$0.00
Call Option$0.0000
Alpha Strike Multiple: 1.00x0.0000
Constant pre-start hedge ratio$0.00
Diff: $0.00 (Fixed Strike)Contract Parameters
Forward Start Greeks
| Pre-Start Delta ($\Delta_0$) | 0.0000 |
| Pre-Start Gamma ($\Gamma_0$) | 0.0000 |
| Vega ($\mathcal{V}$) | 0.0000 |
| Theta ($\Theta$ / day) | 0.0000 |
| Rho ($\rho$) | 0.0000 |
Impact of forward moneyness multiple $\alpha$ on contract value and normalized unit pricing.
Value variation as determination date $t_1$ shifts forward toward maturity $t_2$. When $q = 0$, value is flat.
Forward Start Contract Diagnostics
| Component Parameter | Analytical Symbol | Formula / Definition | Model Value |
|---|---|---|---|
| Current Asset Spot | $S_0$ | Market spot price at time $0$ | $100.00 |
| Expected Future Strike | $E[K]$ | $\alpha \cdot S_0 e^{(r-q)t_1}$ | $100.00 |
| Effective Active Horizon | $\tau = t_2 - t_1$ | Time elapsed between strike lock and expiration | 1.00 yrs |
| Dividend Discount Factor | $e^{-q t_1}$ | Yield drag from time $0$ to determination $t_1$ | 1.0000 |
| Normalized 1-Dollar Option | $c_{\text{unit}}$ / $p_{\text{unit}}$ | $c_{\text{BS}}(1, \alpha, t_2 - t_1)$ | $0.0000 |
Spot ($S_0$) vs. Volatility ($\sigma$) Grid
Alpha Multiple ($\alpha$) vs. Start Date ($t_1$) Grid
Forward Start Option Theory & Mathematical Formulation
1. Rubinstein (1991) Closed-Form Solution
Consider an option with strike determination date $t_1$ and expiration $t_2$. At date $t_1$, the strike is set to $K = \alpha S(t_1)$. At time $t_1$, the option is a standard European option with spot $S(t_1)$, strike $\alpha S(t_1)$, and remaining maturity $\tau = t_2 - t_1$.
Because Black-Scholes call and put payoffs are linearly homogeneous in price and strike, we factor out $S(t_1)$:
Taking the risk-neutral expectation at time $0$, where $E^\mathbb{Q}[S(t_1)] = S_0 e^{(r-q)t_1}$, and discounting back by $e^{-r t_1}$:
2. The Dividend Yield and Greek Paradox
If the underlying pays no dividends ($q = 0$), then $e^{-q t_1} = 1$, and:
Remarkably, the value of the forward start option does not depend on when the strike is set ($t_1$), only on the duration between start and expiry ($\tau = t_2 - t_1$)!
Furthermore, because $V_0$ is strictly linear in $S_0$, the pre-start Gamma is identically zero: $\Gamma_0 = \frac{\partial^2 V_0}{\partial S_0^2} = 0$. The holder faces pure Delta risk equal to $e^{-q t_1} c_1$ with no curvature risk prior to $t_1$.
Cliquet & Ratchet Notes
A cliquet option is a series of consecutive forward start options. At each reset date, the gain $(S_{i+1}/S_i - 1)^+$ is locked in, often subject to a local cap and floor, protecting accrued gains from subsequent market downturns.
Executive Compensation
Corporate compensation plans often commit to granting options at future annual intervals with strikes set at future market price ($\alpha = 1.0$). Forward start option models allow accounting and HR teams to accurately budget fair value upfront.
Delta Hedging Dynamics
Hedging a forward start option requires holding a static position in the underlying stock of $\Delta = e^{-q t_1} c_1$ until $t_1$. At date $t_1$, when the strike is fixed, dynamic rebalancing begins as the option develops standard vanilla Gamma.
Forward Start Option Mastery Quiz
1. If the underlying asset pays no continuous dividends (q = 0), what happens to the price of an ATM forward start call as the start date t1 is delayed (keeping active duration t2 - t1 constant)?