We consider the classical Black-Scholes model with single risky asset that follows a geometric Brownian motion:

$d{S}_t\={\mathrm{rS}}_t\mathrm{dt}+{\mathrm{\σS}}_t{\mathrm{dW}}_t\,$$t\ge 0\,$

where ($W_t\,t\ge 0$) is a standard Brownian motion, $\sigma \ge 0$ is the constant volatility, $r\ge 0$ is the constant risk-free rate and $S_0\ge 0$ is the initial asset price.

Under these conditions, for any $t\ge 0\,$ the stock price $S_t$ is given by the following formula:

$S_t\=S_0{e}^{\left(r-\frac{{\sigma }^2}{2}\right)t+{\mathrm{\σW}}_t}$

We consider a security with time to maturity $T$ and the payoff function:

$P:=\left(S\,K\right)\to \{\begin{array}{cc}1& S\>K\\ 0& S\le K\end{array}\:$

Payoff of the form $P\left(S\right)\=\{\begin{array}{cc}1& S\>K\\ 0& S\le K\end{array}$ corresponds to a digital call options with strike price, $K$.

We will consider several methods for computing the price of this security.

Parameters

$\mathrm{\_EnvStatisticsRandomVariableName}≔\mathrm{\Φ}\:$

$U≔\mathrm{Statistics}:-\mathrm{RandomVariable}\left(\mathrm{LogNormal}\left(\left(r-\frac{{\mathrm{\σ}}^2}{2}\right)\cdot T\, \mathrm{\σ}\cdot \sqrt{T}\right)\right)\:$

${S}_T$ can be represented in the form:

$S_T≔S_0\cdot U\:$

where $\mathrm{\Phi }$ is a lognormal random variable with parameters $T\left(r-\frac{{\sigma }^2}{2}\right)$ and $\sigma \sqrt{T}$.

The price of this option can be computed as the discounted expected payoff of the option:

${\ⅇ}^{-rT}𝔼\(P\(S_T\)\)$

$\mathrm{P\_analytic}≔P\left(S_T \, K\right)$

$\mathrm{V\_analytic}≔{\ⅇ}^{\left(-r\cdot T\right)}\cdot \mathrm{Statistics}:-\mathrm{ExpectedValue}\left(\mathrm{P\_analytic}\right) assuming r \> 0\, \mathrm{\σ} \> 0\, S_0\>0\, K\>0\, T\>0$

Analytic Price

We can use the analytic result to study the various market sensitivities. For example, we can symbolically compute the delta of our option.

$\mathrm{\Δ}≔\mathrm{diff}\left(\mathrm{V\_analytic}\, S_0\right)$

Here is a formula for the Gamma:

$\mathbf{local} \mathrm{\Gamma } ≔ \mathrm{factor}\left(\mathrm{diff}\left(\mathrm{\Delta }\, S_0\right)\right)$

We can also use the symbolic formula to plot the option price as a function of the parameters.

Alternatively, we can estimate the expectation using Monte Carlo simulation to compute the option price.

The discrete-time version of the model is:

$S_{t+h}\=S_t\cdot \mathrm{\Φ}$

where $h\=\frac{1}{N}$, and $\mathrm{\Φ}$ is drawn from the lognormal distribution with parameters $\left(r-\frac{s^2}{2}\right)h$ and $\mathrm{\sigma }\sqrt{h}$.

We can use this expression to generate a sample path for the price of our risky asset.

Simulating Stock Prices

Note: We know the distribution of the final stock price. To compute the option price, we need only to simulate the final stock price, and not the whole stock path.

We can verify the above analytic result using Monte Carlo simulation.

Monte Carlo Simulation

Finally, we can use the Black-Scholes differential equations to compute the option price.

$\mathrm{DE}≔\mathrm{diff}\left(V\left(S\,t\right)\,t\right)\+r\cdot S\cdot \mathrm{diff}\left(V\left(S\,t\right)\,S\right)\+\frac{1}{2}\cdot {\mathrm{\σ}}^2\cdot S^2\cdot \mathrm{diff}\left(V\left(S\,t\right)\,S\,S\right)\=r\cdot V\left(S\,t\right)$

The key boundary condition is:

$\mathrm{BC1}≔V\left(S\,T\right)\=P\left(S\,K\right)$

Another obvious condition:

$\mathrm{BC2}≔V\left(0\,t\right)\=0$

Finally, if $S_t\gg K$ for some $t<T$, then it holds with a high probability that $S_T\gg K$. Our option will thus be exercised and produce cash flow ${P\&lpar;S}_T-K\&rpar;\mathrm{\&approx;P}\left(S_T\right)$

$\mathrm{BC3}≔V\left(2\cdot 100\&comma;t\right)\&equals;1$

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