Black Scholes Formula Puns

Disclaimer: I am long SPRT and this is not advice to buy or sell the stock or its derivatives.

The Black-Scholes Formula is:

https://preview.redd.it/1r45otuxd5k71.png?width=727&format=png&auto=webp&s=9511cd1d6bd0fb618d4e12da0aa59f03da31e9e2

C = Price of call option (what we will pay to buy a call)

N = Cumulative normal distribution function

S sub t = Current stock price

K = Option strike price

e = Base of the natural log function (approx 2.72)

r = Risk-Free interest rate (the annualized continuously compounded rate on a safe asset – like a US treasury bill – with the same maturity date of the option)

t = Time to expiration of option, expressed in years, i.e., 0.0833 for a call expiring in 30 days

ln = The natural logarithm

σ = Standard deviation of the annualized continuously compounded rate of return of the stock, or volatility

Without diving deep into the meaning of most of these big words or statistics, it's important to notice that the call option price is simply the difference between the minuend [N(d1)S] and subtrahend [N(d2)Ke^(-rt)] and it is heavily influenced by the volatility of the security which underlies the option. The minuend takes N(d1) and multiplies it by the price of the stock, and the subtrahend takes N(d2) and multiplies it by the strike price of the call discounted to today based on the time to expiration and the current risk-free interest rate. The key is identifying that when sigma (volatility) increases, the difference between d1 and d2 becomes greater, such that the minuend increases and the subtrahend decreases. This equates to a greater call option price.

Here is a quick calculation with Excel using the 39 strike 17 Sept. 2021 SPRT option when the stock is at $28 and IV is at 400%:

note t = time to expiration in years, so t = 21/365, because there are 21 days until 9/17

If everything were kept the same hold volatility, changed to its value last Monday of 187%, we'd get:

Notice that the price of the option is less than one-third of its price when IV is at 400%

Finally, when IV is calculated by using the options prices as input, it is expressed as a percentage of stock price and indicates a one standard deviation move over

what is the point of using the equations of Black and Scholes and other statistical things if in everyone use them and the options are already priced in with these formulas?

Example: call price on Tesla at $ 400 in May = X price.

This price of the call option is already "priced in" using Black and Scholes equations ... so what is the point of doing all the statistical and fancy math behind if the options (especially if in the short term) are already well priced in ?

The Black-Scholes formula is the following:

Fair Value = N(d1)*S - N(d2)K * e^(-rT)

d1 =( ln(S/K) + (r + sigma^2/2)t ) / sigma * sqrt(T)

d2 = d1 - sigma*sqrt(T)

So, say I have all inputs (S, r, sigma and T) and also the result Fair Value, could I isolate the exercise price K ? I am not sure how to do it considering I have to take the cumulative density function of the normal distribution (N) of the d1 and d2 values, which is where the K would be. Could someone help? Thank you!

I'm preparing for a final interview with an investment bank; and trying to improve one of my weeknesses - the greeks. To me its not clear why people would talk so much about the partial derivatives of a function and not the shape of a function itself; in fact i find it easier to think in terms of shape primarily and derivartives secondarily - so i've been making graphs in matlab, however as I am new to quant mahs I am sure i am covering well trod ground.

I was wondering if anyone has produced graphs of what the BS formula looks like as strike prices/time/volatility interest tend towards 0 and infinity? Or if anyone has made graphs of its level sets?

I think i get the greeks enough to answer interview questions about them - but not enough to feel like I really know them if that makes sense.

I have tried to reverse the BS Formula solving for implied volatility given everything else, but have run into some roadblocks. (Mainly, it seems to solve like a polynomial giving me two equations). Has anyone had a similar thought and how have they gotten around it?

Any resources or tutorials to the application-specific calculation of Black-Scholes formula would be highly appreciated! I'm currently trying to check my work for option calculations and it's hard to find a place to do so. Using NYMEX natural gas prices, implied annualized volatility estimates, and interest rate curves to calculate at-the-money option prices; if that helps.

The dissertation's mission was to prove that the binomial model converges to the Black and Scholes formula not only for a single moment in time (with CLT) but as a whole process.

So if V_t \forall t is the process of value of a contract through the Binomial model then its distribution converges to V_t given by Black and Scholes. It is useful for barrier options and I was thinking arguing briefly about risk management products or insurance products that work similarly. Thank you!

I'm trying to understand these two values, but everywhere I look just gives their definition, without explaining what they represent. Can anyone give an explanation? Thanks!

Hello fellow actuaries

I was doing some practice problems on the Black-Scholes Formula, and an interesting case came up.

The problem asked to evaluate the price of a call option on a stock, given the time, strike price, risk free rate, forward price, volatility, and continuous interest rate. This seems like a pretty straightforward plug and compute type of problem, until I saw that the solution did not use the stock-specific Black-Scholes formula, but the general formula (where they omit the risk free rate and the dividends in the calculation for d_1 and d_2)

I looked at other problems involving pricing stock options where the solution used the stock-specific B.S. formula, and could not discern why the problem used the general form. Is there a rule that we would follow to determine whether we want to use the general formula or the asset specific formula? So far, the only patterns I've found are:

1. If the asset pays discrete dividends, we should use the general formula

2. If the problem does not give you the risk free rate and you have no way of calculating it from the information give, use the general formula.

I do not have access to a phone, but I will upload a picture of the photo and solution ASAP. In the meantime, I would very much appreciate any clarification on this topic. Thank you all!

The headline is a bit of a simplification, but these links have more of the detail.

http://business.time.com/2007/02/05/ed_thorp_explains_the_new_hedg/

http://bfi.cl/papers/Haug%20Taleb%202007%20-%20Why%20we%20have%20never%20used%20the%20Black-Scholes-Merton%20option%20pricing%20formula.pdf

http://edwardothorp.blogspot.com/2008/06/edward-thorp-black-scholes-model-nobel.html

Black Scholes works on some pretty flimsy assumptions. I understand that predicting the future is no easy task, but does anyone have any recommended readings for models that use less assumptions?

Edit: In particular addressing the assumption of Brownian motion for prices.

It was in my book and it said that i didn't need to really know it, but it also had questions that referenced it in the solution.

Sitting for MFE in three weeks and while I do have the B-S formula memorized and practiced, there are a bunch of other formulas that involve N(d1) and N(d2) (or N(-d1), N(-d2)) individually and it would be much easier to derive them knowing the intuition behind them. I'm having a hard time finding a good explanation out there on the internets that breaks it down into simple terms. This AO thread is somewhat helpful (http://www.actuarialoutpost.com/actuarial_discussion_forum/showthread.php?t=187740) but still not getting me all the way there. Maybe this could be a helpful exercise to others sitting in March as well?