How Much Is an Op­tion Worth?

Con­sid­er an at-the-money call op­tion with a strike price of $50. The un­der­ly­ing as­set is cur­rent­ly trad­ing at $50 per share. As­sume it’s a Eu­ro­pean-style op­tion. One trad­er wants to take the long side of the con­trac­t. An­oth­er trad­er wants to take the short side. How can they agree on a fair price?

The in­trin­sic val­ue of the op­tion at the time of ex­pi­ra­tion is a func­tion of two things: the strike price of the op­tion con­tract and the mar­ket price of the un­der­ly­ing as­set when the op­tion ex­pires. You can com­pute the val­ue of the op­tion at ex­pi­ra­tion us­ing this equa­tion:

Since the mar­ket price of the un­der­ly­ing can fluc­tu­ate over time, the price of the un­der­ly­ing at the time of ex­pi­ra­tion can­not be known in ad­vance. To val­ue an op­tion, you must some­how guess what the price of the un­der­ly­ing might be when the op­tion ex­pires.

Av­er­ag­ing Pre­dic­tions

Let’s say the sell­er of the op­tion con­tract pre­dicts the price of the un­der­ly­ing will be trad­ing at $40 per share at ex­pi­ra­tion. The buy­er, on the oth­er hand, spec­u­lates that the un­der­ly­ing will be trad­ing at $60 per share at ex­pi­ra­tion. Us­ing the for­mu­la above, we can com­pute the in­trin­sic val­ue of the op­tion at each of the an­tic­i­pat­ed out­comes:

The sell­er rea­sons that of­fer­ing the op­tion at any price greater than ze­ro would be to his ad­van­tage since, ac­cord­ing to his pre­dic­tion, the op­tion will ex­pire worth­less. The buy­er fig­ures that pur­chas­ing the op­tion at any price be­low $10 is an op­por­tu­ni­ty for prof­it. The two traders de­cide that the fairest price can be ob­tained by tak­ing the av­er­age val­ue of the op­tion for the two pre­dict­ed out­comes:

In valu­ing the op­tion, the two trad­ers ef­fec­tive­ly give equal weight to the prob­a­bil­i­ty of each out­come oc­cur­ring. The trad­er who pre­dicts the cor­rect out­come prof­its from the trans­ac­tion, while the trader who pre­dicts in­cor­rect­ly takes a loss.

Dis­crete Dis­tri­b­u­tions

In a more re­al­is­tic sce­nar­i­o, there might be many pos­si­ble out­comes. Some out­comes might have a high­er prob­a­bil­i­ty of oc­cur­ring than oth­er­s. Con­sid­er an­oth­er ex­am­ple with the fol­low­ing set of pos­si­ble out­comes:

Us­ing the for­mu­la above again, we can com­pute the in­trin­sic val­ue of the op­tion at each of the pos­si­ble out­comes:

Now let’s as­sume the prob­a­bil­i­ty of each out­come is this:

We can com­pute the ex­pect­ed val­ue of the op­tion as a weight­ed av­er­age:

This ap­proach lets us mod­el a pre­dic­tion as a prob­a­bil­i­ty mass func­tion across a range of val­ues. The pre­dic­tion might be based on his­tor­i­cal price ac­tion or it might sim­ply be an ar­bi­trary be­lief cho­sen at our dis­cre­tion.

Con­tin­u­ous Dis­tri­b­u­tions

In re­al mar­ket­s, prices don’t al­ways snap neat­ly to $10 in­cre­ments. Tick sizes can be as small as a pen­ny or even small­er. It might be bet­ter to mod­el a price pre­dic­tion as a con­tin­u­ous prob­a­bil­i­ty dis­tri­b­u­tion. Let’s con­sid­er a set of pos­si­ble out­comes that can span across a con­tin­u­ous range of val­ues:

Prices can’t fall be­low ze­ro, so the range of pos­si­ble val­ues has a low­er bound at ze­ro. As be­fore, some out­comes might have a high­er prob­a­bil­i­ty of oc­cur­ring than oth­er­s. The cu­mu­la­tive dis­tri­b­u­tion func­tion is the prob­a­bil­i­ty that, when the op­tion ex­pires, the price of the un­der­ly­ing is less than or equal to a giv­en val­ue:

From prob­a­bil­i­ty the­o­ry, we know that the prob­a­bil­i­ty den­si­ty func­tion is just the de­riv­a­tive of the cu­mu­la­tive dis­tri­b­u­tion func­tion:

If we can come up with a prob­a­bil­i­ty den­si­ty func­tion that mod­els our pre­dic­tion, we can com­pute the ex­pect­ed val­ue of the op­tion by in­te­grat­ing over the range of pos­si­ble out­comes:

Here the low­er lim­it is ze­ro be­cause the price of the un­der­ly­ing can nev­er fall be­low ze­ro. The up­per lim­it ap­proach­es in­fin­i­ty if the den­si­ty func­tion is un­bound­ed. If us­ing a bound­ed den­si­ty func­tion, you can set the lim­its ac­cord­ing­ly. Let’s look at a con­crete ex­am­ple. Sup­pose we mod­el our pre­dic­tion as a sim­ple tri­an­gu­lar dis­tri­b­u­tion:

Since the prob­a­bil­i­ty den­si­ty is ze­ro for all val­ues out­side the range span­ning from $20 to $80, we can ig­nore val­ues out­side of this range. Fur­ther­more, since we’re on­ly con­sid­er­ing at-the-money call op­tions with a strike price of $50 in this con­tex­t, we know the in­trin­sic val­ue of the op­tion up­on ex­pi­ra­tion is al­ways ze­ro if the un­der­ly­ing is trad­ing be­low $50. Plug­ging in the den­si­ty func­tion and sim­pli­fy­ing:

Fol­low­ing this ap­proach, you can plug in any prob­a­bil­i­ty dis­tri­b­u­tion func­tion that mod­els the volatil­i­ty of the un­der­ly­ing as­set price.

Oth­er Con­sid­er­a­tions

The ex­am­ples above present a few sim­ple mod­els for es­ti­mat­ing the val­ue of an op­tion con­trac­t. In these ex­am­ples, the val­ue of an op­tion de­pends large­ly on how wide or how nar­row the price of the un­der­ly­ing is ex­pect­ed to fluc­tu­ate. How­ev­er, there are oth­er fac­tors to take in­to con­sid­er­a­tion when es­ti­mat­ing the val­ue of an op­tion con­trac­t. Here are a few of them:

• Choos­ing a prob­a­bil­i­ty dis­tri­b­u­tion that best ap­prox­i­mates price fluc­tu­a­tions
• Eval­u­at­ing price changes on a log­a­rith­mic scale vs. a lin­ear scale
• How the pas­sage of time af­fects the val­ue of an op­tion
• Com­put­ing the val­ue of Amer­i­can-style op­tions on div­i­dend-pay­ing stocks
• Het­eroskedas­tic vari­ances, price jump­s, and oth­er such haz­ards

There are oth­er, more so­phis­ti­cat­ed op­tion pric­ing mod­els avail­able that take these and oth­er fac­tors in­to con­sid­er­a­tion. I might ex­plore some of these top­ics in fu­ture post­s.