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(New page: == Black-Scholes option pricing model - Convergence of binomial == * Black-Scholes option pricing formula: <br> The value <math>C<math> of a European call option at time <math>t=0</math> ...)
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<math>r</math>      Continuously compounded risk-free interest <br>
<math>r</math>      Continuously compounded risk-free interest <br>
<math>\sigma</math> Annual standard deviation of the returns of the stock <br>
<math>\sigma</math> Annual standard deviation of the returns of the stock <br>
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<math>t<math>      Time to expiration in years <br>
+
<math>t</math>      Time to expiration in years <br>
<math>\Phi(d_i)</math>  Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math>  <br>
<math>\Phi(d_i)</math>  Cumulative probability at <math>d_i</math> of the standard normal distribution <math>N(0,1)</math>  <br>
* Binomial convergence to Black-Scholes option pricing formula: <br>
* Binomial convergence to Black-Scholes option pricing formula: <br>
The binomial formula converges to the Black-Scholes formula when
The binomial formula converges to the Black-Scholes formula when
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the number of periods <math>n<math> is large.  In the example below we value the call option using the binomial formula for different values of <math>n</math> and also using the Black-Scholes formula.  We then plot the value of the call (from binomial) against the number of periods <math>n<math>.  The value of the
+
the number of periods <math>n</math> is large.  In the example below we value the call option using the binomial formula for different values of <math>n</math> and also using the Black-Scholes formula.  We then plot the value of the call (from binomial) against the number of periods <math>n</math>.  The value of the
call using Black-Scholes remains the same regardless of <math>n</math>.  The data used for this example are:
call using Black-Scholes remains the same regardless of <math>n</math>.  The data used for this example are:
<math>S_0=\$30, \ E=\$29,\ R_f=0.05, \sigma=0.30,\
<math>S_0=\$30, \ E=\$29,\ R_f=0.05, \sigma=0.30,\

Revision as of 15:48, 3 August 2008

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 Time to expiration in years
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 is large. In the example below we value the call option using the binomial formula for different values of n and also using the Black-Scholes formula. We then plot the value of the call (from binomial) against the number of periods n. The value of the call using Black-Scholes remains the same regardless of 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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