FX Numeraire Switching and the Call–Put Price Relationship
Summary
The note explains how changing the pricing numeraire transforms a call on an exchange rate into a put on its reciprocal. The derivation starts with the domestic-currency discounted expectation of the call payoff, rewrites the payoff by factoring out the exchange rate, and uses the exchange-rate-adjusted foreign discount process as a change-of-measure density. After changing to the foreign measure, the transformed payoff is expressed as a put on the reciprocal exchange rate with reciprocal strike. The resulting price includes both the initial spot conversion and the strike multiplier.
This relationship clarifies why simply converting a quoted option value by the spot rate can give a mismatch: the strike factor also matters, and currency denomination and measure affect the interpretation of prices and deltas. The source frames the issue through an example and a risk-neutral expectation argument. Its formula relies on consistent definitions of the exchange-rate direction, domestic and foreign discounting, and payoff units; the example’s zero-rate assumption does not remove the need to track those units.
Key ideas
- Changing the numeraire changes the probability measure used to value the payoff.
- An exchange-rate call can be represented as a put on the reciprocal rate and strike.
- The transformed price carries a spot conversion factor and a strike multiplier.
- Currency denomination affects how prices and deltas should be interpreted.
- Careful rate, quote-direction, and unit conventions are essential when applying the identity.
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Full text
# Numeraire flipping and currency FX options
# Numeraire flipping and currency FX options
In Dynamic Hedging by N. Taleb at pag. 435 is stated that:
> Numeraire flipping consists of switching the unit in which the numeraire is expressed from base currency to counterasset. Thus, a call on S with strike K, risk neutral rate rd and counterasset rate d can be priced as a put on 1/S with strike 1/K, risk neutral rate d and counterasset rate rd.
Since I can't obtain the price of a call as stated in the "numerarie flipping principle", I am not able to figurate out by myself the 3 critical points reported below. Could someone explain to me, with some math if possible, what is happening?
Please, let me know if more details are needed. Thanks for the help.
Edit
B&S formula: If I understand right, I should get the same price by calculating, assuming zero rates for the domestic and foreign rates:
$$Call(S,K,T) = SN(d1)-KN(d2)=\frac1KN(-d2')-\frac1SN(-d1')=Put(\frac1S,\frac1K,T)$$
Example: S=5 USD-EUR (1 USD = 5 EUR), K=5.6 USD-EUR, TTM=180, vol=15.7%. I get call=0.0460 and put=0.0016
The call is priced in EUR so I should divide by 5, the spot rate? I get call=0.0092. Still different.
What am I missing? Am I violating something by taking zero interest rates?
Edit 2
This part is moved to another more specific question. I keep it also there since there is some comments about it and is related to the flipping principle.
From this principle the author in the book highlights some critical points:
- The delta of the call and put obtained through this principle will be different (p. 435)
- A bet in dollars for a dollar based person on USD-DEM is different in price from the translation into German marks of a bet in German marks on USD-DEM of the same strike and expiration (p. 286).
- The price of a bet for a dollar-based person, which is N(d2), is different from the price of the bet for the person based in DEM (for example) as this latter will be N(d1) (p. 286).
## Answer by Rylan (score 3, accepted)
https://quant.stackexchange.com/a/80464
Looking at the example above after the edits. The short answer is you need to multiply by $K$ as well as the FX rate.
For the longer answer: Lets let $D^e, D^u$ be the discount processes in the different currencies. Like your example, S(t) refers to the price in EUR of one USD. Let's We consider the EUR-denominated call on one USD $C(t)$. I'll calculate it at time $t=0$ to make the notation a bit lighter.
$$C(0) = \mathbb{E}^e(D^e(T)(S(T) - K)^+)$$
"multiplying by one" strategically, this gives
$$C(0) = \mathbb{E}^e\Big(D^e(T)\frac{D^u(T)}{D^u(T)}(S(T) - K)^+\Big)$$
Factoring out $S(T)$ from the positive part expression we get
$$C(t) = \mathbb{E}^e\Bigg(D^e(T)\frac{D^u(T)}{D^u(T)}S(T)\Big(1 - \frac{K}{S(T)}\Big)^+\Bigg)$$
Note that $\frac{S(t)D^u(t)}{D^e(t)}$ is an $e-$martingale (an arbitrage argument ensures that the European market risk-neutral drift of $S(T)$ is the difference between the EUR and USD rates.), so we can use it to define a new measure, which we'll call the $u$ measure.
Again multiplying strategically by one, we get
$$C(0) = \frac{S(0)D^u(0)}{D^e(0)}\frac{\mathbb{E}^e\Bigg(D^u(T)\frac{S(T)D^e(T)}{D^u(T)}\Big(1 - \frac{K}{S(T)}\Big)^+\Bigg)}{\mathbb{E}^e\Big(\frac{S(T)D^u(T)}{D^e(T)}\Big)}$$
Changing the measure gives us:
$$C(0) = \frac{S(0)D^u(0)}{D^e(0)}\mathbb{E}^u\Bigg(D^u(T)\Big(1 - \frac{K}{S(T)}\Big)^+ \Bigg)$$
and factoring out the $K$ we get
$$C(0) = \frac{KS(0)D^u(0)}{D^e(0)}\mathbb{E}^u\Bigg(D^u(T)\Big(\frac{1}{K} - \frac{1}{S(T)}\Big)^+ \Bigg)$$
Which simplifies to
$$K \times S(0) \times \text{USD-denominated put}$$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.