Bounds and Prices of Currency Cross-Rate Options

San-Lin ChungNational Taiwan University

Co-authored with Yaw-Huei WangNational Central University

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

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?

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.

),min(),()0,1max( vuvuCvu

Contributions and Main Findings

Our option pricing bounds are useful because (1) 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;

(2) they are portfolios of the dollar-rate options and hence provide potential hedging instruments for cross-rate options; and,

(3) they can be applied to generate bounds on deltas.

Contributions and Main Findings

We found that:1. there are persistent and stable relationships

between the market prices and the estimated bounds of the cross-rate options

2. 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

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.

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

),,0,,( €/$£/$min

)(£/$£/€$

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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

0

€/$£/$)(€/$£/$min )),Pr(min(),,0,,( dxxKSSeTtKSSCall tTr

,))(),((

),Pr(

€/$£/$0

)(

0

€/$£/$)(

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dxxKSxSe

KSS

tTr

tTr

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.

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1)()( ****€/$£/$ KFKF

KSS)(1)( xFxF ii

)(xFi

Lower bound

• Let K* be a constant which solves

Then the lower bound of the cross-rate option price in dollars is as follows:

)()( **€/$£/$ KFKF

KSS

. ),,,(),,,(

,

)()( ),,,( ),,,(

*£/$'€/$

)(€/$)(£/$

*

'€/$*£/$

£/€$

€£

€/$£/$

otherwiseTtKSCallTtKSKCall

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Kufor

uFuFifTtKSCallKTtKSCall

CalltTr

ttTr

t

KSS

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, 0.2

5, 0.37, 0.5, 0.63, 0.75 and 0.9.• Time to maturity: 1 month

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.

Volatility smile

Average Implied Volatilities

0.08

0.09

0.10

0.11

0.12

0.13

90 75 63 50 37 25 10

Delta

Impl

ied

Vol

atili

ty

USD/GBP USD/EUR EUR/GBP

Persistent relationship between themarket price and our bounds

The explanation power of

our pricing bounds • Model 1:

• For Model 1,– β1 and β2 are positive and highly significant.

– R2 :72%-77%

tttt LBUBcMIV 21

The explanation power ofour 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.

ttttt CorrLBUBcMIV 321

Estimates for Model 1 & 2

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%)

Cross-rate options pricing

Delta bounds

Robustness checks

• Methods for dollar-rate RNDs– Using log-normal mixtures produce almost the sam

e 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)

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.