3.7.4 Compound, Chooser, Quanto, Exchange & Spread Options

Basic Prerequisites: Multivariable Calculus & Optimization and Black–Scholes–Merton & Feynman–Kac.

1. Intuition & Practical Objective

These exotics extend BSM along two new axes:

Time - compound (option on an option) and chooser (call-or-put later) add a decision date $t_{1}$ before the underlying’s expiry.
A second asset / currency - exchange (Margrabe), spread (Kirk), quanto and foreign-equity options trade one asset against another or translate across currencies.

The practical objective: the bivariate normal CDF $M (a, b; ρ)$ is the new mathematical primitive (chooser/compound are four-term combinations; two-asset options carry the correlation $ρ$ explicitly). Every closed form here is “BSM in one or two dimensions, with a critical price level $I$ (or a correlation-adjusted volatility) solved for.” Verified anchors: compound put-on-call $= 21.1964$ , simple chooser $= 6.1071$ , complex chooser $= 6.0507$ , Margrabe $= 1.5260$ , quanto $= 5.3280$ , foreign-equity $= 8.3056$ , Kirk spread $= 2.1670$ , complex chooser $I = 51.1158, w = 6.0508$ (Haug 4.27).

2. Mathematical Ground Truth & Derivations

2.1 The bivariate normal primitive

M (a, b; ρ) = P (X \leq a, Y \leq b), (X, Y) \sim N (0, (1 ρ ρ 1)) = \int_{- \infty}^{a} φ (x) N (\frac{b - ρ x}{1 - ρ ^{2}}) d x .

This one-dimensional integral form is exactly how it is computed here (adaptive Simpson) - no external library needed. Haug recommends the Genz algorithm for production accuracy (§13.3.1); the integral form reproduces all anchors to 3–4 dp.

2.2 Compound options (Haug §4.13) - a call on a call/put

For a put-on-call (eq 4.29, verified): the critical level $I$ solves $c_{BSM} (I, X_{1}, T_{2} - t_{1}) = X_{2}$ (the spot at which the underlying call is worth the compound strike $X_{2}$ ), and

p_{c a ll} = X_{1} e^{- r T_{2}} M (z_{2}, - y_{2}; - ρ) - S e^{(b - r) T_{2}} M (z_{1}, - y_{1}; - ρ) + X_{2} e^{- r t_{1}} N (- y_{2}),

with $z_{1}, z_{2}$ (function of $X_{1}$ , $T_{2}$ ), $y_{1}, y_{2}$ (function of $I$ , $t_{1}$ ), $ρ = t_{1} / T_{2}$ . Compound-option put–call parity (Haug eqs 4.32–4.33): $c_{c a ll} + X_{2} e^{- r t_{1}} = p_{c a ll} + c_{BSM} (S, X_{1}, T_{2})$ .

2.3 Choosers (Haug §4.12)

Simple chooser - at $t_{1}$ you pick a call or put (both strike $X$ , expiry $T_{2}$ ). Cleanest verified form is the decomposition (the literal transcription’s last-term exponent is ambiguous in the source; the decomposition reproduces the book’s $6.1071$ ):

chooser = c (S, X, T_{2}) + e^{(b - r) (T_{2} - t_{1})} p (S, X e^{- b (T_{2} - t_{1})}, t_{1}) .

Complex chooser - at $t_{1}$ you choose between a call $(X_{c}, T_{c})$ and a put $(X_{p}, T_{p})$ ; four bivariate-normal terms with correlations $ρ_{1} = t_{1} / T_{c}$ , $ρ_{2} = t_{1} / T_{p}$ , and a critical level $I$ solving $c_{BSM} (I, X_{c}, T_{c} - t_{1}) = p_{BSM} (I, X_{p}, T_{p} - t_{1})$ .

2.4 Exchange (Margrabe, Haug §5.7) and spread (Kirk §5.8)

Margrabe - the option to exchange asset 2 for asset 1: $C = Q_{1} S_{1} e^{(b_{1} - r) T} N (d_{1}) - Q_{2} S_{2} e^{(b_{2} - r) T} N (d_{2})$ with the net volatility $σ = σ_{1}^{2} + σ_{2}^{2} - 2 ρ σ_{1} σ_{2}$ and $d_{1} = [ln (Q_{1} S_{1} / Q_{2} S_{2}) + (b_{1} - b_{2} + \frac{1}{2} σ^{2}) T] / (σ T)$ . Note: no $X$ - the exchange option needs only two assets and the correlation. This is why a quanto / spread reduces to it.

Kirk spread - payoff $(S_{1} - S_{2} - X)^{+}$ approximated by treating $S_{2} e^{(b_{2} - r) T} + X e^{- r T}$ as a single “asset” with weight $F = S_{2} e^{(b_{2} - r) T} / (S_{2} e^{(b_{2} - r) T} + X e^{- r T})$ and vol $σ = σ_{1}^{2} + (F σ_{2})^{2} - 2 ρ σ_{1} F σ_{2}$ .

2.5 Quanto & foreign-equity (Haug §5.16)

Quanto (fixed-rate foreign equity, Derman–Karasinski–Wecker / Reiner): payoff in domestic currency of a foreign asset, struck at a fixed exchange rate $E_{p}$ :

c = E_{p} [S^{*} e^{(r_{f} - r - q - ρ σ_{S} σ_{E}) T} N (d_{1}) - X^{*} e^{- r T} N (d_{2})], d_{1} = \frac{ln ( S ^{*} / X ^{*} ) + ( r _{f} - q - ρ σ _{S} σ _{E} + \frac{1}{2} σ _{S}^{2} ) T}{σ _{S} T} .

The term $- ρ σ_{S} σ_{E}$ is the quanto adjustment - it shifts the foreign drift by the covariance of the asset with the FX rate. Foreign equity struck in domestic currency (Reiner) instead uses the combined vol $σ_{ES} = σ_{S}^{2} + σ_{E}^{2} + 2 ρ σ_{E} σ_{S}$ with $d_{1} = [ln (E S^{*} / X) + (r - q + \frac{1}{2} σ_{ES}^{2}) T] / (σ_{ES} T)$ and $c = E S^{*} e^{- qT} N (d_{1}) - X e^{- r T} N (d_{2})$ .

3. Computational Implementation - the bivariate engine

Stdlib only. $M (a, b; ρ)$ via the one-dimensional integral form; compound/chooser/Margrabe/quanto all reproduced.

4. Failure Modes & First-Principles Breakdowns

Correlation is the hidden input. Margrabe, quanto and spread all carry $ρ$ ; the closed forms look like BSM but the effective volatility is correlation-dependent ( $σ_{1}^{2} + σ_{2}^{2} - 2 ρ σ_{1} σ_{2}$ ). A wrong $ρ$ misprices the whole contract - and correlations are notoriously unstable (see Implied Volatility Surfaces).
Quanto adjustment sign. The drift shift is $- ρ σ_{S} σ_{E}$ in the quanto (domestic payoff at fixed FX); confuse it with the foreign-equity case ( $+ ρ$ in the combined vol) and you flip the sign of the entire covariance correction.
Critical-level root finding. Compound and complex choosers need $I$ solved from a BSM equality; a sloppy root finder (or using the wrong side of the equation) invalidates every bivariate term. The $I$ here is verified ( $538.3165$ , $51.1158$ ) - but re-solve it, don’t hardcode.
$M (a, b; ρ)$ is the real cost. These closed forms hide a bivariate-normal evaluation; a naive/incorrect bivariate routine silently corrupts all four terms. Use the integral form or Genz, and validate against a known anchor before production.

5. References

Haug, The Complete Guide to Option Pricing Formulas
Hull, Options, Futures, and Other Derivatives
Shreve, Stochastic Calculus for Finance II
Glasserman, Monte Carlo Methods in Financial Engineering

6. Connected Graph Bridges

Back: 03 · Lookbacks & Asians
Forward: 05 · Failure Modes · Index Hub
Base: Black–Scholes–Merton & Feynman–Kac · Stochastic Calculus & Itô
Sibling: Heston & SABR (the correlation/vol input)

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