Black-Scholes

Assume the stock price $S_t$ is an Ito-process, satisfying Geometric Brownian Motion dynamics.

$d{S}_t=\mu S_{t}dt+\sigma S_{t}d{B}_t$

Using Ito’s lemma, the natural logarithm of $S_t$ satisfies Arithmetic Brownian motion dynamics:

$dln(S_t)=(\mu -\frac{{\sigma }^2}{2})dt+\sigma d{B}_t$

We can solve this without Ito Integrals because the coefficients of the infinitesimals are constant. The solution is: $S_t=S_0\exp \{(\mu -\frac{{\sigma }^2}{2})t+\sigma B_t\}$

Implementation

When implementing this for option pricing, recall the discounted risk-neutral payout expectation formula, where the option payout at time $t$ is a function of the realized path of the stock process up to time $t$:

$V_0=e^{-rT}{\mathbb{E}}^{\mathbb{Q}}(\text{payout}((S_t{)}_{t\in [0,T]}))$

We can use this closed form solution along with the discounted risk-neutral to construct a MC Algorithm

Also under $\mathbb{Q}$, $\mu =r$ in our GBM solution (Not sure how to prove this), so we can say: $S_t=S_0\exp \{(r-\frac{{\sigma }^2}{2})t+\sigma B_t\}$

Black Scholes MC Algorithm

Given $n$: number of simulation runs, $N:$ number of time discretizations suth that for a maturity time horizon $T$, $T/N=\Delta t$.

1. Discretize time s.t. we only simulate at $0<t_1<t_2<\cdots <t_N$
2. For $i=1,\dots n$ (number of simulation runs):
   • Sample a discretized path $S_1,\dots S_N$ corresponding to $t_1,\dots t_N$.
   • Compute the payout in simulation run $i$, say $V_N^j$. Unlike with binomial models we don’t need to know $V_t$ for $t<N$.
3. Return the estimate ${\hat{V}}_0=e^{-rT}\frac{1}{n}{V}_N^i$ (the discounted expected value of the average payout at maturity.)

This can be used for path-dependent (Asian) or independent options (European, rainbow?) but, as the algorithm is, the options should be exercised at maturity i.e. no american-type options (you probably just modify it a bit like with algo 2.3)

Sources of error for Monte Carlo:

• Discretization error as we sample only at $t_1<t_2<\cdots <t_N$ and not continuosuly (impossible on a finite resource computer)
• Sampling error or MC error, due to only sampling $n$ paths. We can’t ever sample all uncountably many paths.