Unfortunately, “I used a calculator” doesn’t tell me much that’s remotely useful.
I used a Monte Carlo simulation with a mean effective annual return of 2%, a volatility (standard deviation) of (continuously-compounded) returns of 30%, and statistically independent returns each day for 180 days.
Doesn’t the BSM assume a risk-neutral approach? Stocks are expected to have 0% return over the time-period, with a return volatility of x%. The risk free rate is a time-premium for the option price.
It stopped being an option equation when we got the implied stock price at 50-delta. Now we need to find the probability of the stock going above $20 from $19.36 in 180 days with a return volatility of 30%.
I used an online calculator as it’s not on top of my head, but I’m sure the formula for such is quite simple. Something along the lines of solving for the critical value on a normal distribution chart.
If I recall, which I might not, the price at time 0 is such that for either the up move or the down move (whichever actualizes) your portfolio returns exactly the risk-free rate. Remember, if you use the tree to price an option, and it doesn’t match what is available in the market, there exists an arbitrage opportunity (I think they gave this example).
Think of it this way, you go to an options trader, and tell him according to a current spot price of $19.36 and 180 days remaning to maturity, you only have a 2% chance finishing above $20. How would that sound? Either the stock is overly-bearish (breaking the symmetrical normality distribution), or the volatility is close to zero.
I’m not saying the 4x.xx% is completely accurate, but eyeballing it looks pretty sensical compared to only 2%. You are using the 2% risk free rate as a step-wise predictor for stock return (which shouldn’t be a problem given the small number), but you are most likely not including the 30% return volatility into the equation. We only need 3.3% upside to cross the barrier, while you have an annualized 30% return volatility. Your overall premise of the delta argument is on-point, but the numbers are exaggerated. You should try revising the model again, there’s probably something you’ve missed.
I honestly couldn’t care less about how it would sound; just because someone’s a trader doesn’t mean that he understands the underlying mathematics enough to make a judgment about the probability.
I’m more concerned with whether it’s true or not, not how it sounds to anyone.
Starting at $19.36 and returning 2% (EAY), you expect a price of $19.56 in 180 days. That’s well _ below _ the $20 strike price.
In fact, I am. The daily return is given a normal distrubution with an annual mean of 2.0% and an annual standard deviation of returns of 30%. This is then converted into a daily return using r(daily) = (1 + r(annual))^(1/360) − 1. (Actually, it’s r(daily) = EXP(r(annual))^(1/360) − 1, because the volatility is on the continuously compounded return, not the effective annual return.)
That’s 3.3% in 6 months, or 6.69% annually. That’s a lot of sigmas away from 2%.
How ARE you incorporating return volatility? Because a mean of 2% with a 30% sigma on stock return would imply a 95% chance of annual return falling between ~[-58% and 62%], while we only need 6.69% on the upside!
I’m not sure what exactly you’re doing on the model, but most likely has to do with the sigma input. I guess it uses 30% dispersion on the mean return, and not on the stock price,
Let’s go back to Level II, and remember what the volatility of continuously compounded returns means.
In an annual binomial tree, the 1-period up rate (u) and the 1-period down rate (d) are related by the formula:
u = d × e^(2_σ_)
When we say that we have an annual volatility of 30%, it doesn’t mean that we have rates between −58% and +62%, it means that we have 1-year rates between (roughly) 1.48% (= 2% × e^−0.3) and 2.70% (= 2% × e^+0.3).
Perhaps you should review binomial interest rate trees at Level II; it seems that you’ve forgotten the basics.
Aha, I do not disagree (double negative?) with what you’ve said. But you missed the point of the BSM.
The sigma is the volatility of the underlying asset, not the volatility of the underlying’s return! It’s the square root of the quadratic variation of the stock’s log price process, small, but big difference!
What your model did is essentially assume a volatility of nearly zero!
There is no such thing as “the volatility of the underlying asset.” Assets do not have volatility.
Asset _ prices _ have volatility.
Asset _ returns _ have volatility.
Assets do not have volatility.
And it appears that you’ve misunderstood the B-S-M model: σ is the volatility of the _ continuously compounded return _ on the underlying asset.
I just looked at the Monte Carlo simulation I’d run. The (simulated) continuously compunded return ranges from a low of −1.30 to a high of +1.31. So it seems that I have the correct probability distribution on the returns (−130% to +131%; that’s ±4.35_σ_).
“For future reference, x is the initial stock price, σ is referred to as the volatility of the stock, and, because E(Xt) = e(α+σ2/2)t , we call µ ≡ α + σ2/2 the expected rate of return on the stock.”
Black, Merton, and Scholes — Their Central Contributions to Economics, Darrell Duffie_,_ December 22, 1997
And the original 1973 paper:
The Pricing of Options and Corporate Liabilities, the Journal of Political Economy, Vol. 81, No. 3 (May - Jun., 1973), pp. 637-654
_________________________________________________
We previously agreed that underlying follows the risk free rate, it would not make sense to have a variance on that ‘risk-free’ expected return, but a variance on the stock’s price instead.
I don’t know what to tell you. The definition of σ is the _ standard deviation of the continuously compounded returns _ of the stock. If you think that it should be a standard deviation of the stock’s price, you’ll have to take it up with Myron Scholes and Robert Merton; you cannot take it up with Fischer Black as he died in 1995.
I just gave you the original Black-Scholes paper, Merton refined it.
In any case, I can’t find any other source that says otherwise, perhaps you could? I’m not sure from where your definition is sourced. I guess I’ll have to dust off L2 books if nessecary.
By the way, you do realize that an annual standard deviation of returns of 30% means that the daily standard deviation of returns is about 30% / √360 = 1.58%, yes?
