Difference between revisions of "Programming/Kdb/Labs/Option pricing"
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Similarly, if the payoff is given by <math>V(S, T) = P(S, T) =: \max\{K - S, 0\}</math>, in other words, the option is a '''European put option''', then the value of the option at time <math>t</math> is given by the '''Black-Scholes formula for the European put''': | Similarly, if the payoff is given by <math>V(S, T) = P(S, T) =: \max\{K - S, 0\}</math>, in other words, the option is a '''European put option''', then the value of the option at time <math>t</math> is given by the '''Black-Scholes formula for the European put''': | ||
<center><math> | <center><math> | ||
P(S_t, t) = e^{-r\tau} [K N(-d_2) - | P(S_t, t) = e^{-r\tau} [K N(-d_2) - F_t N(-d_1)]. | ||
</math></center> | </math></center> |
Revision as of 22:40, 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 for the European call:
where is the time to maturity, is the forward price, and
and
Similarly, if the payoff is given by , in other words, the option is a European put option, then the value of the option at time is given by the Black-Scholes formula for the European put: