M.4.6 Advanced Extensions

Basic Prerequisites: 05 · Girsanov & Risk-Neutral.

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1. Intuition & Practical Objective

These three results are the multipliers that turn stochastic calculus into a pricing theory. Feynman–Kac declares that the PDE picture and the expectation picture are the same number - solve a parabolic PDE ⇐> take a discounted expectation along an SDE. Martingale Representation (MRT) says that in a Brownian world every martingale is an Itô integral - the mathematical license for replicating any derivative with a delta position. The Fundamental Theorems of Asset Pricing then assemble it: no arbitrage ⇔ a martingale measure exists, and completeness ⇔ that measure is unique.

Practical objective: understand the complete toolkit that (a) prices a derivative either by PDE or by $\mathbb{Q}$-expectation, as convenient; (b) makes “every $T$-claim is hedgeable in a complete one-BM market” rigorous; (c) tells you exactly when that falls apart (extra randomness, jumps, missing assets) - the theoretical reason real markets need stochastic-vol and jump models.

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2. Mathematical Ground Truth & Derivations

2.1 Feynman–Kac (Björk Props 5.5/5.6; Shreve II §4.5)

Let $X$ solve $d{X}_t=\mu (t,X_t)dt+\sigma (t,X_t)d{W}_t$. If $F(t,x)$ solves the parabolic PDE

$$F_t+\mu F_x+\frac{1}{2}{\sigma }^2{F}_{xx}-rF=0,F(T,x)=\Phi (x),$$

then (with discounting, Björk Prop 5.6)

$$F(t,x)=e^{-r(T-t)}{\mathbb{E}}_{t,x}[\Phi (X_T)].$$

Applied to the risk-neutral dynamics ($\mu =r$, $r$ the drift) this says exactly: the BSM PDE solution equals the discounted $\mathbb{Q}$-expectation of the payoff. The two §05-pricing routes - PDE and expectation - coincide by construction. Nondiscounted form ($r=0$) is Björk Prop 5.5; multidimensional form Prop 5.8.

2.2 Martingale Representation Theorem (Shreve II Thm 5.3.1; Björk Thm 11.2; Shreve I §18)

Let $\mathcal{F}(t)$ be the filtration generated by a BM $W$. Then every $\mathcal{F}$-martingale $M$ has the representation

$$M(t)=M(0)+{\int }_0^t\Gamma (u)dW(u)$$

for a unique adapted process $\Gamma$ (in $L^2$ if $M$ is square-integrable). Multidimensional version (Shreve II Thm 5.4.2): $M(t)=M(0)+{\int }_0^t\Gamma (u)\cdot dW(u)$. Key accumulation for hedging: combining MRT with Girsanov (Shreve II Cor 5.3.2), for any $\mathcal{F}(T)$-claim $V$, the $\mathbb{Q}$-martingale $M(t)=\tilde{\mathbb{E}}[\tilde{D}(T)V(T)\mid \mathcal{F}(t)]$ has a representation whose integrand $\Gamma$ is matched by choosing the delta $\Delta (t)=\Gamma (t)/(\sigma (t)D(t)S(t))$ - so the claim is replicated (Shreve II §5.3.2). This requires $\sigma \ne 0$ (the BM randomness must be in the stock) and $\mathcal{F}$ generated by $W$ (no extra randomness).

2.3 The Fundamental Theorems (Shreve II Thm 5.4.7/5.4.9; Harrison–Pliska; Björk 10.9/10.17)

Model: $m$ stocks driven by $d$ BMs, $d{S}_i=S_i({\alpha }_{i}dt+{\sum }_j{\sigma }_{ij}d{W}_j)$. The market price of risk equations

$${\alpha }_i(t)-R(t)=\sum _{j=1}^d{\sigma }_{ij}(t){\Theta }_j(t),i=1,\dots ,m$$

(m equations, $d$ unknowns - one per BM, not per stock) drive the classification (Shreve II §5.4):

Market-price-of-risk system	Risk-neutral measure	Market
unique solution	unique (complete)	no arbitrage + complete (every claim hedgeable)
no solution	none	arbitrage exists
multiple solutions	multiple	no arbitrage but incomplete

• First FT (Shreve II 5.4.7): no arbitrage ⇔ a risk-neutral measure exists (the MPR system is solvable).
• Second FT (Shreve II 5.4.9): market complete ⇔ the risk-neutral measure is unique (⇔ the $m\times d$ volatility matrix has full rank / invertible square case $m=d$).

The meta-theorem (Björk 8.3.1) encapsulates the counting: with $M$ risky assets and $R$ random sources, arbitrage-free ⇔ $M\le R$ (fewer/equal assets than risks), complete ⇔ $M\ge R$ (enough assets to hedge), both ⇔ $M=R$. BSM has $M=R=1$ ⇒ complete and arbitrage-free; a jump process adds a distinct random source so one stock can no longer hedge it.

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3. Computational Implementation - verify the bridges

Three stdlib checks: (a) Feynman–Kac - the $\mathbb{Q}$-expectation of a payoff equals the BSM PDE solution (MC vs closed form); (b) martingale representation - delta-replicating a written call, the discrete hedge P&L closes to $\approx 0$ at expiry (the replication story made concrete); (c) the completeness counting rule.

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4. Failure Modes & First-Principles Breakdowns

1. MRT needs the BM filtration. Both MRT and completeness/hedging fall apart if the filtration is larger than that generated by $W$ (e.g. an independent Poisson, an untraded factor). Replicating an $\mathcal{F}(T)$-claim then requires a hedge $h$ with an extra, unhedgeable source of randomness - the deep root of market incompleteness and the reason models with jumps/SV need completion choices (Shreve II §5.3; Björk §14.2).
2. $\sigma =0$ destroys completeness. If the stock has no diffusion, the BM randomness is not in the tradable; you cannot hedge with something that has no volatility. The MRT-integrand-matching $\Delta =\Gamma /(\sigma DS)$ divides by $\sigma$ (Shreve II §5.3.2).
3. Feynman–Kac under the physical measure is mispricing. The expectation must be along the risk-neutral drift ($r$), not $\mu$. Running the SDE with drift $\mu$ and plugging into $e^{-rT}\mathbb{E}[\Phi ]$ breaks the PDE↔expectation correspondence (05 · Girsanov).
4. Confusing “prices fixed” with “model correct.” Completeness fixes the price of every claim, but only within that model’s distributional assumptions. It guarantees hedgeability, never realism - it is the precondition for the smiles/jumps/SV extensions, not a vindication of BSM (see BSM 05 · Failure Modes).

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5. References

• Björk, Arbitrage Theory in Continuous Time
• Shreve, Stochastic Calculus for Finance II
• Shreve, Stochastic Calculus for Finance I
• Glasserman, Monte Carlo Methods in Financial Engineering

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6. Connected Graph Bridges

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