Implied interest rate pricing models
The main objective and result of this paper is to formulate a model for interest rates which is consistent with market prices of a set of vanilla swaptions (it is supposed that instruments in the market are divided into liquid vanilla products which are readily available at a known market price, and illiquid exotic products). It was done for swaptions of a common maturity date, \(T\), and under the assumption of a single factor model. But to describe the main result mathematically we need some notations. Let \(D_{tS}\) be the time-\(t\) value of a pure discount bond (vanilla pr.) paying unity at time \(S\) and consider a derivative which, at time \(T\), pays an amount \(V_T= V_T(D_{TS_0}, D_{TS_1},\dots, D_{TS_N})\) for a finite set of times \(T= S_0 < S_1 <\cdots <S_N\). As is known in the theory, given a suitable model for the evolution of the bond prices, the time-\(t\) value of the derivative (option, e.g.) is given by \(V_t= N_t\mathbb {E_N}[V_T N_T^{-1}\mid \mathcal F_t]\); here \(N_t\) is any suitable numeraire process, \(\{\mathcal F_t\}\) is the filtration generated by the processes \((N_u, D_{uS_i}, u\leq t, i= 1,\dots ,N)\), and \(\mathbb {E_N}\) denotes expectations in the measure \(\mathbb N\), in which all numeraire re-based assets, \(D_{tS_i}N_t^{-1}\), are \(\{\mathcal F_t\}\)-martingales. Then under some assumptions (all the discount bond prices at time \(T\) can be expressed as a monotone function of a univariate random variable \(r\), all interest rates are positive), and working with \(D_{tT}\) as numeraire, it was shown that the value of a European derivative is given by \[ V_t= D_{tT}\mathbb {E_F}[V_T(D_{TS_0}(r), D_{TS_1}(r),\dots,D_{TS_N}(r))\mid \mathcal F_t]; \] here \(\mathbb F\) is the forward measure, in which \(D_{TS}/D_{tT}\) is an \(\{\mathcal F_t\}\)-martingale for all \(S\). Now we are able to formulate the main result, using 1) the value of the fixed rate which gives the swap (first product considered) zero value and is given by \(y_t^N= (D_{tS_0} - D_{tS_N})(\sum_{j=1}^N \alpha_jD_{tS_j})^{-1}\) and 2) the time-\(t\) value \(V_t^i(K_i)\) of a receiver's swaption (second product used for calibration) with a final payment date \(S_i\) for the underlining swap and with strike \(K_i\); \(\alpha_j= S_j - S_{j-1}.\) Subject to minor and natural regularity conditions on the swaption prices \(V_0^i(K_i)\), the set of equations \[ \frac {\partial V_0^{i-1}(K_{i-1})}{\partial K_{i-1}}= (1+\alpha_iK_i) \frac {\partial V_0^i(K_i)}{\partial K_i} - \alpha_i(V_0^i(K_i)+ \lambda),\;\;\;i=1,2,\dots,N, \tag{1} \] admit a unique solution \((K_1, K_2,\dots,K_N)(\lambda)\) for each \(\lambda \in (0, D_{0T})\). Furthermore, for each \(i=1, 2,\dots,N\), \[ \lambda= D_{0T}\mathbb F(y_T^i < K_i(\lambda)), \tag{2} \] and under some additional assumption there is the extra property \[ \{r: y_T^i(r)<K_i(\lambda)\} = \{r: y_T^j(r)<K_j(\lambda)\}, \qquad \forall i,j. \tag{3} \] The authors state that relations (1-3) completely specify the distribution of \((y_T^1, y_T^2,\dots, y_T^N)\) and hence \((D_{TS_0}, D_{TS_1},\dots, D_{TS_N})\) in the forward measure. Option valuation then reduces to numerically performing a single one-dimensional integral.
scientific article; zbMATH DE number 1160488