速寫 Day35

今天又是懶懶日。

Plain vanilla interest rate swap: the swap arrangement agrees to pay a periodic fixed rate on a notional principal in exchange of a periodic floating rate over the tenor of the swap.
Net potential saving by entering into a swap: difference on difference (the difference between the difference on fixed rate and the difference on floating rate)
1%=100bps

\(V_{swap}=Bond_{fixed}-Bond_{floating}\), we can value the fixed-rate bond using the spot rate curve.
However, the value of a floating-rate bond will be equal to the notional amount at any of its periodic settlement dates when the next payment is set to the market (floating) rate.
Reset date: the floating-rate portion has a value equal to the notional amount.

The floating-rate cash flows are based on expected forward rates via the LIBOR based spot curve.

\(V_{swap}(USD)=B_{USD}-(S_0\times B_{EUR})\), where \(S_0=\)spot rate in USD per EUR.

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A martingale is a zero-drift stochastic process, which means that the coefficient of \(dt\) should be zero.
<> In other words, a martingale has the convenient property that its expected value at any future time is equal to its value today.
<> That is, \(E(\theta_T)=\theta_0\), where \(\theta\) follows a martingale if its process has the form \(d\theta =\sigma dz\)

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Define: \(\phi =\dfrac{f}{g}\), where \(\phi\) is the relative price of \(f\) with respect to \(g\). (It can be thought of as measuring the price of \(f\) in units of \(g\) rather than dollars.)
The security price \(g\) is referred to as the numeraire.
Equivalent martingale measure result shows that, when there are no arbitrage opportunities, \(\phi\) is a martingale when setting the market price of risk which equals to the volatility of \(g\).
Recall: \(dS_t=\mu S_tdt+\sigma S_tdB_t\)
Change the expression: \(df=\mu fdt+\sigma fdz\)
Actually, the complete formula is: \(df=(r+\lambda \sigma )fdt+\sigma fdz\), where \(\lambda \) is referred as the market price of risk.
<> In other words, in the traditional risk-neutral world, the market price of risk \(\lambda =0\). If we set \(g\) equal to the money market account, it grows at rate \(r\) so that \(dg=rgdt\). (The drift of g is stochastic, but the volatility of g is zero.)

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Using Ito’s lemma, we can conclude that: \(d(\dfrac{f}{g})=(\sigma_f-\sigma_g)\dfrac{f}{g}dz\), this shows that \(\dfrac{f}{g}\) is a martingale and proves the equivalent martingale measure result.
We will refer to a world where the market price of risk is the volatility \(\sigma_g\) of \(g\) as a world that is forward risk neutral with respect to \(g\).
\(f_0=g_0E_g(\dfrac{f_T}{g_T})\), where \(E_g\) denotes the expected value in a world that is forward risk neutral with respect to \(g\). (\(FRN_{wrtg}\))

我會繼續努力的。