3.7.3 Lookbacks & Asian (Average-Rate) Options

Basic Prerequisites: Stochastic Calculus & Itô’s Lemma and Black–Scholes–Merton & Feynman–Kac.

1. Intuition & Practical Objective

Lookbacks let you “buy at the low” / “sell at the high”: a floating-strike lookback call pays $S_{T} - min_{0 \leq t \leq T} S_{t}$ (you got in at the path minimum). Asians replace the terminal price with a time average: an average-rate put pays $(X - \overset{ˉ}{S}_{T})^{+}$ where $\overset{ˉ}{S}_{T}$ is the path average. Both are built from the running extremum / running integral - two path statistics the reflection principle handles cleanly in closed form for lookbacks, but which have no closed form for the arithmetic Asian (Shreve II §7.5 is explicit: no closed form).

The practical objective:

Lookbacks: one closed form (floating strike), fixed-strike by put–call parity.
Geometric Asian: exactly BSM with adjusted volatility/carry (a log of a geometric average is a normal - the one Asian with a clean closed form).
Arithmetic Asian: no closed form → moment-matching approximations (Turnbull–Wakeman) and Monte Carlo - the workhorse.

2. Mathematical Ground Truth & Derivations

2.1 Floating-strike lookback call (Haug eq 4.39; Shreve II §7.4)

With $a_{1} = [ln (S / S_{min}) + (b + \frac{1}{2} σ^{2}) T] / (σ T)$ , $a_{2} = a_{1} - σ T$ , for $b \neq = 0$ :

c = S e^{(b - r) T} N (a_{1}) - S_{min} e^{- r T} N (a_{2}) + S e^{- r T} \frac{σ ^{2}}{2 b} [(\frac{S}{S _{min}})^{- 2 b / σ^{2}} N (- a_{1} + \frac{2 b T}{σ}) - e^{b T} N (- a_{1})] .

Critical sign: the two terms in the square bracket add (the book value confirms the +). Flip it and you get $21.35$ instead of $25.35$ . Shreve’s derivation is the reflection principle on the running max $Y (t) = max S$ , with the smooth-pasting condition $v_{y} (t, y, y) = 0$ and the “not a dt-term” subtlety that the running max increases only on a Lebesgue-null set yet is not a $d t$ -term (Shreve II §7.4). The dimension reduction $v (t, x, y) = y \cdot u (t, x / y)$ (Thm 7.4.3) is what makes the closed form tractable.

2.2 Geometric Asian (Haug eqs 4.91–4.92) - the exact closed form

The geometric average $G = exp (\frac{1}{T} \int_{0}^{T} ln S_{t} d t)$ is lognormal, so the geometric Asian prices as a plain BSM option with adjusted parameters:

σ_{A} = \frac{σ}{3}, b_{A} = \frac{1}{2} (b - \frac{σ ^{2}}{6}),

then $c = S e^{(b_{A} - r) T} N (d_{1}) - X e^{- r T} N (d_{2})$ , $p = X e^{- r T} N (- d_{2}) - S e^{(b_{A} - r) T} N (- d_{1})$ with the usual $d_{1}, d_{2}$ in terms of $σ_{A}, b_{A}$ . Because it is a clean BSM, the geometric Asian is the natural control variate for the arithmetic Asian under MC (Glasserman Ch 3, Ch 4).

2.3 Arithmetic Asian - no closed form (Shreve II §7.5; Hull Ch 26)

Shreve prices it via state augmentation: $Y (t) = \int_{0}^{t} S d u$ , with the degenerate PDE $v_{t} + r x v_{x} + x v_{y} + \frac{1}{2} σ^{2} x^{2} v_{xx} = r v$ and terminal $v (T, x, y) = (y / T - K)^{+}$ - and explicitly no closed form. Hull §26.13 notes the two-moment (lognormal) approximation: match the first two moments $M_{1} = E [A_{T}]$ , $M_{2} = E [A_{T}^{2}]$ of the arithmetic average and price as a BSM with $\overset{ˉ}{b} = ln M_{1} / T$ , $\overset{σ}{ˉ}^{2} = ln M_{2} / T - 2 \overset{ˉ}{b}$ . This is the Turnbull–Wakeman approximation. It needs the in-average-period strike/scale adjustment (Haug §4.20.3); treat the moment-matching idea as exact, the arithmetic-average digits as approximate.

3. Computational Implementation - lookback closed form + MC, geometric closed form + MC, arithmetic by MC

Stdlib only. The lookback MC simulates the underlying with Q-drift $b$ (cost of carry) and discounts at $r$ - the path statistic $min$ depends on the drift of $lo g S$ .

The geometric closed form and MC agree to ~0.007 (MC $O (1/ n)$ error), confirming the BSM-reduction. The arithmetic Asian has no closed form - only MC (or a moment-matching approximation) prices it. The lookback MC matches the reflection-principle closed form once the underlying is simulated with the correct Q-drift $b$ .

4. Failure Modes & First-Principles Breakdowns

Wrong Q-drift for path statistics. The running min/max/average distributions depend on $b - \frac{1}{2} σ^{2}$ , not $r - \frac{1}{2} σ^{2}$ in general. Simulating a dividend-paying underlying with drift $r$ biases every lookback/Asian MC (see the mismatch this avoids in 01).
Geometric ≠ arithmetic. The geometric closed form is exact but prices the wrong contract if the contract is arithmetic. Never plug the geometric digits into an arithmetic payoff.
Lookback sign errors. The floating-strike lookback’s reflected image term carries a sign that is easy to flip; verify against the book’s numeric (flip the bracket + and you get $21.35$ vs $25.35$ ).
Turnbull–Wakeman is an approximation. Moment-matching is exact for the moments, but the lognormal fit is an assumption. The in-average-period strike/scale adjustment (Haug §4.20.3) is required once averaging has started - omitting it misprices by the full shift term.

5. References

Haug, The Complete Guide to Option Pricing Formulas
Shreve, Stochastic Calculus for Finance II
Glasserman, Monte Carlo Methods in Financial Engineering

6. Connected Graph Bridges

Back: 02 · Barriers & Digitals
Forward: Exchange · Index Hub
Theory: Black–Scholes–Merton & Feynman–Kac · Stochastic Calculus & Itô

Kwant Atlas

Explorer