Quanto Factor as the Fixed Exchange Rate in a Quanto Forward

Summary

The document clarifies the meaning of the quanto factor in a discounted valuation expression for a forward payoff converted into another currency. The factor is a fixed exchange rate agreed in advance, rather than an exponential adjustment based on correlation and volatility. For example, a payoff linked to a euro-denominated equity index but paid linearly in US dollars uses a constant exchange rate with EUR/USD units to convert the index-linked amount.

The explanation also points out a terminology issue: the displayed expression is discounted, so it represents the present value of the forward contract rather than its forward value. The answer is conceptual and supplies no derivation, calibration procedure, or numerical example. Its main practical takeaway is to distinguish a contractual quanto conversion rate from stochastic FX exposure and to track the currency units in the payoff.

Key ideas

• The quanto factor is a fixed exchange rate set in advance for converting the payoff into another currency.
• Its units reflect the conversion direction, such as EUR/USD for a euro-linked payoff paid in dollars.
• A discounted forward valuation expression gives a present value, despite potentially ambiguous naming.
• The explanation is conceptual and does not provide a calibration method.

Tags

# Quanto Factor - FX Forward


# Quanto Factor - FX Forward












In the paper Quanto Options by Uwe Wystup the "quanto factor" $Q$ is used to describe forwards/options when quanto-ed into a different currency, e.g.

$$\text{Quanto Forward Value} = Q \cdot e^{-r_Q T}\cdot \phi \cdot (S_0 \cdot e^{\tilde\mu T}-K)$$

where $K$ denotes the strike, $T$ the expiration time, $\phi=\pm1$ the usual long-short indicator, $S_0$ the underlying and $Q$ the quanto factor.

The paper never explicitly says (or at least I don't see it), what $Q$ is. Do we have something like $$Q = e^{\rho \cdot \sigma_{1} \cdot \sigma_{2}}$$ ?

## Answer by Randor (score 1)

https://quant.stackexchange.com/a/60702

as noob2 wrote, Q is the pre-agreed on (ie fixed in advance) fx rate , ie it is the guaranteed exchange rate. eg if your underlying were the DAX index and you wanted your payoff to be linear in the DAX, but in USD , then Q would be some constant number and it's units would be EURUSD , and so the "Quanto Forward Value" of your equation would be in USD.

PS, the "Quanto Forward Value" is a bit of an ambiguous name since it is discounted, ie it is a spot value, not forward!. It is the Spot value (ie present value) of a forward contract.

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