Facing the stock option market – Pricing

By admin, January 17, 2016

The stock options pricing method may sounds weird, but it has some reasonable rules to following order to accomplish this task.

The general function related to the stock option price formation is:

P = p(X,T,S,V)

Where:
P – Price or premium. Similar to insurance market concept
X – Strike price. The less the X value, the higher will be the Call price. The higher the price, the lower will be the Put price. X is known in advance.
T – Time or lifetime. In Brazilian market, the stock options usually live (are active) from 1 to 3 months. The more the time goes, the lesser will be the price.
S – Spot or present price.
V – Volatility. The volatility index of the stock related to the specific stock option. Stock options loves volatility.
Finalization:It has two ways of working, both valid in Brazil.

• European – when you need to wait until the deadline to finalize
• American – When you can finalize at any time, usually when the option is ITM (in the money).

The Bovespa model is mostly based on European model.

 

Identifying liquidity signals

 

Stock options are sold in multiples of 100, exactly as the ordinary stocks.
Rule 1: Strike near to the spot or present value;
Rule 2: No liquidity in series far or quite near to the deadline. Considering that you have 3 months to operate, the best month is the second and the 10 following days in some cases;
Rule 3: Look into the Bovespa table to identify the greatest volumes. Look also inside the Bovespa theoretical wallet.

 

Pricing

 

When working with stock options, probability comes to the table as a tool.
If a share has a uniform distribution, as seen in a dice game, a probability would work like following:

 

 

 

 

 

Once the share options are divided in Puts (you win when share value falls) and Calls (you make profit when the share value rises), so we have the scenario:

 

 

 

 

 

Going ahead and considering that the extremes might be less probable:

 

 

 

 

 

And just for explore, let’s see the opposite probability: Consider that the extremes are more probable.

 

 

 

 

 

Normal distribution

Considering the normal distribution, aka Gauss distribution, we will have most probability near the u average and consider the standard deviations as you can see in the wikipedia article https://en.wikipedia.org/wiki/Standard_deviation
-4 standard deviation – (fi or s) -0.1 %
-3 standard deviation – (fi or s) -2.1 %
-2 standard deviation – (fi or s) -13.6 %
-1 standard deviation – (fi or s) -34.1 %
1 standard deviation – (fi or s) 34.1 %
2 standard deviation – (fi or s) 13.6 %
3 standard deviation – (fi or s) 2.1 %
4 standard deviation – (fi or s) 0.1 %
 The standard deviations can occur for both sides – positive and negative

 

Fat tail normal distribution

In the stock market, not all papers in some moments in time might be considered mathematically under a normal distribution. So you can consider the fat tail normal distribution. To find out more, you can see the Wikipedia related topic.
https://en.wikipedia.org/wiki/Fat-tailed_distribution

 

Conditions for pricing

1 – Identify the infinite possible prices;
2 – Attach a probability for each price;
3 – Consider the basic variables;
4 – Efficient market – perfect advising. No irrationality nor asymmetric information is allowed.
And finally, you will be created the pricing conditions.
Remembering:
Strike price – X
Time life – T
Actual price – S
Volatility – V
Opportunity cost i (usually consider the main interest tax of the market)

The Black and Scholes model

These guys were awarded with a Nobel Prize in economics in 1997.
Examples: A share raised R$ 2,00
A Call will raise R$ 1,00
Call Sensibility:
+ When the S is great or rising
+ High volatility
+ More time to deadline
+ Interest rate (less relevant in developed markets)
Put Sensibility:
– When the S is great or rising
+ High volatility
+ More time to deadline
+ Strike price
– Interest rate (less relevant in developed markets)

The Black and Scholes equation can be found in the
https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_equation

And based on that, it is possible to you build your own Excel worksheet to discover any valid price for the stock options. But you must know the historic volatility index for an amount of time. When related to stock options, from 1 to 3 months, and commodities, a year will be the minimum.
You can download your worksheet here.
Consider the time on the Black and Scholes spreadsheet in days.
If you are about to negotiate shares in Bovespa, you will need the volatility work sheet – only in Portuguese – found in the Bovespa website.
http://www.bmfbovespa.com.br/cias-listadas/volatilidade-ativos/BuscaVolatilidadeAtivos.aspx?Idioma=pt-br

Conclusions

You can consider the Black and Scholes model to estimate the stock option price, but not consider it a rule. It is only a reference once it cannot detect the black swans.