# SOCR EduMaterials Activities ApplicationsActivities BlackScholesOptionPricing

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 Revision as of 15:47, 3 August 2008 (view source)Nchristo (Talk | contribs) (New page: == Black-Scholes option pricing model - Convergence of binomial == * Black-Scholes option pricing formula:
The value $C[itex] of a European call option at time [itex]t=0$ ...)← Older edit Revision as of 15:48, 3 August 2008 (view source)Nchristo (Talk | contribs) Newer edit → Line 22: Line 22: $r$      Continuously compounded risk-free interest
$r$      Continuously compounded risk-free interest
$\sigma$ Annual standard deviation of the returns of the stock
$\sigma$ Annual standard deviation of the returns of the stock
- $t[itex] Time to expiration in years + [itex]t Time to expiration in years [itex]\Phi(d_i)$  Cumulative probability at $d_i$ of the standard normal distribution $N(0,1)$
$\Phi(d_i)$  Cumulative probability at $d_i$ of the standard normal distribution $N(0,1)$
* Binomial convergence to Black-Scholes option pricing formula:
* Binomial convergence to Black-Scholes option pricing formula:
The binomial formula converges to the Black-Scholes formula when The binomial formula converges to the Black-Scholes formula when - the number of periods $n[itex] is large. In the example below we value the call option using the binomial formula for different values of [itex]n$ and also using the Black-Scholes formula.  We then plot the value of the call (from binomial) against the number of periods $n[itex]. The value of the + the number of periods [itex]n is large. In the example below we value the call option using the binomial formula for different values of [itex]n$ and also using the Black-Scholes formula.  We then plot the value of the call (from binomial) against the number of periods $n. The value of the call using Black-Scholes remains the same regardless of [itex]n$.  The data used for this example are: call using Black-Scholes remains the same regardless of $n$.  The data used for this example are: [itex]S_0=\$30, \ E=\$29,\ R_f=0.05, \sigma=0.30,\ [itex]S_0=\$30, \ E=\$29,\ R_f=0.05, \sigma=0.30,\

## 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.