Chapter 25 Β· Multidimensional Feynman-Kac

Multiple Assets, Path-Dependent Payoffs & The Universal Recipe

The bridge extends to any number of assets β€” and handles options that care about the whole journey, not just the destination.
πŸ“š Chapter 6 Complete
The real world has more than one asset

Chapter 23 built the Feynman-Kac bridge for one stock. Chapter 24 applied it to one interest rate. The real world β€” and any serious quantitative strategy β€” involves multiple assets moving together. A NIFTY 500 factor model has 500 correlated stocks. A basket option depends on three currencies simultaneously. An Asian option depends on the same stock across every day of the contract. All of these require the multidimensional extension of Feynman-Kac.

One dimension vs two dimensions vs path-dependent β€” three levels of complexity Three panels showing increasing complexity. Left shows a single asset and simple PDE. Middle shows two correlated assets and a 2D PDE with a cross-term. Right shows a single asset with its running average as a second dimension, enabling path-dependent pricing. Ch 23 β€” 1D Multi-asset β€” 2D Asian option β€” 2D trick One stock S Vβ‚œ + rxVβ‚“ + ½σ²xΒ²Vβ‚“β‚“ = rV Standard BSM PDE S₁ and Sβ‚‚ + ρσ₁σ₂x₁xβ‚‚ Β· g_{x₁xβ‚‚} term Cross-diffusion captures correlation S and Y = ∫S du + xΒ·v_y term (running sum grows) History captured as a 2nd variable

Three levels of the Feynman-Kac bridge: 1D (one stock), 2D (correlated assets), and 2D trick (path-dependent options converted to Markovian).

1 Β· The multidimensional PDE β€” assets that talk to each other

When two assets X₁ and Xβ‚‚ are correlated (driven by Brownian Motions with correlation ρ), the Feynman-Kac bridge produces a PDE with an extra term: the cross-diffusion term g_{x₁xβ‚‚}. This term is the mathematical statement that a basket option is not just the sum of two individual options β€” the correlation between them matters.

Multi-Asset Feynman-Kac PDE (Two Correlated Assets)
gβ‚œ + β₁gₓ₁ + Ξ²β‚‚gβ‚“β‚‚ + ½γ₁²gₓ₁ₓ₁ + Β½Ξ³β‚‚Β²gβ‚“β‚‚β‚“β‚‚ + ρ·γ₁·γ₂·gₓ₁ₓ₂ = rΒ·g
β₁gₓ₁ + Ξ²β‚‚gβ‚“β‚‚
Drift terms β€” how each asset pulls the option price in its direction
½γ₁²gₓ₁ₓ₁ + Β½Ξ³β‚‚Β²gβ‚“β‚‚β‚“β‚‚
Self-diffusion β€” each asset's own Gamma term (convexity)
ρ·γ₁·γ₂·gₓ₁ₓ₂
β˜… Cross-diffusion β€” the correlation term. New in 2D. Zero if ρ=0.
Cross-diffusion intuition β€” when two correlated assets both move, the basket option moves more than the sum of individual moves Three scenarios showing correlation rho equals minus 1 where basket value is dampened, rho equals 0 where basket adds linearly, and rho equals plus 1 where basket gains accelerate due to cross-diffusion. ρ = βˆ’1 (Hedged) ρ = 0 (Independent) ρ = +1 (Amplified) When X₁ rises, Xβ‚‚ falls. Cross-term g_{x₁xβ‚‚} is negative β€” dampens basket. Basket cheaper Diversification benefit Moves are unrelated. Cross-term vanishes. Simple sum of PDEs. Sum of individuals No cross-effect Both rise together. Cross-term is positive β€” amplifies the basket. Basket more expensive Correlation premium

Correlation ρ changes the cross-diffusion term's sign and magnitude β€” and therefore the basket option's price. This is the mathematical reason diversification reduces risk.

For a basket option on HDFC Bank and ICICI Bank (correlation β‰ˆ +0.85), the cross-term ρ·γ₁·γ₂·gₓ₁ₓ₂ adds significant value to the option. Pricing a basket option as the sum of two individual options systematically underprices it β€” you are ignoring the cross-diffusion term. The PDE approach automatically includes it; Monte Carlo also captures it by simulating correlated paths.

2 Β· The Asian option β€” solving path-dependency

An Asian option pays the difference between the average stock price over the contract's life and the strike. This is popular with commodity traders because it reduces the impact of manipulation on a single day's closing price. But it creates a fundamental problem: the payoff depends on the entire price history, which seems to violate the Markov Property.

European Option

Payoff = max(S(T) βˆ’ K, 0). Only cares about where the stock ends up. Markov-friendly: the price today is all you need to compute the option value.

h(S(T)) β€” one point in time

Asian Option

Payoff = max((1/T)βˆ«β‚€α΅€S(u)du βˆ’ K, 0). Cares about the whole path. Naively breaks the Markov Property: you need the full history of S to compute the running average.

h((1/T)∫S du) β€” the whole path
πŸ”§ The Markov Fix β€” Turn History into a New Variable
1
Define a new "running sum" variable: Y(t) = βˆ«β‚€α΅— S(u) du. This accumulates the total area under the stock price curve from start to time t.
2
Write the SDE for Y(t). It is trivially simple: dY(t) = S(t) dt. The running sum grows by exactly the current stock price at each instant. No randomness β€” a pure drift term.
3
Now the pair (S(t), Y(t)) is Markov! Knowing these two numbers tells you everything about the future. The past history is captured in Y(t) β€” you no longer need to store the full path.
State = (S(t), Y(t)) β†’ Markov βœ“ β†’ PDE in 2D βœ“
4
Apply the three-step recipe. The resulting PDE gains a new term xΒ·vᡧ (the running sum accumulates at rate S = x), while the terminal condition becomes v(T, x, y) = max(y/T βˆ’ K, 0).
The Asian Option PDE
vβ‚œ + rxΒ·vβ‚“ + ½σ²xΒ²Β·vβ‚“β‚“ + xΒ·vᡧ = rΒ·v
Standard BSM terms: vβ‚œ (theta), rxΒ·vβ‚“ (delta), ½σ²xΒ²Β·vβ‚“β‚“ (gamma) β€” unchanged
New term: xΒ·vᡧ β€” the running sum Y(t) grows at rate x = S(t) per unit time
Terminal condition: v(T, x, y) = max(y/T βˆ’ K, 0) β€” the average payoff

The hedge for an Asian option is still Ξ” = vβ‚“ β€” the derivative with respect to S. But because the PDE is different from BSM, this Ξ” takes different values. An Asian option's delta is typically smaller than a European option's delta with the same strike and maturity β€” because averaging reduces the option's sensitivity to any single day's price move. This is the mathematical reason Asian options are cheaper to hedge.

Numerical note: The Asian PDE has a degeneracy β€” the vᡧ term has no second-derivative in y (no Ξ³Β²vᡧᡧ term). This makes standard finite difference solvers numerically unstable on a grid. Practitioners use the Večer transformation β€” a change of variables that converts the Asian PDE into a standard parabolic PDE that solvers handle cleanly. This is why a custom numerical library is needed for Asian options in production systems.

3 Β· The universal recipe β€” your quant development template

Shreve distils the entire chapter into a four-step universal recipe. Every derivative pricing problem β€” no matter how exotic β€” can be mapped onto this template. This is the starting point for any quant research workflow.

StepActionAsian option exampleBasket option example
1 State variables β€” what matters? S(t) and Y(t) = ∫S du S₁(t) and Sβ‚‚(t)
2 SDEs β€” how do they move? dS = rS dt + ΟƒS dWΜƒ   dY = S dt dS₁ = r S₁ dt + σ₁S₁ dW̃₁   dSβ‚‚ = rSβ‚‚ dt + Οƒβ‚‚Sβ‚‚ dWΜƒβ‚‚ (corr. ρ)
3 PDE β€” apply Feynman-Kac, kill the drift BSM + xΒ·vᡧ term. Terminal: max(y/Tβˆ’K,0) BSM Γ— 2 + ρσ₁σ₂x₁xβ‚‚g_{x₁xβ‚‚} term
4 Hedge β€” match the noise Ξ” = vβ‚“ (hold vβ‚“ shares of S) Δ₁ = gₓ₁, Ξ”β‚‚ = gβ‚“β‚‚ (one delta per asset)

The power of this template: once you can identify the state variables and write their SDEs, the rest follows mechanically. Step 2 is always the hardest β€” getting the SDEs right requires deep understanding of the product. Steps 3 and 4 are then essentially algorithmic. This is why the SDE literacy built in Chapters 1–22 is the foundation of all practical quant work.

Try it β€” Asian vs European option pricer

Simulate stock paths and compute both the European call payoff (final price only) and the Asian call payoff (average price over the path). Observe that the Asian option is always cheaper than the European β€” because averaging reduces extreme outcomes. Watch how the discount grows with higher volatility.

$100
$100
25%
0.70
European / Basket
β€”
Asian / Single
β€”
Discount
β€”
Avg Delta Asian
β€”

Purple = European/Basket payoff distribution  |  Gold = Asian/Single payoff distribution  |  Asian payoffs cluster tighter β€” averaging dampens extremes

The complete book β€” all six parts in one view

This chapter completes the theoretical core of Shreve Volume II. Here is the full arc of the journey, from the first coin flip to the multidimensional Asian option PDE:

Ch 1–8
The Randomness β€” ItΓ΄ Calculus
Random walks β†’ Brownian Motion β†’ GBM β†’ Quadratic Variation β†’ ItΓ΄ Integral β†’ ItΓ΄-Doeblin formula. The mathematical foundation of every model.
Key formula: df = fβ‚œdt + fβ‚“dX + Β½fβ‚“β‚“dt
Ch 9–17
The BSM World β€” Options & Hedging
Markov property β†’ BSM portfolio β†’ BSM PDE β†’ BSM Formula β†’ Greeks β†’ Put-Call Parity β†’ Multivariable ItΓ΄ β†’ Brownian Bridge β†’ Exercises.
Key formula: cβ‚œ + rxcβ‚“ + ½σ²xΒ²cβ‚“β‚“ = rc
Ch 18–22
The Probability Bridge β€” Risk-Neutral Pricing
Girsanov β†’ MRT β†’ Two Fundamental Theorems β†’ Dividends β†’ Forwards & Futures. Pricing by expectation. Alpha is irrelevant. Only Οƒ matters.
Key formula: V(t) = 𝔼̃[e^(βˆ’r(Tβˆ’t)) V(T) | β„±(t)]
Ch 23–25
The Calculus Bridge β€” Feynman-Kac & PDEs
Feynman-Kac theorem β†’ Bond pricing PDEs β†’ Affine yield models β†’ Asian option trick β†’ Universal four-step recipe. Expectation = PDE. Both give the same price.
Key formula: Vβ‚œ + Ξ²Vβ‚“ + Β½Ξ³Β²Vβ‚“β‚“ = rV

Chapter 25 β€” multidimensional Feynman-Kac in four ideas:

The theoretical core of Shreve Volume II is now complete. Chapters 1–8 built randomness. Chapters 9–17 built the BSM world. Chapters 18–22 built the probability bridge. Chapters 23–25 built the calculus bridge. What remains: exotic options (Chapter 7 style), Change of Numeraire techniques, and numerical implementation via Finite Difference Methods.