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Black-Scholes option pricing model - Convergence of binomial

  • Black-Scholes option pricing formula:

The value C < math > ofaEuropeancalloptionattime < math > t = 0 is: 
C=S_0 \Phi (d_1) - \frac{E}{e^{rt}} \Phi(d_2)

d_1=\frac{ln(\frac{S_0}{E})+(r+\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}

d_2=\frac{ln(\frac{S_0}{E})+(r-\frac{1}{2} \sigma^2)t}
{\sigma \sqrt{t}}=d_1-\sigma \sqrt{t}
Where,
S0 Price of the stock at time t = 0
E Exercise price at expiration
r Continuously compounded risk-free interest
σ Annual standard deviation of the returns of the stock
t < math > Timetoexpirationinyears < br > < math > Φ(di) Cumulative probability at di of the standard normal distribution N(0,1)

  • Binomial convergence to Black-Scholes option pricing formula:

The binomial formula converges to the Black-Scholes formula when the number of periods n < math > islarge.Intheexamplebelowwevaluethecalloptionusingthebinomialformulafordifferentvaluesof < math > n and also using the Black-Scholes formula. We then plot the value of the call (from binomial) against the number of periods n < math > .ThevalueofthecallusingBlackScholesremainsthesameregardlessof < math > n. The data used for this example are: Failed to parse (lexing error): S_0=\$30, \ E=\$29,\ R_f=0.05, \sigma=0.30,\ \mbox{Days to expiration}=40 .

  • For the binomial option pricing calculations we divided the 40 days into intervals from 1 to 100 (by 1).
  • The snapshot below from the SOCR Black Scholes Option Pricing model applet shows the path of the stock.
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