7.2 Black-Scholes Formulae for European Options

In this section we are going to use Black-Scholes' equation to compute the price of European options. We keep the notation introduced in the previous chapter. That is, we denote

$\displaystyle C(S,t) = C_{K,T}(S,t) , \hspace*{2cm} P(S,t) = P_{K,T}(S,t)$

the value of a European call respectively put option with exercise price $ K$ and maturity date $ T$ at time $ t \leq T,$ where the underlying, for example a stock, at time $ t$ has a value of $ S_t=S.$ The value of a call option thus satisfies for all prices $ S$ with $ 0<S<\infty$ the differential equation
$\displaystyle rC(S,t) - bS\frac {\partial C(S,t)}{\partial S}-\frac 12\sigma^2S...
...l^2C(S,t)}{\partial S^2}
=\frac {\partial C(S,t)}{\partial t}, \; 0\leq t\le T,$     (7.17)
$\displaystyle C(S,T)=\max \{0,S-K\},\; 0<S<\infty ,$     (7.18)
$\displaystyle C(0,t)=0, \quad \lim \limits_{S\rightarrow\infty}C(S,t) - S = 0,\; 0\leq t\leq
T.$     (7.19)

The first boundary condition (6.18) follows directly from the definition of a call option, which will only be exercised if $ S_T > K$ thereby procuring the gain $ S_T-K.$ The definition of Brownian motion implies that the process is absorbed by zero. In other words, if $ S_t=0$ for one $ t<T$ it follows $ S_T=0.$ That is the call will not be exercised, which is formulated in the first part of condition (6.19). Whereas the second part of (6.19) results from the reflection that the probability that the Brownian motion falls below $ K$ is fairly small if it attained a level significantly above the exercise price. If $ S_t
\gg K$ for a $ t<T$ then it holds with a high probability that $ S_T
\gg K.$ The call will be, thus, exercised and procures the cash flow $ S_T - K \approx S_T$.

The differential equation (6.17) subject to boundary conditions (6.18). (6.19) can be solved analytically. To achieve this, we transform it into a differential equation known from the literature. First of all, we substitute the time variable $ t$ for the time to maturity $ \tau=T-t.$ By doing this, the problem with final condition (6.18) in $ t=T$ changes to a problem subject to an initial condition in $ \tau=0.$ Following, we multiply (6.17) by $ 2/\sigma^2$ and substitute the parameters $ r, b$ for

$\displaystyle \alpha =\frac {2r}{\sigma^2},~\beta =\frac {2b}{\sigma^2} , $

as well as the variables $ \tau, S$ for

$\displaystyle v=\sigma^2(\beta -1)^2\frac{\tau}{2} , \hspace*{2cm} u=(\beta -1)\ln\frac{S}{K}+v .$

While for the original parameters hold $ 0 \leq S < \infty, 0 \leq
t \leq T,$ for now the new parameters it holds

$\displaystyle -\infty < u < \infty, \hspace*{2cm} 0 \leq v \leq \frac{1}{2} \sigma^2(\beta -1)^2 T \stackrel{\mathrm{def}}{=}v_T . $

Finally, we set

$\displaystyle g(u,v) = e^{r \tau} C(S,T-\tau)$

and obtain the new differential equation

$\displaystyle \frac {\partial ^2g(u,v)}{\partial u^2} = \displaystyle \frac {\partial g(u,v)}{\partial v} .$ (7.20)

with the initial condition

$\displaystyle g(u,0) = K\max \{0,e^{\frac u{\beta -1}}-1\} \stackrel{\mathrm{def}}{=}g_0(u), \quad -\infty <u<\infty .$ (7.21)

Problems with initial conditions of this kind are well known from the literature on partial differential equations. They appear, for example, in modelling heat conduction and diffusion processes. The solution is given by

$\displaystyle g(u,v)=\int \limits_{-\infty}^{\infty} \frac{1}{2\sqrt{\pi v}} g_0(\xi )e^{-\frac{(\xi -u)^2}{4v}}d\xi.$

The option price can be obtained by undoing the above variables and parameter substitutions. In the following we denote, as in Chapter 2, by $ C(S,\tau)$ the call option price being a function of the time to maturity $ \tau = T - t$ instead of time $ t.$ Then it holds

$\displaystyle C(S,\tau)=e^{-r \tau}g(u,v)=e^{-r \tau}\int \limits_{-\infty}^{\infty}
\frac{1}{2\sqrt{\pi v}}g_0(\xi )e^{-\frac{(\xi -u)^2}{4v}}d\xi.$

Substituting $ \xi =(\beta -1)\ln(x/K)$ we obtain the original terminal condition $ \max \{
0,x-K\}.$ Furthermore, replacing $ u$ and $ v$ by the variables $ S$ and $ \tau$ we obtain

$\displaystyle C(S,\tau)$ $\displaystyle =$ $\displaystyle e^{-r\tau}\int \limits_0^{\infty}\max ( 0,x-K)
\frac{1}{\sqrt{2\pi }\sigma
\sqrt{\tau}x}$  
    $\displaystyle \exp\left\{-\frac{[\ln x-\{\ln S +(b-\frac{1}{2}\sigma^2)
\tau\}]^2}{2\sigma ^2 \tau}\right\}dx.$ (7.22)

In the case of Brownian motion $ S_T - S_t$ is lognormally distributed, i.e.  $ \ln (S_T - S_t)$ is normally distributed with parameters $ (b-\frac{1}{2}\sigma^2)\tau$ and $ \sigma ^2 \tau.$ The conditional distribution of $ S_T$ given $ S_t=S$ is therefore lognormal as well but with parameters $ \ln S +
(b-\frac{1}{2}\sigma^2)\tau$ and $ \sigma ^2 \tau.$ However, the integrant in equation (6.22) is except for the term $ \max(0,x-K)$ the density of the latter distribution. Thus, we can interpret the price of a call as the discounted expected option payoff $ \max ( 0,S_T-K),$ which is the terminal condition, given the current stock price $ S:$

$\displaystyle C(S,\tau) = e^{-r\tau} \mathop{\text{\rm\sf E}}[ \max ( 0,S_T-K) \, \vert \, S_t = S] .$ (7.23)

This property is useful when deriving numerical methods to compute option prices. But before doing that, we exploit the fact that equation (6.22) contains an integral with respect to the density of the lognormal distribution to further simplify the equation. By means of a suitable substitution we transform the term in an integral with respect to the density of the normal distribution and we obtain

$\displaystyle C(S,\tau)=e^{(b-r)\tau}S\Phi (y+\sigma\sqrt {\tau}) - e^{-r\tau}K\Phi (y),$ (7.24)

where we use $ y$ as a shortcut for

$\displaystyle y=\frac {\ln\frac SK +(b-\frac {1}{2}\sigma^2)\tau}{\sigma\sqrt {\tau}}.$ (7.25)

$ \Phi $ denotes the standard normal distribution

$\displaystyle \Phi(y)=\frac 1{\sqrt {2\pi}}\int_{-\infty}^ye^{-\frac {z^2} 2}dz .$

Equations (6.24) and (6.25) are called the Black-Scholes Formulae. For the limit cases $ S \gg K$ and $ S=0$ it holds: The corresponding Black-Scholes Formula for the price $ P(S,\tau)$ of a European put option can be derived by solving Black-Scholes differential equation subject to suitable boundary conditions. However, using the put-call parity (Theorem 2.3) is more convenient:

$\displaystyle P(S,\tau)=C(S,\tau)-Se^{(b-r)\tau}+Ke^{-r\tau} .$

From this and equation (6.24) we obtain

$\displaystyle P(S,\tau)=e^{-r\tau}K\Phi (-y)-e^{(b-r)\tau}S\Phi (-y-\sigma\sqrt {\tau}).$ (7.26)

As we see the value of European put and call options can be computed by explicit formulae. The terms in equation (6.24) for, say the value of a call option, can be interpreted in the following way. Restricting to the case of a non dividend paying stock, $ b=r,$ the first term, $ S\Phi (y+\sigma
\sqrt{\tau}),$ represents the value of the stock which the option holder obtains when he decides to exercise the option. The other term, $ e^{-r\tau}K\Phi (y),$ represents the value of the exercise price. The quotient $ S/K$ influences both terms via the variable $ y.$

Deriving Black-Scholes' differential equation we saw in particular that the value of a call option had been duplicated by means of bonds and stocks. The amount of money invested in stocks was $ \frac {\partial C}{\partial S}S$ with $ \frac{\partial C}{\partial
S}$ being the hedge ratio. This ratio, also called delta, determines the relation of bonds and stocks necessary to hedge the option position. Computing the first derivative of Black-Scholes' formula in equation (6.24) with respect to S we obtain

$\displaystyle \frac{\partial C(S,t)}{\partial S}=\Phi (y+\sigma \sqrt{\tau}) .$

Thus the first term in equation (6.24) reflects the amount of money of the duplicating portfolio invested in stocks, the second term the amount invested in bonds.

Since the standard normal distribution can be evaluated only numerically, the implementation of Black-Scholes' formula depending on the standard normal distribution requires an approximation of the latter. This approximation can have an impact on the computed option value. To illustrate we consider several approximation formulae (see for example Hastings (1955))


a.) The normal distribution can be approximated in the following way:

$\displaystyle \Phi (y)\approx 1-(a_1t+a_2t^2+a_3t^3)e^{-\frac {y^2}2},$   where

\begin{displaymath}
\begin{array}{ll}
\displaystyle t= (1+by)^{-1},\quad &b=0.33...
...401209,\quad &a_2=-0.04793922,\\
a_3=0.373927817.&
\end{array}\end{displaymath}

The approximating error is independently of $ y$ of size $ {\mathcal{O}}(10^{-5})$.
8893 SFENormalApprox1.xpl

b.)

$\displaystyle \Phi (y)\approx 1-(a_1t+a_2t^2+a_3t^3+a_4t^4+a_5t^5)e^{-\frac {y^2}2},$   where

\begin{displaymath}
\begin{array}{lll}
\displaystyle t=(1+by)^{-1},\quad &b=0.23...
...70687,\\
a_4=-0.726576013,\quad &a_5=0.530702714.&
\end{array}\end{displaymath}

The error of this approximation is of size $ {\mathcal{O}}(10^{-7})$. 8899 SFENormalApprox2.xpl

c.) An approximation of the normal distribution, with error size $ {\mathcal{O}}(10^{-5})$ is given by:

$\displaystyle \Phi (y)\approx 1-\frac 1{2(a_1t+a_2t^2+a_3t^3+a_4t^4+a_5t^5)^8},$   where

\begin{displaymath}
\begin{array}{lll}
a_1=0.099792714,\quad &a_2=0.044320135,\q...
...699203,\\
a_4=-0.000098615,\quad &a_5=0.00581551.&
\end{array}\end{displaymath}

8905 SFENormalApprox3.xpl

d.) Finally we present the Taylor expansion:

$\displaystyle \Phi (y)$ $\displaystyle \approx$ $\displaystyle \frac 12+\frac 1{\sqrt {2\pi}}\left(y-\frac {y^ 3}{1!2^13}+\frac
{y^5}{2!2^25}-\frac {y^7}{3!2^37}+\cdot\cdot\cdot\right
)$  
  $\displaystyle =$ $\displaystyle \frac 12+\frac 1{\sqrt {2\pi}}\sum_{n=0}^{\infty}(-1)^n\frac {y^{ 2n+1}}{n!2^n(2n+1)}.$  

By means of this series the normal distribution can be approximated arbitrarily close depending on the number of terms used in the summation. Increasing the number of terms increases as well the number of arithmetic operations.
8908 SFENormalApprox4.xpl

Table 6.1 compares all four approximation formulae. The Taylor series was truncated after the first term whose absolute value is smaller than $ 10^{-5}.$ The last column shows the number of terms used.

Table 6.1: Several approximations to the normal distribution
$ x$ norm-a norm-b norm-c norm-d iter
1.0000 0.8413517179 0.8413447362 0.8413516627 0.8413441191 6
1.1000 0.8643435425 0.8643338948 0.8643375717 0.8643341004 7
1.2000 0.8849409364 0.8849302650 0.8849298369 0.8849309179 7
1.3000 0.9032095757 0.9031994476 0.9031951398 0.9031993341 8
1.4000 0.9192515822 0.9192432862 0.9192361959 0.9192427095 8
1.5000 0.9331983332 0.9331927690 0.9331845052 0.9331930259 9
1.6000 0.9452030611 0.9452007087 0.9451929907 0.9452014728 9
1.7000 0.9554336171 0.9554345667 0.9554288709 0.9554342221 10
1.8000 0.9640657107 0.9640697332 0.9640670474 0.9640686479 10
1.9000 0.9712768696 0.9712835061 0.9712842148 0.9712839202 11
2.0000 0.9772412821 0.9772499371 0.9772538334 0.9772496294 12



Table: Prices of a European call option for different approximations of the normal distribution 8913 SFEBSCopt1.xpl
Stock price $ S_t$   230.00 EUR  
Exercise price $ K$   210.00 EUR  
Time to maturity $ \tau = T - t$   0.50000    
Continuous interest rate $ r$   0.04545    
Volatility $ \sigma$   0.25000    
No dividends        
  norm-a norm-b norm-c norm-d
Option prices 30.74262 30.74158 30.74352 30.74157


Table 6.2 shows the price of a particular European call option computed by means of the four approximations presented above.