Quantitative Finance & Derivatives Geske (1979) Compound Option Model

Compound Option Pricing Lab (Geske 1979)

Value options on options under Robert Geske's bivariate analytical model. Analyze Call-on-Call, Put-on-Call, Call-on-Put, and Put-on-Put with critical asset boundary S*, Greeks, and multi-stage corporate real options.

Industry Presets:
Compound Option Value
$7.946
7.95% of underlying spot
Critical Stock Price (S*)
$92.240
Exercise threshold at T₁ (V_opt = K₁)
Underlying Vanilla Value
$11.121
Black-Scholes vanilla price at T₀
Effective Gearing / Leverage
6.67x
(Delta × S) / Compound Price

Compound Contract Parameters

Call-on-Call

Underlying Asset Parameters

$100.00
25.0%
2.0%
5.0%

Strikes & Two-Tier Expirations

$4.00
Paid at T₁ to exercise compound
$100.00
Vanilla strike on asset at T₂
0.50 yrs
First decision date
1.00 yrs
Terminal vanilla expiry (T₂ > T₁)

Analytical Formula Breakdown

Component Variable Value Analytical Interpretation
Compound Price $7.946 Theoretical Geske (1979) option value
Critical Price (S*) $92.240 Root solved: V_underlying(S*, K₂, T₂-T₁) = K₁
Vanilla Option (T₂) $11.121 Standalone Black-Scholes underlying option
Compound Put-Call Parity $0.0000 Exact zero residual: CoC - PoC = BS - K₁e^(-rT₁)
Bivariate Correlation (ρ) 0.7071 Analytical link ρ = √(T₁ / T₂)
Upper Limits (a₁, a₂) 0.3648 / 0.1880 Standardized limits for T₁ exercise boundary
Lower Limits (b₁, b₂) 0.1800 / -0.0700 Standardized limits for terminal strike K₂
Bivariate Integral M(a₁, b₁; ρ) 0.5188 Joint probability under asset numeraire
Bivariate Integral M(a₂, b₂; ρ) 0.4432 Joint probability under cash numeraire
Marginal CDF Φ(a₂) 0.5746 Probability of compound exercise at T₁

Compound Risk Greeks Suite

Delta (Δ = ∂C/∂S)
+0.5297
Gamma (Γ = ∂²C/∂S²)
0.0183
Vega (𝒱 per 1% vol)
+0.3692
Theta (Daily Calendar Decay)
-$0.0189 / day
Rho (Sensitivity per 1% r)
+0.4371
Underlying Delta (BS Vanilla)
+0.5602

Dynamic Visualizer & Critical Boundary

Interactive payoff curve comparing the compound option against the standalone Black-Scholes underlying option, highlighting the critical exercise threshold S*.

5×5 Sensitivity Grid: Spot Price (S) vs. Volatility (σ)

5×5 Sensitivity Grid: Strike K₁ vs. Expiry Horizon T₁

Mathematical Formulation & Economic Foundations

1. The Geske (1979) Compound Option Framework

A compound option gives the holder the right to buy or sell another option. Let $T_1$ be the expiration date of the compound option, $K_1$ its exercise price, $T_2$ ($T_2 > T_1$) the expiration date of the underlying option, and $K_2$ the exercise price of the underlying option.

At time $T_1$, the compound call option on an underlying call will be exercised if and only if the market value of the underlying call at $T_1$ exceeds the strike price $K_1$:

V_{\text{call}}(S_{T_1}, K_2, T_2 - T_1) \ge K_1 \iff S_{T_1} \ge S^*

Here, $S^*$ is the critical asset price threshold, determined as the unique root solving $V(S^*, K_2, T_2 - T_1) = K_1$.

2. Closed-Form Analytical Formulas

Using the bivariate standard normal distribution function $M(a, b; \rho)$ where the correlation between Brownian motions over intervals $[0, T_1]$ and $[0, T_2]$ is $\rho = \sqrt{T_1 / T_2}$, Robert Geske derived the exact pricing formulas:

  • Call on Call: $C_{\text{call}} = S e^{-q T_2} M(a_1, b_1; \rho) - K_2 e^{-r T_2} M(a_2, b_2; \rho) - K_1 e^{-r T_1} N(a_2)$
  • Put on Call: $P_{\text{call}} = -S e^{-q T_2} M(-a_1, b_1; -\rho) + K_2 e^{-r T_2} M(-a_2, b_2; -\rho) + K_1 e^{-r T_1} N(-a_2)$
  • Call on Put: $C_{\text{put}} = -S e^{-q T_2} M(-a_1, -b_1; \rho) + K_2 e^{-r T_2} M(-a_2, -b_2; \rho) - K_1 e^{-r T_1} N(-a_2)$
  • Put on Put: $P_{\text{put}} = S e^{-q T_2} M(a_1, -b_1; -\rho) - K_2 e^{-r T_2} M(a_2, -b_2; -\rho) + K_1 e^{-r T_1} N(a_2)$

3. Standardized Integration Limits & Parity

The limits of integration are specified relative to the critical threshold $S^*$ and terminal strike $K_2$:

a_1 = \frac{\ln(S/S^*) + (r - q + \frac{1}{2}\sigma^2)T_1}{\sigma \sqrt{T_1}}, \quad a_2 = a_1 - \sigma \sqrt{T_1}
b_1 = \frac{\ln(S/K_2) + (r - q + \frac{1}{2}\sigma^2)T_2}{\sigma \sqrt{T_2}}, \quad b_2 = b_1 - \sigma \sqrt{T_2}

Compound Put-Call Parity: Just as in vanilla options, compound options satisfy strict no-arbitrage relationships:

C_{\text{call}} - P_{\text{call}} = C_{\text{BS}}(S, K_2, T_2) - K_1 e^{-r T_1}
C_{\text{put}} - P_{\text{put}} = P_{\text{BS}}(S, K_2, T_2) - K_1 e^{-r T_1}

4. Industry Applications & Staged Real Options

Biopharmaceutical Drug Trials: Undertaking a Phase 1 or Phase 2 clinical trial is essentially purchasing a call-on-call. Phase 2 costs $K_1$ and produces the option to invest $K_2$ in commercial manufacturing if drug efficacy is proven.

Venture Capital Staged Financing: Seed and Series A financing tranches give investors options to extend subsequent capital in Series B and C as operational milestones are de-risked.

Natural Resources Exploration: Leasing an oil field gives the exploration firm the option to drill exploratory wells ($K_1$), which in turn unlocks the option to construct commercial pipelines ($K_2$).

Compound Option Mastery Quiz

Unsubmitted

Frequently Asked Questions

A compound option is an option on an option. The contract grants the holder the right, upon payment of an initial exercise strike price K1 at time T1, to buy or sell another underlying vanilla option having strike K2 and subsequent maturity T2 (where T2 > T1). Formulated by Robert Geske in 1979, compound options provide high leverage and downside protection.

The four configurations are: (1) Call-on-Call (CoC), the right to buy a call; (2) Put-on-Call (PoC), the right to sell a call; (3) Call-on-Put (CoP), the right to buy a put; and (4) Put-on-Put (PoP), the right to sell a put. Each has distinct payoff profiles and risk sensitivities.

The critical asset price S* is the unique underlying stock price at first expiration T1 at which the market value of the underlying vanilla option exactly equals the compound option exercise price K1. For Call-on-Call options, the compound option is exercised if S(T1) > S*; for Put-on-Call options, it is exercised if S(T1) < S*.

Compound option theory is widely used in corporate capital budgeting to value multi-stage investments. Examples include pharmaceutical drug development (where Phase 1 trials buy the option to enter Phase 2 and 3), venture capital staged financing rounds, patent commercialization, mining exploration, and corporate capital structure with subordinated debt.

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