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101 Concepts for the Level I Exam

Essential Concept 80: The BSM Model


The inputs to the BSM model are:

  • Price of underlying stock, S
  • Continuously compounded risk-free rate, r
  • Time to maturity in years, T
  • Strike price, X
  • Volatility of the underlying in annual percentage terms, σ

The BSM model for non-dividend paying stock is:

c = SN(d1) – e–rTXN(d2)

p = e–rTXN(–d2) – SN(–d1)

where:

\mathrm{d}}_{\mathrm{1}}\mathrm{=}\frac{\mathrm{ln}\mathrm{}\mathrm{(S/X)+(r+}{\mathrm{\sigma }}^{\mathrm{2}}\mathrm{/2)T}}{\mathrm{\sigma }\sqrt{\mathrm{T}}}

d${}_{2}$ = d${}_{1}$ $\mathrm{-}$ $\sigma$$\sqrt{\mathrm{T}}

Using the above inputs, the BSM model can be used to predict:

  • Call option price, c
  • Put option price, p

Instructor’s Note:

The following tips will help you remember the formulas.

c = SN(d1) – e–rTXN(d2)

p = e–rTXN(–d2) – SN(–d1)

  • A call option is of the form S- X, whereas a put option is of the form X – S.
  • The present value of strike price X is obtained by multiplying it by e–rT
  • d1 is associated with S, whereas d2 is associated with X
  • For call options we use positive values of d1 and d2. Whereas, for put options we use negative values of d1 and d2.