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...y the firms, since it can offer the firms a flexible way in hedging risks. A swap is a kind of financial derivative between two counterparties which exchange, usually, one stream of cash flows against another cash stream. The dates when to pay the cash flows and the way to calculate them are demonstrated in the swap agreements. Interest rate swaps, foreign currency swaps, equity swaps, commodity swaps and credit default swaps make up the five basic swaps which are determined by an interest rate, foreign exchange rate, equity price, commodity price or some other underlying assets respectively. [1] 1.1 Swaps (1) Interest rate swaps The interest rate swaps are the simplest interest rate derivative. In the contract, one party exchanges a loan at a fixed rate of interest, which is called swap rate, for a loan at a floating rate during a given period. In general, the notional principle is not exchanged between the two counterparties. The interest swaps have advantages such as refinancing the debt of the firm and reducing the risk of interest rate fluctuations. For example, one party B, specifically a bank, possesses assets which yield returns at a floating rate referring to LIBOR (the Londo...

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..., not only the interest payments but also the principal can be exchanged on the two equal loans. [3] It takes advantage of hedging against exchange rate fluctuations. Generally, three kinds of payment flows are involved in a currency swap. First, the two parties get the amount of cash in certain currency they need respectively by the exchange of two different currencies with each other. Second, they pay each other the interests that demonstrate the interest rates level in the home country periodically according to the contract. Finally, the principle is re-exchanged and the swap is completed. [6] (3) Commodity swaps It is increasingly common that commodity swaps are used as an important tool in the energy industries and agriculture. Commodity swaps apply the principle of the interest rate swaps to the commodity prices. For a given quantity of some commodity, like crude oil for example, Party A makes payments to Party B at a fixed price per unit during a certain period. Meanwhile, Party B offers cash flows to Party A at a floating price per unit. By means of the commodity swap, the users of the commodity can control the cost at a desirable level; they have to bear some kind of risks su...

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... comparative advantages of the different firms. For instance, Party A has a comparative advantage in fixed-rate asset markets, while Party B takes a comparative advantage in floating-rate markets. Due to a interest rate swap introduced to the credit markets, transforming a fixed rate capital flow into a floating rate one comes true or vice versa. Therefore, the comparative advantages can be exploited to reduce the interest rate risks effectively. (2) Currency swaps 7 Currency swaps, introduced in the 1970s due to foreign exchange controls in Britain, have been an important tool for financing. [2] In a currency swap contract, Party A makes predetermined payments periodically to Party B in one currency like U.S. dollars, meanwhile, Party B pays a certain amount in another currency like Euros. However, in this kind of agreements between two parties, not only the interest payments but also the principal can be exchanged on the two equal loans. [3] It takes advantage of hedging against exchange rate fluctuations. Generally, three kinds of payment flows are involved in a currency swap. First, the two parties get the amount of cash in certain currency they need respectively by the exchange o...

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...fore, the comparative advantages can be exploited to reduce the interest rate risks effectively. (2) Currency swaps 7 Currency swaps, introduced in the 1970s due to foreign exchange controls in Britain, have been an important tool for financing. [2] In a currency swap contract, Party A makes predetermined payments periodically to Party B in one currency like U.S. dollars, meanwhile, Party B pays a certain amount in another currency like Euros. However, in this kind of agreements between two parties, not only the interest payments but also the principal can be exchanged on the two equal loans. [3] It takes advantage of hedging against exchange rate fluctuations. Generally, three kinds of payment flows are involved in a currency swap. First, the two parties get the amount of cash in certain currency they need respectively by the exchange of two different currencies with each other. Second, they pay each other the interests that demonstrate the interest rates level in the home country periodically according to the contract. Finally, the principle is re-exchanged and the swap is completed. [6] (3) Commodity swaps It is increasingly common that commodity swaps are used as an important tool...

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...me debt issuers need the right to withdraw a certain swap in future, while others require the chance to enter into a prespecified swap later. Obviously, we can regard swaptions as calls or puts on coupon bonds. There are two forms of swaption, the payer swaption and receiver swaption. A payer swaption grants the holder of the swaption the right to get into a swap contract where they pay the fixed leg and receive the floating leg, while the receiver swaption gives the holder a chance which is not obligatory to enter into a swap contract where they pay the floating leg instead of the fixed leg. [5] Not only can a swaption hedge the holders against risks, it also benefits the buyers, compared to the ordinary swaps. The swaption will be exercised as declared in the contract between two counterparties, in the condition that the strike rate of the swaption is more attracting than the present market swap rate. In detail, the swaption gives the holder the benefit of the relatively lower agreed-upon strike rate than the market rates, with the flexibility to enter into the current market swap rates and vice versa. 9 Typically, such a contract includes the terms: (1) the price of the swaption (2...

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... (Surely we can approximate the pricing with the help of discrete-time models as well). Given a derivative, we can obtain a valuation formula either by martingales or by partial differential equations (PDEs). Now we just talk about the latter method-- PDE approach. In that way, first we make the descriptions of the state variables with a stochastic differential equation (SDE) and then acquire a PDE, with the coefficients of the given SDE included. On the other hand, following the processes with jumps, the state variables can also be got. Naturally, the additional integral terms will come out. [7] Constructing the stochastic processes, we can model the behavior of financial market. Partial differential equations act as a valuable part in the process of mathematical modeling. Feynman-Kac formula establishes the relationship between the stochastic analysis and PDEs. 2.3.1 Feynman-Kac Formula Ϝ (t, x) + μ(t, x) Ϝ + σ2(t,x) =0, F(T, x) = Φ(x) Provided F is a solution to the above boundary value problem on [0,T]. The formula presents us a probabilistic representation of solutions to PDEs, which has something to do with general SDEs. Consequently, by solving the corresponding PDE, the proble...

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...be done by choosing an appropriate short rate model arbitrarily introduced in Section 4, for instance, the Ho-Lee Model. Finally we can obtain a PDE which describes the dynamics of Li. By choosing some numerical method like Monte Carlo, we can 26 simulate Li in discrete time because LIBOR rate is a discrete market rate instead of the continuously time market rate like instaneously interest rate. However, in my thesis, we mainly focus on the performance of the discount factors in two different cross currency swap valuation methods. Therefore, for simplicity, we just directly cite the data from [8]. Of all types of cross currency swaps, the floating to floating cash flows type is called a basis swap. That is, a basis swap can be regarded as a contract in which two floating rate bonds are exchanged. Since the two principals cannot be equal all the time because of the change in the exchange rate. Then, how can the market manage the problem of making basis swaps to be fair? The answer is the marker introduces so-called cross currency basis spread. The cross currency basis spreads usually refer to a liquidity benchmark, e.g. USD LIBOR. The discount factors are used to discount the cash flow...