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Forward-Starting Options Require a Joint Distribution Across Dates

Article Quant Q&A · Author: R. Rayl
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Summary

The document asks whether vanilla options at two maturities can replicate a payoff based on the ratio of an underlying’s later value to its earlier value. The response identifies the contract as a forward-starting option and explains why prices of ordinary calls across strikes and maturities are insufficient on their own: they reveal marginal distributions at each date, but not how the values at the two dates move together.

Valuation therefore needs information about the joint distribution, or an assumption that supplies the intertemporal dependence. The answer describes expected forward volatility as an equivalent way to express the missing information and notes that correlation between dates affects that volatility. A copula could be used to impose dependence and integrate the payoff when a model-based price is needed. No specific model, calibration, market data, or numerical valuation is provided, so the result depends on assumptions about the relationship between the two observations.

Key ideas

  • Vanilla option prices across maturities determine marginal distributions, not the joint distribution across dates.
  • A payoff based on the ratio of future values is a forward-starting option.
  • Pricing requires an assumption or estimate for intertemporal dependence between the underlying values.
  • Forward volatility can express the dependence relevant to the option’s value.
  • A copula can impose a joint distribution, but the resulting price depends on that modeling choice.

Tags

OptionsDerivatives pricingVolatilityStatistics
Full text
# Ratio of the same process at different times


# Ratio of the same process at different times












Suppose I have an adapted process $X_t$. I have available option prices on $X_t$ for a range of strikes and maturites. In particular, I have

$$ C_0(K, T_1) = D(T_1)\mathbb{E}_Q[(X_{T_1} - K)_+], $$

and

$$ C_0(K, T_2) = D(T_2)\mathbb{E}_Q[(X_{T_2} - K)_+] $$ where $D(t)$ is the discounting factor (and I am assuming determinisitc IR). I want to synthesise an option with payoff $$ \bigg( \frac{X_{T_2}}{X_{T_1}} - K^\prime \bigg)_+ $$ at time $T_2$ (assuming $T_1$ < $T_2$). Is this possible using the available option prices?

Additional Info:

- I am aware of Margrabe's formula for the pricing of an option on the ratio of two different assets, but I don't think this can be applied here?

- Is there any literature on options like this? It is hard to search for something when you don't know the technical name.

Thanks.

## Answer by Arshdeep (score 4, accepted)

https://quant.stackexchange.com/a/57143

This is a forward starting option. It's price cannot be determined by the vanillas you have because vanillas only determine the marginal distributions of the random variables - whereas to price this one, you need to somehow produce a joint distribution (correlate $X_{T1}$ and $X_{T2}$).

An equivalent way of doing the same thing is the following. You need to be able to have a good forecast of the forward volatility (i.e., what price would a vanilla option with maturity $T2$ trade on date $T1$?). The expected value of this volatility will determine the price of this option. Note that this is equivalent to specifying the joint distribution in paragraph one, because talking in terms of forward volatility is the same as talking in terms of (intertemporal) correlation. High correlation means low forward vol and vice versa.

Edit: In case you just want to spit out some price without caring too much about fair value, you may correlate these by a copula, and integrate the payoff.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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