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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> | ||
- | <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 | ||
- | 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 15:50, 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:
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.