Probability of closing price

[lukaluke]

Hi,

Could you help me to tackle this problem? I don't see how to do it!

A stock has a beta of 2.0 and a specific daily volatility of 0.02. Suppose that yesterday's closing price was 100 and today the market goes up by 1%. What's the probability of today's closing price being at least 103?

Thanks for your help.

 

[GoIllini]

*Specific* Daily volatility of 0.02 meaning an annualized vol of 0.02*sqrt(252)= 32%? I'm also going to assume these are arithmetic returns but we took the beta using logged returns.

Ok, so this is fairly straightforward assuming it's just a traditional stats problem. Given that the market is up 1% and we have a beta of 2, the expected value of the stock is 101*101 following a lognormal model=102.1. The systematic volatility is already accounted for- leaving only the unsystematic/specific/idiosyncratic volatility, which you state is 2%.

The classical assumption is that this idiosyncratic risk is distributed lognormally. Some french-speaking sticklers out there will tell me I need to correct the drift in the lognormal distribution (thus applying Girsanov and Ito's Lemma), and that there's also the drift due to rates. I'm not going to do that here, and simply assume this is just a stats problem:

P(px>103)=P(~Lognormal(mu=0, sigma=0.02)>103/102.1)

P(px>103)=P(~N(mu=0,sigma=0.02)>ln(103/102.1))
P(px>103)= 1-pnorm(ln(103/102.1)/0.02)

Where pnorm= the CDF of the standard normal.

This is just the probability that the normal distribution comes out about 0.43 standard deviations above 0. So that's about ~33%.


So it's really pretty straightforward. Lognormal and Normal distributions have certain additive properties. A normal plus a normal is a normal. A normal conditional on another normal is a normal. Same with lognormal distributions when you are in multiplicative mode.

There's some ambiguity here as to whether we're dealing with logged returns and what exactly our beta means. Most people use a log returns model for stock prices, but you can do this arithmetically, too. (In which case, you have an expected stock price of 102 with a standard deviation of 2. This would make the probability of the stock price going above 103 a 0.5 standard deviation event. But as you can see, the idiosyncratic risk doesn't scale with the systematic returns and it's theoretically possible to get a negative stock price.)

EDIT: I hope this isn't a stochal problem. If you're some polytechnique student, sorry for insulting your intelligence and underestimating this as a mere stats problem.

 

[moretodo]

This question is not a stochal problem. It seems pretty easy. Applying CAPM, its obvious that the stock is expected to be at 102 (this assumes rf to be 0 since your not given that information in the question). Using the daily volatility and assuming the asset random variable to either follow a normal or lognormal distribution (I believe both assumptions are reasonable. This does not involve option pricing where perhaps assuming a lognormal dist. in line with BS would be prudent), you should be able to determine the probability by applying the Cumulative distribution function.

Any stochastic effects over a day are pretty much insignificant.

 

[GoIllini]

This question is not a stochal problem. It seems pretty easy. Applying CAPM, its obvious that the stock is expected to be at 102 (this assumes rf to be 0 since your not given that information in the question). Using the daily volatility and assuming the asset random variable to either follow a normal or lognormal distribution (I believe both assumptions are reasonable. This does not involve option pricing where perhaps assuming a lognormal dist. in line with BS would be prudent), you should be able to determine the probability by applying the Cumulative distribution function.

Any stochastic effects over a day are pretty much insignificant.

OP is from France, he's posting on quantnet and likely has a STEM background (in France), so he may know more about what he's doing on this than either of us do.

Stochal was never my strong suit, but I believe the drift correction for the lognormal distribution is 1/2*sigma^2. That's 2%*2%*0.5=0.0002, 1% of the SD, or about 2-2.5% of the final answer's position on the normal distribution. I would argue it could matter a bit.

http://en.wikipedia.org/wiki/Girsanov_theorem (use this to get a lognormal model to be a martingale)
http://en.wikipedia.org/wiki/Itō's_lemma (use this to calculate the drift introduced by the volatility combined with the lognormal model's convexity)

These two theorems and their application to the lognormal model explain why the black-scholes algebraic formula has a square volatility term right next to the rates term:

http://en.wikipedia.org/wiki/Black-Scholes#Black.E2.80.93Scholes_formula

It's not big, but I think the stochastic drift effect is pretty noticeable on most options quotes. This is why the French-speaking stochal geniuses get paid the big bucks while working in exotics pricing, and this is why I'm not a pricing quant.

 

[lukaluke]

Hi,

Thank you for your help.
Your approach and the stochastic one, where we make the assumption that the stock price follows a geometric brownian motion, are similar.
My problem was that I wasn't very familiar with CAPM model. And I didn't see how to use the beta.
My guess is to use your approach with the CAPM model to determine an estimation (expected value) of the returns (which is normally distributed, if the stock prices are normally distributed). We have that:
$$E[R] = r + \beta (E[R_{\text{market}}] - r)$$
where $E[R_{\text{market}}] = 1\%$. Then I can make the assumption that $r = 0$, and get: $E[R]=2\%$.
Which eventually gives me a result slightly different from yours... :S

What do you think of this approach?