Lesson 6.5: Fixing r - Stochastic Interest Rates (Vasicek & CIR)

Welcome to Lesson 6.5, the final lesson of our 'Advanced Models' module. We have successfully 'fixed' two of the three major bugs in the original Black-Scholes model:

We fixed constant volatility ( $\sigma$ ) with the Heston Model (Lesson 6.3).
We fixed continuous paths with the Merton Jump-Diffusion Model (Lesson 6.4).

But both of those models still have one fundamental flaw: they assume the risk-free interest rate (r) is a constant, known number (e.g., 5% forever).

The Third "Bug": Constant Interest Rates

This assumption is a catastrophic failure for the largest financial market on Earth: Fixed Income (Bonds).

A "zero-coupon bond" is a simple contract: "I will pay you \$1 at time $T$." Its price is the present value of that \$1. In a constant-r world, its price today is just:

P(t, T) = e^{-r(T-t)}

If $r$ is constant, this formula's value never changes (except for a tiny, predictable "roll" as $t$ moves). This model predicts that bond prices are not volatile.

This is completely wrong. In the real world, bond prices are extremely volatile, because interest rates themselves are volatile. To price bonds, we must give $r$ its own SDE.

Part 1: The "Fix" (The "Thermostat" Analogy)

What kind of SDE should the interest rate ( $r_t$ ) follow?

Can it be GBM (like a stock)? No. A stock can (in theory) grow to infinity. Interest rates can't. If $r_t$ went to 50%, the economy would collapse. Rates tend to be "pulled back" to some long-term average.

This "pull-back" effect is called Mean Reversion. This is the exact same concept we used for volatility in the Heston model.

The "Dog on a Leash" Analogy:

$\theta$ (theta): A stake in the ground. This is the "long-term average rate" (e.g., 3%).
$r_t$ : A dog on a leash, wandering randomly. This is the "current short rate."
$\kappa$ (kappa): The "stiffness of the leash." This is the "speed of mean reversion."

When the dog ( $r_t$ ) wanders too far from the stake ( $\theta$ ), the leash ( $\kappa$ ) pulls it back. The SDE for this process is called an Ornstein-Uhlenbeck process.

Part 2: Model 1 - The Vasicek Model (1977)

The first and simplest SDE for interest rates was the Vasicek Model. It is an Ornstein-Uhlenbeck process.

The Vasicek Model SDE

dr_t = \kappa (\theta - r_t) dt + \sigma dW_t

Let's translate this. The change in the interest rate ( $dr_t$ ) is the sum of two parts:

$\kappa (\theta - r_t) dt$ (The "Drift" / "The Pull"):
- $\theta$ is the long-term mean (the "stake," e.g., 3%).
- $r_t$ is the current rate.
- $(\theta - r_t)$ is the "error." If $r_t$ is 5% (it's "too high"), this term is negative, and the drift pulls the rate down.
- $\kappa$ is the speed of that pull.
$\sigma dW_t$ (The "Diffusion" / "The Jiggle"):
- $\sigma$ is a constant volatility (e.g., 0.5%).
- $dW_t$ is the random engine.

The "So What?" (Pros vs. Cons):

Pro: It's simple. Because $\sigma$ is constant, this is a Gaussian Process (like our original "Failed Model #2" for stocks). This makes it mathematically easy to solve, and we can get a clean, simple formula for bond prices.
Con (The Big Bug): It has the exact same bug as "Failed Model #2." The jiggle ( $\sigma$ ) is constant. If the rate is $r_t = 0.1\%$ and it gets a big random shock (a "3-sigma event" of, say, -0.6%), the new rate becomes 0.1% - 0.6% = -0.5%.

This is a huge problem. Negative interest rates are (mostly) unrealistic and can break many financial models.

Part 3: Model 2 - The CIR Model (The "Better" Fix)

How do we "fix" the negative rate bug? We need to use the exact same trick we used in the Heston model: make the "jiggle" proportional to the rate itself.

This is the Cox-Ingersoll-Ross (CIR) Model (1985).

The CIR Model SDE

dr_t = \kappa (\theta - r_t) dt + \sigma \sqrt{r_t} dW_t

This is a tiny change, but it has a massive impact.

The Drift: $\kappa (\theta - r_t) dt$
- This is identical to the Vasicek model. It's still a "thermostat" pulling the rate back to its long-term average $\theta$ .
The Diffusion: $\sigma \sqrt{r_t} dW_t$
- This is the "magic" fix. The "jiggle engine" $\sigma dW_t$ is multiplied by $\sqrt{r_t}$ .
- The "Aha!" Moment: As the interest rate $r_t$ gets closer and closer to 0, the term $\sqrt{r_t}$ also gets closer to 0.
- This means the random "jiggle" itself runs out of fuel as it approaches the zero boundary. The volatility of the rate shrinks as the rate shrinks. This creates a "natural barrier" that (if $\kappa$ and $\theta$ are high enough) makes it impossible for the rate to become negative.

The "So What?" (Pros vs. Cons):

Pro: It fixes the negative rate problem. It's a much more realistic and stable model.
Con: It's harder. It's a "square-root process" (non-Gaussian), so the "pure math" formulas for pricing bonds are much more complex. (But for Monte Carlo, it's just as easy! You just code this new SDE).

Part 4: The "Real" Payoff (How to Price a Bond)

Now that we have a random SDE for our interest rate $r_t$ , how do we actually price a zero-coupon bond?

A zero-coupon bond (ZCB) is a contract that pays you \$1 at a future time $T$. Its price today is $P(t, T)$.

We must use Risk-Neutral Valuation (Lesson 6.1). The price today is the "expected payoff" discounted back.

But what's the discount factor? It's not $e^{-r(T-t)}$ anymore, because $r$ is constantly jiggling between $t$ and $T$. The total discount is the "sum" (or integral) of all the random rates along the path:

\text{Total Discount Factor} = e^{-\int_t^T r_s ds}

This is a random variable! The total discount factor is not known today.

Therefore, the bond's price today must be the expected value (the average) of all possible discount factors that could happen, calculated in the risk-neutral world ( $\mathbb{Q}$ ).

The "Holy Grail" of Fixed Income

This is the fundamental formula for all bond pricing:

P(t, T) = \mathbb{E}^{\mathbb{Q}} \left[ e^{-\int_t^T r_s ds} \mid \mathcal{F}_t \right]

Deconstructing this "Scary" Formula

This formula is terrifying for a beginner. Let's translate it piece by piece.

$\int_t^T r_s ds$ : This is the "Total Accumulated Interest" along one specific random path that the interest rate $r_s$ takes from now ($t$) until the bond matures ($T$). Since the path is random, this total interest is a random number.
$e^{-\int_t^T r_s ds}$ : This is the "Total Discount Factor" for that one specific path. This is also a random number.
$\mathbb{E}^{\mathbb{Q}} \left[ \dots \right]$ : This is the "Expected Value" (the average). This is where Monte Carlo (Lesson 6.2) comes in. We run thousands of possible $r_s$ paths:
- Path 1 (Rates spike): Total Discount Factor is $0.85$ (worth 85 cents)
- Path 2 (Rates stay low): Total Discount Factor is $0.95$ (worth 95 cents)
- Path 3 (Rates crash): Total Discount Factor is $0.99$ (worth 99 cents)
- ...
$P(t, T)$ (The Price): The bond's price today is the average of all those random discount factors. This is why bond pricing is so computationally intensive. This is what "Fixed Income Quants" do all day.

Congratulations! (The End of the Main Course)

You have officially completed the core curriculum of Stochastic Calculus for Quant Finance.

We started with a single, "broken" idea (the bell curve) and ended by modeling the three most important, complex, and random things in finance.

Let's review the "bugs" in the original Black-Scholes model and how we FIXED them:

The Bug: $\sigma$ (volatility) is constant.
The Fix (Lesson 6.3): Stochastic Volatility (Heston Model). We gave $\sigma_t$ its own mean-reverting SDE. This fixed the "volatility smile."
The Bug: The stock path $S_t$ is continuous (no jumps).
The Fix (Lesson 6.4): Jump-Diffusion (Merton Model). We added a second random engine (a Poisson Process, $dN_t$ ) to our SDE to model "crashes." This fixed "fat tails."
The Bug: $r$ (interest rate) is constant.
The Fix (Lesson 6.5): Stochastic Interest Rates (Vasicek/CIR). We gave $r_t$ its own mean-reverting SDE. This allowed us to price bonds.

You now have a complete, professional-level "toolkit" for modeling any random process. You've seen that SDEs are just "building blocks" that allow quants to build models that get closer and closer to reality.

Lesson 6.4: Fixing "No Jumps" - Jump-Diffusion (The Merton Model)