Lesson 4.4: The "No-Free-Lunch" Argument & The Final PDE

Welcome to the final lesson of Module 4. This is the 'summit' of our derivation. We have done all the difficult stochastic calculus, and now all that's left is a simple, elegant economic argument.

In Lesson 4.3, we achieved something incredible. We built a "delta-hedged" portfolio, Π\Pi, and proved that its change, dΠd\Pi, is 100% deterministic and objective.

Our 'Cured' Portfolio (from Lesson 4.3)

Equation (A)

dPi = left( - rac{partial V}{partial t} - rac{1}{2}sigma^2 S_t^2 rac{partial^2 V}{partial S^2} ight) dt

Our portfolio's change is now 100% risk-free. It has no dWtdW_t term and no subjective μ\mu term. It's as safe as a government bond. This simple fact is the key.

Part 1: The "No-Arbitrage" Principle (The "No-Free-Lunch" Rule)

This is the central economic argument of all financial engineering.

Arbitrage is the fancy word for a "free lunch" or a "money pump." It's a trading strategy that gives you a guaranteed profit with zero risk.

The "Two Bank Accounts" Analogy

Imagine you have two bank accounts that are both 100% risk-free (e.g., they are both insured by the government).

  • Bank Account 1: This is a standard savings account. It pays the risk-free interest rate, rr. (Let's say r=5%r = 5\%).
  • Bank Account 2: This is our "magic portfolio" Π\Pi. We just proved it is 100% risk-free.

The Question: What interest rate must our portfolio Π\Pi pay?

The "No-Free-Lunch" Answer: It must pay the same risk-free rate, rr.

Why?

  • If Π\Pi paid more than rr (e.g., 10%), you would perform arbitrage. You'd borrow $1 billion from Bank Account 1 (owing 5%) and put it all in your Π\Pi portfolio (earning 10%). You would make a 5% profit ($50 million) with zero risk. This is a "free lunch," and in a real market, this opportunity would disappear in a nanosecond as everyone did it.
  • If Π\Pi paid less than rr (e.g., 1%), you would do the reverse. You would "short" the Π\Pi portfolio (agreeing to pay 1%) and put all the money in Bank Account 1 (earning 5%). You'd make a 4% risk-free profit.

Conclusion: In a fair, stable market, any two risk-free assets must have the same return. Therefore, our portfolio Π\Pi *must* grow at the risk-free rate, rr.

Part 2: The Derivation (The Final Step)

The "No-Arbitrage" Equation

The change in our portfolio (dΠd\Pi) must be equal to the value of our portfolio (Π\Pi) earning the risk-free rate rr over a tiny time step dtdt.

dΠ=r⋅Π⋅dtd\Pi = r \cdot \Pi \cdot dt

This is Equation (B).

Step 1: Substitute the value of Π

From Lesson 4.1, we know the value of our portfolio is Π=−V+ΔSt\Pi = -V + \Delta S_t.

dΠ=r(−V+ΔSt)dtd\Pi = r \left( -V + \Delta S_t \right) dt

Step 2: Substitute the value of Δ

From Lesson 4.2, we know our "magic hedge" value is Δ=∂V∂S\Delta = \frac{\partial V}{\partial S}.

dΠ=r(−V+(∂V∂S)St)dtd\Pi = r \left( -V + \left(\frac{\partial V}{\partial S}\right) S_t \right) dt

Step 3: Simplify

Distribute the rr and dtdt.

dΠ=(−rV+rSt∂V∂S)dtd\Pi = \left( -rV + r S_t \frac{\partial V}{\partial S} \right) dt

This is our final, simplified version of Equation (B).

Part 3: The Final Equation (Setting A = B)

We have two different, non-random equations for the exact same thing (dΠd\Pi).

From Calculus (Eq. A):

dPi = left( - rac{partial V}{partial t} - rac{1}{2}sigma^2 S_t^2 rac{partial^2 V}{partial S^2} ight) dt

From Economics (Eq. B):

dPi = left( -rV + r S_t rac{partial V}{partial S} ight) dt

The only way both of these can be true is if the parts inside the brackets are equal.

−∂V∂t−12σ2St2∂2V∂S2=−rV+rSt∂V∂S- \frac{\partial V}{\partial t} - \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} = -rV + r S_t \frac{\partial V}{\partial S}

Now, we just rearrange the formula to make it "clean." Let's move all the terms to the right-hand side, which makes them all positive.

0=∂V∂t+rSt∂V∂S+12σ2St2∂2V∂S2−rV0 = \frac{\partial V}{\partial t} + r S_t \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} - rV

The Black-Scholes-Merton PDE (The 'Holy Grail')

∂V∂t+rSt∂V∂S+12σ2St2∂2V∂S2−rV=0\frac{\partial V}{\partial t} + r S_t \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} - rV = 0

Part 4: The "So What?" (We Have Our "Price Tag")

We Have Our "Price Tag"

We started with a random process (dVdV) that was "poisoned" by a subjective guess (μ\mu). By using Itô's Lemma and a "magic" portfolio, we have created a non-random (deterministic) equation that:

  1. Has no dWtdW_t term. The randomness is gone.
  2. Has no μ\mu term. The subjective guess about the stock's return is gone.

This equation gives a single, objective price VV for the option that only depends on things we can all agree on: St,t,r,σS_t, t, r, \sigma.

We have successfully "tamed" randomness and turned a "bet" into a "science."

Translating to "The Greeks" (from Lesson 3.3)

We can rewrite our PDE using the financial "Greek" names for the derivatives:

Θ⏟∂V∂t+rStΔ⏟rSt∂V∂S+12σ2St2Γ⏟12σ2St2∂2V∂S2−rV⏟Risk-free growth=0\underbrace{\Theta}_{\frac{\partial V}{\partial t}} + \underbrace{r S_t \Delta}_{r S_t \frac{\partial V}{\partial S}} + \underbrace{\frac{1}{2}\sigma^2 S_t^2 \Gamma}_{\frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2}} - \underbrace{rV}_{\text{Risk-free growth}} = 0

This equation is a fundamental law of finance. It says that for a perfectly hedged portfolio, the "profit" from Time Decay (Theta) must be exactly balanced by the "costs" and "profits" from Gamma and earning interest.

What's Next?

    We have completed the derivation. We have found the problem we need to solve.

    But this PDE is a complex equation; it's not a number. It's like having the blueprint for a car, but not the car itself.

    Our final module, Module 5, is all about The Solution. How do we solve this PDE to get the final, famous Black-Scholes-Merton formula that gives us an actual dollar price for VV? We will also learn how to use this formula to manage risk by calculating all the "Greeks."

Up Next: Module 5: The Solution & The Greeks