Why a Forward Price Has a Geometric Brownian Motion Exponent

Summary

The document asks why a forward contract price is expressed both as the spot price multiplied by the deterministic financing factor and as an exponential stochastic process containing a volatility adjustment and Brownian motion. It focuses on reconciling the forward pricing relation with the apparent randomness in the underlying asset price.

The excerpt gives the questioned formulas but provides no answer or derivation. It offers no empirical evidence and leaves assumptions about the spot process and probability measure unstated. As a result, it serves as a conceptual prompt about how stochastic spot dynamics carry through to forward prices, rather than a complete explanation of forward valuation.

Key ideas

• The document compares a forward price relation based on spot and deterministic interest with a stochastic exponential representation.
• It asks where the volatility adjustment and Brownian term in the exponent come from.
• The excerpt does not provide a derivation or identify the assumptions behind the formulas.
• The relationship depends on the model for the underlying spot price and the pricing framework.

Tags

# Price of a Forward Contract


# Price of a Forward Contract












I have the following,

> Let ${F_t,t\geq0}$ be the price process of the forward contract on the risky asset with maturity $T' > 0$. Since interest rates are deterministic, we have $$F_t=S_t\ e^{r(T^\prime-t)}\ =F_0\ e^{-\frac{1}{2}\sigma^2t+\sigma B_t}$$

Why do we have $-\frac{1}{2}\sigma^2t+\sigma B_t$ in the exponent? Where did it come from? If the price of the underlying $S_t$ is stochastic, why don't we have a stochastic term in the middle equation $S_t\ e^{r(T^\prime-t)}$?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.