Three prices of three risks: A real world measure IR-FX hybrid model

Transcription

1 Three prices of three risks: A real world measure IR-FX hybrid model Alexander Sokol* Head of Quant Research, CompatibL *Includes material from a recent paper by Hull, Sokol, and White and Risk Magazine, October, 2014 WBS Fixed Income Conference, Paris October 8, 2015 Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 1 / 49

2 Outline For large multi-currency portfolios, the evolution of FX rate is frequently the primary risk driver At the long time horizons of interest in limit and certain capital calculations, the FX rate must be modelled jointly with the interest rates The real world drift of the FX rate is determined by market prices of the two interest rate risks, and the idiosyncratic risk of the FX rate. Somewhat counterintuitively, at long time horizons it is the market price of the two IR risks what makes the biggest impact on the real world FX drift. Using Hull-Sokol-White calibration of the market price of risk in IR, we calibrate the real world model of long dated FX. The model is shown to be consistent with a wide range of historical data on FX. Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 2 / 49

3 1. Introduction to the Real-World Measure 1. Introduction to the Real-World Measure 2. Risk Premium of Interest Rates in IR-FX 3. Risk Premium in Idiosyncratic FX Risk Driver 4. IR-FX Model in Risk-Neutral Measure 5. IR-FX Model in Real-World Measure 6. Numerical Results 7. Summary Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 3 / 49

4 1. Introduction to the Real-World Measure How significant is the choice of measure? Risk must be managed in terms of real world probabilities The risk neutral probability is often described as the combination of the real world probability and the risk premium (the excess return the market participants require for taking risk) We can also think of the risk neutral probability distribution as having increased (a) (b) probability of worse outcomes for the investor relative to their actual (real world) probability Pricing contingent claims assuming higher proportion of adverse outcomes than is actually likely to occur is how the market participants express risk aversion RW Worse outcomes RW X! RN t RN Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 4 / 49

5 1. Introduction to the Real-World Measure Can risk neutral probabilities be used in place of the real world probabilities? A risk manager is looking for the actual (real world) probability of a potential loss However, for most firms is often much more practical to calculate the risk neutral probability Calculating risk neutral probability can rely on the established infrastructure for derivatives valuation and CVA Such replacement would only be valid if the difference between measures is moderate It makes sense to estimate the typical magnitude of the difference between real world and risk neutral expectations and quantiles and ask: Is it less than 10%? Could it be as high as 20%? Could it reach 100%? The specific answer depends on the risk factor and time horizon, but overall it appears to be a much larger difference than we should feel comfortable with in substituting the risk neutral probability for the real world one Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 5 / 49

6 1. Introduction to the Real-World Measure An example from the interest rate world Consider the probability distribution of a short term interest rate (e.g. LIBOR) at 30y horizon It is well known that market price of risk is the reason why long term yields are, on average, higher than short term yields In risk neutral measure, we advance toward the forward For an economy where the average short rate is 4% and the average 30y forward is 8%, the effect of measure change at 30y horizon is x2 change in the expected LIBOR rate Should a risk manager consider acceptable the absolute error of 4% or relative error of x2 in interest rate level when modeling cashflows of a swap portfolio? Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 6 / 49

7 1. Introduction to the Real-World Measure Where and why do we need the real world probability We do not need it for valuation or XVA because these are prices (under certain assumptions) We do not need it for market risk but only because of the short time horizon - at short timescales the volatility is much higher than the drift, so the drift is a small correction However, once we encounter the need to compute a quantile at medium (even as low as 1 year) or long time horizon, we are no longer able to do this in the risk neutral measure Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 7 / 49

8 1. Introduction to the Real-World Measure Calculations which require real world measure 1 Regulatory Capital - medium time horizon (1 year) for the default capital charge, IRC, and CRM; depending on the IMM model choice, also long time horizon (to maturity) for the CVA capital charge 2 Regulatory Liquidity - medium time horizon (1 year) for NSFR, long time horizon (to maturity) for the contractual maturity mismatch profile 3 PFE - potential future exposure for limit management 4 Insurance - modeling insurance reserves requires real world, not risk neutral probabilities. 5 Pensions - Pension funds are required to model the probability of a shortfall in reserves. 6 KVA - Capital calculation in real world measure is now part of valuation Generally, any risk measure based on a quantile for a long time horizon requires real world measure modeling Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 8 / 49

9 1. Introduction to the Real-World Measure Can the real world drift be estimated from the time series directly? For equities, a reasonable estimate for the real-world drift can be obtained from the historical price trend directly There are two reasons why this approach does not work for the interest rates 1 The presence of mean reversion, which confines each constant maturity length interest rate to a finite band 2 The long timescale on which the rate variations tend to occur. With the typical timescale of variation being around one decade, no conclusions about the drift can be drawn from studying the historical trend. Short rate (%) USD AUD GBP Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 9 / 49

10 1. Introduction to the Real-World Measure Local price of risk Local price of risk is a new technique for estimating the real world drift for rate-like risk factors, originally developed by Hull, Sokol, and White for interest rate models It was subsequently used by Sokol to calculate the real world drift of CDS In this presentation, we will use local price of risk for the interest rate component in long dated FX models The local price of risk is a way to model the differing amounts of market price of risk in different segments of the yield curve Once estimated, it can be used to calculate the excess drift due to the market price of interest rate risk that should be added to the risk neutral drift in order to obtain the real world drift. Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 10 / 49

11 2. Risk Premium of Interest Rates in IR-FX 1. Introduction to the Real-World Measure 2. Risk Premium of Interest Rates in IR-FX 3. Risk Premium in Idiosyncratic FX Risk Driver 4. IR-FX Model in Risk-Neutral Measure 5. IR-FX Model in Real-World Measure 6. Numerical Results 7. Summary Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 11 / 49

12 2. Risk Premium of Interest Rates in IR-FX Model selection For clarity, in this presentation the approach will be illustrated on the example of a one factor short rate SDE such as HW1F, CEV, CIR++, etc. With minor modifications, it can be applied to other models, including: Two factor short rate SDE such as HW2F, G2++, etc. For each risk driver of the model, we will have a separate local price of risk. Yield curve models such as LMM (with some modifications) Stochastic volatility (with one more state variable) Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 12 / 49

13 2. Risk Premium of Interest Rates in IR-FX Joint Measure Model for Interest Rates In risk neutral measure, the short rate follows this process: dr = µ(r, t)dt + σ(r, t)dz This is the "risk neutral side" of the joint measure model. In the real world measure, we have the same volatility but there is an additional drift term: dr = (µ(r, t) + λ(r, t)σ(r, t))dt + σ(r, t)dz This is the "real world side" of the joint measure model. The parameter λ(r, t) is the local price of risk, which in the most general form of the model should be an additional stochastic variable. By convention, λ < 0 if the investors are risk averse. Because the two "sides" of the joint measure model are referencing the same state variable (the short rate r), we can compute market prices as a function of r using the risk neutral "side" and then assign them to the same value of r on the in real world "side". Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 13 / 49

14 2. Risk Premium of Interest Rates in IR-FX Estimating λ from the shape of the yield curve The procedure of estimating the local price of risk from the slope of the short end of the curve is well known (Stanton, Cox and Pedersen, Ahmad and Wilmott) and can be summarized as follows. We know that the yield curve is upward sloping most of the time. This means that in risk neutral measure the short rate has upward drift. In risk neutral measure, we advance toward the forward Of course, in real world measure the drift averaged over a sufficiently long time horizon must be zero. Unlike stocks or FX, the interest rates move within a stable range. The average yield curve slope at t 0 then makes it possible to estimate λ. Note that only the long term regression can be used. At any given time, some of the slope may arise because the traders expect the rates to move in a certain direction. For example, when the yield curve is downward sloping, it does not mean that the investors suddenly became risk seeking. It is more likely that they simply expect the rates to fall. Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 14 / 49

15 2. Risk Premium of Interest Rates in IR-FX Estimate from the short end of the curve For the estimate of λ based on the slope of the short end of the yield curve (maturities less than 1y were considered in all of the publications listed on the previous slide), most authors agree on the value around λ 1 for major currencies. If the normal vol is about 1%, the difference between risk neutral and real world rate will grow 1% per year, or 30% of difference (in absolute terms) between risk neutral and real world short rate accumulating over a 30y horizon. Even if somewhat reduced by mean reversion, this is an absurd number which shows that the value of λ 1 cannot be used over long time horizons. The root cause of this problem is in assuming that the market price of risk is a constant Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 15 / 49

16 2. Risk Premium of Interest Rates in IR-FX In fact, the local price of risk is a term structure which we can estimate from the regression of swap rates of different maturity The rates of different maturity are correlated within the one factor short rate model, but not in reality Because the actual yield curve has multiple risk drivers which blend in different ways at different maturity, the drift at time t (the local price of risk, λ(t)) should be a term structure Time dependent λ(t) can be obtained from first principles in a stationary multifactor short rate or LMM model where each risk driver has its own market price of risk We will estimate λ from the regression of bond prices and interest rates of all maturities The calibration procedure must take into account convexity and mean reversion. Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 16 / 49

17 2. Risk Premium of Interest Rates in IR-FX Regression of 30y vs. 10y swap rate for USD X axis is 10y swap rate Y axis is 30y swap rate (c) y swap rate (%) Fit Historical r flat y swap rate (%) Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 17 / 49

18 2. Risk Premium of Interest Rates in IR-FX Regression of 10y vs. 5y swap rate for USD X axis is 5y swap rate Y axis is 10y swap rate (b) y swap rate (%) Fit Historical r flat y swap rate (%) Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 18 / 49

19 2. Risk Premium of Interest Rates in IR-FX Regression of 10y vs. 5y swap rate for MXN X axis is 5y swap rate Y axis is 10y swap rate (b) Rate (%) Fit 5y-10y Rate (%) Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 19 / 49

20 2. Risk Premium of Interest Rates in IR-FX Regression of 5y vs. 2y swap rate for USD X axis is 2y swap rate Y axis is 5y swap rate (a) y swap rate (%) Fit Historical r flat y swap rate (%) Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 20 / 49

21 2. Risk Premium of Interest Rates in IR-FX Regression of 5y vs. 2y swap rate for MXN X axis is 2y swap rate Y axis is 5y swap rate (a) Rate (%) Fit 2y-5y Rate (%) Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 21 / 49

22 2. Risk Premium of Interest Rates in IR-FX Regression of 5y vs. 2y swap rate for BRL X axis is 2y swap rate Y axis is 5y swap rate (a) Rate (%) Fit 2y-5y Rate (%) Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 22 / 49

23 2. Risk Premium of Interest Rates in IR-FX Dependence of the estimated local price of risk on bond maturity using data from different time periods for USD Unlike the earlier estimates of constant λ 1 from the short end of the yield curve, our calibration procedure produces time dependent λ(t) which drops to λ 0.2 for long simulation horizons The calibration of λ is highly stable and nearly the same in two very different historical periods 0.00 Maturity (years) λ T Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 23 / 49

24 2. Risk Premium of Interest Rates in IR-FX Expected interest rates in both measures for USD When calibrated to e.g. the USD markets as of Dec 2013, the long term difference in short rate average reaches 1.5% as early as t =10y For a typical interest rate portfolio, such rate difference which can easily change exposure from positive to negative and will have a large effect on risk, limit, and liquidity measures 6 5 Expected Short Rate, % Risk-Neutral E(r) Real-World E(r) Years Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 24 / 49

25 3. Risk Premium in Idiosyncratic FX Risk Driver 1. Introduction to the Real-World Measure 2. Risk Premium of Interest Rates in IR-FX 3. Risk Premium in Idiosyncratic FX Risk Driver 4. IR-FX Model in Risk-Neutral Measure 5. IR-FX Model in Real-World Measure 6. Numerical Results 7. Summary Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 25 / 49

26 3. Risk Premium in Idiosyncratic FX Risk Driver When the analogy between FX and equity can be misleading Modelling FX is superficially similar to modelling equity, the latter being a simple textbook example often used to illustrate the basic notions of quant finance. This analogy may often be misleading, as it does not reflect the symmetric nature of FX trading. In modelling equity, there is no doubt as to what is the riskless asset is: cash. Relative to cash, equity is a risky asset that is more likely to lose much of its value than make a large gain. Most risky assets have negative volatility skew (out-of-the-money puts are more expensive than out-of-the-money calls) and grow faster than the risk-free rate. Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 26 / 49

27 3. Risk Premium in Idiosyncratic FX Risk Driver Safe-haven currency pairs The situation is not so clear for FX when both currencies in the FX pair are safe-haven currencies. The informal sounding term safe-haven currency is an established notion in economics. It encompasses the currencies that are reliable stores of value, likely to appreciate against non-safe-haven currencies in a crisis. The list of safe-haven currencies is usually taken to include the US dollar, euro, Swiss franc, pound sterling, Japanese yen, Singapore dollar, Canadian dollar and Australian dollar. For our purposes, we will use a non-subjective definition of safe-haven currency based on market implied data for FX options. Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 27 / 49

28 3. Risk Premium in Idiosyncratic FX Risk Driver Direction of FX skew The same FX position can be viewed from either currency s side as a contingent claim on the other currency. Consider an FX pair between two safe-haven currencies neither of which can be considered in any way superior to the other. To an investor domiciled in either of the two countries, the local currency would be the riskless asset and the other currency would be the risky asset. Two groups of investors, each domiciled in different countries and trading the same FX pair, would have the opposite views on what is a drop in value versus a gain in value for an FX trade. It follows that the volatility skew of such FX pair will be, on average, zero. By examining the historical record,? found that indeed for many currency pairs the sign of the volatility skew changes from time to time. The differences in sign reflect periodic changes in the market perception of which currency of the pair is safer at a given moment in time. Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 28 / 49

29 3. Risk Premium in Idiosyncratic FX Risk Driver In FX trading, domestic cash is not always the risk-free asset In an asymmetric currency pair, most FX market participants see one of the currencies as a significantly safer store of value than the other (the risky currency ). In extreme cases, even the investors for whom the risky currency is domestic may perceive the other currency as being a safer store of value. Consider an investor in a country that is subject to considerable economic and political risks, and where the domestic currency holdings have perhaps in the past lost value through inflation and a gradual FX rate slide or a formal devaluation. In the mind of this investor, the domestic currency is anything but safe: the safer asset is the safe-haven currency, which is the foreign currency relative to the investor s domicile. In a pair where one currency is seen as safer than the other, the FX skew will usually be in the direction that makes out-of-the-money puts more expensive, seen from the side of the transaction where the safe-haven currency is the money currency and the risky currency is the dealt currency. Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 29 / 49

30 3. Risk Premium in Idiosyncratic FX Risk Driver Implied volatility skew of asymmetric FX pairs The implied volatility skew for two safe-haven currency pairs and two asymmetric currency pairs. As expected, the asymmetric pairs have more pronounced skew. 12 EUR/USD 12 GBP/USD Log vol (%) 8 4 Log vol (%) p 25p... 25c 5c Delta 0 5p 25p... 25c 5c Delta Log vol (%) MXN/USD p 25p... 25c 5c Delta Log vol (%) BRL/USD p 25p... 25c 5c Delta Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 30 / 49

31 3. Risk Premium in Idiosyncratic FX Risk Driver Historical realized volatility skew of asymmetric FX pairs For the two asymmetric pairs, depreciation of the risky currency caused its historical realized volatility to increase as it was seen even riskier ; appreciation caused its historical realized volatility to decrease. For the two safe-haven currency pairs, the historical realized volatility does not have a pronounced dependence on the FX rate. EUR/USD GBP/USD Historical log vol (%) Historical log vol (%) FX spot FX spot MXN/USD BRL/USD Historical log vol (%) Historical log vol (%) FX spot FX spot Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 31 / 49

32 4. IR-FX Model in Risk-Neutral Measure 1. Introduction to the Real-World Measure 2. Risk Premium of Interest Rates in IR-FX 3. Risk Premium in Idiosyncratic FX Risk Driver 4. IR-FX Model in Risk-Neutral Measure 5. IR-FX Model in Real-World Measure 6. Numerical Results 7. Summary Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 32 / 49

33 4. IR-FX Model in Risk-Neutral Measure Long-dated FX Most FX option pricing models use the measure in which the interest rates are deterministic. This is not equivalent to assuming that the rates are deterministic in the real world; the model simply incorporates their volatility into the risk-neutral probabilities of FX after the measure change. This approach only works well for short maturities, where most of the trading volume in FX options is concentrated. To model FX over long time horizons, the model must consider the joint dynamics of the FX rate and the two interest rates of the FX pair. The models for FX that expressly include interest rate volatilities are called long-dated FX models. Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 33 / 49

34 4. IR-FX Model in Risk-Neutral Measure Construction of long-dated FX model The model for long-dated FX combines two interest rate models, one for each currency of the pair, with an additional process for the changes in FX unrelated to the interest rates ( idiosyncratic FX volatility ). The influence of interest rates on FX dynamics has to do with the no-arbitrage condition between the FX spot and FX forward. The FX forward can be replicated by the combination of the FX spot and two zero bonds (one in each currency). For short time horizons, the main contributor to the changes in FX rate is the volatility term in Equation 1. As the time horizon becomes longer, additional FX volatility from the influence of the interest rates on FX drift becomes increasingly important. Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 34 / 49

35 4. IR-FX Model in Risk-Neutral Measure Risk neutral model for the FX process Following the prevailing market practice in long-term portfolio simulation, we will not model FX volatility as stochastic in the primary model and only include its volatility skew via the CEV process for the FX rate: ds(t) S(t) ( ) β(t) 1 S(t) = (rd(t) rf(t)) dt + ν(t) dw (1) L(t) where β(t) is the CEV power, L(t) is the CEV scale, r d is the domestic rate and r f is the foreign rate. LSV and other stochastic volatility models can be implemented as trade-specific and external models after the primary model is constructed. The CEV process for the FX was used is a popular choice for modelling long-dated cross-currency swaps and other hybrid products. This gives us an added measure of confidence in the primary model we selected. Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 35 / 49

36 5. IR-FX Model in Real-World Measure 1. Introduction to the Real-World Measure 2. Risk Premium of Interest Rates in IR-FX 3. Risk Premium in Idiosyncratic FX Risk Driver 4. IR-FX Model in Risk-Neutral Measure 5. IR-FX Model in Real-World Measure 6. Numerical Results 7. Summary Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 36 / 49

37 5. IR-FX Model in Real-World Measure Three prices of three risks in long dated FX The FX rate is subject to three distinct sources of market risk: 1 The market risk in the domestic interest rate; 2 The market risk in the foreign interest rate; and 3 The idiosyncratic risk in the FX rate itself. At short time horizons, the idiosyncratic risk driver of the FX rate dominates because it acts on the rate via the volatility factor ( t), while the interest rates act via the drift ( t). At long time horizons, the situation reverses: the two interest rates contribute a larger share of the overall volatility of FX than the idiosyncratic factor. Using the interest rates projected in the real world measure is the most important real-world drift adjustment for FX. This adjustment accounts for the market price of interest rate risk in FX. The market price of the remaining risk in FX, the risk due to the idiosyncratic volatility component, results in an excess real-world drift relative to the interest rate differential. Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 37 / 49

38 5. IR-FX Model in Real-World Measure Risk-neutral no-arbitrage constraint between the FX spot, FX forward and two zero bonds The risk-neutral drift in FX is determined by a no-arbitrage constraint between the FX spot, FX forward and two zero bonds in the respective currencies. If the FX drift is not the same as the interest rate differential, there will be arbitrage between investing in the spot and investing in the forward and the two zero bonds. In the risk-neutral measure, the FX rate climbs up (or down ) the forward curve with the passage of time. If one interest rate is higher than the other, the lower yielding currency will appreciate over time relative to the higher yielding currency, such that the investment in the interest rate market in each country creates the same value over time otherwise arbitrage would be present. Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 38 / 49

39 5. IR-FX Model in Real-World Measure Estimation of the drift adjustment due to the market price of idiosyncratic risk in FX To estimate the drift adjustment due to the market price of idiosyncratic risk in FX, we first need to eliminate the part of the drift related to the interest rates. This can be done by comparing the annually observed spot rate with the 1y forward rate observed 1y earlier: Spot(t) Forward(t 1y, t) 1 (2) If the world were risk-neutral, this quantity would average to zero. In the presence of risk aversion, if the foreign currency is the riskier asset, Spot(t) will be consistently higher than Forward(t 1y, t). The historical average of their difference can then be used to estimate the market price of idiosyncratic FX risk. The riskier currency of the pair is expected to appreciate faster than the safer currency, just like equity grows faster than the risk free rate. We expect that the excess drift will be higher in magnitude for risky currencies compared to safe heaven currencies. Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 39 / 49

40 5. IR-FX Model in Real-World Measure Cross-sectional average of the excess drift due to idiosyncratic FX risk Due to the small length of the available time series, the excess drift cannot be estimated individually by currency. However it can be estimated by currency group (cross-sectional average). The drift is reported relative to USD Drift Y.Strong 1Y.Medium 1Y.Weak Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 40 / 49

41 5. IR-FX Model in Real-World Measure Variation of excess drift due to idiosyncratic FX risk within the strong currency group As expected, within the strong (safe haven) currency group, the excess drift changes sign. This is consistent with the change in skew in safe haven currency pairs, as the higher level of risk switches from one currency to the other MAX AED AUD CAD CHF EUR GBP JPY NOK SEK SGD Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 41 / 49

42 5. IR-FX Model in Real-World Measure Variation of excess drift due to idiosyncratic FX risk within the medium currency group Within the medium currency group, the excess drift always leads to faster appreciation of the riskier currency in USD terms relative to the risk-neutral drift. The impact of the estimation uncertainty resulting from the variation within the group is acceptable overall because at long time horizons this entire drift is much smaller than the drift resulting from the interest rate risk MAX CNY CZK MXN NZD PHP RUB SAR Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 42 / 49

43 6. Numerical Results 1. Introduction to the Real-World Measure 2. Risk Premium of Interest Rates in IR-FX 3. Risk Premium in Idiosyncratic FX Risk Driver 4. IR-FX Model in Risk-Neutral Measure 5. IR-FX Model in Real-World Measure 6. Numerical Results 7. Summary Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 43 / 49

44 6. Numerical Results RW Densities MXN/USD Probability RN FX Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 44 / 49

45 6. Numerical Results Quantiles MXN/USD RN 80% RW 80% MXN/USD Actual 0.02 RN 20% RW 20% 0 0 5y 10y 15y 20y 25y 30y Time Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 45 / 49

46 6. Numerical Results Densities BRL/USD Probability RN RW FX Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 46 / 49

47 6. Numerical Results Quantiles BRL/USD Actual 0.3 BRL/USD RN 80% RN 20% 0 0 5y 10y 15y 20y 25y 30y Time RW 80% RW 20% Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 47 / 49

48 7. Summary 1. Introduction to the Real-World Measure 2. Risk Premium of Interest Rates in IR-FX 3. Risk Premium in Idiosyncratic FX Risk Driver 4. IR-FX Model in Risk-Neutral Measure 5. IR-FX Model in Real-World Measure 6. Numerical Results 7. Summary Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 48 / 49

49 7. Summary Summary For large multi-currency portfolios, the evolution of FX rate is frequently the primary risk driver At the long time horizons of interest in limit and certain capital calculations, the FX rate must be modelled jointly with the interest rates The real world drift of the FX rate is determined by market prices of the two interest rate risks, and the idiosyncratic risk of the FX rate. Somewhat counterintuitively, at long time horizons it is the market price of the two IR risks what makes the biggest impact on the real world FX drift. Using Hull-Sokol-White calibration of the market price of risk in IR, we calibrate the real world model of long dated FX. The model is shown to be consistent with a wide range of historical data on FX. Alexander Sokol (CompatibL) Three prices of three risks: A real world measure IR-FX hybrid model 49 / 49