Strongly Path-Dependent Derivatives: When History Matters
Barrier options showed us weak path dependence. The contract cared about the path, but we still solved a two-variable problem. Chapter 24 takes the next step: strong path dependence. Cannot be hidden in boundary conditions. We need an extra variable.
The good news: Wilmott builds a general framework here that handles a huge number of exotic contracts. Learn it once, price Asians, lookbacks, and many others without starting from scratch.
Path-Dependent Quantities as Integrals
Most path-dependent contracts care about a summary of the path: an average, a maximum, time spent above a level. Wilmott’s insight is that many such summaries can be written as an integral:
$$I(t) = \int_0^t f(S, \tau) , d\tau$$
For arithmetic Asians, $f(S, t) = S$. For geometric Asians, $f(S, t) = \ln(S)$. For “time above a level,” $f(S, t) = H(S - L)$ (Heaviside function). Different f functions, same framework.
The Pricing Equation
The option value V(S, I, t) has three independent variables. The stochastic differential equation for I is trivially simple:
$$dI = f(S, t) , dt$$
No randomness. No dW term. The integral changes smoothly. Set up the usual hedging argument, apply Ito’s lemma, and you get:
$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} + f(S,t)\frac{\partial V}{\partial I} - rV = 0$$
Black-Scholes plus one extra term: $f(S,t) \partial V / \partial I$. Because I has no randomness, only a first derivative appears (no second derivative). The I direction behaves “time-like,” which is good news for numerical methods.
Discrete Sampling: The Practical Case
In practice, nobody samples continuously. Contracts specify exact dates: “average of Friday closing prices” or “maximum of monthly readings.” If samples are closely spaced, the continuous approximation works. If they are far apart, discrete sampling matters.
The Updating Rule
Instead of an integral, Wilmott introduces an updating rule. On each sampling date $t_i$:
$$I_i = g(I_{i-1}, S(t_i), t_i)$$
The new value depends only on the old value and the current stock price. I remains a single number.
For a discrete arithmetic average: $A_i = \frac{i-1}{i} A_{i-1} + \frac{1}{i} S(t_i)$
For a discrete lookback maximum: $M_i = \max(M_{i-1}, S(t_i))$
Between Samples: Just Black-Scholes
Between sampling dates, I is constant (dI = 0). The pricing equation is the ordinary Black-Scholes with I as a parameter.
At Sampling Dates: The Jump Condition
When the clock hits sampling time, I updates. The stock price is continuous, no money changes hands, so the option value must be continuous too. This gives the jump condition:
$$V(S, I_{i-1}, t_i^-) = V(S, g(I_{i-1}, S, t_i), t_i^+)$$
Value just before equals value just after, but with updated I. Wilmott calls it a “jump condition” even though the option value itself does not jump. What jumps is the path-dependent variable. The surface V(S, I) appears discontinuous at sampling dates, but along any realized path, value is perfectly continuous.
The Algorithm
Putting it all together, the pricing algorithm for discrete sampling is:
- Start at expiry with the payoff as the final condition
- Solve the Black-Scholes equation backward in time until you reach the next sampling date
- Apply the jump condition to translate the “just after” solution into the “just before” initial condition
- Repeat: solve Black-Scholes backward to the next sampling date
- Apply jump condition again
- Continue until you reach the current time
It is like solving a sequence of two-dimensional Black-Scholes problems, stitched together by jump conditions. Each sampling date is a checkpoint where the I variable updates.
Higher Dimensions and Monte Carlo
Nothing stops you from having multiple path-dependent quantities. Geometric and arithmetic average together? Four dimensions. That is the edge of what PDEs can handle. Beyond that, Monte Carlo wins.
Monte Carlo is simple: simulate the risk-neutral random walk, compute the path-dependent quantity for each path, average the payoffs, discount. Easy to code, works in any dimension. Slower for low-dimensional problems and bad at American-style early exercise.
For PDEs in low dimensions, adding early exercise is almost free: apply $V(S, I, t) \ge P(S, I)$ with delta continuity. A couple of lines in a finite-difference scheme.
The Takeaway
Chapter 24 gives you a general machine for pricing strongly path-dependent derivatives. The recipe is:
- Identify the path-dependent quantity
- Write it as an integral (continuous) or updating rule (discrete)
- Add it as a new state variable to get the pricing equation
- Solve with appropriate boundary/jump conditions
The continuous case gives you a three-dimensional PDE with an extra first-derivative term. The discrete case gives you a sequence of two-dimensional Black-Scholes problems connected by jump conditions. Both approaches are systematic and apply to a wide range of contracts.
The next two chapters apply this framework to the two most important strongly path-dependent options: Asians and lookbacks.
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