This document discusses discount factors and mark-to-market valuation of cross currency swaps. It begins by explaining how discount factors are derived from risk-free bonds and how rates like Libor and OIS are used as proxies. However, it notes that swap rates cannot be directly used as discount factors since they do not guarantee a fixed payment amount at maturity. The document then discusses how to model the cash flows of interest rate swaps and cross currency swaps, and how to calculate stochastic and implied swap rates to value them using mark-to-market approaches.
Swaps explained. Very useful for CFA and FRM level 1 preparation candidates. For a more detailed understanding, you can watch the webinar video on this topic. The link for the webinar video on this topic is https://www.youtube.com/watch?v=JKBKnxM2Nj4
1) Currency swaps involve two companies exchanging loans in different currencies so they can each borrow at lower domestic rates rather than higher international rates. For example, a US and Brazilian company arrange for the US company to borrow Brazilian reals from a Brazilian bank at 5% while the Brazilian company borrows dollars from a US bank at 4%.
2) The companies then swap the loans, so the US company receives dollars and pays 5% interest to the Brazilian bank, while the Brazilian company receives reals and pays 4% interest to the US bank. This allows both companies to access lower domestic borrowing rates rather than higher international rates of 9-10%.
3) Currency swaps also involve an exchange of notional principal amounts
Interest rate and currency swaps allow companies to exchange interest rate and currency cash flows to hedge against risks. There are two main types: interest rate swaps exchange fixed and floating rate payments, while currency swaps involve exchanging loan principals and interest payments in different currencies. Comparative advantages in borrowing costs across currencies can create opportunities for swap banks to arrange deals where all parties receive better rates and the bank earns profits through hedging the swap cash flows in money or forward markets.
This document discusses interest rate parity theory. It begins by defining spot and forward rates. Spot rates are prices for immediate settlement, while forward rates refer to rates for future currency delivery adjusted for cost of carry. Interest rate parity theory states that interest rate differentials between currencies will be reflected in forward premiums or discounts. The theory prevents arbitrage opportunities by making returns equal whether investing domestically or abroad when measured in the home currency. The document provides an example of covered and uncovered interest rate parity. Covered parity involves hedging exchange rate risk while uncovered parity does not. Empirical evidence shows uncovered parity often fails while covered parity generally holds for major currencies over short time horizons.
An interest rate swap involves the periodic exchange of interest payments between two parties, with no exchange of the underlying principal amount. One party pays a fixed interest rate while the other pays a floating rate, typically tied to a reference rate such as LIBOR. Interest rate swaps can be used by companies and investors to hedge against interest rate risk and manage the costs associated with fixed versus floating rate financing. While swaps can be profitable if rates move as anticipated, they also carry risks of losses if rates move adversely.
A swap is an agreement between two counterparties to exchange cash flow streams, such as interest payments or currencies. The main types of swaps discussed are interest rate swaps, currency swaps, and forex swaps. An interest rate swap involves exchanging interest payments, such as a fixed rate for a floating rate. A currency swap exchanges principal and interest payments in different currencies. A forex swap is an agreement to buy one currency now and sell it back in the future at an agreed upon exchange rate.
Swaps explained. Very useful for CFA and FRM level 1 preparation candidates. For a more detailed understanding, you can watch the webinar video on this topic. The link for the webinar video on this topic is https://www.youtube.com/watch?v=JKBKnxM2Nj4
1) Currency swaps involve two companies exchanging loans in different currencies so they can each borrow at lower domestic rates rather than higher international rates. For example, a US and Brazilian company arrange for the US company to borrow Brazilian reals from a Brazilian bank at 5% while the Brazilian company borrows dollars from a US bank at 4%.
2) The companies then swap the loans, so the US company receives dollars and pays 5% interest to the Brazilian bank, while the Brazilian company receives reals and pays 4% interest to the US bank. This allows both companies to access lower domestic borrowing rates rather than higher international rates of 9-10%.
3) Currency swaps also involve an exchange of notional principal amounts
Interest rate and currency swaps allow companies to exchange interest rate and currency cash flows to hedge against risks. There are two main types: interest rate swaps exchange fixed and floating rate payments, while currency swaps involve exchanging loan principals and interest payments in different currencies. Comparative advantages in borrowing costs across currencies can create opportunities for swap banks to arrange deals where all parties receive better rates and the bank earns profits through hedging the swap cash flows in money or forward markets.
This document discusses interest rate parity theory. It begins by defining spot and forward rates. Spot rates are prices for immediate settlement, while forward rates refer to rates for future currency delivery adjusted for cost of carry. Interest rate parity theory states that interest rate differentials between currencies will be reflected in forward premiums or discounts. The theory prevents arbitrage opportunities by making returns equal whether investing domestically or abroad when measured in the home currency. The document provides an example of covered and uncovered interest rate parity. Covered parity involves hedging exchange rate risk while uncovered parity does not. Empirical evidence shows uncovered parity often fails while covered parity generally holds for major currencies over short time horizons.
An interest rate swap involves the periodic exchange of interest payments between two parties, with no exchange of the underlying principal amount. One party pays a fixed interest rate while the other pays a floating rate, typically tied to a reference rate such as LIBOR. Interest rate swaps can be used by companies and investors to hedge against interest rate risk and manage the costs associated with fixed versus floating rate financing. While swaps can be profitable if rates move as anticipated, they also carry risks of losses if rates move adversely.
A swap is an agreement between two counterparties to exchange cash flow streams, such as interest payments or currencies. The main types of swaps discussed are interest rate swaps, currency swaps, and forex swaps. An interest rate swap involves exchanging interest payments, such as a fixed rate for a floating rate. A currency swap exchanges principal and interest payments in different currencies. A forex swap is an agreement to buy one currency now and sell it back in the future at an agreed upon exchange rate.
A currency swap allows two companies to exchange loans denominated in different currencies to better meet their financing needs. In the example, a British company borrows dollars and an American company borrows pounds through a swap bank. This allows each to access comparatively better interest rates in the other's local market. The swap bank profits by exploiting small differences in interest rates between the currencies over time. It can hedge its risk and lock in gains by entering forward currency contracts or money market positions. Overall, currency swaps allow companies and banks to benefit from comparative advantages in borrowing costs across currencies.
The document discusses the foreign exchange market, including its functions, participants, rates, and factors that affect exchange rates. It provides definitions of exchange rates, discusses direct and indirect quotes, and covers concepts like appreciation/depreciation and forward premiums/discounts. Key participants in the forex market include importers/exporters, commercial banks, central banks, and forex brokers. Exchange rates are the price of one currency in terms of another.
Unit 2.2 Exchange Rate Quotations & Forex MarketsCharu Rastogi
This presentation deals with exchange rate quotations, common currency symbols, direct and indirect quotes, American terms, European terms, cross rates, Bid and Ask rates, Mid rate, Spread and its determinants, Spot markets, Forward Markets, Premium and Discounts, various practices of writing quotations, calculating broken period forward rates, Speculation and arbitrage, Forex futures and Currency Options.
The document discusses various types of interest rate swaps and how to value them. It begins by explaining how the standard approach values plain vanilla interest rate and currency swaps by assuming forward rates will be realized. It then describes how this approach can also value swaps with variations like different payment frequencies or floating rates. More complex swaps like LIBOR-in-arrears, CMS, differential, and equity swaps require adjustments to the standard approach for accurate valuation. The document also discusses swaps with embedded options and other unique types of swaps.
A currency swap involves the exchange of principal and interest payments in one currency for the same in another currency at fixed intervals over the contract period. In a currency swap, counterparties can choose to exchange principal at the start and end of the swap or just exchange interest payments. An interest rate swap is an agreement where one party pays a fixed interest rate on a loan while receiving a floating rate, or vice versa, from the other party in order to reduce exposure to interest rate fluctuations. Common types of interest rate swaps include fixed to floating, floating to fixed, and float to float (basis) swaps. Swaps allow parties to achieve their desired interest rate exposure and are customized over-the-counter agreements.
This document provides solutions to end-of-chapter questions for a chapter on interest rate and currency swaps. It includes answers explaining the difference between a swap broker and dealer, the necessary condition for a fixed-for-floating interest rate swap, basic motivations for currency swaps, and how comparative advantage relates to the currency swap market. It also summarizes the various risks confronting swap dealers and some variants of basic interest rate and currency swaps. Sample problems are provided showing how companies can use swaps to lower their borrowing costs.
Currency trading operates 24 hours a day except weekends and involves trading between different currency pairs during overlapping trading sessions centered around European, Asian, and US markets. Currency is traded in micro, mini and standard lots where the lot size determines the monetary value of each pip. The major currency pairs that are most actively traded involve the US dollar, euro, British pound, Japanese yen and others, though there are over 180 available pairs.
This document discusses interest rate swaps, including defining a basic swap transaction, the gains achieved through swaps, pricing and valuation of swaps, risks and applications of swaps. A basic swap involves two parties exchanging interest rate payment obligations, with one party paying a fixed rate and receiving a floating rate, and vice versa. Swaps allow parties to achieve lower financing costs by exploiting differences in borrowing rates available to higher and lower rated entities. Risks include pricing risk, credit risk, and potential systemic risks from unhedged dealer positions.
There are two types of foreign exchange quotations: European and American. European quotations give the price of a currency in terms of units of another currency, while American quotations give the price in terms of dollars per unit of another currency. Direct quotes give the home currency price per unit of foreign currency, while indirect quotes give the opposite. Cross rates can be calculated using exchange rates between two currencies and a third currency. The TT buying rate is used for clean inward or outward remittances, while the bill buying rate factors in exchange and forward margins to account for delays in collection. The bill selling rate adds an exchange margin to the base rate to cover document handling costs.
This document provides an overview of interest rate swaps, caps, and floors. It defines an interest rate swap as an agreement where two parties exchange periodic interest payments, based on a notional principal amount, with one party paying a fixed rate and the other paying a floating rate. The document outlines key terms used in swaps, how swap rates are calculated, and how swaps can be used for asset/liability management. It also discusses related derivatives like swaptions, rate caps and floors, and how these tools can be used by institutional investors to manage interest rate risk.
Currency and interest rate swaps allow companies to exchange interest rate and currency cash flows to hedge risks. Interest rate swaps exchange fixed and floating rate payments in the same currency. Currency swaps exchange interest payments in different currencies to fund projects abroad. Swap banks facilitate these exchanges and make markets in standard swaps, earning small spreads. Swaps benefit all parties by allowing each to borrow in their preferred currency or rate.
This document discusses various methods for translating foreign currency financial statements when consolidating a foreign subsidiary's statements into a parent company's reporting currency. It provides examples of how the current/noncurrent method, monetary/nonmonetary method, temporal method, and current rate method would account for a change in exchange rates from DM3=$1 to DM2=$1. In general, the methods differ in whether they use the historical or current exchange rate to translate non-monetary assets such as inventory and fixed assets, and whether translation gains/losses affect net income.
Determination of Exchange Rate in Forward MarketSundar B N
The document discusses exchange rates and factors that influence them. It begins with definitions of exchange rates and the two main types - spot and forward exchange rates.
It then discusses how exchange rates vary under different monetary standards like the gold standard. The key factors that determine the spot exchange rate are then outlined, including the balance of payments between countries and purchasing power parity.
The document concludes by noting that changes in exchange rates originate from the monetary sector under a gold standard. It also provides a reference for more information on forward exchange rates.
A swap is an agreement between two parties to exchange cash flows over a period of time, where at least one cash flow is determined by a variable such as interest rate, foreign exchange rate, or equity price. The most common type is an interest rate swap, where parties exchange interest payments on a notional principal amount at fixed and floating rates. Swaps allow users to align the risk characteristics of their assets and liabilities.
The document discusses the foreign exchange market and its key functions and participants. It provides information on:
1) The main functions of the forex market are to facilitate the international transfer of funds for trade, tourism, investment and other purposes.
2) There are four main levels of participants - immediate users/suppliers of currencies, commercial banks, forex brokers, and central banks.
3) Exchange rates are determined by the demand and supply of currencies. Factors like inflation rates, interest rates, economic growth, and risk affect the equilibrium exchange rate.
An interest rate future is a futures contract between the buyer and seller to deliver an interest bearing asset, that allows the buyer and seller to lock in the price of the interest bearing asset for a future date.
Interest rate futures are used to hedge against interest rate risk. Investors can use Eurodollar futures to secure an interest rate for money it plans to borrow or lend in the future. This presentation gives an overview of interest rate future product and pricing model. You find more presentations at http://www.finpricing.com/productList.html
A currency swap involves exchanging interest payments on loans denominated in different currencies. Party A borrows in one currency like INR and pays interest in another currency like USD, while Party B does the opposite. This allows companies to hedge currency risk if their cash flows match the swap currency, but currency risk still exists if there are no matching cash flows. Interest rate swaps involve exchanging a fixed interest rate loan for a floating rate loan to reduce funding costs. An example shows how swapping interest rates can lower costs for both parties. Interest rate futures allow hedging of interest rate risk in India using government bond contracts.
Interest rate parity is a theory stating that the interest rate differential between two countries should equal the forward exchange rate premium or discount relative to the spot exchange rate. This establishes a break-even condition where returns on domestic and foreign currency investments are equal after accounting for exchange risk. If interest rate parity is violated, an arbitrage opportunity exists where investors can borrow, invest, and exchange currencies to earn risk-free profits. Kim Deal, a European portfolio manager, should choose to invest in 1-year Japanese yen deposits covered by a 1-year forward contract to hedge exchange risk, as this option provides the highest euro return of €352,005 compared to €352,000 from euro deposits.
1) The document discusses models for pricing corporate bonds, specifically comparing a reduced form default model to the author's proposed model.
2) In the author's model, the date-t bond price is a random variable between the minimum and maximum price on date t, rather than a single number. This models the bond price as the present value of the recovery rate assuming default occurs at maturity.
3) With the bond price as a random variable, the recovery rate can be assumed to be a non-random constant, reducing the default problem to finding the unknown recovery rate and default probability. Equations for the first and second moments of the bond price can then be derived.
1) The document discusses constructing multiple swap curves to price financial products consistently with different swap markets, including those with and without collateral.
2) It explains how to construct swap curves for a single currency market based on interest rate swaps alone. It then expands this to incorporate cross-currency swaps by deriving separate discounting and index curves.
3) The method is further expanded to consistently incorporate basis spreads observed in tenor swaps between different tenors (e.g. 3-month and 6-month rates) into the construction of discounting and multiple index curves.
A currency swap allows two companies to exchange loans denominated in different currencies to better meet their financing needs. In the example, a British company borrows dollars and an American company borrows pounds through a swap bank. This allows each to access comparatively better interest rates in the other's local market. The swap bank profits by exploiting small differences in interest rates between the currencies over time. It can hedge its risk and lock in gains by entering forward currency contracts or money market positions. Overall, currency swaps allow companies and banks to benefit from comparative advantages in borrowing costs across currencies.
The document discusses the foreign exchange market, including its functions, participants, rates, and factors that affect exchange rates. It provides definitions of exchange rates, discusses direct and indirect quotes, and covers concepts like appreciation/depreciation and forward premiums/discounts. Key participants in the forex market include importers/exporters, commercial banks, central banks, and forex brokers. Exchange rates are the price of one currency in terms of another.
Unit 2.2 Exchange Rate Quotations & Forex MarketsCharu Rastogi
This presentation deals with exchange rate quotations, common currency symbols, direct and indirect quotes, American terms, European terms, cross rates, Bid and Ask rates, Mid rate, Spread and its determinants, Spot markets, Forward Markets, Premium and Discounts, various practices of writing quotations, calculating broken period forward rates, Speculation and arbitrage, Forex futures and Currency Options.
The document discusses various types of interest rate swaps and how to value them. It begins by explaining how the standard approach values plain vanilla interest rate and currency swaps by assuming forward rates will be realized. It then describes how this approach can also value swaps with variations like different payment frequencies or floating rates. More complex swaps like LIBOR-in-arrears, CMS, differential, and equity swaps require adjustments to the standard approach for accurate valuation. The document also discusses swaps with embedded options and other unique types of swaps.
A currency swap involves the exchange of principal and interest payments in one currency for the same in another currency at fixed intervals over the contract period. In a currency swap, counterparties can choose to exchange principal at the start and end of the swap or just exchange interest payments. An interest rate swap is an agreement where one party pays a fixed interest rate on a loan while receiving a floating rate, or vice versa, from the other party in order to reduce exposure to interest rate fluctuations. Common types of interest rate swaps include fixed to floating, floating to fixed, and float to float (basis) swaps. Swaps allow parties to achieve their desired interest rate exposure and are customized over-the-counter agreements.
This document provides solutions to end-of-chapter questions for a chapter on interest rate and currency swaps. It includes answers explaining the difference between a swap broker and dealer, the necessary condition for a fixed-for-floating interest rate swap, basic motivations for currency swaps, and how comparative advantage relates to the currency swap market. It also summarizes the various risks confronting swap dealers and some variants of basic interest rate and currency swaps. Sample problems are provided showing how companies can use swaps to lower their borrowing costs.
Currency trading operates 24 hours a day except weekends and involves trading between different currency pairs during overlapping trading sessions centered around European, Asian, and US markets. Currency is traded in micro, mini and standard lots where the lot size determines the monetary value of each pip. The major currency pairs that are most actively traded involve the US dollar, euro, British pound, Japanese yen and others, though there are over 180 available pairs.
This document discusses interest rate swaps, including defining a basic swap transaction, the gains achieved through swaps, pricing and valuation of swaps, risks and applications of swaps. A basic swap involves two parties exchanging interest rate payment obligations, with one party paying a fixed rate and receiving a floating rate, and vice versa. Swaps allow parties to achieve lower financing costs by exploiting differences in borrowing rates available to higher and lower rated entities. Risks include pricing risk, credit risk, and potential systemic risks from unhedged dealer positions.
There are two types of foreign exchange quotations: European and American. European quotations give the price of a currency in terms of units of another currency, while American quotations give the price in terms of dollars per unit of another currency. Direct quotes give the home currency price per unit of foreign currency, while indirect quotes give the opposite. Cross rates can be calculated using exchange rates between two currencies and a third currency. The TT buying rate is used for clean inward or outward remittances, while the bill buying rate factors in exchange and forward margins to account for delays in collection. The bill selling rate adds an exchange margin to the base rate to cover document handling costs.
This document provides an overview of interest rate swaps, caps, and floors. It defines an interest rate swap as an agreement where two parties exchange periodic interest payments, based on a notional principal amount, with one party paying a fixed rate and the other paying a floating rate. The document outlines key terms used in swaps, how swap rates are calculated, and how swaps can be used for asset/liability management. It also discusses related derivatives like swaptions, rate caps and floors, and how these tools can be used by institutional investors to manage interest rate risk.
Currency and interest rate swaps allow companies to exchange interest rate and currency cash flows to hedge risks. Interest rate swaps exchange fixed and floating rate payments in the same currency. Currency swaps exchange interest payments in different currencies to fund projects abroad. Swap banks facilitate these exchanges and make markets in standard swaps, earning small spreads. Swaps benefit all parties by allowing each to borrow in their preferred currency or rate.
This document discusses various methods for translating foreign currency financial statements when consolidating a foreign subsidiary's statements into a parent company's reporting currency. It provides examples of how the current/noncurrent method, monetary/nonmonetary method, temporal method, and current rate method would account for a change in exchange rates from DM3=$1 to DM2=$1. In general, the methods differ in whether they use the historical or current exchange rate to translate non-monetary assets such as inventory and fixed assets, and whether translation gains/losses affect net income.
Determination of Exchange Rate in Forward MarketSundar B N
The document discusses exchange rates and factors that influence them. It begins with definitions of exchange rates and the two main types - spot and forward exchange rates.
It then discusses how exchange rates vary under different monetary standards like the gold standard. The key factors that determine the spot exchange rate are then outlined, including the balance of payments between countries and purchasing power parity.
The document concludes by noting that changes in exchange rates originate from the monetary sector under a gold standard. It also provides a reference for more information on forward exchange rates.
A swap is an agreement between two parties to exchange cash flows over a period of time, where at least one cash flow is determined by a variable such as interest rate, foreign exchange rate, or equity price. The most common type is an interest rate swap, where parties exchange interest payments on a notional principal amount at fixed and floating rates. Swaps allow users to align the risk characteristics of their assets and liabilities.
The document discusses the foreign exchange market and its key functions and participants. It provides information on:
1) The main functions of the forex market are to facilitate the international transfer of funds for trade, tourism, investment and other purposes.
2) There are four main levels of participants - immediate users/suppliers of currencies, commercial banks, forex brokers, and central banks.
3) Exchange rates are determined by the demand and supply of currencies. Factors like inflation rates, interest rates, economic growth, and risk affect the equilibrium exchange rate.
An interest rate future is a futures contract between the buyer and seller to deliver an interest bearing asset, that allows the buyer and seller to lock in the price of the interest bearing asset for a future date.
Interest rate futures are used to hedge against interest rate risk. Investors can use Eurodollar futures to secure an interest rate for money it plans to borrow or lend in the future. This presentation gives an overview of interest rate future product and pricing model. You find more presentations at http://www.finpricing.com/productList.html
A currency swap involves exchanging interest payments on loans denominated in different currencies. Party A borrows in one currency like INR and pays interest in another currency like USD, while Party B does the opposite. This allows companies to hedge currency risk if their cash flows match the swap currency, but currency risk still exists if there are no matching cash flows. Interest rate swaps involve exchanging a fixed interest rate loan for a floating rate loan to reduce funding costs. An example shows how swapping interest rates can lower costs for both parties. Interest rate futures allow hedging of interest rate risk in India using government bond contracts.
Interest rate parity is a theory stating that the interest rate differential between two countries should equal the forward exchange rate premium or discount relative to the spot exchange rate. This establishes a break-even condition where returns on domestic and foreign currency investments are equal after accounting for exchange risk. If interest rate parity is violated, an arbitrage opportunity exists where investors can borrow, invest, and exchange currencies to earn risk-free profits. Kim Deal, a European portfolio manager, should choose to invest in 1-year Japanese yen deposits covered by a 1-year forward contract to hedge exchange risk, as this option provides the highest euro return of €352,005 compared to €352,000 from euro deposits.
1) The document discusses models for pricing corporate bonds, specifically comparing a reduced form default model to the author's proposed model.
2) In the author's model, the date-t bond price is a random variable between the minimum and maximum price on date t, rather than a single number. This models the bond price as the present value of the recovery rate assuming default occurs at maturity.
3) With the bond price as a random variable, the recovery rate can be assumed to be a non-random constant, reducing the default problem to finding the unknown recovery rate and default probability. Equations for the first and second moments of the bond price can then be derived.
1) The document discusses constructing multiple swap curves to price financial products consistently with different swap markets, including those with and without collateral.
2) It explains how to construct swap curves for a single currency market based on interest rate swaps alone. It then expands this to incorporate cross-currency swaps by deriving separate discounting and index curves.
3) The method is further expanded to consistently incorporate basis spreads observed in tenor swaps between different tenors (e.g. 3-month and 6-month rates) into the construction of discounting and multiple index curves.
A swap is an agreement between two parties to exchange interest payments on a set notional principal over a period of time. In a plain vanilla interest rate swap, one party pays a fixed interest rate while the other pays a floating rate, typically linked to LIBOR. The value of the swap to each party is determined by discounting the expected cash flows using the current LIBOR curve. Midway through the contract, the value depends on how interest rate expectations have changed since inception.
The document discusses several key concepts in finance including:
1) Finance involves allocating resources across time through borrowing, lending, and investing. Markets provide information to compare returns and risks of different investments.
2) Interest rates reflect the exchange between present and future resources, with higher rates translating to a steeper slope and greater future resources needed to exchange for present amounts.
3) Net present value, internal rate of return, and other concepts are used to evaluate investments based on discounted cash flows.
The document discusses several key concepts in finance including:
1) Finance involves allocating resources across time through borrowing, lending, and investing. Markets provide information to compare returns and risks of different investments.
2) Interest rates reflect the exchange between present and future resources, with higher rates translating to a steeper slope and greater future resources needed for a present amount.
3) Net present value, internal rate of return, and other concepts are used to evaluate investments based on discounted cash flows.
This document discusses no arbitrage pricing theory and market risk. It begins by defining no arbitrage pricing as having a zero initial and expiration value. However, it notes that this definition does not guarantee a zero expiration value when holding coupon payments. It then introduces the concepts of present value and forward value, and defines no arbitrage prices that set the present and forward values equal to zero. However, it notes that this introduces market risk, as forward rates are random variables. It concludes by providing examples of interest rate swap valuation and defining market risk probabilities.
Options Pricing The Black-Scholes ModelMore or .docxhallettfaustina
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Options Pricing: The Black-Scholes ModelMore or less, the Black-Scholes (B-S) Model is really just a fancy extension of the Binomial Model.
(Fancy enough, however, to win a Nobel Prize…).
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How B-S extends the Binomial Model1. Instead of assuming two possible states for future exchange rates, and thus returns (i.e., “up” and “down”), B-S assumes a continuous distribution of returns, R, so that returns can take on a whole range of values.
Binomial B-S
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How B-S extends the Binomial ModelIn fact, exchange rate returns are approximately normally distributed, so this is a “reasonable” assumption:
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How B-S extends the Binomial Model2. Instead of just one time period, B-S assumes multiple time periods and that the time between periods is instantaneous (i.e., continuous).
(See lecture)
Also, the time between periods t=0, t=1, t=2, etc. shrinks to zero, so that spot rate is changing at every instant.
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How B-S extends the Binomial ModelThis is more realistic, since actual currency trades take place on a second-to-second, nearly continuous basis.
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How B-S extends the Binomial ModelIt turns out that these two extensions are enough to make the math very hard. Thus, deriving the B-S model is no easy task.
The most important thing to recognize is that despite the above complications, the basic underlying approach of the B-S model remains the same…
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How B-S extends the Binomial Model3. Create a replicating portfolio and price the option using a no-arbitrage argument.Calculate NS and NB: Now, since these are constantly changing over time, this process is called “dynamic hedging”.Replicating portfolio:It turns out that it is possible to use a combination of foreign currency and USD, and now in addition, options themselves, to form a riskless portfolio (i.e., return is known for sure).No-arbitrage: Riskless portfolios must have the same price as risk-free securities, otherwise arbitrage is possible. Use this fact to figure out c.
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The Black-Scholes Options Pricing FormulaPutting the above all together, we get the Black-Scholes formula for pricing a European call option on foreign currency:
where
and S, X, T as before
r = domestic risk-free rate, r* = foreign risk-free rate
s = volatility of the foreign currency (sd of returns).
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The Black-Scholes Options Pricing Formula
Also, N(x) = Prob that a random variable will be less than x under the standard normal distribution (i.e., cumulative distribution function).Calculate in EXCEL using “=NORMSDIST(x)”.
represents discounting when interest rates are continuously compounded, so basically it corresponds to:
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7_Analysing and Interpreting the Yield Curve.pptMurat Öztürkmen
The document discusses analysing and interpreting the yield curve. It covers the importance of the yield curve, constructing the curve from discount functions, theories like the expectations hypothesis and liquidity preference theory, the formal relationship between spot and forward rates, interpreting the shape of the curve, and fitting the curve from market data. Specifically, it notes the challenges of fitting the curve given a lack of liquid market data inputs and impact of bid-offer spreads, and recommends using a non-parametric interpolation method like the Svensson model to produce a smoother forward curve.
The document discusses forward volatility agreements (FVAs). It defines an FVA as a volatility swap contract where the buyer and seller agree to exchange a straddle option at a future date based on a specified volatility level. The key motivation for trading FVAs is that it allows investors to speculate on future volatility levels. The document provides details on pricing and hedging FVAs, including using volatility gadgets and forward start straddle options to isolate exposure to future local volatility.
This document provides an overview of interest rate swaps, caps, and floors. It defines an interest rate swap as an agreement where two parties exchange periodic interest payments, based on a notional principal amount, with one party paying a fixed rate and the other paying a floating rate. The document outlines key terms used in swaps like notional amount, counterparties, and reset frequency. It also explains how swap rates are quoted in the market and calculated, and how swaps can be used by institutional investors to manage assets and liabilities. Finally, it briefly introduces related financial agreements like swaptions, caps, and floors.
This document provides an overview of interest rate swaps, caps, and floors. It defines an interest rate swap as an agreement where two parties exchange periodic interest payments, based on a notional principal amount, allowing one party to pay a fixed interest rate and receive a floating rate, and vice versa. The document discusses how swap rates are quoted in the market, how swap payments are calculated, and how swaps can be used by institutional investors for asset/liability management. It also explains what interest rate caps and floors are and how they relate to options.
The document discusses the uncovered interest rate parity condition, which states that expected returns from investing in two currencies should be equal when interest rates and exchange rates adjust to equilibrium. It presents the basic equation and explains how it relates the interest rate differential to expected exchange rate changes. The condition is examined under fixed exchange rates, where it suggests capital flows will pressure a central bank defending a peg. Risk and liquidity premia are introduced to account for why investors currently hold a volatile currency like nubits.
The document discusses interest rates and bond pricing. It defines different types of rates like Treasury rates and LIBOR. It also covers concepts like continuously compounded rates, zero rates, bond yields, and the bootstrap method for constructing a yield curve from market data. Forward rates represent the future implied rates on the current yield curve. The document also discusses bond duration and convexity as well as theories of the term structure.
The document discusses interest rates and bond pricing. It defines different types of rates like Treasury rates and LIBOR. It also covers concepts like continuously compounded rates, zero rates, bond yields, and the bootstrap method for constructing a zero curve from market data. Forward rates represent the future implied rates along the current yield curve. Forward rate agreements allow parties to exchange interest payments at a fixed rate versus the future market rate. Bond duration and convexity measure sensitivity to yield changes.
The document discusses interest rates and bond pricing. It defines different types of rates like Treasury rates and LIBOR. It also covers concepts like continuously compounded rates, zero rates, bond yields, and the bootstrap method for constructing a yield curve from market data. Forward rates represent the future implied rates on the current yield curve. The document also discusses bond duration and convexity as well as theories of the term structure.
This chapter discusses mechanics, duration, and hedging strategies related to interest rate futures. It covers the mechanics of Treasury bond and note futures contracts, including delivery options that provide value to short positions. Eurodollar and Treasury bill futures are also discussed. Duration is introduced as a measure of how long, on average, a bondholder must wait to receive cash flows from a bond. The chapter derives the duration formula and provides an example calculation. It discusses limitations of duration related to assumptions of parallel yield curve shifts and ignores convexity for large shifts.
The document discusses various models for pricing interest rate derivatives such as options on bonds, caps, floors, and swaptions. It describes Black's model, which assumes the underlying rate is lognormally distributed. Black's model is then applied to value European bond options, caplets, and swaptions. The relationships between these derivatives and how their values are impacted by shifts in the yield curve are also covered.
A new derivation of the Black Scholes Equation (BSE) based on integral form stochastic calculus is presented. Construction of the BSE solution is based on infinitesimal perfect hedging. The perfect hedging on a finite time interval is a separate problem that does not change option pricing. The cost of hedging does not present an adjustment of the BS pricing. We discuss a more profound alternative approach to option pricing. It defines option price as a settlement between counterparties and in contrast to BS approach presents the market risk of the option premium.
equity, implied, and local volatilitiesIlya Gikhman
This document discusses connections between stock volatility, implied volatility, and local volatility in option pricing models. It provides an overview of the Black-Scholes pricing model, which assumes stock volatility is known. However, implied volatility estimated from market option prices does not match the true stock volatility. The local volatility model develops implied volatility as a function of underlying variables to better match market prices, without relying on an assumed stock process.
This document discusses pricing models for American option contracts. It begins by outlining the standard model, which values American options based on the moment that guarantees maximum option value. However, the author proposes an alternative view, where the optimal exercise time is when the underlying asset reaches its maximum value on [0,T]. Exercising at this maximum value ensures a payoff equal to the selling price, avoiding arbitrage. The document formalizes this idea using concepts like risk-neutral probabilities and derivations of put-call parity relationships to define fair option prices.
This document discusses pricing models for American options. It specifies that American options can be exercised at any time prior to maturity, unlike European options which can only be exercised at maturity. The value of an American option is defined as the expected value of the European option price using the random exercise time. American options can be decomposed into their European counterpart plus an early exercise premium. Determining the optimal early exercise time is formulated as finding the stopping time that maximizes the expected discounted payoff over the lifetime of the contract. References for further reading on pricing American options are also provided.
We use stochastic methods to present mathematically correct representation of the wave function. Informal construction was developed by R. Feynman. This approach were introduced first by H. Doss Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques. Communications in Mathematical Physics
October 1980, Volume 73, Issue 3, pp 247–264.
Primary intention is to discuss formal stochastic representation of the Schrodinger equation solution with its applications to the theory of demolition quantum measurements.
I will appreciate your comments.
This document discusses issues with the derivation of the Black-Scholes equation and option pricing model. It highlights two popular derivations of the Black-Scholes equation, noting ambiguities in the original derivation. It proposes defining the hedged portfolio over a variable time interval to address these ambiguities. The document also notes drawbacks of the Black-Scholes price, including that it only guarantees a risk-free return over an infinitesimal time period and does not reflect market prices which may incorporate other strategies.
A short remark on Feller’s square root condition.Ilya Gikhman
This document presents a proof of Feller's square root condition for the Cox-Ingersoll-Ross model of short interest rates.
The CIR model describes the dynamics of the short rate r(t) as a scalar SDE with parameters k, θ, and σ.
The theorem states that if the Feller condition 2kθ > σ^2 is satisfied, then there exists a unique positive solution r(t) on each finite time interval t ∈ [0, ∞).
The proof uses Ito's formula and Gronwall's inequality to show that as ε approaches 0, the probability that the solution falls below ε approaches 0 as well.
1. The document presents a new approach to proving comparison theorems for stochastic differential equations (SDEs) using differentiation of solutions with respect to initial data.
2. It proves that if the drift term of one SDE is always greater than or equal to the other, and their initial values satisfy the same relation, then the solutions will also satisfy this relation for all time.
3. Two methods are provided: the first uses explicit solutions, the second avoids this by showing the difference process cannot reach zero in finite time based on its behavior.
In this short notice, we present structure of the perfect hedging. Closed form formulas clarify the fact that Black-Scholes (BS) portfolio which provides perfect hedge only at initial moment. Holding portfolio over a certain period implies additional cash flow, which could not be imbedded in BS pricing scheme, and therefore BS option price cannot be derived without additional cash flow which affects BS option price.
1. The document discusses the pricing of variance swaps using risk neutral valuation. It defines variance swaps as transactions where the payout is based on the difference between realized variance and a prespecified strike variance.
2. It derives a formula for the strike variance that equates it to the risk-neutral expected value of the integrated variance process over the swap period, where the expectation is calculated using Black-Scholes option prices.
3. The document explains that variance swaps allow parties to hedge differences between estimates of ex-ante variance derived from option prices and ex-post variance calculated from realized stock returns over the swap period.
Abstract. Regulations of the market require disclosure of information about the nature and extent of risks arising from the trades of the market instruments. There are several significant drawbacks in fixed income pricing modeling. In this paper we interpret a corporate bond price as a random variable. In this case the spot price does not a complete characteristic of the price. The price should be specified by the spot price as well as its value of market risk. This interpretation is similar to a random variable in Probability Theory where an estimate of the random variable completely defined by its cumulative distribution function. The buyer market risk is the value of the chance that the spot price is higher than it is implied by the market scenarios. First we quantify credit risk of the corporate bonds and then consider marked-to-market pricing adjustment to bond price. In the case when issuer of the corporate bond is the counterparty of the bond buyer counterparty and credit risks are coincide.
There are several significant drawbacks in derivative price modeling which relate to global regulations of the derivatives market. Here we present a unified approach which in stochastic market interprets option price as a random variable. Therefore spot price does not complete characteristic of the price in stochastic environment. Complete derivatives price includes the spot price as well as thevalue of market risk implied by the use of the spot price. This interpretation is similar to the notion of therandom variable in Probability Theory in which an estimate of the random variable completely defined by its cumulative distribution function
Critical thoughts about modern option pricingIlya Gikhman
1. The document discusses issues with the Black-Scholes pricing model and proposes an alternative approach. Specifically, it argues that the B-S equation does not necessarily hold at each point in time and adjustments are needed when moving between time intervals.
2. The alternative approach defines option price stochastically based on possible future stock prices at expiration. Price is set to 0 if stock price is below strike, and a fraction of the in-the-money amount otherwise. This defines a distribution for option price based on the underlying stock distribution.
3. In the alternative approach, option price has an associated market risk defined by the probabilities of the traded price being above or below the stochastically defined price.
This document discusses the construction of riskless derivatives portfolios as proposed by Black and Scholes. It summarizes Black and Scholes' approach and then argues that their portfolio is not truly riskless, as it takes on risk at each discrete time interval. Specifically, the portfolio requires reconstruction at each time point to eliminate risk, and in the limit of infinitesimally small time intervals, the portfolio retains risk at all times. The document makes a similar argument against the claim that portfolios of multiple derivatives can be constructed to be riskless.
1) The document outlines drawbacks in the Black-Scholes option pricing theory, including mathematical errors in its derivations. Specifically, the assumption that a hedging portfolio eliminates risk is incorrect as a third term was omitted from the change in the portfolio value.
2) It also discusses issues with the local volatility adjustment concept, noting that transforming the constant diffusion coefficient to a local volatility surface does not actually explain the smile effect observed in options data.
3) While local volatility aims to match implied volatilities observed in the market, the theory suggests the local volatility surface should actually be equal to the original constant diffusion coefficient.
1) The document discusses pricing models for derivatives such as options and interest rate swaps. It introduces concepts such as local volatility, which models implied volatility as a function of strike price and time to maturity.
2) Black-Scholes pricing is based on the assumption of a perfect hedging strategy, but the document notes this is formally incorrect as the hedging portfolio defined does not satisfy the required equations.
3) Local volatility presents the option price as a function of strike and time to maturity, with the diffusion coefficient estimated from option price data, whereas Black-Scholes models the price as a function of the underlying and time, with volatility as an input.
Market risk and liquidity of the risky bondsIlya Gikhman
This document discusses modeling the effect of liquidity on risky bond pricing using a reduced form approach. It begins by presenting a simplified model where default can only occur at maturity. It then extends this to a discrete time approximation for default occurrence. The key concepts discussed are:
- Defining bid and ask prices for risk-free and corporate bonds to model liquidity spread
- Using a single price framework and extending it to account for liquidity spread
- Modeling the corporate bond price as a random variable based on default/no default scenarios
- Defining market and spot prices of bonds and the associated market risks for buyers and sellers
- Estimating the recovery rate and default probability given observations of spot prices over time
Lecture slide titled Fraud Risk Mitigation, Webinar Lecture Delivered at the Society for West African Internal Audit Practitioners (SWAIAP) on Wednesday, November 8, 2023.
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Cross currency swap
1. 1
MARKET RISK AND MARK-TO-MARKET VALUATION OF THE CROSS CURRENCY
SWAP.
Ilya I. Gikhman
6077 Ivy Woods Court Mason,
OH 45040, USA
Ph. 513-573-9348
Email: ilyagikhman@yahoo.com
JEL : G12, G13
Key words. Discount factor, risk free bond, Libor, OIS, market risk, mark-to-market valuation, interest
rate swap, cross currency swap, stochastic Libor model.
Abstract. In this paper we discuss some popular notions of the fixed income pricing. We pay more
attention to formal side of the use such notions as discount factor and mark-to-market valuation of the risk
free cross currency swap.
Introduction. In the first section of this paper, we present arguments for using swap rate as a discount
factor. In the next sections, we continued development of the two pricing concept which was introduced
in [1]. One price of Interest Rate Swap contract is based on the implied swap rate known as swap fixed
rate or swap spread. In our interpretation it is an estimate of the other price that is based on the ‘market’
swap rate. This price is defined for each admissible market scenario. Such point of view interprets market
price as random process. Swap spread then is an estimate of the most probable or most expected outcome
of the random spread. The difference between deterministic swap spread and stochastic spread specifies
market risk of the swap rate. The contemporary finance deals with only implied swap rate and ignores the
market risk that stems from stochastic setting of the problem. In Cross Currency Swap section, we
construct stochastic and implied spreads. The problem is similar to one that was studied in [2], [3] where
the value of the spot spread was presented. On the other hand, we changed the setting format similar to
[3] and the stochastic and deterministic market implied basis of the swap is calculated. In [3] the two
discounting curves method was introduced. One discount curve is used for the presenting implied forward
rate, which is an estimate of the realized future rate. Other discount curve is served as discount all cash
flows. We pay attention to the latter discount factor that should guarantee swap value at par.
2. 2
0. Discount. First, we begin with remarks on discounting problem. Discount factor from date T to
date t, t ≤ T is a currency rate which presents at date t the value of the one unit of the currency at T.
There is no need to assume existing of other currencies and their discount rates. Last time the primary
discount rate in US has been changed from the government bond rates to Libor and then to OIS rates.
Initially discount factor comes to the existence with the risk free bond concept. One pays B ( t , T ) at t to
get one dollar at T. In this case the bond price B ( t , T ) < 1 at t is interpreted as discounted factor from
date T to date t. One can interpret bond purchase as a deal when government borrows B ( t , T ) from
investors at t and return them one dollar at T. In this case, the interest paid by the government is also
referred to as to funding cost. Coupon bond that pays periodic coupon c at dates t j , j = 1, 2, … n
can be represented as a cash flow
CF =
n
1j
c χ ( t = t j ) 1 ( t j ) + F χ ( t = T ) 1 ( T )
Here, 1 ( t j ) denotes one dollar at the date t j , t n = T and F is called face value of the bond. The price
of the coupon bond at t 0 is interpreted as present value, PV 0 of the CF. In the single currency
environment the unique way to present one dollar in a future time is the risk free bond. In this case PV
can be interpreted as a functional defined on discrete time functions as
PV 0 χ ( t = t j ) 1 ( t j ) = B ( t 0 , t j ) 1 ( t 0 )
j = 1, 2, … n. Hence,
B c ( t 0 , T ) = PV 0 CF =
n
1j
[ c B ( t 0 , t j ) + F B ( t 0 , T ) ] 1 ( t 0 )
Libor rate L ( t , T ) can also be interpreted as other type of the dollar denominated discount factor outside
of the USA. Regardless of the way of its construction, the rate can be used to get a fixed amount of
currency in a future moment. Indeed, in order to get one unit of a currency at T one should invest at t the
amount
[ 1 + L ( t , T ) ] – 1
in the Libor contract that expired at T. If Libor is set that there is no arbitrage opportunity between
funding in Libor and T-bond. Investors can use one of these two rates depending on location. One cannot
use bond rate outside of USA and Libor rate if there is no Libor contracts on a particular market. There is
also an economic difference between bond’s interest rate and Libor. The bond’s interest is USA
government borrowing rate while Libor is the market borrowing rate formed by investor demand for
trading. This economic difference between two primary rates represents itself in pricing of liquidity and
credit risks.
One of the latest developments in the fixed income market is a practice to use a swap rate as a discount
factor. In contrast to above two rates a swap rate does not provide fixed amount of dollars at a particular
date in the future and therefore it could not be used as a PV or discount factor. To illustrate this idea
consider overnight index swap (OIS). OIS rate is the rate of a fix-for-floating single currency swap
3. 3
denominated in dollars. The swap maturity is usually between one week and one year period. There are
no intermediate transactions between counterparties and the settlement of the OIS is netted value at
maturity of the swap, which represents the difference between daily compounded fixed and floating
overnight rates. The floating rate of OIS is Fed Funds Rate, FFR. This rate is charged for lending funds
by primary US financial institutions. The FFR of a current business day is issued next business day in the
morning. Here we present a formal definition of the OIS contract. Define fixed and floating interests due
to exchange at OIS maturity T by formulas
OIS fixed = N S ois ( T – t )
OIS floating = N {
n
1j
[ 1 + ff ( j ) d j ] – 1 }
Here, OIS fixed is the interest over lifetime of the swap, N is notional principal of the OIS, S ois is the fixed
interest rate, ff ( j ) is the floating federal fund rate for the j period, ( T – t ) is lifetime of the swap in days
over 365 days in the year format, d j is number of days for the j term in 365 days format. The settlement
value of the OIS is the absolute value of the difference between the two values above.
Therefore, there is no fixed amount at maturity of the OIS. In other words S ois rate fails to answer: what is
the OIS’s present value at t of the $1 at a future moment T. Hence, the OIS or other swap rate could not
be interpreted as a discount rate. One can roll over $1 over a particular period by applying market fixed
rate but there is no fixed amount at t is waiting investors at maturity. The use of a rate from OIS-family as
discount factor for a an instrument might facilitate its valuation but this argument is not a sufficient to
interpret it as a discount rate. More correctly we can say that the OIS rate is a spot estimate of the risky
yield of the daily compounded FFR. As far as the future overnight FFR values are unknown at initiation
the swap contract holds a market risk with probably a small standard deviation. On the other hand, a FFR
but not a OIS rate is a convenient rate for mark-to market (MtM) daily valuations. We return to MtM
valuation later in this paper.
The heuristic idea to low discount rate can be viewed as a way to increase impression on market
participants of a better market stability that in turn may effect on trade volumes and from the other hand
as well as increase availability for funding. This would improve liquidity of the market. Hence, printing
money policy for main currencies can considered as a temporary support of the market nevertheless a
long term effect of QE policy leads to the destruction of the economy in the sense of changes prices either
for the goods or for the currencies. Other observations suggest that additional money do not are
distributed uniformly. More cash are coming to more economically developed countries while less
developed countries arrive at worst economic situation.
1. Interest Rate Swap, IRS. In this section, we outline floating-for-fixed IRS valuation. Suppose
for example that floating rate is a Libor rate. It is a common to use notation L ( t , T ) for the Libor at date
t which expired at T. This notation should be considered as a function of two variables t and T that is
determined for any fixed T and t [ 0 , T ]. This definition corresponds to an assumption that a Libor
contract can be bought and sold during lifetime of the contract. Let counterparty A pays fixed rate
payments and receives floating rate payments from counterparty B at a sequence of reset dates of the
swap. Let t = t 0 denote initiation date of the swap. Assume that A makes fixed rate payments of N C at
dates s 1 < s 2 < … < s m = T to counterparty B and counterparty B makes floating rate payments
4. 4
N L ( t j – 1 , t j ) to A at t 1 < t 2 < < … < t n ≤ T. Then from the A perspective the cash associated
with IRS can be written in the form
CF A ( L ) = N [
n
1j
L ( t j – 1 , t j ) χ ( t = t j ) – C
m
1i
χ ( t = s i ) ] (1)
N is a notional principal of the swap. IRS valuation problem is a determination value of the fixed rate C.
CF A is a stepwise function that changes its values at t = { t j , s i } , j = 0, 1, … n , i = 1, 2, … m.
Rates L ( t j – 1 , t j ) are unknown its values at initiation date t = t 0 and they can be modeled by a random
process [4]. The standard valuation approach for IRS valuation is that the present values, PVs of fixed and
floating legs of the swap should be equal. This approach implies that
PV CF A ( L ) = 0 (2)
Introduce an appropriate discount factor D ( t , T ). Then from (1) and (2) it follows that
n
1j
L ( t j – 1 , t j , ) D ( t , t j ) – C p
m
1i
D ( t , s i ) = 0
and therefore
C p ( ) =
)s,t(D
)t,t(D)ω,t,t(L
i
m
1i
jj1j
n
1j
(3)
Random variable C p = C p ( t , ) depends on market scenario . It represents market fair value of the
IRS for each scenario at t. Besides market price, there exists spot price. This price is applied for trades.
Spot price calculation uses implied market rates l ( t j – 1 , t j , t ) calculated at date t
l ( t j – 1 , t j ; t ) = ]1
)tt()t,t(L1
)tt()t,t(L1
[
1
1-j1-j
jj
j
where Δ j = t j – t j – 1 as estimates of the future random rates L ( t j – 1 , t j , ). Construction of the
cash flow CF A ( l ) follows (1) in which rate L is replaced by its implied estimate l . Thus, we arrive at
the formula
c p =
)s,t(D
)t,t(D)t;t,t(
i
m
1i
jj1j
n
1j
l
(4)
For example formulas
5. 5
P { C p ( ) > c p }, M seller = E C p ( ) χ { C p ( ) > c p },
V seller = E [ C p ( ) - c p ] 2
χ { C p ( ) > c p }
represent market risk, mean of the losses and correspondent volatility of the swap. Latter formulas are
primary characteristics of the risk management of the IRS valuation based on PV reduction. These
formulas use distribution of the random variable C p ( ). This variable is defined by the formula (3) that
is a weighted sum of the random variables L ( t j – 1 , t j , ). Hence in order to present distribution of the
random spread C p ( ) we need to know distribution of the future rates L ( t j – 1 , t j , ) , j = 2, 3, … .
We can use [4] either implied forward rate to present or other model for presentation the future rate
L ( v , v + h , )
l ( v , v + h ; s ) = l ( v , v + h ; t ) +
s
t
( u ) l ( v , v + h ; u ) d u +
+
s
t
λ ( u ) l ( v , v + h ; u ) d w ( u )
Note that given current date t and putting s = v we note that l ( v , v + h ; v ) = L ( v , v + h , ).
Other approach [4] is a direct model of the future rate. The model can be presented in the form
L ( v , v + h , ) = L ( t , t + h ) +
v
t
( u ) L ( u , u + h , ) d u +
+
v
t
( u ) L ( u , u + h , ) d w ( u )
where coefficients are deterministic or random functions, which satisfy some standard conditions.
Historical data usually used for estimation of the unknown coefficients of the equations. The solution of
the general linear stochastic differential equation can be presented in closed form [5]. Having the
distribution of the random variables L ( t j – 1 , t j , ), j = 2, 3, … one can calculate introduced above
market risk characteristics.
2. Cross Currency Swap, CCS.
The Problem Setting. Now we apply discount concept for valuation of floating-for-floating risk free
cross currency swap (CCS). Let t = t 0 denote initiation date and t 1 < t 2 < … < t n = T be a sequence
of reset dates of the swap. At initiation date the exchange of the two principals takes place. Counterparty
A sends to counterparty B amount of N 0 denominated in USDs and receives from B the equivalent
amount of N 1 of denominated in euro. During lifetime of the contract counterparty A pays B payments in
EUR based on 3M ( Libor + α ) and receives payments in USD based on 3M Libor from B. At the end of
the contract the initial amounts of N j , j = 0 , 1 are returned to their original owners. From the
counterparty B perspective cash flow associated with the CCS transactions can be presented in the form
6. 6
CF B = [ N 0 1 0 ( t 0 ) – N 1 1 1 ( t 0 ) ] χ ( t = t 0 ) +
n
1j
{ N 1 [ L 1 ( t j – 1 , t j ) + α ] 1 1 ( t j ) –
– N 0 L 0 ( t j – 1 , t j ) 1 0 ( t j ) } χ ( t = t j ) + [ N 0 1 0 ( T ) – N 1 1 1 ( T ) ] χ ( t = T )
where low index 0 is assigned to USD and index 1 to the EUR. Denote L k ( t , T ) date t Libor rate for
j-currency with expiration at T , and 1 j ( t ) , j = 0 , 1 denotes one unit of j-currency. The exchange rate
q ( t ) at t is defined by equality
1 0 ( t ) = q ( t ) 1 1 ( t )
and therefore N 1 = N 0 q ( t 0 ). Note that cash flow CF B should be considered as a formal definition of
the CCS contract. The cash flow from the A perspective is CF A = – CF B . The CCS pricing problem is
derivation of the swap basis α that is also often referred to as to the cross currency swap spread. The cash
flow can be rewritten in equivalent form as
CF B = N 0 { [ 1 0 ( t 0 ) – q ( t 0 ) 1 1 ( t 0 ) ] χ ( t = t 0 ) +
+
n
1j
{ q ( t 0 ) [ L 1 ( t j – 1 , t j ) + α ] 1 1 ( t j ) – (5)
– L 0 ( t j – 1 , t j ) 1 0 ( t j ) } χ ( t = t j ) + [ 1 0 ( T ) – q ( t 0 ) 1 1 ( T ) ] χ ( t = T ) }
As far as the value N 0 does not effect on basis value without loss of generality one can put N 0 = 1.
Valuation. Ignoring credit and counterparty risks we note that first term on the right hand side (5) is
equal to zero. Similar to it the third term does not equal to zero as far as it represents transaction at T and
in general q ( T ) ≠ q ( t 0 ). Current approach to CCS valuation first defines primary and secondary
currencies. Then primary the USD Libor is used as primary discount rate for both sides of the contract
valuation. In case USD and EUR payments should be converted into correspondent to USD cash flow.
Then USD discount rate would be applied for calculations the PV equality of the swap legs. The EUR is
considered less liquid than USD and therefore it is chosen as secondary currency.
Consider the CCS defined by the cash flow (5). During the lifetime of the contract the Libor rates
L k ( t j – 1 , t j ) , k = 0 , 1 over the future periods [ t j – 1 , t j ] , j = 2, …, n are unknown at initiation date
t = t 0 and we assume that these rates can be interpreted as random variables. This assumption implies
that historical data are interpreted as a statistical population. This statistical point of view is called
randomization of the model. Using PV valuation we replace random future rates by its market implied
forward rates that known at t 0. Replacement of the random rates by its market estimates always presents
market risk. The inverse exchange of the original notional sums N j , j = 0, 1 occurs at maturity. In
formula (5) it is expressed by the third term. These final transactions is subject exchange rate risk. It
stems from the fact that exchange rate at t 0 in general does not equal to the real exchange rate at T, i.e.
equality
N 0 1 0 ( t 0 ) = N 1 1 1 ( t 0 )
7. 7
is true at initiation date t 0 and it does not true at the swap maturity T. Hence, the valuation model should
keep in mind two types of the risks exchange and market risks. Each leg of the CCS deals with two
different currencies. Two currencies of the swap do not have equal liquidities and the credit ratings. From
B perspective the value of the market risk of the transaction at date t j , j = 1, 2, … n can be expressed in
the form
{ q ( t 0 ) [ L 1 ( t j – 1 , t j , ω ) + α ] 1 1 ( t j ) – L 0 ( t j – 1 , t j , ω ) 1 0 ( t j ) } χ ( Ω m B ( t j )) χ ( t = t j ) +
(1)
+ [ q ( t 0 ) 1 1 ( T ) – 1 0 ( T ) ] χ ( Ω q B ( T ) ) χ ( t = T )
Here, Ω m B ( t j ) = { ω : L 1 ( t j – 1 , t j , ω ) + α < L 0 ( t j – 1 , t j , ω ) $( t j ) } , j = 1 , 2 , … , n – 1.
Then Ω q B ( T ) = { ω : q ( t ) < q ( T , ω ) } represents the market scenarios for which cost in USD of
the one euro at T is lower than at initiation date t , i.e. 1 1 ( T ) < 1 1 ( t ).
Pricing problem from B perspective is two folds problem. One is to estimate the basis α and the second
is the value of the market risk that is the value of the chance that the basis value implied by the scenario
will be lower than initially estimated , i.e. P { : α ( ) < α }.
Risky scenario for A is a scenario when A pays higher price than spread value. The benchmark approach
presents the spot estimate α as the market implied value. Let us consider implied value of the swap in
stochastic market. Market price is the price assigned to the swap at the date t for each observable scenario.
The first step of the valuation problem is a model for the future rates L k ( t j – 1 , t j , ω ) and future
exchange rate q ( T , ω ). The second step is calculation of the spot price and presenting its risk analysis.
The benchmark approach uses the equation PV CF B = 0 to present a value of the basis swap value
α ( · , ω ). As far as the cash flow (5) consists from the two currencies the present value reduction should
be made cautiously. Pricing derivatives consist from fragmentation of the complex onto simple blocks.
There are different ways to present PV of the cash flow CF B . First, we need to choose single currency
for representation. Usually USD is used for swap valuation. Consider USD PV reduction of the one EUR
at t j , j = 1, 2, … n cash flow. Bearing in mind equalities 1 1 ( t j ) = q – 1
( t j ) 1 0 ( t j ) and the fact that
exchange rate at a future moments are unknown at initiation assume that exchange rate is a random
process q ( t ) = q ( t , ω ) , t ≥ t 0 . Valuation approach corresponds to the strategy of receiving a
payment in EUR and converting it immediately in USD. Such point leads us to the USD denominated
cash flow (1)
n
1j
{ q ( t 0 ) q – 1
( t j ) [ L 1 ( t j – 1 , t j , ω ) + α ] – L 0 ( t j – 1 , t j , ω ) } 1 0 ( t j ) χ ( t = t j ) +
(1)
+ [ q ( t 0 ) q – 1
( T ) – 1 ] 1 0 ( T ) χ ( t = T )
No arbitrage principle applied for a market scenario ω implies that PV 0 of the cash flow (1) leads to
the equation
8. 8
n
1j
{ q ( t 0 ) q – 1
( t j ) [ L 1 ( t j – 1 , t j , ω ) + α ] – L 0 ( t j – 1 , t j , ω ) } D 0 ( t 0 , t j ) +
+ [ q ( t 0 ) q – 1
( T ) – 1 ] D 0 ( t 0 , T ) = 0
It follows that
α ( · ; t 0 , ω ) = [
n
1j
q ( t 0 ) q – 1
( t j , ω ) B 0 ( t , t j ) ] – 1
{
n
1j
[ L 0 ( t j – 1 , t j , ω ) –
(6)
– q ( t 0 ) q – 1
( t j , ω ) L 1 ( t j – 1 , t j , ω )] D 0 ( t 0 , t j ) + [ 1 – q ( t 0 ) q – 1
( T , ω ) ] D 0 ( t 0 , T ) }
Equality (6) represents the market basis that is determined for each market scenario ω. At initiation of the
swap contract date t sellers and buyers draw a market estimate of the α ( · ; t , ω ). It is a market practice
to use market implied prices as the estimates of the contract components. The date t estimates of the
random future exchange rate and Libor rate q ( t j , ω ) , L ( t j – 1 , t j , ω ) are the non-random date t
forward exchange rate contract q ( t j , t ) and implied forward Libor rate l ( t j – 1 , t j , t ) correspondingly.
Then the market implied swap basis is an estimate of the random basis α ( · ; t , ω ). It can be presented in
the form
αˆ ( · , t 0 ) = [
n
1j
q ( t 0 ) q – 1
( t j , t ) B 0 ( t 0 , t j ) ] – 1
{
n
1j
{ [ l 0 ( t j – 1 , t j , t 0 ) –
(6)
– q ( t 0 ) q – 1
( t j , t 0 ) l 1 ( t j – 1 , t j , t 0 ) ] D 0 ( t 0 , t j ) + [ 1 – q ( t 0 ) q – 1
( T , t 0 ) ] D 0 ( t 0 , T ) }
Scenarios for which { ω : α ( · ; t , ω ) > α ( · , t ) } correspond to the risk of the counterparty A. For this
scenarios A pays a higher price than it is implied at t. Note that the basis valuation represented above is
applied by PV reduction
PV 0 ( t 0 ) CF B = 0
for valuations of the basis of the swap.
3. Mark-to-Market (MtM) valuation of the CCS with zero chance of default.
Given PV reduction of the cash flows let us outline the essence of the MtM valuation. Consider the first
trade day t 0 + 1 after swap initiation at t 0 and suppose that t 0 + 1 < t 1. At the date t 0 + 1 new
values of the implied Libor l k ( t j – 1 , t j , t 0 + 1 ), k = 0, 1 and forward exchange rates q ( t j , t 0 + 1 ),
j = 1, 2, … n are coming up that do not coincide with the corresponding data of these rates calculated at
previous date t 0 . New data effect on swap spread value αˆ ( t 0 + 1 ) which can go up or down with
respect to αˆ ( t 0 ). The market implied value of the CCS at the date t 0 + 1 can be presented in the form
9. 9
V B ( t 0 + 1 ) =
n
1j
{ q ( t 0 ) q – 1
( t j , t 0 + 1 ) [ l 1 ( t j – 1 , t j , t 0 + 1 ) + αˆ ( t 0 + 1 ) ] –
– l 0 ( t j – 1 , t j , t 0 + 1 ) } D 0 ( t 0 + 1 , t j ) + [ q ( t 0 ) q – 1
( T , t 0 + 1 ) – 1 ] D 0 ( t 0 + 1 , T )
Recall that αˆ ( t 0 + 1 ) is calculated at t 0 + 1 and V B ( t 0 + 1 ) = V B ( t 0 + 1 , αˆ ( t 0 + 1 )) = 0.
The date-t 0 hypothetical forward transactions at t 0 + 1 is based on the basis spread αˆ ( t 0 ) can be
presented by the formula
N 1 [ L 1 ( t 0 , t 0 + 1 ) + αˆ ( t 0 ) ] 1 1 ( t 1 ) – N 0 L 0 ( t 0 , t 0 + 1 ) 1 0 ( t 1 ) =
= N 0 { q ( t 0 ) q – 1
( t 0 + 1 , t 0 ) [ L 1 ( t 0 , t 0 + 1 ) + αˆ ( t 0 ) ] – L 0 ( t 0 , t 0 + 1 ) } 1 0 ( t 0 + 1 )
while the real value of this transaction at the date t 0 + 1 is
N 0 { q ( t 0 ) q – 1
( t 0 + 1 ) [ L 1 ( t 0 , t 0 + 1 ) + αˆ ( t 0 ) ] – L 0 ( t 0 , t 0 + 1 ) } 1 0 ( t 1 )
The difference between realized value at date t 0 + 1 and its date t 0 estimate is equal to
δ ( t 1 , ) = N 0 q ( t 0 ) [ q – 1
( t 0 + 1 ) – q – 1
( t 0 + 1 , t 0 ) ] [ L 1 ( t 0 , t 0 + 1 ) + αˆ ( t 0 ) ]
The value δ ( t 1 , ) and date t 0 + 1 new values of the implied forward rates l k ( t j – 1 , t j , t 0 + 1 ),
k = 0, 1 will effect on change of the spread value at t 0 + 1. It is sufficient to make MtM calculations for
one of the cash flow either for ‘in’ or ‘out’. Assuming that t 0 + 1 < t 1 we note that
PV ( k ) ( CF B ,in ) =
n
1j
q ( t 0 ) q – 1
( t j , k ) [ l 1 ( t j – 1 , t j , k ) + αˆ ( k ) ] D 0 ( k , t j ) +
+ q ( t 0 ) q – 1
( T , k ) D 0 ( k , T ) ,
where k = t 0 , t 0 + 1. Define inflow and outflow to counterparty B. Ignoring chance of default the
inflow and outflow to B can be represented correspondingly in the forms
CF B , in =
n
1j
q ( t 0 ) q – 1
( t j ) [ L 1 ( t j – 1 , t j , ω ) + α ] 1 0 ( t j ) χ { t = t j } +
+ q ( t 0 ) q – 1
( T ) 1 0 ( T ) χ { t = T }
CF B , out =
n
1j
L 0 ( t j – 1 , t j , ω ) 1 0 ( t j ) χ { t = t j } + 1 0 ( T ) χ { t = T }
Let
Δ PV ( t 0 ) ( CF B ,in ) = PV ( t 0 + 1 ) ( CF B ,in ) – PV ( t 0 ) ( CF B ,in ) > 0 (7)
10. 10
This represents a market scenario for which A should make larger hypothetical payment than it was
scheduled at t 0 . In this case counterparty A should make the first payment of Δ PV ( t 0 ) ( CF B ,in ) in
MtM account. The value of the payment represents risk exposure of the counterparty A. If the value (7) is
negative then the first MtM payment should be made by B. On the next day t 0 + 2 we repeat similar
calculation to present the value
Δ PV ( t 0 + 1 ) ( CF B ,in ) = PV ( t 0 + 2 ) ( CF B ,in ) – PV ( t 0 + 1 ) ( CF B ,in )
If Δ PV ( t 0 + 1 ) ( CF B ,in ) > 0 then A should add new payment Δ PV ( t 0 + 1 ) ( CF B ,in ) to MtM
account. To present value of the MtM account at t 0 + 2 we need to adjust the past date t 1 value of the
MtM account Δ PV ( t 0 ) ( CF B ,in ) for [ t 0 , t 0 + 1 ] period. If D 0 ( t 0 , t 0 + 1 ) represents primary
discount rate then date t 0 + 2 value of the past MtM account is
D 1
0
( t 0 , t 0 + 1 ) Δ PV ( t 0 ) ( CF B ,in )
Assume that Δ PV ( t 0 + 1 ) ( CF B ,in ) < 0. Then counterparty B due to MtM payment at the date
t 0 + 2. Consider the difference
Δ PV ( t 0 + 1 ) ( CF B ,in ) – D 1
0
( t 0 , t 0 + 1 ) Δ PV ( t 0 ) ( CF B ,in )
If the difference is positive then counterparty B should make the payment equal to the above value in
MtM account. If the difference is negative then
D 1
0
( t 0 , t 0 + 1 ) Δ PV ( t 0 ) ( CF B ,in ) – Δ PV ( t 0 + 1 ) ( CF B ,in )
should be returned to A. Similar calculations continue on next steps. Let t j , j = 1, 2, … n be a reset date
and Q ( t j – 1 ) denote the value of the MtM account at t j – 1 date. Consider the difference
Δ PV ( t j – 1 ) ( CF B ,in ) = PV ( t j ) ( CF B ,in ) – PV ( t j – 1 ) ( CF B ,in )
The value of the MtM account at the date t j is equal to
MtM act ( t j ) = Q ( t j – 1 ) D 1
0
( t j – 1 , t j ) + Δ PV ( t j – 1 ) ( CF B ,in ) (8)
If two terms on the right hand side (8) are positive then party A should pay to MtM account the sum of
Δ PV ( t j – 1 ) ( CF B ,in ). On the other hand if Δ PV ( t j – 1 ) ( CF B ,in ) < 0 and MtM act ( t j ) > 0
the sum Δ PV ( t j – 1 ) ( CF B ,in ) should be withdrawn from MtM account and return to party A. In case if two
terms on the right hand side (9) are negative then B should add the sum Δ PV ( t j – 1 ) ( CF B ,in ) to
MtM account.
Recall that in our construction we assumed that CCS with zero chance of default. In case of positive
chance of default MtM account should not be netted as MtM account should protect against
counterparties default.
11. 11
Appendix.
Recall that PV concept came up with bonds valuations. By definition PV is an amount paid at initiation of
contracts in order to receive $1 at bond’s maturity. On the other hand an IRS is a contract for exchange
future rate payments N L ( t j – 1 , t j , ) unknown at initiation for the determined at t fixed rate payments
N c. Assume for example that t 1 < s 1. In this case counterparties obtain their payments and invest them
immediately at risk free interest rate D – 1
( t j , T ) , j = 1, 2, ... . Counterparty A account at maturity T
is equal to
FV A ( L ) =
n
1j
L ( t j – 1 , t j , ) D – 1
( t j , T ) – C f
m
1i
D – 1
( s i , T )
Note that the values D ( s i , T ) do not known at t and these values we assume can be interpreted as
random variables. No arbitrage principle extended on pricing for admissible market scenarios states that
counterparty starting with 0 account at initiation date t should arrive at 0 at maturity regardless of the
market scenario . Applying this idea, we arrive at the market spread value
C f ( ) =
)ω,T,s(D
)ω,T,t(D)ω,t,t(L
i
1
m
1i
j
1
j1j
n
1j
(3′)
Denote D ( t j , T ; t ) date t implied forward rate over the time period [ t j , T ]. Then we can define the
implied date-t spot value of the cash flow FV A ( l ) and calculate the spread implied by the market
c f =
)t;T,s(D
)t;T,t(D)t,t,t(
i
1
m
1i
j
1
j1j
n
1j
l
(4′)
One can expect that in the most scenarios c p c f . Assume for example that c p > c f . Then forward
value reduction suggests that the fixed leg rate c p is overvalued. Indeed, date t market analysis
demonstrates that swap spread that makes equal fixed and floating cash flows equal at T is less than
applied for the swap contract by using PV reduction. On the other hand inequality c p < c f implies the
opposite situation that suggests that floating leg payments dominates over the fixed one. A complete
valuation of the IRS cannot be reduced to a single FV or PV of the cash flows. Let us consider additional
characteristic of the IRS spread. Define market and market implied rate of return of the floating leg
RR ( t , ) =
)ω(PV
)ω(FV
- 1 =
)t,t(D)ω,t,t(L
)ω;T,t(D)ω,t,t(L
jj1j
n
1j
j
1
j1j
n
1j
- 1
12. 12
rr ( t ) =
PV
FV
- 1 =
)t,t(D)t,t,t(
)t;T,t(D)t,t,t(
jj1j
n
1j
j
1
j1j
n
1j
l
l
- 1
These are market estimates of the real rate of return of the IRS. The case c p > c f ( c p < c f )
corresponds to rr ( t ) < 0 ( rr ( t ) > 0 ). In stochastic environment investment can be characterized by
its PV and its expected market rate of return. Hence, in case rr ( t ) < 0 market suggests that that fixed
payment of the IRS is overstated while rr ( t ) > 0 means the opposite. The market estimate and
corresponding statement regarding value of the IRS’s spread is subject of risks. In the favorable for the
fixed rate payer’s case c p < c f the market risk can be defined as P { RR ( t , ) < rr ( t ) }. In the
favorable for fixed rate receiver case c p > c f the market risk can be measured by the probability
P { RR ( t , ) > rr ( t ) }.
Remark. The notation L ( t , T ) , 0 ≤ t ≤ T < + ∞ use above inexplicitly admitted that Libor contracts
can be traded over lifetime of the contract by using Libor rate. Indeed, for a particular contract with
expiration at T the function L ( t , T ) is defined for any t, t ≤ T. This function specifies the value of the
Libor contract at any time prior to T. On the other hand the real Libor rates are existed only for the fixed
maturities h equal to 1 day, 1 week, 2 weeks, and j = 1, …, 12 months. In such setting Libor rates are
defined with the help of the function L ( t , t + h ) = L h ( t ). Formulas (1) – (4) hold as they are take
place also for the fixed t and T. The stochastic equation for implied forward rate
l ( v , v + h ; s ) = l h ( v ; s ) , s > t which presents the future rate L h ( v , ) = L ( v , v + h , )
can be written as
l h ( v ; s ) = l h ( v ; t ) +
s
t
( u ) l h ( v ; u ) d u +
+
s
t
λ ( u ) l h ( v ; u ) d w ( u )
Here, l ( v , v + h ; t ) = l h ( v ; t ) is deterministic number known at t. On the other hand we can use the
above direct model to define future Libor rate
L h ( v , ) = L h ( t ) +
v
t
( u ) L h ( u , ) d u +
+
v
t
( u ) L h ( u , ) d w ( u )
Denote
13. 13
c p ( h ) =
)s,t(D
)t,t(D)t;t,t(
i
m
1i
jj1jh
n
1j
l
Putting c p ( h ) = c p ( h 1 ) – c p ( h 2 ) , h 1 > h 2 we note that
c p ( h ) =
)s,t(D
)t,t(D)t;t,t(
)s,t(D
)t,t(D)t;t,t(
i
m
1i
jj1j2
n
1j
i
m
1i
jj1j1
n
1j
2
2
1
1
ll
Here l j = l jh . We used for simplicity the same sequence of reset dates for the shorter rate l 2 as for the
longer rate l 1 . This is implied market estimate of the basis spread. Investment in basis spread is risky.
The value of the risk from buyer perspective is measured by the probability
P { C f ( h , ) < c p ( h ) }
where C f ( h , ) = C f ( h 1 , ) - C f ( h 2 , ) and
C p ( h q , ) =
)s,t(D
)t,t(D)ω,t,t(L
i
m
1i
jj1jq
n
1j
q
q
q = 1, 2. It presents a value of the chance that buyer pays more than it implies by the market scenario.
Here we present alternative approach to CCS valuation. We call two investments are equal at a moment of
time if their instantaneous rates of return are equal and if two investments are equal at any moment on a
time interval then we called them equal at this interval [1].
Payment received at t j and denominated in EURs should be at t j converted in USD and invested in US
Treasury bonds. Alternatively they can be invested in EUR denominate bonds and then converted into
USDs at the swap maturity T. In either case the pricing should specify the lowest swap spread. Note that
it is possible that the pricing solution of the CCS can be obtained by converting a EUR payment
immediately in USD while other EUR payment is better to invest in EUR bond and convert it at a later
time or at the maturity. This remark highlight the fact that spread calculation is the sufficiently risky
procedure.
The PV denominated in USD at t and FV at T can written in the forms
14. 14
PV ( t ) ( CF B ,in ) =
n
1j
q ( t 0 ) q – 1
( t j , ω ) [ L 1 ( t j – 1 , t j , ω ) + α ] D 0 ( t , t j ) +
+ q ( t 0 ) q – 1
( T , ω ) D 0 ( t , T )
FV ( T ) ( CF B , in ) =
n
1j
q ( t 0 ) q – 1
( t j , ω ) [ L 1 ( t j – 1 , t j , ω ) + α ] F 0 ( t j , T, ω ) +
+ q ( t 0 ) q – 1
( T , ω )
PV ( T ) ( CF B , out ) =
n
1j
L 0 ( t j – 1 , t j , ω ) D 0 ( t , t j ) + D 0 ( t , T )
FV ( T ) ( CF B , out ) =
n
1j
L 0 ( t j – 1 , t j , ω ) F 0 ( t j , T , ω ) + 1
Here D 0 ( t , T ), F 0 ( t , T ) denote an appropriate USD discount and forward rates. One should presents
the models for the future rates D , l , q , F and their market implied forward rates should be modeled to
present a numerical solution of the swap pricing problem. The equation representing equality of the rates
of return of the market implied cash flows can be written as
)CF(VP
)CF(VF
in,B
in,B
=
)CF(VP
)CF(VF
out,B
out,B
(9)
Solving (9) for α ( · , ω ) we arrive at the market stochastic market swap basis values. Thus
α ( · , ω ) =
=
)ω,T,t(F)ω,t(q)t(q)ω,T,t(Λ)t,t(D)ω,t(q)t(q
])t,t(D)ω,T,t(F)ω,T,t(Λ[)ω,t,t(L)ω,t(q)t(q
j0j
1
0
n
1j
0j00j
1
0
n
1j
j00j00j1-j1j
1
0
n
1j
–
)ω,T,t(F)ω,t(q)t(q)ω,T,t(Λ)t,t(D)ω,t(q)t(q
])T,t(D)ω,T,t(Λ[)ω,t(q)t(q
j0j
1
0
n
1j
0j00j
1
0
n
1j
00j
1
0
where
15. 15
Λ 0 ( t 0 , T , ) =
1)t,t(D)ω,t,t(L
1)ω,T,t(F)ω,t,t(L
j00j1-j0
n
1j
j0j1-j0
n
1j
Next applying market implied deterministic estimates we can get implied by the market value α ( · , t ).
The valuation formula is similar to above formula in which random rates q ( t j , ), L k ( t j – 1 , t j , ),
F k ( t j , T , ) are replaced by its market estimates q ( t j , t 0 ) , l k ( t j – 1 , t j , ), f k ( t j , T , t 0 ),
k = 0, 1. We used notation D 0 ( t , t j ), j = 1, 2, … n for USD discount factor. The real world examples
of the discount factor are US T-bond or Libor rates. In section 1 we argued that OIS rate by its
construction does not guarantee at t = t 0 a fixed amount at any future moments t j , j = 1, 2, … n.
16. 16
Bibliography.
1. Gikhman I. Alternative Derivatives Pricing: Formal Approach. LAP LAMBERT Academic
Publishing, 2010, p. 164.
2. R. Jarrow, S Turnbull. Derivatives Securities, South-Western College Publishing, 2ed. 2000, 684.
3. W. Boenkost, W. Schmidt. Cross Currency Swap Valuation, 2005, 11.
4. I. Gikhman. Fixed Rates Modeling.
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2287165.
5. I. Gikhman , A. V. Skorohod. Stochastic Differential Equation Springer-Verlag 1972.