Difference between revisions of "Programming/Kdb/Labs/Option pricing"

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* <math>q</math> is the annualized (continuous) dividend yield.
* <math>q</math> is the annualized (continuous) dividend yield.


The solution of this equation depends on the '''payoff''' of the option &mdash; the terminal condition. In particular, if at the time of expiration, <math>T</math>, the payoff is given by <math>V(S, T) = C(S, T) =: \max\{S - K, 0\}</math>, in other words, the option is a '''European call option''', then the value of the option at time <math>t</math> is given by the '''Black-Scholes formula'''
The solution of this equation depends on the '''payoff''' of the option &mdash; the terminal condition. In particular, if at the time of expiration, <math>T</math>, the payoff is given by <math>V(S, T) = C(S, T) =: \max\{S - K, 0\}</math>, in other words, the option is a '''European call option''', then the value of the option at time <math>t</math> is given by the '''Black-Scholes formula''':
<center><math>
C(S_t, t) = e^{-r\tau} [F_t N(d_1) - K N(d_2)]
</math></center>
where <math>F = S_t e^{(r - q)\tau}</math> is the forward price and
<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]
</math></center>
and
<center><math>
d_2 = d_1 - \sigma\sqrt{\tau}.
</math></center>

Revision as of 22:34, 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

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and