Black–Scholes Call Pricing Formula and a Missing Discount Factor
Summary
The answer explains that the questioned expression is a European call price under geometric Brownian motion with constant volatility and risk-neutral drift, rather than a special result from the Heston model. It relates the complementary error function in the expression to the standard normal cumulative distribution, recovering the familiar Black–Scholes form. The drift may incorporate the risk-free rate and dividend yield under the stated setup.
It also identifies a missing discount factor in the source formula: the strike-related payoff must be discounted over the time to maturity. The corrected expression discounts the call value and uses two standardized terms, one for the asset contribution and one for the strike contribution. The response points readers toward standard continuous-time finance texts for a fuller derivation, but does not provide the requested step-by-step derivation. Its correction assumes the model's constant-coefficient, risk-neutral pricing framework; it is not itself a Heston-model derivation.
Key ideas
- The expression is a European call valuation under constant-coefficient geometric Brownian motion.
- The risk-neutral drift for a dividend-paying stock can be expressed using the risk-free rate and dividend yield.
- The complementary error function terms convert to standard normal cumulative probabilities.
- The stated formula needs discounting over the period to maturity.
- The result belongs to the Black–Scholes framework and does not derive a Heston pricing equation.
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# Analytical Solution for Heston Model
# Analytical Solution for Heston Model
What is the rationale of equation 7 in this paper? Could you please provide a step-by-step demonstration of this equality?
## Answer by LocalVolatility (score 5, accepted)
https://quant.stackexchange.com/a/31923
This equation is unrelated to the Heston model. It is simply the value of a European call under the a constant coefficient geometric Brownian motion, i.e. the Black and Scholes (1973) model. Here $\nu$ is the constant volatility and $\mu$ is the risk-neutral drift of the asset. For a stock you could for example have $\mu = r - q$ where $r$ is the risk-free interest rate and $q$ is the dividend yield.
You find a derivation of this formula in almost any introductory book on continuous time finance - e.g. Shreve's "Continuous Time Finance II" or Musiela and Rutkowksi's "Martingale Methods in Financial Modelling".
Note that there is one mistake in the formula: the authors forget to discount the call price. In their notation it should read
\begin{equation} C_0 = \frac{1}{2} \color{red}{e^{-r \left( T - t_0 \right)}} \left( S_0 e^{\mu \left( T - t_0 \right)} \mathrm{erfc} \left( -\frac{d_+}{\sqrt{2}} \right) - K \mathrm{erfc} \left( -\frac{d_-}{\sqrt{2}}\right) \right), \end{equation}
where
\begin{equation} d_\pm = \frac{1}{\sqrt{\nu \left( T - t_0 \right)}} \left( \ln \left( \frac{S_0}{K} \right) + \left( \mu \pm \frac{1}{2} \nu \right) \left( T_0 - t \right) \right) \end{equation}
Usually you find this formula expressed in terms of the cumulative normal distribution function $\mathcal{N}(x)$ instead of the complementary error function $\mathrm{erfc}(x)$. The connection between the two is
\begin{eqnarray} \frac{1}{2} \mathrm{erfc} \left( -\frac{d_\pm}{\sqrt{2}} \right) & = & \frac{1}{2} \left( 1 - \mathrm{erf} \left( -\frac{d_\pm}{\sqrt{2}} \right) \right)\\ & = & \frac{1}{2} \left( 1 + \mathrm{erf} \left( \frac{d_\pm}{\sqrt{2}} \right) \right)\\ & = & \mathcal{N} \left( d_\pm \right). \end{eqnarray}
You then get the more familiar expression
\begin{equation} C_0 = e^{-r \left( T - t_0 \right)} \left( S_0 e^{\mu \left( T - t_0 \right)} \mathcal{N} \left( d_+ \right) - K \mathcal{N} \left( d_- \right) \right). \end{equation}Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.