Applications to Option Pricing¶
The Feynman-Kac formula transforms every option pricing problem into a PDE, and every PDE back into an expectation. This page applies the framework to concrete financial derivatives: European options (the Black-Scholes formula as a direct Feynman-Kac computation), barrier options (Feynman-Kac on restricted domains), and the connection to American options (free boundary problems and variational inequalities).
Related Content
- Feynman-Kac Formula -- the general theorem
- Discounted Feynman-Kac -- the discounting mechanism
- Proof Sketch -- the martingale argument
European Options: Black-Scholes via Feynman-Kac¶
Setup¶
Under the risk-neutral measure \(\mathbb{Q}\), the stock price follows:
A European derivative with payoff \(g(S_T)\) at maturity \(T\) has price:
By the Feynman-Kac formula, \(V\) satisfies:
Deriving the Black-Scholes Formula¶
For a European call with payoff \(g(S) = (S - K)^+\), we compute the expectation directly.
Step 1: Under \(\mathbb{Q}\), \(S_T = S\exp\!\left((r - \sigma^2/2)(T-t) + \sigma\sqrt{T-t}\,Z\right)\) where \(Z \sim N(0,1)\).
Step 2: The discounted expectation is:
Step 3: The integrand is positive when \(z > z^*\), where:
Step 4: Split and evaluate:
where:
Verification
One can verify that \(V = S\Phi(d_1) - Ke^{-r(T-t)}\Phi(d_2)\) satisfies the Black-Scholes PDE by direct differentiation. The computation is involved but straightforward. The key identities are:
European Put¶
By the same method (or by put-call parity):
Put-call parity (a model-independent no-arbitrage relation):
Digital (Binary) Option¶
Payoff \(g(S) = \mathbf{1}_{\{S > K\}}\) (pays \(\$1\) if in the money):
Barrier Options via Feynman-Kac¶
Barrier options modify the domain of the PDE by introducing absorbing boundaries. The Feynman-Kac formula applies to the killed diffusion -- the process conditioned on not hitting the barrier.
Down-and-Out Call¶
Payoff: \((S_T - K)^+\) if \(\min_{t \leq s \leq T} S_s > B\) (barrier \(B < K\)).
Feynman-Kac formulation: The price satisfies the Black-Scholes PDE on the restricted domain \(S > B\):
Probabilistic representation:
where \(\tau_B = \inf\{s > t : S_s \leq B\}\) is the first hitting time of the barrier.
Closed-Form Solution¶
The reflection principle for Brownian motion (after the log-transformation \(X = \log S\)) gives:
where \(V_{\text{BS}}\) is the standard Black-Scholes price. The second term is the correction for paths that cross the barrier.
In-Out Parity
For any barrier option:
This is an immediate consequence of the partition of paths into those that hit the barrier and those that do not.
Up-and-Out Put¶
Domain: \(S < B\) with \(V(t, B) = 0\).
The PDE is the same Black-Scholes equation, but now solved on \((0, B)\) instead of \((0, \infty)\).
Double Barrier Options¶
With barriers \(B_l < S < B_u\), the PDE domain is the finite interval \((B_l, B_u)\):
The Green's function on this bounded domain can be expressed via the spectral decomposition (eigenfunction expansion) or the method of images (infinite series of reflected Gaussians).
Connection to American Options¶
American options can be exercised at any time \(t \leq T\), which fundamentally changes the mathematical structure. The Feynman-Kac formula does not directly apply because the exercise time is a stopping time chosen by the holder.
The Optimal Stopping Formulation¶
The American option price is:
where the supremum is over all stopping times \(\tau\).
The Variational Inequality¶
Instead of a PDE with equality, the American option satisfies a variational inequality:
This encodes two complementary conditions:
- In the continuation region (\(V > g\)): \(\partial_t V + \mathcal{L}V - rV = 0\) (the standard Feynman-Kac PDE holds)
- In the exercise region (\(V = g\)): The option is exercised immediately
The Free Boundary¶
The optimal exercise boundary \(S^*(t)\) separates the two regions:
- For an American put: exercise when \(S \leq S^*(t)\) (put is deep in the money)
- For an American call on a dividend-paying stock: exercise when \(S \geq S^*(t)\)
At the free boundary, two matching conditions hold:
Early Exercise Premium
The American option price decomposes as:
The early exercise premium is the additional value from the right to exercise before maturity. For a non-dividend-paying call, this premium is zero (it is never optimal to exercise early), so \(V^{\text{Am}} = V^{\text{Eu}}\).
Exotic Options and Extensions¶
Asian Options¶
The payoff depends on the average price: \(g = \left(\frac{1}{T}\int_0^T S_s\,ds - K\right)^+\).
The average \(A_t = \frac{1}{t}\int_0^t S_s\,ds\) is not Markovian in \(S_t\) alone. To apply Feynman-Kac, introduce \(A_t\) as an additional state variable:
This is a 2D PDE in \((S, A)\) -- no closed-form solution exists in general.
Lookback Options¶
The payoff depends on the running maximum: \(g = S_T - \min_{0 \leq s \leq T} S_s\).
Again, the state space must be augmented with the running minimum \(M_t = \min_{s \leq t} S_s\):
with the boundary condition \(\partial_M V(t, S, M)\big|_{M=S} = 0\).
Options on Multiple Assets (Rainbow Options)¶
For a basket of \(d\) assets with correlated dynamics:
the pricing PDE is \(d\)-dimensional:
For \(d > 3\), finite difference methods become impractical (curse of dimensionality), and Monte Carlo -- justified by the Feynman-Kac expectation representation -- becomes the method of choice.
The PDE-Monte Carlo Duality¶
The Feynman-Kac formula establishes a computational duality: every option pricing problem can be attacked from either the PDE side or the expectation (Monte Carlo) side.
| Approach | Method | Best For |
|---|---|---|
| PDE | Finite differences on the PDE | Low-dimensional, full grid of prices, Greeks |
| Monte Carlo | Simulate paths, average payoff | High-dimensional, path-dependent, complex payoffs |
| Analytical | Solve PDE or evaluate expectation in closed form | When possible (GBM, affine models) |
The Feynman-Kac Principle in Practice
When designing a pricing method:
- Write the PDE (identify the generator, discounting, boundary conditions)
- Write the expectation (identify the SDE, discount factor, payoff)
- Choose the more efficient approach based on dimensionality and complexity
Example: Complete Pricing Workflow¶
Problem¶
Price a European put with \(S_0 = \$100\), \(K = \$100\), \(r = 5\%\), \(\sigma = 20\%\), \(T = 1\) year.
Feynman-Kac PDE¶
Terminal: \(V(1, S) = (100 - S)^+\). Boundary: \(V(t, 0) = 100\,e^{-0.05(1-t)}\), \(V(t, \infty) = 0\).
Feynman-Kac Expectation¶
Analytical Solution (Black-Scholes)¶
Summary¶
| Derivative | PDE Domain | Boundary Conditions | Feynman-Kac Representation |
|---|---|---|---|
| European | \((0, \infty)\) | Far-field asymptotics | \(e^{-r\tau}\,\mathbb{E}[g(S_T)]\) |
| Down-out barrier | \((B, \infty)\) | \(V(t, B) = 0\) | \(e^{-r\tau}\,\mathbb{E}[g(S_T)\,\mathbf{1}_{\tau_B > T}]\) |
| Double barrier | \((B_l, B_u)\) | \(V = 0\) at both | Killed diffusion on bounded domain |
| American | \((0, \infty)\) with free boundary | Smooth pasting | \(\sup_\tau \mathbb{E}[e^{-r\tau}g(S_\tau)]\) |
| Asian | \((S, A)\) augmented | --- | Path-dependent average |
The Feynman-Kac formula unifies all of derivatives pricing under a single mathematical framework. European options give the simplest application, barrier options demonstrate the power of domain restrictions, and American options show the limits of the standard formula -- requiring the extension to optimal stopping and variational inequalities.
See Also¶
- Feynman-Kac Formula -- the foundational theorem
- Discounted Feynman-Kac -- the discounting mechanism
- Proof Sketch -- the martingale derivation
- Boundary Value Problems -- types of boundary conditions
- Free vs Bounded Domains -- Green's functions with barriers
Exercises¶
Exercise 1. For a European put option with payoff \(g(S) = (K - S)^+\), write the Feynman-Kac probabilistic representation and the corresponding Black-Scholes PDE with terminal and boundary conditions. What are the boundary conditions as \(S \to 0\) and \(S \to \infty\)?
Solution to Exercise 1
Feynman-Kac probabilistic representation: For a European put with payoff \(g(S) = (K - S)^+\):
Black-Scholes PDE:
Terminal condition: \(P(T, S) = (K - S)^+\)
Boundary conditions:
- As \(S \to 0\): The put is deep in the money with certainty, so \(P(t, 0) = Ke^{-r(T-t)}\) (the discounted strike).
- As \(S \to \infty\): The put is far out of the money with certainty, so \(P(t, S) \to 0\).
Exercise 2. A down-and-out call with barrier \(B < S_0\) and strike \(K > B\) has price \(V(t, S) = e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}[(S_T - K)^+\mathbf{1}_{\{\tau_B > T\}} | S_t = S]\). Write the PDE domain and boundary conditions. Explain why the PDE is solved on \((B, \infty)\) rather than \((0, \infty)\) and how the killing at the barrier is implemented.
Solution to Exercise 2
PDE domain: The down-and-out call is priced on \(S \in (B, \infty)\) with \(t \in [0, T]\).
Boundary conditions:
- Terminal: \(V(T, S) = (S - K)^+\) for \(S > B\)
- Knockout: \(V(t, B) = 0\) for all \(t \in [0, T]\)
- Far-field: \(V(t, S) \to S - Ke^{-r(T-t)}\) as \(S \to \infty\) (the option behaves like a forward)
Why the domain is \((B, \infty)\): Once the stock price hits the barrier \(B\), the option is immediately worthless (knocked out). The process is effectively "killed" at \(\tau_B = \inf\{s > t : S_s \leq B\}\). In the Feynman-Kac framework, the indicator \(\mathbf{1}_{\{\tau_B > T\}}\) restricts the expectation to paths that survive above \(B\) for the entire interval \([t, T]\).
Implementation of killing at the barrier: The Dirichlet boundary condition \(V(t, B) = 0\) enforces the knockout. When solving the PDE numerically via finite differences, the grid starts at \(S = B\) (not \(S = 0\)), and the boundary value \(V = 0\) is imposed there at every time step. This is equivalent to absorbing the probability mass at \(B\): any path reaching the barrier is removed from the expectation.
Exercise 3. For a double barrier option with lower barrier \(B_l\) and upper barrier \(B_u\), the domain is \((B_l, B_u)\). If the payoff is \(g(S) = (S - K)^+\) and both barriers are knock-out, write the boundary conditions at \(S = B_l\) and \(S = B_u\). How does the solution domain affect the Feynman-Kac expectation?
Solution to Exercise 3
Boundary conditions for the double barrier knockout call:
- Lower barrier: \(V(t, B_l) = 0\) for all \(t \in [0, T]\) (knockout when the stock falls to \(B_l\))
- Upper barrier: \(V(t, B_u) = 0\) for all \(t \in [0, T]\) (knockout when the stock rises to \(B_u\))
- Terminal: \(V(T, S) = (S - K)^+\) for \(B_l < S < B_u\)
Effect on the Feynman-Kac expectation: The solution domain is the bounded interval \((B_l, B_u)\), and the Feynman-Kac representation becomes:
where \(\tau_{B_l}\) and \(\tau_{B_u}\) are the first hitting times of the lower and upper barriers. Only paths that remain entirely within \((B_l, B_u)\) during \([t, T]\) contribute to the expectation. The bounded domain means the Green's function decays faster than on \((0, \infty)\), and the option price is strictly less than the corresponding single-barrier or vanilla option.
Exercise 4. Explain the difference between the Feynman-Kac representation for European options (\(e^{-r\tau}\mathbb{E}[g(S_T)]\)) and the optimal stopping formulation for American options (\(\sup_\tau \mathbb{E}[e^{-r\tau}g(S_\tau)]\)). Why does the American option require a free boundary problem rather than a standard PDE?
Solution to Exercise 4
European option: The price is a fixed expectation over all paths at the terminal time:
The exercise time is fixed at \(T\), and \(V\) satisfies the standard Feynman-Kac PDE \(\partial_t V + \mathcal{L}V - rV = 0\) with equality throughout the domain.
American option: The holder chooses an optimal stopping time \(\tau \in [t, T]\):
Why a free boundary is needed: At each time \(t\), the holder compares the exercise value \(g(S)\) with the continuation value \(\mathbb{E}^{\mathbb{Q}}[e^{-r\Delta t}V^{\text{Am}}(t + \Delta t, S_{t+\Delta t})]\). There exists a critical stock price \(S^*(t)\) separating:
- Continuation region (\(V^{\text{Am}} > g\)): The Feynman-Kac PDE holds with equality.
- Exercise region (\(V^{\text{Am}} = g\)): It is optimal to exercise immediately; \(V^{\text{Am}} = g(S)\).
The boundary \(S^*(t)\) between these regions is not known a priori -- it must be determined as part of the solution. This is why the problem is a free boundary problem rather than a standard PDE with fixed boundary conditions. The standard Feynman-Kac formula does not apply directly because the terminal time is replaced by an optimally chosen random time.
Exercise 5. For a European call in the Black-Scholes model, verify that the solution \(V(t,S) = S\Phi(d_1) - Ke^{-r(T-t)}\Phi(d_2)\) satisfies the terminal condition \(V(T, S) = (S - K)^+\) by computing \(\lim_{t \to T^-} V(t, S)\) for the cases \(S > K\) and \(S < K\).
Solution to Exercise 5
We need to show \(\lim_{t \to T^-} V(t, S) = (S - K)^+\).
As \(t \to T^-\), we have \(\tau = T - t \to 0^+\), so \(d_1 = \frac{\log(S/K) + (r + \sigma^2/2)\tau}{\sigma\sqrt{\tau}}\) and \(d_2 = d_1 - \sigma\sqrt{\tau}\).
Case \(S > K\): \(\log(S/K) > 0\), so as \(\tau \to 0^+\), \(d_1 \to +\infty\) and \(d_2 \to +\infty\). Therefore \(\Phi(d_1) \to 1\) and \(\Phi(d_2) \to 1\):
Case \(S < K\): \(\log(S/K) < 0\), so as \(\tau \to 0^+\), \(d_1 \to -\infty\) and \(d_2 \to -\infty\). Therefore \(\Phi(d_1) \to 0\) and \(\Phi(d_2) \to 0\):
Case \(S = K\): \(\log(S/K) = 0\), so \(d_1 = (r + \sigma^2/2)\sqrt{\tau}/\sigma \to 0^+\) and \(d_2 \to 0^-\). Therefore \(\Phi(d_1) \to 1/2\) and \(\Phi(d_2) \to 1/2\):
Exercise 6. An Asian option has payoff \(g = (\frac{1}{T}\int_0^T S_s\,ds - K)^+\). Explain why the standard one-dimensional Feynman-Kac approach is insufficient and an augmented state variable \(A_t = \int_0^t S_s\,ds\) is needed. Write the two-dimensional PDE that the option price satisfies.
Solution to Exercise 6
The payoff \(g = \left(\frac{1}{T}\int_0^T S_s\,ds - K\right)^+\) depends on the entire path of \(S\) through its time-average. The price \(V\) at time \(t\) depends not only on the current stock price \(S_t\) but also on how much of the average has already been accumulated.
Why one dimension is insufficient: The Feynman-Kac formula requires the state to be Markovian. The stock price \(S_t\) alone does not determine the value of the Asian option because the accumulated average \(\frac{1}{T}\int_0^t S_s\,ds\) is needed. Two scenarios with the same \(S_t\) but different past averages have different option values.
Augmented state variable: Define \(A_t = \int_0^t S_s\,ds\), so \(dA_t = S_t\,dt\). The pair \((S_t, A_t)\) is Markovian, and \(V = V(t, S, A)\).
Two-dimensional PDE: Applying the Feynman-Kac formula to the augmented state \((S_t, A_t)\):
with terminal condition \(V(T, S, A) = \left(\frac{A}{T} - K\right)^+\). The \(S\,\partial_A V\) term arises from the drift \(dA_t = S_t\,dt\) of the auxiliary variable. There is no second-order derivative in \(A\) because \(A_t\) has no diffusion component.
Exercise 7. A digital (binary) barrier option pays \(\$1\) if \(S_T > K\) and the stock never crosses the barrier \(B < K\) during \([0, T]\). Write the Feynman-Kac representation, identify the PDE domain, and describe the terminal and boundary conditions.
Solution to Exercise 7
Feynman-Kac representation: The digital barrier option pays \(\$1\) if \(S_T > K\) and the barrier \(B\) is never breached:
where \(\tau_B = \inf\{s > t : S_s \leq B\}\).
PDE domain: \(S \in (B, \infty)\), \(t \in [0, T]\). The PDE is the standard Black-Scholes equation on this restricted domain:
Terminal condition: \(V(T, S) = \mathbf{1}_{\{S > K\}}\) for \(S > B\). This is a step function: \(V(T, S) = 1\) if \(S > K\), and \(V(T, S) = 0\) if \(B < S \leq K\).
Boundary conditions:
- Knockout at barrier: \(V(t, B) = 0\) for all \(t\) (the option is worthless once the barrier is hit)
- Far-field: \(V(t, S) \to e^{-r(T-t)}\) as \(S \to \infty\) (deep in-the-money digital converges to the discounted \(\$1\) payment, and the barrier is irrelevant)