Author(s): Douglas T. Breeden and Robert H. Litzenberger. Source: The . ( ). They use the Black-Scholes model in a sequential manner to estimate. The approach of Breeden-Litzenberger is being used to estimate tail risks and risk neutral densities in practice. 2. Time spreads of interest. The Breeden and Litzenberger result. .. Breeden & Litzenberger allows us to convert the implied volatility function into an implied risk-neutral density. Shimko Breeden, D. T., & Litzenberger, R. H. (). Prices of.

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Exchange Rate Risk Premium: The objective of this paper is to analyze what are the main determinants of the exchange rate risk premium ERP. The empirical case is conducted for the daily Mexican peso-USD for a sample period from until C22, C53, C58, G10, G It is recognized that the analysis of exchange rate dynamics has become a crucial financial topic, especially, in a world becoming more integrated in its international financial markets.

Exchange rate movements through time may influence consumption and investment decisions from economic agents.

### EconPapers: Prices of State-contingent Claims Implicit in Option Prices

Some of these observed fluctuations in currencies throughout the world may be related to several economic factors. Among these there are risk factors, which may be consider relevant components of exchange rate variations. Within this concept the exchange rate risk premium is defined as the expected future spot exchange rate minus the current forward exchange rate Hakkio and Sibert: The difference between these two prices may justify an additional amount of money that an investor will demand in order to be compensated for an expected depreciation of a currency, in which the investor holds a long position.

Another definition is that of a monetary amount that is not in line with fundamental equilibrium conditions in exchange rate markets Frankel and Chinn: It is important to point out that the ERP is not a standard concept in the relevant financial literature, so there are several ways to quantify it.

For different reasons, it may be important to some economic agents to quantify the previously defined exchange rate premium ERP. Again, this risk measure is a relevant one to look for, especially, for agents taking economic or policy decisions based upon current financial information from Foreign Exchange FX markets. These agents could be financial FX investors, risk managers, hedge fund managers, individual investors, policy makers, among others. In addition, from a more fundamental analysis, academic researchers could also find relevant to understand in more detail the ERP, including its dynamics from theoretical or empirical concepts.

The purpose of the present research paper is twofold. One is to estimate the ERP for the Mexican peso-USD exchange rate using novel methods applying the estimation of Risk-neutral densities, which include financial derivatives options and forward prices.

The other one is to analyze what are the main drivers of the previously mentioned ERP. These econometric models were chosen in order to provide the estimations that will show the possible statistical relationship between the variables under study. According to the estimations and the results presented in this current research document, it appears that the ERP for the exchange rate under study has been decreasing since the end of the last Great Recession.

The VAR estimations are consistent qualitatively speaking with the results obtained from the linear regression models. It is then considered an additional contribution to apply a method in the present research document, which up to date, is breeven commonly applied for the Mexican peso USD exchange rate.

Further research about the theoretical justification of these empirical analyses are suggested for future understanding of this topic. Most of these results are in line with the literature that shows that uncertainty in financial markets specifically financial market volatility are the main factors that impact risk premiums including those for exchange rates.

The layout of the rest of this research paper is as follows. Section 2 presents a formal definition of the ERP and then details about the applied methodology used to estimate the ERP including the econometric models are given.

In Breedn 3 details about the data and regression analysis are presented. In this section there is a detailed explanation of the Risk-Neutral density concept, which is relevant in order to estimate the ERP.

11978 in the RiskNeutral density explanation the Volatility Function Technique method is also presented. This latter method is used mainly for exchange rate options, given that, as it will be shown, it considers options strategy information in order to extract the expectations of the FX.

Lastly, the procedure related to the estimation of the ERP is presented.

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The RND is a set of expectations, which are estimated from traded option prices and are presented in a form of a statistical density i. The idea to estimate RNDs implied by option prices was first postulated by Breeden and Litzenberger The main reason to do this was the belief that there is a rich source of forward-looking financial information in derivatives markets.

A way to extract this information is by estimating an implicit probability distribution of an asset from the observed prices of option contracts on such an litzenbwrger, which are traded in these financial markets.

This of course does not mean strictly speaking that investors are assumed to be risk-neutral. The RND can be litzenbwrger for any asset including currencies. Breeden and Litzenberger brreeden that the RND can be extracted from the prices of call or put options, which are usually traded in the market.

In particular the value of a call option is defined as. Then, breeden calculating the second partial derivative of the call price function c X, Twith respect to the exercise price X we obtain. The problem with defining f X is the assumption that the call price function is continuous for the range of exercise prices.

As it is known, this is not realistic given that in practice only some prices in discrete time are available or observed. Considering this limitation, Shimko proposed an interpolation method using the available exercise prices. In subsequent research, Malz proposed to interpolate across implied volatilities using the framework of Garman and Kohlhagen: In this case, the delta has to pass through at least three points of the volatility smile as it will be explained in more detail in Subsection 2.

This is because traders trade quoting implied volatility as a function of delta. Therefore, there is always hard data available for implied volatility, which can be used for a smoother interpolation. RNDs estimations not only give a point estimate forecast about the moments of a specific underlying asset, but also provide information about the whole asset price distribution expected by the market.

Hence, extracting a RND provides information about market sentiment. For example, if an exchange rate shows RNDs with skewness that are systematically positive through time, the interpretation is that the market is expecting one of the currencies to depreciate or keep depreciating in the near future. The volatility function technique VFT was originally postulated by Malz He extended the idea proposed by Shimkoin which, the application of interpolation methods to exercise prices allows to recover the RND.

Several studies have applied the VFT. After an extensive comparison with other estimation methods, they concluded that the VFT approach shows better goodness-of-fit and stability of the parameters. However, they found that the tails of the RNDs were significantly unstable for the methods they analyzed VFT and the mixture of lognormals.

Similarly, Benavides and Mora found qualitatively similar results for both methods, but applied for the Mexican Peso-USD exchange rate.

This concept is not innocuous given that an unstable tail could make difficult the Value-at-Risk analysis. The principal difference between Benavides and Mora paper and the present research document is that, the former, does not estimates the ERP whilst for the latter, there is a methodological procedure to estimate the ERP. Among other methods, he applied the VFT method and concluded that there is a tradeoff between goodness-of-fit accuracy and stability of the parameters.

To understand this procedure with more detail it is convenient to refer to the equivalent Black and Scholes BS model for exchange rates, which was postulated by Garman and Kohlhagenhenceforth, GK. The underlying assumptions of this option lltzenberger model for exchange rates are the following: While some of these assumptions appear quite strong, there is still interest to apply such models to estimate RNDs.

This is because these approximations provide useful information about market litzenebrger that could give us some feedback about dynamic features of a specific financial asset, litzenbrger around an economic event.

The GK model is therefore:. If observed option prices in the market are used instead of the theoretical ones, it is possible to implicitly litzenbergsr the probability distribution that was relevant for the agents when they traded the options.

In this research project, implied RNDs are derived from nearby to expiration over-the-counter three month option contracts on the MXN-USD exchange rate for each trading day. Thus, instead of assuming a standard cumulative normal N xas it is shown in Equations 4 and 5 above, the RND is implicitly extracted from the model using the observed option values with the additional variables.

With the estimated call and put prices, the RND can be extracted by applying the previously defined Breeden and Litzenberger approach. The approach of Malz estimates a RND by interpolating the smile curve. Specifically, interpolation can be carried out in terms of the implied volatilities determined from market expectations. The considered implied volatilities are: For exchange rates these were taken from market traders. The implied volatilities from the above mentioned option strategies for a 25 delta call and put option can be theoretically obtained as follows.

Let rr be defined as. Once this curve is obtained a transformation is performed in which the implied volatility can be expressed in terms of exercise price X and not in terms of the delta.

Thus, the definition of the delta function is now as follows:. Equation 9 is substituted into Equation 8 and then Equation 10 below is obtained. Equation 9 is substituted into Equation 8 and then Equation 10 below is obtained, where a1 is equal to. To estimate the density function for the underlying asset the Breeden and Litzenberger result is applied here. Thus, by substituting these expressions into Equation 3, it is then possible to estimate the probability function for the underlying asset, which is expressed as follows:.

Finally, by using different values of Xit is then possible to extract the RND through option prices. As explained in the introduction, the idea behind quantifying the ERP is related to that amount of money investors should be compensated given systematic depreciations of a currency they hold as a long position. This last component is related to market inefficiencies in which, there is a skewness toward depreciation of one of the currencies.

Basically, the ERP is that bias towards the depreciation of one of the currencies that in most of the cases, is the one of the emerging economy with respect to the developed one.

The rest of the notation is the same as stated above. It is known that the standard formula for a currency forward, given no-arbitrage pricing is Hull: By combining the last two equations it is possible to obtain. It can be seen from the above expression that the implied depreciation on forward contracts is determined by the difference between internal and external interest rates. This shows why the probability functions of risk-neutral densities shown for some currencies e.

This implies that the proposed premium corresponds to the additional yield that should be received by a risk-neutral investor by maintaining an asset denominated in a weak currency pesos with a bias to its depreciation that offers a lower expected return in foreign currency US dollars. Graph 1 shows graphic details about the ERP extraction. Own estimations with data from Bloomberg and UBS. As it can be observed in that figure, the ERP can be estimated as the percentage difference of the mean values of the previously mentioned probability densities lognormal and the RNDs.

Thus, ERP can be extracted implicitly from information about exchange rate option prices.

## Prices of State-contingent Claims Implicit in Option Prices

It is important to point out, that the estimation of the ERP is based in a parsimonious approach related to interest rate parity. It is understood that there may be other risk premiums related to Equation 12, possibly related to currency liquidity or counterparty risk, however, since the Mexican peso is a relatively highly traded currency there are Mexican peso futures contracts at the Chicago Mercantile Exchangeit is assumed that litzenberer liquidity risk is relatively small.

Also, it is assumed that counterparty risk is close to zero given that in the wholesale market for the Mexican peso, most of the trading is carried out by well-established financial institutions. Most of the data for the estimations bbreeden from the Bloomberg database.