Abstract
So far, with only few exceptions (e.g. Sect. 8.3.3), we have considered interest rates as being deterministic or even constant. This directly contradicts to the simple existence of interest rate options.
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Notes
- 1.
As we will see later, e.g. in Sect. 30.1 ff, there is more beyond Black-Scholes.
- 2.
The Black-76 model for simple interest rate options can be derived as a special case of the more complex Heath-Jarrow-Morton term structure model.
- 3.
This approach is to be differentiated from market rate models (which are also used to be called Brace-Gatarek-Musiela models or (BGM models for short). For such models, forward interest rates over longer periods (e.g. the 3-months LIBOR rate), but distinct start dates, are modeled (see Sect. 14.13 and [19]).
- 4.
In spot rate models, every interest rate instrument can be interpreted as a derivative V on the underlying S(t) = r(t) or \(S(t)=\ln r(t)\).
- 5.
For convex function f and stochastic variable X, Jensen’s inequality states that f(E[X]) ≤E[f(X)]. The above equation follows from the convexity (all points f(x) with a < x < b lie below a straight line through points f(a) and f(b)) of function f(x) = 1∕(1 + ax).
- 6.
In practice, a FRA pays out at maturity (i.e. at the beginning of the interest rate period) the present value of a virtual future payment at the end of the period. This reduces the credit default risk. If, as we did here, credit default risk is neglected, or if the trade is sufficiently collateralized, this does not have a significant impact.
- 7.
An In-Arrears Swap is simply a portfolio consisting of such FRAs.
- 8.
More traditional models, known as equilibrium models, will not be investigated here. Our discussion will be restricted to arbitrage-free pricing methods.
- 9.
Or for financial instruments whose payoff profile is independent of the short rate, such as a zero bond, for which f(i, j) = 1 for t + (i + j) Δt = T and f(i, j) = 0 otherwise.
- 10.
For m = i we have G i,n(i − 1, j) = 0 and for n = j we have G m,j(i, j − 1) = 0.
- 11.
- 12.
Substituting the expectation into the definition of the variance defined as the expectation of the squared deviation from the expectation gives
$$\displaystyle \begin{aligned} \text{Var}\left[ x\right] & =\text{E}\left[ \left(x-\text{E}\left[ x\right] \right) ^{2}\right] \\ & =p\left(x_{u}-\text{E}\left[ x\right] \right) ^{2}+(1-p)\left(x_{d}-\text{E}\left[ x\right] \right) ^{2}\\ & =p\left(x_{u}-px_{u}-(1-p)x_{d}\right) ^{2}+(1-p)\left(x_{d} -px_{u}-(1-p)x_{d}\right)^{2}\;. \end{aligned} $$Multiplying out and collecting terms yields the desired expression.
- 13.
Here we use the property \(\exp \left \{ ax\right \} =(\exp \left \{ x\right \} )^{a}\) of the exponential function with \(a=e^{2b(i-1,j)\sqrt {\delta t}}.\)
- 14.
To solve a non-linear equation of the form f(x) = 0, the Newton-Raphson method uses the following iteration to find the points were the function f equals zero: having an estimate x i for a zero of f, a better estimate is obtained from the formula
$$\displaystyle \begin{aligned} x_{i+1}=x_{i}-f(x_{i})\left(\left. \frac{\partial f}{\partial x}\right\vert {}_{x=x_{i}}\right) ^{-1}\;. \end{aligned}$$We usually start the procedure with a rough estimate x 0 and iterate until the difference between x i+1 and x i is sufficiently small for the required purpose. The iteration sequence converges if
$$\displaystyle \begin{aligned} \left\vert f\frac{\partial^{2}f\,/\partial^{2}x}{\left(\partial f\,/\partial x\right) ^{2}}\right\vert <1 \end{aligned}$$holds in a neighborhood of the zero of f. This can always assumed to be the case in our applications.
- 15.
At an interest rate of 6% this would correspond to a relative volatility of 14%.
- 16.
As emphasized in Eq. 14.55, the forward rate for the caplet period τ within linear compounding serves as the underlying of a cap in the payoff profile. This has already been accounted for in the term structure model through Eq. 14.55. To obtain the correct forward rate as input for the Black-76 model in the Excel workbook TermStructureModels.xlsm from the download section [50] from the current term structure (which holds for continuous compounding), we first determine the forward discount factors using Eq. 2.7. From those discount factors the desired forward rates for linear compounding are given by \(r=\left ( B^{-1}-1\right ) /\tau \).
- 17.
Alternatively, the volatilities can be (simultaneously) calibrated using a least squares fit. We then minimize the sum of the quadratic differences between the calculated and the traded option prices by varying the volatilities.
- 18.
Naturally, this procedure can be extended if a “caplet-price-surface”, i.e. caplet prices with different times to maturity and different strikes, is available to obtain a calibrated volatility surface as a function of time and moneyness (the relative or absolute difference between underlying and strike).
- 19.
Though, in some cases, market makers quote both, prices and volatilities, at the same time.
- 20.
Doing a move with its associated probability only ensures a very effective sampling of the phase space (the set of all possible values of the simulated variables): phase space regions (values of the simulated variables) which have low probabilities (and therefore contribute only little to the desired averages of whatever needs to be measured by the simulation) are only visited with low probability (i.e. rarely) while phase space regions with high probabilities (which contribute a lot to the desired averages) are visited with high probability (i.e. often). Because of this feature importance sampling is heavily used in thousands of Monte-Carlo applications, especially in physics, meteorology and other sciences which rely on large-scale simulations.
- 21.
This is not to be confused with the definition of stationary time series in Chap. 32.
- 22.
Ho-Lee and Hull-White are often applied for time-dependent volatilities. Their inventors, however, originally assumed constant volatilities.
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Deutsch, HP., Beinker, M.W. (2019). Interest Rates and Term Structure Models. In: Derivatives and Internal Models. Finance and Capital Markets Series. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-22899-6_14
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