Option-Implied Currency Risk Premia - Princeton University

More documents

Recommendations

Info

To derive the CGF for the corresponding time-changed increments we rely on Theorem 1 in Carr and Wu (2004). In our setup, the time-change is controlled by pre-determined state variable, S t ∈ {Z t , Y i t }, which allows the model to have nonidentically distributed innovations over time. Theorem 1 of Carr and Wu (2004) states that for a generic time change, T , the cumulant generating function of the time-changed Lévy process, L T , is given by k T [k L [u]], where k L [u] is the cumulant generating function of the non-time-changed process and k T [u] is the cumulant generating function of the time-change. In our case, the time-change variables are fixed within the measurement interval (i.e. they follow a degenerate stochastic process with zero drift and volatility), such that: [ ] k L j [u] = k St k S L j [u] = k t t L j [u] · S t t (A.6) Consequently, the second and third cumulants of the time-changed increments are given by: κ 2 L j S t = (( ) ) 1 − η j t + κ 2 X · S t = S t (A.7a) κ 3 L j S t = ( 0 + κ 3 X) · St (A.7b) Unlike in a more traditional stochastic volatility, both the variance and skewness of the total time-changed Lévy increment are time-varying and governed by the state variable, S t . The variance is given by: κ 2 L j S ) t − 3 κ 3 · (κ 2 2 κ L j S L = j √ 3 X St . t S t = S t , and the skewness is given by: 39
References [1] Ang, A. and J. Chen, 2010, Yield Curve Predictors of Foreign Exchange Returns, Columbia University, working paper. [2] Backus, David K., Silverio Foresi and Chris Telmer, 2001, Affine Term Structure Models and the Forward Premium Anomaly, Journal of Fianance, 56(1), p. 279-304. [3] Bakshi, Gurdip, Carr, Peter, and Liuren Wu, 2008, Stochastic Risk Premiums, Stochastic Skewness in Currency Options, and Stochastic Discount Factors in International Economies, Journal of Financial Economics, 87, p. 132-156. [4] Bakshi, Gurdip, Nikunj Kapadia and Dilip Madan, 2003, Stock Return Characteristics, Skew Laws and the Differential Pricing of Individual Equity Options, Review of Financial Studies, 16(1), p. 101-143. [5] Barro, Robert, 2006, Rare Disasters and Asset Markets in the Twentieth Century, Quarterly Journal of Economics, 121, p. 823-866. [6] Black, Fischer and Myron Scholes, 1973, The pricing of options and corporate liabilities, Journal of Political Economy, 81, p. 617-654. [7] Brandt, Michael, Cochrane, John, Santa-Clara, Pedro, 2006. International risk sharing is better than you think, or exchange rates are too smooth, Journal of Monetary Economics, 53, p. 671698. [8] Breeden, Douglas and Robert Litzenberger, 1978, Prices of state-contingent claims implicit in option prices, Journal of Business, p. 621-651. [9] Brunnermeier, Markus, Stefan Nagel and Lasse Pedersen, 2009, Currency Crashes and the Carry Trade, NBER Macroeconomics Annual 2008, 23(1), pp. 77-100. [10] Burnside, Craig, Martin Eichenbaum, Isaac Kleshchelski and Sergio Rebelo, 2011, Do Peso Problems Explain the Returns to the Carry Trade?, Review of Financial Studies, 24(3), p. 853-891. [11] Caballero, Ricardo J., and Joseph Doyle, 2012, Carry Trade and Systemic Risk: Why Are FX Options So Cheap?, NBER Working Paper No. 18644. [12] Campbell, John Y., 1987, Stock Returns and the Term Structure, Journal of Financial Economics, 18(2), 373399. 40
Page 1 and 2: Option-Implied Currency Risk Premia
Page 3 and 4: a closed-form expression for the ge
Page 5 and 6: We conduct a similar analysis for r
Page 7 and 8: the determination of currency risk
Page 9 and 10: Y j t parameter can be interpreted
Page 11 and 12: where S ji t is the currency exchan
Page 13 and 14: where V g t = κ2 L g and S g t =
Page 15 and 16: coupon bond in country i; otherwise
Page 17 and 18: 2 Data and Model Calibration We cal
Page 19 and 20: How does the identification intuiti
Page 21 and 22: account for the fact that the avera
Page 23 and 24: √ δ requires loadings to range f
Page 25 and 26: volatilities (blue) and their fitte
Page 27 and 28: isk premium can be computed on the
Page 29 and 30: 3.2.1 Slope (carry) factor To const
Page 31 and 32: 3.2.2 Short U.S. dollar factor Lust
Page 33 and 34: Figure 7 contrasts the ability of o
Page 35 and 36: in fact, largely orthogonal to fact
Page 37 and 38: etter than under Specification I. T
Page 39: A Cumulant Generating Functions of
Page 43 and 44: [27] Ghysels, Eric, Pedro Santa-Cla
Page 45 and 46: Figure 2. Cross-Sectional Relation
Page 47 and 48: Figure 4. Returns to Empirical Fact
Page 49 and 50: Figure 6. Option-Implied Currency R
Page 51 and 52: Table I Calibrated Model Parameters
Page 53 and 54: Table III Model Short Reference Cur
Page 55 and 56: Table V Option-Implied Currency Ris
Page 57 and 58: Table A.I State-variable Dynamics a
Page 59 and 60: Table A.III Option-Implied Currency

Option-Implied Currency Risk Premia - Princeton University

Create successful ePaper yourself

Delete template?

Save as template?