Lesson 6.2: The "Computer Way" - Monte Carlo Methods

Our new "shortcut" formula, $V = \mathbb{E}^{\mathbb{Q}}[\dots] \cdot (\text{discount})$, just replaced one hard problem (solving a PDE) with a *new* hard problem: How do we calculate the "Expected Payoff" $\mathbb{E}^{\mathbb{Q}}[\dots]$?

• For a Simple Option: (like a standard call, $\text{Payoff} = \max(S_T - K, 0)$) This is *possible*, but it's a very hard PhD-level statistics problem. We (as Black and Scholes) "solved" it, and the answer is the Black-Scholes formula.
• For a Complex Option: What about an **"Asian Option"**? This is a very common option whose payoff depends on the *average* stock price over its whole life.
  $$\text{Payoff} = \max\left( \text{Average}(S_t) - K, 0 \right)$$
  How do we calculate the "expected average price" of a random path?

At this point, "pure math" fails.

1. The PDE Method (Module 4) is broken. The payoff doesn't just depend on $S_t$ and $t$. It depends on the *entire path history*. This "path-dependency" makes the PDE unsolvable.
2. The RNV Method (Lesson 6.1) is also broken. There is no clean, elegant formula for $V$. We can't "solve" for $\mathbb{E}^{\mathbb{Q}}[\text{Average}(S_t)]$.

We are stuck. We have a beautiful theory, but we can't get a price.

Let's go back to Module 0. How do you find the "expected value" (the average) of a simple 6-sided die?

• Method 1 (Pure Math): You know the "formula" for the outcomes.
  $$\mathbb{E}[\text{Die Roll}] = (1 + 2 + 3 + 4 + 5 + 6) \cdot \frac{1}{6} = 3.5$$
  This is what we did for the Black-Scholes formula. It's clean, but only works for simple problems.
• Method 2 (Brute Force): What if you didn't know the "formula"? You could just:
  1. Roll the die 1,000,000 times.
  2. Write down every result: $\{3, 1, 6, 4, 1, 5, \dots\}$
  3. Calculate the simple *average* of that list.
  Because of the Law of Large Numbers, this "brute force" average will get *extremely* close to the true, pure-math answer of 3.5.

This is the "wow" moment. **We can do the exact same thing for our "impossible" Asian Option!**

We don't need to *solve* the "expected value" formula. We can *simulate* it 1,000,000 times and find the average. This "brute force" simulation is the **Monte Carlo Method**.

This is the "how-to" guide. This is what quants *actually* code.

Our Goal: Price our "Asian Option" $\to V = e^{-r(T-t)} \cdot \mathbb{E}^{\mathbb{Q}}[ \max(\text{Avg}(S) - K, 0) ]$

Step 1: Get the "Magic" SDE (The Risk-Neutral SDE)

We *must* run our simulation in the "magic" risk-neutral world (from Lesson 6.1), where the drift $\mu$ is replaced by the risk-free rate $r$. Our stock model is:

Step 2: Discretize the Path (The "Random Walk" Formula)

We can't code a *continuous* path. We have to make it a step-by-step "random walk" (like our Drunkard from Lesson 1.3). This is called the **Euler-Maruyama method**.

We know from Lesson 1.3 that the "typical size" of $dW_t$ is $\sqrt{dt}$. So, we can *replace* the abstract $dW_t$ with a concrete, code-able formula:

where $\Delta t$ is our tiny time-step (e.g., 1 day) and $Z$ is a random number from $\mathcal{N}(0, 1)$ (the standard bell curve, which computers are great at generating).

Let's plug this into our SDE:

This is our **coding formula**. It tells us how to get from $S_t$ to $S_{t+\Delta t}$.

Step 3: Simulate one *full path* (Path 1)

Let's say $S_0 = 100$, $K=100$, $r=5\%$, $\sigma=20\%$, $T=1$ year, and we use 252 steps (trading days).

• $S_0 = 100$
• For $S_1$: Generate a random $Z_1$ (e.g., +0.5). Calculate $S_1 = S_0 + rS_0\Delta t + \sigma S_0 Z_1 \sqrt{\Delta t}$.
• For $S_2$: Generate a *new* random $Z_2$ (e.g., -1.2). Calculate $S_2 = S_1 + rS_1\Delta t + \sigma S_1 Z_2 \sqrt{\Delta t}$.
• ...repeat 252 times...
• We now have one full, random path: $\{100, 101.2, 99.8, 103.4, \dots, 114.5\}$

Step 4: Calculate the Payoff for Path 1

We have our path. Now we calculate its *average* price.

Now, we calculate the payoff for this one path:

We store this number.

Step 5: Repeat (The "Monte Carlo" part)

Now, we do Steps 3 and 4 again, 1,000,000 times. We get a giant list of 1,000,000 payoffs:

Step 6: Find the "Expected Value"

By the Law of Large Numbers, the "expected value" is just the *simple average* of this giant list.

Step 7: Get the Final Price (Discount)

This $4.32 is the average payoff in the *future*. We need its value *today*. We just discount it using our "Finance 101" formula.

We've done it. We've found the fair price for our "impossible" option.

This "brute force" computer method is the *standard* for pricing complex derivatives at every major bank.

• Pro 1: It can price *anything*.Payoff depends on the average? No problem. Payoff depends on the maximum price? No problem. Payoff depends on 3 different stocks and the weather in London? As long as you can *write the code for the payoff*, you can price it.
• Pro 2: It works with *complex models*.The PDE method only works for our simple GBM model. What if $\sigma$ is *also* random (our next lesson)? For Monte Carlo, this is *easy*! Your code just gets one more line:
  1. Simulate the random $\sigma_t$ for this step.
  2. Simulate the random $S_t$ for this step, *using* the $\sigma_t$ you just found.
  This flexibility is why MC is the default tool for modern quants.
• Con 1: It's slow. It requires massive computing power to run millions of paths.
• Con 2: It's an *approximation*. The PDE gives an *exact* answer. MC just gives an average that gets *closer* to the true price the more paths you run.