"Putting VaR to Work" - calculating a call option price using Black Scholes

[Ryan S]

I'm going crazy trying to understand what I'm not doing right here, and I think it's my unfamiliarity with using or understanding Black Scholes model. But anyway, would appreciate if anyone can help.

In the first paragraph under the subsection "Nonlinear Derivatives" Allen gives an example of an option's price according to the Black-Scholes option pricing formula. The section begins as "The primary example for a nonlinear derivative is an option. Consider for example an..."

risk free rate=5%
T=.5 0r 6 months
Non dividend paying stock
price =$100
Strike=$100
volatility=20% per annum

value of the call option = $6.89. I simply cannot get this calc and I don't know why, I'm thinking its bc of the above reasons and maybe I'm assuming N to be 1.

I'm having issues moving ahead with understanding the chapter when I can't do a simple calculation right...

Any help appreciated.

Thanks,
Ryan

 

[David Harper CFA FRM]

Hi Ryan,

I input into this BSM XLS, I do get 6.89, see https://www.dropbox.com/s/uhlgksuy1rvzl7c/0313_linda_allen_BSM_p86.xlsx

• d1 = [LN(100/100) + (5% + 20%^2/2)*0.5] / [20%*sqrt(0.5)] = [0+ (5% + 20%^2/2)*0.5] / [20%*sqrt(0.5)] = 0.2475
• N(d1) = 0.5977; consistent with her $Delta (although I consider delta unitless so i don't agree with $0.59)
• N(d2) = 0.5422 = N(0.2475 - 20%*sqrt(0.5)) = N(0.1061)
• c = 100*0.5977 - 100*exp(-5%*0.5)*0.5422 ~= 6.89. I hope that helps!

 

[Ryan S]

Thanks David, appreciate the fast response. I was not plugging my d1 and d2 values into the Z table to get the N(d1) and N(d2) probabilities. What is not frustrating is I'm failing to make the connection of calculating the first part (d1 and d2) to why I need to plug these numbers into a cumulative Z table. I am seeing how the numbers are calculated and how the final answer is derived, but it's more robotic than anything now in my head. Maybe I need to let it settle in and it will click, but if you can guide me to how to develop my intuition on this I'd be very grateful. I'm assuming I need to step back and look at what BSM is telling me, and how this falls into the normal distribution assumptions?

Ryan

 

[David Harper CFA FRM]

Hi Ryan,

A normal distribution gets to be used merely because the BSM model makes that assumption It's a limiting assumption; e.g., the existence of implied volatility smiles/smirks/frowns basically implicate the assumption as unrealistic. The BSM assumes future asset price, S(t), are lognormal, which is the same as assuming the log returns, LN[S(0)/S(t)] are normally distributed; and just LN[S(t)] is normally distributed. It is a convenience, like the other really unrealistic assumption that volatility is constant, which enables an elegant solution.

In regard to an intuition for BSM, books have been written about it. Personally, I like to build on minimum value--i.e., minimum value of a call = S - K*exp(-rt)--and "wrap in" N(d1) as delta, and N(d2) as the risk-neutral probability of exercise. As I have written many times here in the forum, it is not difficult to see the intuition of d2 and N(d2). The intuition of BSM (i.e., a call option has the value of a replicating portfolio with two positions, a delta share of stock plus short a bond) is a little harder. Here are some links:

• Old post on BSM; e.g., c = Stock*delta - Discounted Strike*(prob of striking at expiration in the RISKNEUTRAL world).
  
  
  
• My youtube video on Intuition behind BSM
  
  
  
• How d2 in BSM becomes PD in Merton
  
  
  
• My thread on Merton model https://forum.bionicturtle.com/threads/merton-model-a-summary-of-the-issues.5646/
  
• But also, Hull's Chapter 14 (BSM) is really good

I hope that's a good start!