Estimating Option Prices

[options-george]

hi, i have a question about option prices.

I am looking at trading DIA options. On Friday DIA closed at 111.96.

My question is this:
Assume I buy any of the DIA 111, 112 or 113 May call options.

I downloaded an option value calculator here:
http://spreadsheets.google.com/pub?...Og&single=true&gid=8&range=A1:G25&output=html

Example DIA 112 Call
Value if DIA @ 113 = $2.03
Value if DIA @ 112 = $1.47
Value if DIA @ 111 = $1.02


It seems to me that the appreciation of the option value for a $1 rise is greater ($0.56)than the depreciation of the option value for a $1 fall ($0.45). Is this correct? Assume that the options are traded intra-day so that time decay is minimal.

Would this mean that I a $0.45 loss is as likley as a $0.56 gain?

Look forward to any comments that people may have,
George

 

[Martinghoul]

There are two phenomena that you're potentially observing there:
a) the fundamental zero-boundedness property of the log-normal distribution;
b) the shape of the skew that reflects the expectation of the mkt regarding the future direction of the mkt.

 

[options-george]

There are two phenomena that you're potentially observing there:
a) the fundamental zero-boundedness property of the log-normal distribution;
b) the shape of the skew that reflects the expectation of the mkt regarding the future direction of the mkt.

Thanks for your reply. I won't say that I understand completely what you are saying but I think I am getting your general drift.

In regard to (b) - using my example in the inital post - would this imply that the market has a bearish expectation?

I am trying to determine whether, assuming a given directional prediction for a stock or index, I would be better off with using options or with using CFD's - taking into account commissions, spreads and price behavior of the derivative instruments, win rate, likely loss, likely profit and expected return per trade.

At this point the options route seems to give the better returns, mainly because a $1 unfavorable movement in the underlying triggers a derivate loss less than the gain triggered by a $1 favorable movement.

 

[Martinghoul]

In general, decomposing a) from b) is a non-trivial task...

However, I just realized that I may have been barking up the wrong tree a bit. Having actually looked at the spreadsheet, it appears that you keep vol constant when calculating the value of the option under the two scenarios. If that is in fact the case (pls do confirm), forget I ever mentioned b). It's silly of me to mention skew in a flat-vol BSM world. It's, therefore, all about a), i.e. the fundamental property of the log-normal distribution.

 

[options-george]

In general, decomposing a) from b) is a non-trivial task...

However, I just realized that I may have been barking up the wrong tree a bit. Having actually looked at the spreadsheet, it appears that you keep vol constant when calculating the value of the option under the two scenarios. If that is in fact the case (pls do confirm), forget I ever mentioned b). It's silly of me to mention skew in a flat-vol BSM world. It's, therefore, all about a), i.e. the fundamental property of the log-normal distribution.

about the IV levels -
according to the option chain on Options Xpress the IV levels (with DIA at 111.96) were as follows at the end of Friday:
111 calls 12.2 (ITM)
112 calls 11.6 (ATM)
113 calls 11.2 (OTM)

I will assume that that IV will be 11.6 on the ATM strike. Is this a valid assumption to make, i.e. that IV will be 11.6 at the relevant ITM strike assuming that the move happens quickly, IV will be more on the ITM strikes, and less on the OTM strikes.
So the 112 calls would have 11.6 IV at 112, 12.2 IV at 113 and 11.2 IV at 111.

Using these figures, I get the following theoretical option values
112 calls (with DIA @ 111, IV=11.2, OTM) = $0.88
112 calls (with DIA @ 112, IV=11.6, ATM) = $1.37
112 calls (with DIA @ 113, IV=12.2, ITM) = $2.00

Other inputs - interest rate 0.7%, Div Yield 1%, 26 days to expiration

Thus for a $1 unfavorable movement I get a loss of $0.49, though for a $1 favorable movement I get a profit of $0.63 (before providing for transaction costs).

Going back to your earlier comments, does this mean that the option market-makers are bearish on DIA?

Look forward to your next reply

 

[Martinghoul]

Right, so there you have it...

In order to see the decomposition between a) and b) do the same exercise as above, but with constant vols. In a constant vol world, you will be seeing only a), whereas doing what you did above shows you the likely impact from both a) and b).

Looking at the skew for these, it seems that its shape is actually implying that the mkt has a somewhat bearish view, but it's important to note that is the risk-neutral expectation. Keep in mind that equity/index smile is normally downward sloping, at least post 1987. That means that the risk-neutral distribution you may imply from option prices isn't actually the true mkt expectation. I suspect that this is exactly the case here.

 

[options-george]

Right, so there you have it...

In order to see the decomposition between a) and b) do the same exercise as above, but with constant vols. In a constant vol world, you will be seeing only a), whereas doing what you did above shows you the likely impact from both a) and b).

Looking at the skew for these, it seems that its shape is actually implying that the mkt has a somewhat bearish view, but it's important to note that is the risk-neutral expectation. Keep in mind that equity/index smile is normally downward sloping, at least post 1987. That means that the risk-neutral distribution you may imply from option prices isn't actually the true mkt expectation. I suspect that this is exactly the case here.

ok, i re-did the analysis this time using IV of 11.6 for all situations and I got:

112 calls (with DIA @ 111, IV=11.6, OTM) = $0.92
112 calls (with DIA @ 112, IV=11.6, ATM) = $1.37
112 calls (with DIA @ 113, IV=11.6, ITM) = $1.93

$1 fall causes $0.45 loss and $1 rise causes $0.56 gain - that brings the numbers a little closer together. So is this impact due solely from (a)?

Would it be okay for you to elaborate on your earlier comments regarding 'risk neutral distribution' and 'equity/index smile is downward sloping' - unfortunately these are terms/concepts that I am not familiar with. (note i understand the 'smile' concept - the plotting of IV levels against various strikes right?)

 

[Martinghoul]

Yes, using the same IV yields the impact resulting solely from a).

The point about risk-neutral distribution is this, in simple terms.

Firstly, assume no single mkt participant has a particular preference, in terms of assets, risks, strategies, etc. That would imply that the mkt option prices offer a perfect reflection of the mkt expectations of the price of the underlying asset at expiry. In this case, the probability distribution implied by option prices (risk-neutral distribution) is the true distribution.

Now, let's change things a bit and assume that equity mkt investors are actually risk-averse. That means they care about preventing losses a lot more than they care about enhancing gains. This would, in turn, imply that they would pay a proportionately higher price for insurance (OTM puts), even though they don't think the underlying will ever get there. In this case, the implied risk-neutral distribution is emphatically not the true distribution, because it reflects not only the mkt expectation of the price of the underlying, but also the risk preferences of investors.

Does this make sense?

 

[options-george]

Right, this actually does make some sense to me :smart:

Ok so assuming that investors are more keen on OTM puts and are thus willing to pay up for them this inflates their price - and thanks to Put-Call Parity this also inflates the price of the ITM calls - i.e. the calls at the same strike level as the puts .

So looking at my numbers from earlier, I could hypothesize that the higher prices at the lower strikes are due to a mix of people being keen to buy insurance on the lower strikes and/or a bearish risk-neutral distribution. Is that correct?

Could you also be so kind to elaborate on the 'equity/index smile downward slope' concept?

 

[Martinghoul]

Right, this actually does make some sense to me :smart:

Ok so assuming that investors are more keen on OTM puts and are thus willing to pay up for them this inflates their price - and thanks to Put-Call Parity this also inflates the price of the ITM calls - i.e. the calls at the same strike level as the puts .

So looking at my numbers from earlier, I could hypothesize that the higher prices at the lower strikes are due to a mix of people being keen to buy insurance on the lower strikes and/or a bearish risk-neutral distribution. Is that correct?

Could you also be so kind to elaborate on the 'equity/index smile downward slope' concept?

Yes, you got it... The skew reflects true expectations and/or risk premium. Decomposing the two is sort of the unattainable holy grail of finance.

As to the slope, it's really just saying the same thing, i.e. that in the world of equities and indices the systematic bid for OTM puts (or ITM calls) implies a particular shape for the skew (i.e. plot of IV as a function of strike). Like this (image courtesy of LIveVol):

 

[options-george]

okay that's great. thank you so much for your explanations. I got some key insight from our discussion today. this should help a lot.

 

[options-george]

a follow-up question - now looking at XOM options (Exxon Mobil).
The IV increases as the strikes rise.

XOM @ 69.24

65 calls IV 14.6
67.5 calls IV 14.8
70 calls IV 15.2

Presumably this means that the risk-neutral distribution has a bullish bias?

 

[Martinghoul]

I hesitate to say yes, since there could be one of several things going on. I don't follow specific equities and I don't want to speculate. If you want, I can look tomorrow and see if I can come up with something better.

 

[Martinghoul]

I don't see the same thing you're seeing. I am looking at the shape of the skew for XOM May options and it looks "normal" (see attached for Jun expiry).

 

[options-george]

I don't see the same thing you're seeing. I am looking at the shape of the skew for XOM May options and it looks "normal" (see attached for Jun expiry).

Thanks for taking another look at that.

Ok, that worries me a little - it implies that I am either misinterpreting the IV figures that Options Xpress gives me, or that the OX information is simply wrong.

I remember Think or Swim had a very good options platform a few years ago - would you recommend that or any other options platform? what is the product that you are using? What are you actually trading (if you are happy to share)?

 

[Martinghoul]

NP...

I really don't know anything about ThinkOrSwim, as I haven't used it. The screenshot I have attached is from Bloomberg. If I were you, I'd invest a little time and write my own Black-Scholes prices in Excel. It's easy to do and you'll have complete control over the implementation and all the parameters.

I don't trade stocks, so I am hardly an expert. My area is rates, mostly EUR and GBP.