1. Bounds and Prices of Currency Cross-Rate Options
San-Lin Chung
National Taiwan University
Co-authored with Yaw-Huei Wang
National Central University

2. Motivation
Target:
Option market’s information is believed to be more efficient
Thus we want to price (or provide pricing bounds) cross-rate options using the information of option markets
Problem:
The historical information of cross-rate option prices is lack or not reliable because cross-rate option markets are either much less liquid or inexistent

3. Motivation Observations:
Options on dollar-rates are usually traded with satisfactory liquidity
There is a triangular relationship between the three currencies.
Conjectures:
Is it possible to bound or to price cross-rate options using the price information of dollar-rate options?

4. Motivation Solutions:
This paper derives pricing bounds of a currency cross-rate option using the option prices of two related dollar rates via a copula theory - Fréchet bounds.

5. Contributions and Main Findings
Our option pricing bounds are useful because
they are general in the sense that they do not rely on the distribution assumptions of the state variables or on the selection of the copula function;
they are portfolios of the dollar-rate options and hence provide potential hedging instruments for cross-rate options; and,
they can be applied to generate bounds on deltas.

6. Contributions and Main Findings
We found that:
there are persistent and stable relationships between the market prices and the estimated bounds of the cross-rate options
our option pricing bounds (obtained from the market prices of options on two dollar rates) and the historical correlation of two dollar rates are highly informative for determining the prices of the cross-rate options

7. Option pricing bounds
The arbitrage-free approaches:
Different portfolio compositions of the underlying stock and options should not produce the existence of a dominant portfolio.
Merton (1973), Garman (1976), Levy (1985) etc.
The linear programming methods:
A linear programming problem with a discrete state space.
Ritchken (1985), Ritchken and Kuo (1989), etc.
Most of the previous methods investigate the price behavior of the options and/or their underlying assets.

8. Methodology A €/£ option with strike K
= a European option to buy £1 for €K
= an option to exchange KS$/€ dollars for S$/£ dollars (under $ measure)
Payoff in dollars = max(S$/£ - KS$/€, 0)
= S$/£ - max[min(S$/£, KS$/€), 0]
Minimum option

9. Methodology
Let Pr denote the probability, Fi(u) the CDF, and r the risk-free rate under $ measure,
Using the Frechet bounds to bound the price of the minimum option and thus the exchange option

10. Upper bound
Proposition 1. The upper bound of the cross-rate option price in dollars is as follows:
where K** is a constant satisfying that
is the cumulative distribution function.

11. Lower bound Let K* be a constant which solves
Then the lower bound of the cross-rate option price in dollars is as follows:

12. Data
Options prices quoted as Black-Scholes implied volatilities for the $/£, $/€ and €/£ exchange rates.
Source: A confidential file of OTC option price mid-quotes, supplied by the trading desk of an investment bank.
Period: from 15 March 1999 to 11 January 2001
Seven exercise prices for each day. Delta: 0.1, 0.25, 0.37, 0.5, 0.63, 0.75 and 0.9.
Time to maturity: 1 month

13. Empirical designs for generating the bounds
Use the observed market prices of options on $/£ and $/€ and a distribution specification to estimate their risk-neutral densities (RND)
We use GB2 distribution to estimate the RND.
We then determine K* and K** and price the dollar-rate options with strikes K* and K** using the estimated RND.

14. Volatility smile

15. Persistent relationship between the market price and our bounds

16. The explanation power of our pricing bounds
Model 1:
For Model 1,
β1 and β2 are positive and highly significant.
R2 :72%-77%

17. The explanation power of our pricing bounds and correlation
Model 2:
Correlations: DCC (Engle, 2002)
β3 is negative and highly significant.
Corr ↑=> Var(€/£) ↓=> call price ↓
All R2 (83%- 87%) are raised (about 10%) substantially.

18. Estimates for Model 1 & 2

19. Cross-rate options pricing
Given the estimated parameters, we can infer the current implied volatility for options on €/£ from current market prices of dollar-rate options and the DCC correlation of two spot dollar-rates.
The average errors are smaller than the bid-ask spread in the OTC market. (0.32%-0.36%).
The volatility of the errors are very small as well. (0.29%-0.33%)

20. Cross-rate options pricing

21. Delta bounds

22. Robustness checks Methods for dollar-rate RNDs Sample selections
Using log-normal mixtures produce almost the same bounds.
Sample selections
Two evenly divided sub-samples.
Same pattern
Volatility, skewness and kurtosis levels
Regressions based on an AR(1) model
No clear evidence for the relationships between pricing errors and volatility, skewness or kurtosis. (BKM2003)

23. Concluding Remarks & Contributions
Provide an alternative for looking at the option bounds for currency cross-rates.
Derive pricing bounds for currency cross-rate options, which are very general and do not rely on the distribution assumptions of the state variables.
Our pricing bounds are of economic meanings because they are composed by portfolios of the dollar-rate options and spot dollar-rates.
Provide an effective approach for inferring prices of currency cross-rate options.