Difference between revisions of "Programming/Kdb/Labs/Option pricing"
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where <math>F = S_t e^{(r - q)\tau}</math> is the forward price and | where <math>F = S_t e^{(r - q)\tau}</math> is the forward price and | ||
<center><math> | <center><math> | ||
d_1 = \frac{1}{\sigma\sqrt{\tau}} \left[ \ln\left(\frac{S_t}{K} + (r - q + \frac{1}{2}\sigma^2)\tau\right) \right] | d_1 = \frac{1}{\sigma\sqrt{\tau}} \left[ \ln\left(\frac{S_t}{K} + \left(r - q + \frac{1}{2}\sigma^2\right)\tau\right) \right] | ||
</math></center> | </math></center> | ||
and | and |
Revision as of 22:35, 17 June 2021
Recall the celebrated Black-Scholes equation
Here
- is a time in years; we generally use as now;
- is the value of the option;
- is the price of the underlying asset at time ;
- is the volatility — the standard deviation of the asset's returns;
- is the annualized risk-free interest rate, continuously compounded;
- is the annualized (continuous) dividend yield.
The solution of this equation depends on the payoff of the option — the terminal condition. In particular, if at the time of expiration, , the payoff is given by , in other words, the option is a European call option, then the value of the option at time is given by the Black-Scholes formula:
where is the forward price and
and