Black-Scholes Pricing Surface

[karafrylee]

If I were to plot a graph of option price as a function of stock and strike prices (that is, a 3D pricing surface with stock and strike prices as the independent variables and option price as the dependent variable), what good would the graph be for traders/investors? What observations would they be interested in?

Thanks.

 

[Tsotne]

Isn't the spot and strike a bit confusing combination of variables to use for the option price graph?! It is more natural (at least as I have come across in most of books) to graph the option value in terms of time to expiry and spot.

 

[Devdeep]

What @Tsotne has mentioned is indeed the most natural one as you get variation of the value of the options both with Spot and Time.

@karafrylee... It is possible that what you are suggesting could be redundant... 'coz with everything else being the same, increasing spot is kind of equivalent to decreasing strike. Suppose the spot=100 and strike=100 at the moment. Along one of your axes, you are showing, for example, what happens when strike is at 90. This information can be obtained from the other axis where it is predicting what would be the option value when the spot is at 110 (assuming same vol etc.).

Am I missing something?

 

[Tsotne]

That's exactly what I meant, spot itself talks about the strike, so the best way is to choose the independent variables which themselves are independent from each other, so you get more accurate and reasonable picture. In option, the most information is obtained when you have graphed time along with spot...

Suppose the spot=100 and strike=100 at the moment. Along one of your axes, you are showing, for example, what happens when strike is at 90. This information can be obtained from the other axis where it is predicting what would be the option value when the spot is at 110 (assuming same vol etc.).

Am I missing something?

As I understood, here you showing that one axis is enough to get the information about the other. So while graphing BS for strike and spot, you are obtaining the duplicate information from the independent axes.

 

[Devdeep]

As I understood, here you showing that one axis is enough to get the information about the other. So while graphing BS for strike and spot, you are obtaining the duplicate information from the independent axises.

Yes... exactly... apart from the fact that I would use axes as a plural of axis as opposed to axises... :P

 

[Tsotne]

:D Grammar mistake in Black-Sholes Edited

 

[Xiaoyong]

What @Tsotne has mentioned is indeed the most natural one as you get variation of the value of the options both with Spot and Time.

@karafrylee... It is possible that what you are suggesting could be redundant... 'coz with everything else being the same, increasing spot is kind of equivalent to decreasing strike. Suppose the spot=100 and strike=100 at the moment. Along one of your axes, you are showing, for example, what happens when strike is at 90. This information can be obtained from the other axis where it is predicting what would be the option value when the spot is at 110 (assuming same vol etc.).

Am I missing something?

You are right about getting information using only one axis, but the example you give is not exactly right.
Under the BS-framework, V(S,K) = K*V(S/K,1) = K/K'*V(S/(K/K'),K'). So once you get all the option prices of different spot with one common strike, you can derive option prices with other strikes, of course with all other parameters fixed.

 

[Devdeep]

You are right about getting information using only one axis, but the example you give is not exactly right.
Under the BS-framework, V(S,K) = K*V(S/K,1) = K/K'*V(S/(K/K'),K'). So once you get all the option prices of different spot with one common strike, you can derive option prices with other strikes, of course with all other parameters fixed.

Thanks!

 

[Xiaoyong]

Thanks!

No prob

 

[karafrylee]

Thanks for the useful insights. I actually came across this from Wolfram's demo: http://demonstrations.wolfram.com/ExploringTheBlackScholesFormula/
But then again, this demo includes every combination of variables. I just thought if they allow us to explore stock and strike against option, there must be some significant use for it, right?

 

[Tsotne]

Thanks for the useful insights. I actually came across this from Wolfram's demo: http://demonstrations.wolfram.com/ExploringTheBlackScholesFormula/
But then again, this demo includes every combination of variables. I just thought if they allow us to explore stock and strike against option, there must be some significant use for it, right?

It just gives you the option value for stated combination of variables no matter how unreasonable they are. It's a simple calculation. In reality when the underlying price increases for the option, leaving strike as it was before increase in spot is not correct. While exploring graphs, all the variables should be moving taking into account the dependence on another variable. So if you want to calculate the effect of spot price increase on the value of option, this is not too problematic since the graph gives the insight leaving all others unchanged.