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Consistent foreign exchange options

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Consistency in foreign exchange derivatives is being discussed in the below note where we look at the problem from the probability measure perspective. We review option valuation from both sides of the FX contract and conclude that investors' preferences are subject to different probability measures when the FX rate inverts. Following this we prove the validity of Siegel's paradox.

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Introduction

Foreign exchange options are the oldest options in the market with a long history of trading. As such, they have been deeply-researched and are well-understood. Nevertheless, we return to this topic to look at the product consistency, since this may still not be entirely clear. We review this consistency from both - domestic and foreign perspectives and show what adjustments are required to ensure the options are arbitrage-free when the investor's position changes

Foreign exchange options - 1st currency measure

Foreign e change options are financial contracts on FX rate -i.e. rate of exchange of currency 1 fro currency 2. GBP/USD or EUR/USD are examples of such currency pairs. Options are essentially contracts on the future spot FX rate. We will demonstrate the exposition to this subject using EUR/USD exchange rate. This is the rate that sets the exchange equation of $X = [Euro]1

A reader familiar with the equity derivatives market will immediately spot the similarity between these two products. If the equity growth rate under the risk-neutral measure is the risk-free rate r, the equity pays a continuous dividend yield q and the price process is assumed log-normal, this is identical to the FX when we express r and the USD risk-free rate and q as the equivalent EUR risk-free rate.

Looking at his from the USD-perspective, we can express the EUR/USD FX process as:

$$dF = F (r-q) dt + σ F dW$$

This is a well-known log-normal process for the exchange rate where F represents the EUR/USD rate, [Sigma] is the FX rate volatility and W represents a Wiener process under the USD-measure.

Pricing option on this future rate is trivial - this is an option buy [Euro] 1 for K USD and time T. Therefore from the USD-perspective, the option pays: Max[0,F-K] where K is the strike exchange rate. Pricing this option in Mathematica is easy - we build the standard Ito Process for initial value F0.

ipUSD = Refine[
ItoProcess[{(r - q)*F, \[Sigma]*F}, {F, F0}, t], {\[Sigma] > 0, 
F[t] > 0, t > 0}];
{Mean[ipUSD[t]], Variance[ipUSD[t]]}

{E^((-q + r) t) F0, E^(-2 q t + 2 r t) (-1 + E^(t $[Sigma]^2)) F0^2}

The option premium from the USD-perspective is an expectation of the above Ito Process.

usdOpt = Exp[-r*t]*
Expectation[Max[F[t] - K, 0], F \[Distributed] ipUSD, 
Assumptions -> 
F0 > 0 && K > 0 && \[Sigma] > 0 && t > 0 && r > 0 && q > 0] // 
Simplify

-(1/2) E^(-r t) (-2 E^((-q + r) t) F0 + 
E^((-q + r) t)
F0 Erfc[(t (-2 q + 2 r + \[Sigma]^2) + 2 Log[F0] - 2 Log[K])/(
2 Sqrt[2] Sqrt[t] \[Sigma])] + 
K Erfc[(t (2 q - 2 r + \[Sigma]^2) - 2 Log[F0] + 2 Log[K])/(
2 Sqrt[2] Sqrt[t] \[Sigma])])

Foreign exchange options - 2nd currency measure

Now we touch upon a part that is less clear - what if the option buyer (seller) thinks from the the EUR-perspective? This is quite legitimate as option buyers or sellers can have different preferences when entering into the option contract. How do we ensure that the option contract is consistent from each side-perspective?

Let's spell out the EUR investor position by replicating the USD investor side

  • EUR riskless process is dP = P q dt and not dB = B r dt representing USD process
  • The exchange-rate is now 1/F and not F
  • When SDE for the exchange rate from the USD-point of view is the one above, then for the process 1/F this becomes - using Ito lemma:

    f = 1/F;
    ip02 = Refine[
    ItoProcess[{(r - q)*F, \[Sigma]*F, f}, {F, F0}, t], {\[Sigma] > 0, 
    F[t] > 0, t > 0, r > 0, q > 0}];
    ipEUR = ItoProcess[ip02] // Simplify     
    
    ItoProcess[{{(-q + r) F[t], (q - r + \[Sigma]^2)/
    F[t]}, {{\[Sigma] F[t]}, {-(\[Sigma]/F[t])}}, \[FormalX]1[
    t]}, {{F, \[FormalX]1}, {F0, 1/F0}}, {t, 0}]
    

The inverted FX rate (USD/EUR) produces different Ito Process than the one observed on the USD-side. This is clear from the definition below:

$$d(1/F) = (1/F) (q-r+σ^2) dt -σ (1/F) d W$$

Our objective is to find probability measure under which the FX option priced in the first section from the USD-perspective will be identical to the one priced from the EUR-perspective. Let's take all tradable components of the trade: (i) USD risk-free discount factor B , (ii) FX rate EUR/USD F and (iii) EUR discount factor P. Based on this we define:

  • USD-risk-free process converted to EUR: B/F
  • Discounted value of the above : B/ (F P)
    So, we need a multi-dimensional Ito process to model B/(F P)

    ip03 = Refine[
        ItoProcess[{{0, r B, q P}, {F \[Sigma], 0, 0}, 
          B/(P F)}, {{F, B, P}, {F0, B0, P0}}, t], {\[Sigma] > 0, r > 0, 
         q > 0, t > 0}] // Simplify;
    ipEUR2 = ItoProcess[ip03]
    
    ItoProcess[{{0, r B[t], q P[t], (-q B[t] + r B[t] + \[Sigma]^2 B[t])/(
       F[t] P[t])}, {{\[Sigma] F[t]}, {0}, {0}, {-((\[Sigma] B[t])/(
         F[t] P[t]))}}, \[FormalX]1[t]}, {{F, B, P, \[FormalX]1}, {F0, B0,
        P0, B0/(F0 P0)}}, {t, 0}]
    

From the above Ito Formula, we extract two coefficients - drift and volatility of B/(F P) and create new ItoProcess that reflects the changes when FX inversion occurs.

Flatten[ipEUR2[[1]]];
dr = %[[4]] /. {F[t] -> 1, B[t] -> 1, P[t] -> 1}
vl = %%[[8]] /. {F[t] -> 1, B[t] -> 1, P[t] -> 1}
ItoProcess[{dr F, vl F}, {F, F0}, t];
ipEUR3 = ItoProcess[%]

-q + r + \[Sigma]^2
-Sigma
ItoProcess[{{(-q + r + \[Sigma]^2) F[t]}, {{-\[Sigma] F[t]}}, 
  F[t]}, {{F}, {F0}}, {t, 0}]  

It is quite clear that the inverted FX rate process USD/EUR is indeed different to the one observed in the EUR/USD case.

In order to prove this consistency, we need to show that FX call option on EUR/USD from EUR point of view is identical to the one priced from the USD-perspective. So, we need to prove that:

E^(-r t) Subscript[E, USD] ( Max[ Subscript[F, t]-K,0]) = E^(-q t) Subscript[F, 0]   Subscript[E, EUR] (  (1/Subscript[F, t])  Max[1/Subscript[F, t]-K,0] )

This is because the expectation of the option payoff has to be converted back into EUR. All we need to price this option is use the following expectation:

eurOpt = F0 Exp[-q t] Expectation[Max[F[t] - k, 0]/F[t], 
    F \[Distributed] ipEUR3, 
    Assumptions -> 
     F0 > 0 && k > 0 && \[Sigma] > 0 && t > 0 && r > 0 && q > 0] // 
  Simplify

1/2 E^(-(q + r) t) (E^(r t) F0 - E^(q t) k + 
   E^(r t) F0 Erf[(
     t (-2 q + 2 r + \[Sigma]^2) + 2 Log[F0] - 2 Log[k])/(
     2 Sqrt[2] Sqrt[t] \[Sigma])] + 
   E^(q t) k Erf[(t (2 q - 2 r + \[Sigma]^2) - 2 Log[F0] + 2 Log[k])/(
     2 Sqrt[2] Sqrt[t] \[Sigma])])

To finalise this exercise, we compute both option premiums:

usdNum = usdOpt /. {F0 -> 1.35, t -> 0.5, K -> 1.36, \[Sigma] -> 0.2, 
   r -> 0.01, q -> 0.012}
eurNum = eurOpt /. {F0 -> 1.35, t -> 0.5, k -> 1.36, \[Sigma] -> 0.2, 
   r -> 0.01, q -> 0.012}
usdNum - eurNum // Chop

0.070452

0.070452

0

Both option premium are the same. This proves they are consistent.

Siegel's paradox

In the context of the above discussion, it is worth mentioning Siegel's paradox as it directly links the FX processes to probability measures. Let's start again with the definition of FX evolution from the USD-perspective . Under the USD probability measure (USD risk-neutral process) we showed earlier that this was:

$$dF = F (r-q) dt + σ F dW$$

The expected future FX rate - the FX Forward at time t is an expectation of Subscript[F, t] under the USD measure:

usdExp = Expectation[F[t], F \[Distributed] ipUSD, 
Assumptions -> F0 > 0 && \[Sigma] > 0 && t > 0 && r > 0 && q > 0] //
Simplify 
E^((-q + r) t) F0

Let's look now at EUR-investor point of view. (S)he can do similar calculation and under her/his neutral measure the USD/EUR process follows:

$$d(1/F) = (1/F) (q-r) dt + (1/F) σ dW$$

So, the forward rate of 1/F (EUR per USD) is:

eurExp2 = 
Expectation[1/F[t], F \[Distributed] ipEUR3, 
Assumptions -> F0 > 0 && \[Sigma] > 0 && t > 0 && r > 0 && q > 0] //
Simplify 

E^((q - r) t)/F0

This seems logical, since inverted FX is simply :1/F. Here lies the problem: since 1/F is essentially a convex function, by Jensen's inequality:

(E[F])^-1 < E [F^-1]

when both expectations are taken w.r.t to same probability measure = i.e. calculated with the same distribution and F is non-constant. This runs contrary to our assertion above where we outlined the conditions for consistency - i.e. different probability measure.

Siegel's paradox is simply a statement confirming that the spot rate inversion does not extrapolate to the forward space and the forward FX rate in general cannot be an unbiased estimate of future spot FX rate. At least not simultaneously for both sides of the contract due to 'convexity' effect in the inverted FX function. This is due to the Jensen's inequality statement above. If the property holds for the USD-investor, it cannot be true for the EUR investor and vice-versa since their forward expectation are subject to different probability measures.

We prove this on the simple case - define standard Ito process and then take the expectations for for F and 1/F

ip05 = Refine[
ItoProcess[{(r - q)*F, \[Sigma]*F}, {F, F0}, t], {\[Sigma] > 0, 
F[t] > 0, t > 0, r > 0, q > 0}];
usdFwrd = 
Expectation[F[t], F \[Distributed] ip05, 
Assumptions -> F0 > 0 && \[Sigma] > 0 && t > 0 && r > 0 && q > 0] //
Simplify
eurFwrd = 
Expectation[1/F[t], F \[Distributed] ip05, 
Assumptions -> F0 > 0 && \[Sigma] > 0 && t > 0 && r > 0 && q > 0] //
Simplify

E^((-q + r) t) F0
E^(t (q - r + \[Sigma]^2))/F0

We see that the FX forwards are different as their are taken from different probabilities (with different mean and variance). The forward of 1/F depends also on volatility whereas F does not. Let's show the validity of Jensen's inequality: 1/ Subscript[F, USD] and Subscript[F, EUR]

fxMeanDiff = 1/usdFwrd - eurFwrd // Simplify
-((E^((q - r) t) (-1 + E^(t \[Sigma]^2)))/F0)

Since the above quantity is negative, this shows that indeed

(E[Subscript[F, t]])^-1 < E[Subsuperscript[F, t, -1]]

Plot[fxMeanDiff /. {F0 -> 1.35, r -> 0.01, q -> 0.0045, 
   t -> 0.5}, {\[Sigma], 0.1, 0.3}, 
 PlotLabel -> 
  Style["Jensen's inequality and FX forward rates", {15, Bold}, Blue],
  PlotStyle -> {Thick, Red}]

enter image description here

Jensen's inequality effects increases with volatility. On the other hand, the only instance when both forwards will be consistent w.r.t the same probability occurs when [Sigma]=0. Since this is never the case, we conclude that Siegel's paradox holds.

Conclusion

The objective of this note was to present the FX derivatives - forwards and options from different perspective. Whilst the FX spot market is reasonably simple, derivatives are more complicated, especially when we start looking at them from each contractual perspective. Change of probability measure, and hence different probabilities are required to ensure consistency. Existence of Siegel's paradox proves this.

Change of probability measure is handled implicitly by Mathematica once the FX process is correctly defined. The same applies to proving of Siegel's paradox.

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Thanks again Igor for your interesting discussion here on FX options. However in your triple Ito Process for the model B/(F P) given by ip03, you have F with zero drift. However from the first equation in the article we have that F has a drift of (r - q)*F. This changes the value of drift of B/(F P) so that instead of being (r-q+sigma^2) B[t]/(F[t] P[t]), it is now reduced to sigma^2 B[t]/(F[t] P[t]). This has run-on effects on the values of ipEUR2, ipEUR3 and eurOpt. Next the third equation in the article establishing consistency for both sides of the FX call option on EUR/USD is incorrect in that the Radon-Nikodym derivative

dUSD/ dEUR = Exp[(r-q) t]*F0/F[t]

hence the equation should be

E^(-r t) Subscript[E, USD] ( Max[ Subscript[F, t]-K,0]) = E^(-q t) Subscript[F, 0]   
Subscript[E, EUR] (  (1/Subscript[F, t])  Max[Subscript[F, t]-K,0] ) 

and this is also consistent with your definition of eurOpt. With all of these changes we get a difference between the EUR and USD versions of the FX option of -6 10^(-4), small but not zero, but this is consistent with Jensen's inequality.

I may have missed something in the subtlety of the arguments but the SDE definition of F[t] is explicitly defined at the outset. Thanks again for any help in explaining this discrepancy

Michael

Hello Michael Thank you again for your comments and observations. 1) I start from zero-drift to show how the real drift is actually obtained. In FX market this is usually a function of domestic and foreign rate. This is described both from USD and EUR investor points of view

2) Consistency equation you refer to is merely a description of non-arbitrage condition for FX options when investor position changes. LHS of the equation is implemented through the usdOpt formula (call option on USD/EUR) whereas the RHS is calculated in eurOpt which is the option on the inverted FX rate. RHS is particular was done this way to highlight the case we were dealing with the inverted rate.

To assess the consistency, pls look at both option premium formulas - usdOpt and eurOpt.. As I have shown, the premiums are the same , hence no arbitrage exists when investor position changes. This shows formulas are consistent.

Hope this answers your queries. Igor

Hello Michael

I will try to be clearer to eliminate any confusion:

1) FX options - 1st currency measure This is the 'standard' FX option pricing formula for USD/EUR - i.e. option on buying €1 for K dollars and this is implemented through usdOpt formula. This is standard lognormal ito process with mean = F[0]Exp[(r-q)T] and volatility = sigma*Sqrt[T]. This is essentially the same B/S formula as the equity option on the stock paying continuous dividend q.

When I talk about the options pricing consistency, the LHS of the equality refers to the above FX option - i.e. usdOpt formula

2) The Ito triplet you refer to in your comment is quite different - essentially unrelated to the above. Therefore it cannot have any run-on effect on usdOpt formula above. This uses standard Ito process ipUSD - different from other processes derived delow.

The entire second section deals with derivation of a probability measure when the investor position changes from being USD to the one governed by EUR. Although I use the same notation, the process for the inverted FX rate is different and it results in a different Ito SDE where both location and scale parameters are different. I started with zero drift for the second derivation of FX process from practical point of view, knowing that the process is martingale and hence the drift is zero. I could have started with any drift - say mu and then drop it the final SDE formulation.

The whole point of section 2 is to show that the Ito SDE for (1/F) is indeed different and requires different probability measure.

The point of the article is to prove that investor's stance (USD or EUR) cannot lead to arbitrage when the option to buy €1 for K USD is priced. LHS of the consistency equation is the usdOpt, the RHS is the eurOpt. To prove they are consistent we need to show the premiums are the same. usdNum - eurNum shows this is the case.

So, any correctness or not should be just based on this relationship - i.e. comparison of standard (well-known) FX option against the same option evaluation from the second currency measure. The rest are just partial steps to come to the final result.

Hope this clarifies my point.

Igor

Thanks very much Igor for this second explanation. I strongly appreciate your going into the details here. Your comment that "Although I use the same notation, the process for the inverted FX rate is different and it results in a different Ito SDE where both location and scale parameters are different. I started with zero drift for the second derivation of FX process from practical point of view, knowing that the process is martingale and hence the drift is zero." This explains the points that I was misunderstanding: that while you are using the same notation, the actual real world process is different in the second Euro-side case and because it is a martingale then it has zero drift.

However I do feel that in order to avoid misunderstanding initially, it would have been good to have included these observations in the original text, so that literal minded mathematicians like myself don't get led astray.

Regards Michael

Thank you, Michael, for your comment. You have actually raised a valid point - I should have made this explicit in the article. I am taking this as a recommendation for my future work, and will try to make it clear from the very beginning.

Thank you for your valuable contribution to this discussion!

Igor

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