Pricing a Forward-Start Spread Option with a Later-Trading Asset

Summary

The document considers a European spread option paying the positive part of one asset’s terminal price minus a fixed multiple of another asset’s price. The second asset trades from the outset, while the first becomes available only at an intermediate date. Under a joint lognormal assumption, the question is how to value the contract at inception.

The proposed approach first prices the remaining option at the intermediate date using the Margrabe exchange-option formula, then discounts the expected value of that price back to today. If the newly listed asset’s price at that date is known, the remaining uncertainty is the other asset’s price, whose stated lognormal distribution can be integrated numerically. The answer assumes a known intermediate-date price for the later-trading asset; if it is not known, an additional estimate or model is required. The document gives a conceptual integration recipe but does not specify model calibration or address alternative assumptions about the asset’s opening price.

Key ideas

• At the later asset’s listing date, the remaining spread payoff can be valued with an exchange-option formula under the stated assumptions.
• The inception value is the discounted expected option value at that intermediate date.
• If one asset’s intermediate price is known, numerical integration over the other asset’s lognormal distribution provides a direct valuation route.
• If the later-trading asset’s opening price is unknown, it must be estimated or modeled.
• The answer does not provide calibration details or discuss alternative opening-price assumptions.

Tags

# Forward Start Spread Options


# Forward Start Spread Options












> Question: We have a spread option with payoff: $\max (P_{T} - HR\times G_T, 0)$, where $P$, $G$ are underlying prices and $HR$ is a constant. At time zero only contract $G$ is available for trading. The contract $P$ will only open trading at $0 < t_1 < T$. What's the (optimal i.e. risk neutral expectation based) price of the contract assuming joint lognormality.

I am a bit confused on where to start with this question. Unless I am incorrect are there different cases to consider before we price the contract? Any suggestions would be appreciated.

## Answer by bhutes (score 3, accepted)

https://quant.stackexchange.com/a/45614

At $t_1$, this payoff can be priced using the Margrabe formula as used for pricing an exchange option.

See Margrabe Formula here

Using the notations in the question and those used the hyperlinked document above -

$Price_{t_1} = P_{t_1}e^{(\mu_P-r)\tau}\Phi(d_+) -HR \times G_{t_1}e^{(\mu_G-r)\tau}\Phi(d_-) \tag{1}$

$Price_0$ is the discounted value of $Price_{t_1}$ using the discount factor $e^{-rt_1}$

$P_{t_1}$ is assumed to be known at time $0$ - e.g. $P$ begins trading at $t_1$ at par, such that we know that $P_{t_1} = 100$. Otherwise $P_{t_1}$ needs to be estimated by other means.

So, the only unknown we are left with in $(1)$ is $G_{t_1}$.

$G_{t_1}$ is random at $t_1$ but it's distribution is known.

Hence, the brute force method would be to find the expectation of $Price_{t_1}$ by numerical integration of $(1)$ over the known probability distribution of $G_{t_1}$ (lognormal distribution).

$Price_0 = e^{-rt_1}\mathbf{E}(Price_{t_1})$

$Price_0 = e^{-rt_1}\int_{-\infty}^{+\infty}Price_{t_1}pdf(G_{t_1}(x))dx$

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