investment is reduced to (50 - 6) = $44. Now if the stock increases, you have ($60 share - $10 call payoff) = $50 (note this is erroneously shown as $54 in Exhibit 25). If the stock decreases, you have ($40 share - $0 call payoff) = $40. Your expected payoff is (50 x .65) + (40 x .35) = $46.5, and your expected return is (46.5/44)-1 = 5.68%.

Why did the expected return take a hit? Because the expected realistic payoff of the call is (10 x .65) + (0 x .35) = $6.5. With a price of $6, the call has an expected return of (6.5/6)-1 = 8.33%. You've sold something that has a higher expected return, given your "knowledge" of the realistic return distribution, than your current position, and that has hurt your portfolio.

In order to enhance your return given the realistic distribution (which in this case is a grossly simplified and not very realistic distribution), the call needs to be priced high enough to have less than a 6% return, no matter what risk-neutral pricing says. (6.5 / 1.06 = $6.13)

I think this is what the example on pages 540-541 is trying to say...I hope I was coherent.

Good night everyone and have fun at the seminar! :toast:

There is one point that you are missing. The return on the stock is not risk free, while selling over priced option and replicating the cheap portfolio is risk free under author's assumption( might not be in the real world due to volatility and transaction cost).

The expected return in the real world does not guarantee you gain any money. In other words, you are expecting 5% return but in the end you can loss 1% no matter what your expected return was. However, when the option is replicated, no matter what, the strategy guarantees an earning between the call and value of the replicating portfolio.

You can not mix a risk free return with a risk demanding risk-premium.

My main point is just for the example not against the idea. The idea is okay, but the example is not good.