## 23 March, 2015

### A Primer in Harmonic Analysis

I picked these problems from Modern Fourier Analysis Vol I - I think that they serve as a good primer for the basic techniques and theorems in harmonic analysis (a subject that I have recently started looking back into in order to deal with some of the techniques used when Levy processes in mathematical finance).
Problem I.  Fix $d\geq1$ and suppose $\psi:(0,\infty)\mapsto[0,\infty)$ is $C^{1}$, non-increasing, and $\int_{\mathbb{R}^{d}}\psi(|x|)\;dx\leq A<\infty.$  Define
$$[M_{\psi}f](x):=\sup_{0<r<\infty}\frac{1}{r^{d}}\int_{\mathbb{R}^{d}}|f(x-y)|\psi\left(\frac{|y|}{r}\right)\;dy$$
and show that $$[M_{\psi}f](x)\leq A[Mf](x)$$
where $M$ is the usual Hardy-Littelwood maximal function.
Solution.  We first observe that the translation invariance of the indicated estimate implies that it is sufficient to prove the case $x=0$ (this can be seen more explicitly by replacing $f$ by $\tau_{x}f$, where $\tau_{x}$ is the translation by $x$ operator, and applying the present case to be proven to see then that the estimate holds for all $x$).   For convenience let us define $\psi_{r}(|y|)=r^{-d}\psi(|y|/r)$.  The radial properties of the terms in the estimate suggest polar coordinates will be useful in dealing with the resultant integrals.  Let us recall that the polar coordinate formula implies as a consequence of itself that
$$\frac{d}{ds}\int_{B(0,s)}f(y)\;dy=\frac{d}{ds}\int_{0}^{s}dt\int_{\partial B(0,t)}f(\omega)\;dS(\omega)=\int_{\partial B(0,s)}f(\omega)\;dS(\omega)=s^{d-1}\int_{S^{d-1}}f(s\omega)\;dS(\omega).$$
In the last term we have used a change of variables in order to place the integration over the the unit sphere (and in particular, to keep the domain fixed).  In order to apply this formula in an integration by parts without causing notational chaos, let us define
$$\alpha(s)=\int_{S^{d-1}}|f(s\omega)|\;dS(\omega)$$
and
$$\beta(s)=\int_{0}^{s}\alpha(t)t^{d-1}\;dt.$$
Note that $\beta(s)$ is majorized by $\omega(d)s^{d}[Mf](0)$ where $\omega(d)$ is the measure of the unit ball in $\mathbb{R}^{d}$.
Let us make a further assumption that $\psi$ is compactly supported on a ball of radius $\delta$ so that $\psi_{r}$ is also compactly supported (and thus bounded) on a ball of radius $r\delta$.  Invoking polar coordinates in the left hand side, setting $x=0$, $\psi(|y|)=\psi(|-y|)$ (in particular, the associativity of convolution), and integration by parts along with the fact that $\beta(0)=0=\psi_{r}(r\delta)$, we estimate at last

\begin{align*} \int_{\mathbb{R}^{d}}|f(-y)|\psi_{r}(|y|)\;dy&=\int_{\mathbb{R}^{d}}|f(y)|\psi_{r}(|y|)\;dy\\ &=\int_{0}^{\infty}\psi_{r}(s)s^{d-1}\;ds\int_{\mathcal{S}^{d-1}}|f(s\omega)|\;dS(\omega)\\ &=\int_{0}^{r\delta}\psi_{r}(s)s^{d-1}\int_{\mathcal{S}^{d-1}}|f(s\omega)|\;dS(\omega)\\ &=\int_{0}^{r\delta}\psi_{r}(s)s^{d-1}\alpha(s)\;ds\\ &=\beta(r\delta)\psi_{r}(r\delta)-\beta(0)\psi_{r}(0)-\int_{0}^{r\delta}\beta(s)d\psi_{r}(s)\\ &=\int_{0}^{r\delta}\beta(s)d(-\psi_{r}(s))\\ &\leq[Mf](0)\int_{0}^{\infty}\omega(d)s^{d}d(-\psi_{r}(s))\\ &=[Mf](0)\int_{0}^{\infty}d\omega(d)s^{d-1}\psi_{r}(s)\;ds\\ &=[Mf](0)\int_{0}^{\infty}\psi_{r}(s)\;ds\int_{\partial B(0,s)}\;dS(\omega)\\ &=[Mf](0)\int_{\mathbb{R}^{d}}\frac{1}{r^{d}}\psi\left(\frac{|y|}{r}\right)\;dy\\ &=[Mf](0)\int_{\mathbb{R}^{d}}\psi(|y|)\;dy\\ &=A[Mf](0), \end{align*}
as desired.  To complete the proof, simply take an increasing sequence $\psi_{n}\to\psi$ of compactly supported $C^{1}$ functions.  Since the estimate holds for each $\psi_{n}$, it holds for the limit function $\psi$.

Problem II.  Consider the heat kernel $$G(x,t)=\frac{1}{(4\pi t)^{\frac{d}{2}}}e^{-\frac{|x|^{2}}{4t}}.$$
\mathcal{F}_{t}\}_{t\geq0}.$We take the following preliminary facts for granted, and defer to previous blog posts covering Brownian motion and stochastic integration for proofs. 1. Almost surely, we have the variation formulas$[W]^{1}(t)=+\infty,[W]^{2}(t)=t$and$[W]^{k}(t)=0$for$k\geq3$. 2. Almost surely, we have the convergence of$\lim_{||\Pi_{[0,T]}||\to0}\sum_{i=1}^{n}\Delta(t_{i})(W(t_{i+1})-W(t_{i}))$for any continuous and adapted process$\Delta(t)$. We denote this limit by$\int_{0}^{T}\Delta(t)\;dW(t)$and refer to it as the Ito integral of$\Delta$. The limit is taken in$L^{2}(\Omega).$Theorem (Ito's Lemma). With the notation above, we have for all$T>0\begin{align*}f(W(T),T)-f(W(0),0)=\\\int_{0}^{T}f_{t}(W(t),t)\;dt+\int_{0}^{T}f_{x}(W(t),t)\;dW(t)+\frac{1}{2}\int_{0}^{T}f_{xx}(W(t),t)\;dt.\end{align*} We sometimes write forf=f(W(t),t)$$$df=f_{t}dt+f_{x}dW+\frac{1}{2}f_{xx}dt.$$ Proof. Fix$T>0$and let$\Pi=\{t_{0}=0,t_{1},\ldots,t_{n}=T\}$be a partition of$[0,T]and compute using Taylor's expansion \begin{align*} f(W(T),T)-f(0,0)&=\sum_{i=0}^{n-1}(f(W(t_{i+1}),t_{i+1})-f(W(t_{i}),t_{i}))\\ &=\sum_{i=0}^{n-1}f_{t}(W(t_{i}),t_{i})(t_{i+1}-t_{i})\\ &+\sum_{i=0}^{n-1}f_{x}(W(t_{i}),t_{i})(W(t_{i+1})-W(t_{i}))\\ &+\frac{1}{2}\sum_{i=0}^{n-1}f_{xx}(W(t_{i}),t_{i})(W(t_{i+1})-W(t_{i}))^{2}\\ &+\sum_{i=0}^{n-1}O((t_{i+1}-t_{i})(W(t_{i+1})-W(t_{i})))\\ &+\sum_{i=0}^{n-1}O((t_{i+1}-t_{i})^{2})\\ &+\sum_{i=0}^{n-1}O((W(t_{i+1})-W(t_{i}))^{3})\\ &:= A+B+C+D+E+F.\end{align*} The left hand side is unaffected by taking limits as||\Pi||\to0$, and so we may do so in computing the right hand side terms. Without loss of generality we assume$\Pi$is uniform, so we consider equivalently$n\to\infty.$The regularity of$f$implies that $$A\to\int_{0}^{T}f_{t}(W(t),t)\;dt\;\text{as}\;n\to\infty,$$ the integral being an ordinary Lebesgue (Riemann) integral. By item 2 above we have $$B\to\int_{0}^{T}f_{x}(W(t),t)\;dW(t)\;\text{as}\;n\to\infty,$$ the integral being an Ito integral as discussed here. To deal with$D$,$E$and$F$we estimate $$|D|\ll_{\beta}\sup_{0\leq i\leq n}|W(t_{i+1})-W(t_{i})|\sum_{i=0}^{n-1}(t_{i+1}-t_{i})\ll_{\beta}T\sup_{0\leq i\leq n}|W(t_{i+1})-W(t_{i})|,$$ $$|E|\ll_{\beta}\sup_{0\leq i\leq n}|t_{i+1}-t_{i}|\sum_{i=0}^{n-1}(t_{i+1}-t_{i})\ll_{\beta}T\sup_{0\leq i\leq n}|t_{i+1}-t_{i}|,$$ and $$|F|\ll_{\beta}\sup_{0\leq i\leq n}|W(t_{i+1})-W(t_{i})|\sum_{i=0}^{n-1}(W(t_{i+1})-W(t_{i}))^{2}.$$ Appealing to item 2 above we then conclude (since the maps$t\mapsto t$and$t\mapsto W(t)$are continuous) that $$D,E,F\to0\;\text{as}\;n\to\infty.$$ It remains to establish the limit $$C\to\frac{1}{2}\int_{0}^{T}f_{xx}(W(t),t)\;dt\;\text{as}\;n\to\infty.$$ Intuitively this should be true since$[W]^{2}(T)=T,$a fact that we sometimes write as$dWdW=dt.$However, a rigorous proof requires some effort, and this is precisely the point in the proof (assuming Brownian motion and stochastic integration are covered) that almost every mathematical finance text skips over. (Note that theorem has already been proved in the special case that$f=p(x,t)$, a second degree polynomial; as an example, consider the special case$f(x,t)=\frac{1}{2}x^{2}$in order to compute the Ito integral$\int_{0}^{T}W(t)\;dW(t)$). Because this fact is of interest in and of itself, we isolate the proof that$C\to\frac{1}{2}\int_{0}^{T}f_{xx}(W(t),t)\;dt\;\text{as}\;n\to\infty$in the following lemma. Lemma. Let$f$be a bounded continuous function on$[0,T]$and$\{W(t)\}_{t \geq 0}$a standard one-dimensional Brownian motion. Then almost surely $$\sum_{i=0}^{n-1} f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}\to\int_{0}^{T}f(W(t))\;dt\;\text{as}\;n\to\infty$$ where$n\to\infty$means (WLOG)$\Pi=\{t_{0}=0,t_{1},\ldots,t_{n}=T\}$is a uniform partition of$[0,T]$and$|\Pi| := \max_j |t_j-t_{j-1}|\to0$. Proof. Since$t \mapsto f(W(t))$is (almost surely) continuous, $$\sum_{i=0}^{n-1} f(W_{t_{i}})(t_{i+1}-t_{i}) \to \int_0^T f(W(t))\;dt\;\text{as}\;n\to\infty.$$ Therefore, it suffices to show $$I_n := \sum_{i=0}^{n-1} f(W(t_{i})) \bigg[ (W(t_{i+1})-W(t_{i}))^2 - (t_{i+1}-t_{i}) \bigg] \to 0\;\text{as}\;n\to\infty.$$ At this point it is convenient to define$\Delta t_{i} := t_{i+1}-t_{i}$and$\Delta W_i := W(t_{i+1})-W(t_{i})$. Recalling that$\{W(t)^2-t\}_{t \geq 0}$is a martingale with respect to the canonical filtration$(\mathcal{F}_t)_{t \geq 0}, we compute \begin{align*} &\quad \mathbb{E} \bigg( f(W(t_{i})) f(W(t_{i-1}))[\Delta W_i^2 - \Delta_i]\Delta W_i^2-\Delta_i]\bigg)\\ &= \mathbb{E} \bigg( \mathbb{E} \bigg( f(W(t_{i})) f(W(t_{i-1})) [\Delta W_i^2 - \Delta_i] [\Delta W_i^2-\Delta_i] \mid \mathcal{F}_{t_{i}} \bigg) \bigg) \\ &= \mathbb{E} \bigg( f(W(t_{i})) f(W(t_{i-1})) [\Delta W_i^2-\Delta_i] \underbrace{\mathbb{E} \bigg( \Delta W_i^2 - \Delta_i \mid \mathcal{F}_{t_{i}} \bigg)}_{\mathbb{E}(\Delta W_i^2-\Delta i)=0} \bigg) = 0, \end{align*} and thus $$\mathbb{E}(I_n^2) = \mathbb{E}\left(\sum_{i=0}^{n-1} f(W(t_{i}))^2 (\Delta W_i^2-\Delta_i)^2 \right).$$ (Observe that the cross-terms vanish.) Using thatf$is bounded and$W(t)-W(s) \sim W(t-s) \sim \sqrt{t-s} W(1)we find \begin{align*} \mathbb{E}(I_n^2) &\leq \|f\|_{\infty}^2 \sum_{i=0}^{n-1} \mathbb{E}\bigg[(\Delta W_i^2-\Delta_i)^2\bigg] \\ &= \|f\|_{\infty}^2 \sum_{i=0}^{n-1} \Delta_i^2 (\mathbb{E}(W_1^2)-1)^2 \\ &\leq C |\Pi| \sum_{i=0}^{n-1} \Delta_i = C |\Pi| T \end{align*} forC := \|f\|_{\infty}^2 (\mathbb{E}(W_1^2)-1)^2$. Letting$|\Pi| \to 0$, the claim follows. III. CLARIFICATION OF "ALMOST SURE" CONVERGENCE We assume the reader is familiar with the various lines of convergence in real analysis: pointwise, uniform, almost uniform, in measure/probability,$L^{p}$, etc. This short section is just to help clarify what is meant by almost sure convergence in the context of this and related topics. Statements of convergence involving Brownian motion are almost always established in$L^{2}(\Omega,P)$, which in turn implies convergence in probability because Chebyshev's inequality states for a sequence of random variables$X_{n}$and proposed limit$X$that $$P(|X_{n}-X|\geq\epsilon)\leq\frac{1}{\epsilon^{2}}\mathbb{E}\left[|X_{n}-X|^{2}\right]\to0\;\text{as}\;n\to\infty\;\text{for all}\;\epsilon>0\;\text{fixed}.$$ For example, in the proof of Ito's lemma we really proved that $$\lim_{n\to\infty}\sum_{i=0}^{n-1}f(W(t_{i-1}),t_{i-1})(W(t_{i+1})-W(t_{i}))^{2}=\int_{0}^{T}f(t)\;dt$$ in$L^{2}(\Omega)$, and by consequence, almost surely. To clarify, this means that for almost every sample path, or outcome$\omega\in\Omega$, we have $$\lim_{n\to\infty}X_{n}(\omega):=\lim_{n\to\infty}\sum_{i=0}^{n-1}f(W(t_{i-1}),t_{i-1})(W(t_{i+1})-W(t_{i}))^{2}=\int_{0}^{T}f(t)\;dt.$$ The case is similar to proving things like almost surely$[W,W](t)=t$and almost surely$\int f(t)\;dW(t)$exists in the Ito sense. Click here to read full post » ### Bifurcating Lease Embedded FX Derivatives Section I. Overview Suppose an entity enters into an agreement to lease property and make rental payments each month, but that the fixed notional underlying the lease payments is denominated in some other currency. This introduces an exposure for the lessee (and lessor), since now the lessee must pay (and the lessor receive) a domestic currency equivalent of some fixed amount in a foreign currency - in other words, the actual payment in the functional currency is Foreign Denominated Lease Notional x Exchange Rate, whatever that might be at the time the payment becomes due. If the entity is a corporate entity, accounting regulations require the entity to "bifurcate" the embedded derivative from the contract and account for it as though it was a legitimate derivative, per the rules of derivative accounting. This introduces accounting complexities, but the problem must also of course be solved from a valuation point of view. In this post we consider lease agreements as just described, as well as those with caps and floors on the exchange rate with strikes contractually written into the agreements. Section II. Valuation Methodology For leases determined to have embedded derivatives (from the point of view of the domestic entity), we value the embedded derivative as a strip of component derivatives corresponding to each future cash flow. That is, each cash flow represents the notional (denominated in the foreign currency CUR1) for each component derivative, and value of the lease embedded derivative is the aggregate value of these component derivatives. Our FX convention is CUR1/CUR2 where this rate is the \# units of CUR2 per 1 unit of CUR1 - such a quantity has units [CUR2]/[CUR1]. We refer to CUR2 as the domestic, functional and settlement currency and CUR1 as the foreign, deal and notional currency. Our valuation methodology is based on usual market-practice - in particular, no arbitrage and discounted cash flow principles. For options, we use the additional assumption of no-arbitrage for an asset price following a simple geometric Brownian motion (Black-Scholes-Merton model). Consider a present valuation date$t$, future maturity date$T>t$, future cash flow$N=N(T)$, corresponding strike rate$K=K(T)$, forward rate$F=F_{t}(T)$, discount rate$D=D_{t}(T)$and volatility$\sigma=\sigma_{t}(K,T)$. Let$V=V_{t}(T,N,K,F,D,\sigma)$denote the value of a derivative written on CUR1/CUR2 with the previous parameters. Then our previous assumptions lead us to the following valuation formulas: $$(1)\;\;\;\;V^{\text{fwd}}_{t}(T)=N(T)\cdot(F_{t}(T)-K(T))\cdot D_{t}(T)$$ $$(2)\;\;\;\;V^{\text{call}}_{t}(\Phi(d_{+})F_{t}(T)-\Phi(d_{-})K)\cdot D_{t}(T),$$ and $$(3)\;\;\;\;V^{\text{put}}_{t}(T)=(\Phi(-d_{-})K-N(-d_{+})F_{t}(T))\cdot D_{t}(T).$$ In (2) and (3) we define $$d_{\pm}=\frac{1}{\sigma_{t}(K,T)\sqrt{T-t}}\left[\log\left(\frac{F_{t}(T)}{K(T)}\right)\pm\frac{1}{2}\sigma_{t}(K,T)^{2}(T-t)\right]$$ and $$\Phi(x)=(2\pi)^{-1/2}\int_{-\infty}^{x}e^{\frac{-y^{2}}{2}}\;dy,$$ the standard normal cumulative distribution function. (Note the dependence of$\sigma_{t}$on$(K,T)$is due to the nature of FX option markets exhibiting term structure variation and smiles.'') Section III. Extraction Methodology Section III(a). Specifying the Strike We extract the embedded derivative in accordance to the principle that the stated value of the cash flow at inception of the lease agreement should be such that the value of the embedded derivative at inception is$0$. We appy this principle to the forward component of the embedded derivative, and approximate it by assuming the cancellation between the cap and floor values (because one is a short position and the other is a long position - see below) would net$0$if we assumed that they constitute a forward in combination. This is exactly true from put-call parity when the strikes are the same, but only approximately true if they are different (which they must be since otherwise the combination of the three instruments would net$0$and there would be no embedded derivative). In particular, if the lease agreement has$i=1,2,3,\ldots,n$future cash flows, a cap$\overline{S}=\overline{S}(T_{i})$and a floor$\underline{S}=\underline{S}(T_{i})$, then for each corresponding component derivative we set (where again,$t=0$is the inception date of the lease): $$(4)\;\;\;\;K^{\text{fwd}}(T_{i})=F_{0}(T_{i}),$$ $$(4)\;\;\;\;K^{\text{cap}}(T_{i})=\overline{S}(T_{i}),$$ and $$(4)\;\;\;\;K^{\text{flr}}(T_{i})=\underline{S}(T_{i}).$$ Accounting rules indicate that this is the proper approach from a valuation point of view. Section III(b). Specifying the Derivative - A Decomposition In keeping with our notation, we let$L_{t}(T_{i})$denote the present fair value at time$t$of the future cash flow made at time$T_{i}$. This is always a negative quantity from the entity's point of view. The idea in order to obtain the embedded derivative is to separate the risky portion of this value from the non-risky portion. In particular, we decompose$L_{t}(T_{i})$as $$L_{t}(T_{i})=B_{t}(T_{i})+Z_{t}(T_{i}),$$ where$B_{t}(T_{i})$only depends on$t$through the discounting at time$tD_{t}(T_{i})$(in particular, it is independent of market variables like$F_{t}(T_{i})$), and$Z_{t}(T_{i})$is a function of all random market variables inherent in$L_{t}(T_{i})$. There are an infinite number of ways to structure such a decomposition, but accounting guidance discussed above is equivalent to certain initial and terminal conditions which allow us to uniquely solve for$B(T_{i})$and$Z(T_{i})$. Section III(c). Specifying the Derivative - Forward Only Case If the lease payment$L_{t}(T_{i})$lacks any optional features, then its payoff is $$L_{T_{i}}(T_{i})=-N\cdot S_{T_{i}}$$ and therefore its fair present value for$0<t<T_{i}$is given by $$(7)\;\;\;\;L_{t}(T_{i})=-N(T_{i})\cdot F_{t}(T_{i})\cdot D_{t}(T_{i}).$$ Observe that this quantity has units of CUR2 and this is the present value of what the entity has to pay at time$T_{i}$. Since it depends on the forward rate$F_{t}(T_{i})$, it has an exposure to CUR1/CUR2 and is therefore risky. The ideas previous discussed involves decomposing$L_{t}(T_{i})$into two parts $$L_{t}(T_{i})=B_{t}(T_{i})+Z_{t}(T_{i}),$$ where$B_{t}(T_{i})$only depends on$t$through$D_{t}(T_{i})$(in particular, it is independent of$F_{t}(T_{i})$), and$Z_{t}(T_{i})$is a function of$F_{t}(T_{i}).$The initial condition $$Z_{0}(T_{i})=0$$ and terminal payoff condition $$L_{T_{i}}(T_{i})=B_{T_{i}}(T_{i})+Z_{T_{i}}(T_{i})=N(T_{i})\cdot F_{T_{i}}(T_{i})\cdot D_{T_{i}}(T_{i})=-N(T_{i})\cdot S_{T_{i}}$$ allow us to uniquely solve for the payoff of$B(T_{i})$and$Z(T_{i})$, the principle of rational pricing and the fact that$B_{t}(T_{i})$is a constant in$t$(ignoring discounting) then gives us$L$,$B$, and$Z$for all$0<t<T_{i})$. Indeed, from the terminal condition we have $$Z_{T_{i}}(T_{i})=-N(T_{i})\cdot S_{T_{i}}-B_{T_{i}}(T_{i})$$ and from the initial condition $$B_{0}(T_{i})=L_{0}(T_{i})=-N(T_{i})\cdot F_{0}(T_{i})\cdot D_{0}(T_{i})=-N(T_{i})\cdot K^{\text{fwd}}(T_{i})\cdot D_{0}(T_{i}).$$ Hence (dropping the fwd'' from$K$), $$Z_{T_{i}}(T_{i})=-N(T_{i})\cdot(K(T_{i})-S_{T_{i}}).$$ This shows that the payoff of$Z(T_{i})$is equal to a short position in CUR1/CUR2 with notion$N(T_{i})$. Applying our reasoning above, we discover$Z_{t}(T_{i})$is given by (2). Explicitly, $$(8)\;\;\;\;Z_{t}(T_{i})=-N(T_{i})\cdot(K(T_{i})-F_{t}(T_{i}))\cdot D_{t}(T_{i})$$br /> and $$(9)\;\;\;\;B_{t}(T_{i})=-N(T_{i})\cdot K(T_{i}).$$ Section III(c). Specifying the Derivative - Ranged Forward Case (Caps & Floors) If$L_{t}(T_{i})$has optional features, then the initial condition$Z_{0}(T_{i})$is replaced by the value of these optional features using the strike prices given by (5) and (6). For a ranged forward, we have a cap$\overline{S}(T_{i})$and a floor$\underline{S}(T_{i})$. With our FX convention CUR1/CUR2, the terms cap'' and floor' are really as such from the counter-party's perspective, or from the entity's perspective when considering the value of the overall lease$L_{t}(T_{i})$(a cap and floor on how much the entity has to pay). However, when considering the value of$Z_{t}(T_{i})$from the entity's, the cap$\overline{S}(T_{i})$is an upper-bound on how much CUR2 can weaken against CUR1, hence a floor ($=1/\overline{S}(T_{i})$) on their losses from their short position in the forward component of the embedded derivative. Conversely, the floor$\underline{S}(T_{i})$is a lower-bound on how much CUR2 can strengthen against CUR1 hence a cap ($=1/\underline{S}(T_{i})$) on their gains. The previous paragraph shows that$Z_{t}(T_{i})$is a sum the sum of three distinct derivatives$\sum_{k=1}^{3}Z^{k}_{t}(T_{i})$- a short position in a put option on CUR1/CUR2 struck at$\underline{S}(T_{i})$, a long position in a call option on CUR1/CUR2 struck at$\overline{S}(T_{i})$, and a short position in a forward on CUR1/CUR2 struck at$K(T_{i})=F_{0}(T_{i}).$This can be proved as we did for the case of a forward, where the initial condition is taken to be $$Z_{0}(T_{i})=\sum_{k=1}^{3}Z^{k}_{0}(T_{i})=V^{\text{call}}_{0}(T_{i})-V^{\text{put}}_{0}(T_{i})+\underbrace{V^{\text{fwd}}_{0}(T_{i})}_{=0},$$ as given by (1), (2) and (3), respectively. The terminal condition is $$L_{T_{i}}(T_{i})=\left\{\begin{array}{ll}-N(T_{i})\cdot\overline{S}(T_{i}),&S_{T_{i}}>\overline{S}(T_{i})\\-N(T_{i})\cdot S_{T_{i}},&\underline{S}(T_{i})\leq S_{T_{i}}\leq\overline{S}(T_{i})\\-N(T_{i})\cdot \underline{S}(T_{i}),&S_{T_{i}}<\underline{S}(T_{i}).\end{array}\right.$$ It follows that $$B_{0}(T_{i})=-N(T_{i})\cdot K(T_{i})\cdot D_{0}(T_{i})$$ and hence $$B_{t}(T_{i})=-N(T_{i})\cdot K(T_{i})\cdot D_{t}(T_{i})$$ for all$0<t<T_{i}.$Now, $$Z_{T_{i}}(T_{i})=L_{T_{i}}(T_{i})-B_{T_{i}}(T_{i})=\left\{\begin{array}{ll}K(T_{i})-\overline{S}(T_{i}),&S_{T_{i}}>\overline{S}(T_{i})\\K(T_{i})-S_{T_{i}},&\underline{S}(T_{i})\leq S_{T_{i}}\leq\overline{S}(T_{i})\\K(T_{i})-\underline{S}(T_{i}),&S_{T_{i}}<\underline{S}(T_{i}).\end{array}\right.$$ One verifies easily that this is equal to $$Z_{T_{i}}(T_{i})=-(S_{T_{i}}K(T_{i}))+\max(S_{T_{i}}-\overline{S}(T_{i}),0)-\max(\underline{S}(T_{i})-S_{T_{i}},0)$$ which are the payoff functions of the indicated derivatives. Thus, $$(10)\;\;\;\;Z_{t}(T_{i})=-A+B-C$$ for all$0<t<T_{i}$where$A$is given by (1),$B$by (2) and$C$by (3). Section IV. Lease Modifications (Introduction) In a subsequent post I will elaborate on the bifurcation and valuation of modifications to lease agreements. For now, let us keep in mind the above consider a typical lease cash flow$L_{t}(T)$with a notional of$N$. At the time the lease is entered into, FASB requires bifurcation of any implied derivative$Z$. Suppose$Z$is just an FX forward (short CUR1/CUR2). At inception ($t=0$) the strike is$K_{0}=F_{0}(T)$, the forward rate corresponding to the future time$T$as calculated at time$t=0$. The value of$Z$at any time$0<t<T$is $$Z_{t}(N_{0},K_{0},T)=N_{0}\cdot(K_{0}-\cdot F{t}(T))\cdot D_{t}(T),$$ where$D_{t}(T)$is the discount factor for term$T$at time$t$. This valuation methodology makes the embedded derivative$0$at inception of the lease. Suppose at some time$0<\tau<T$we have the modification$N_{0}\mapsto N_{\tau}<N_{0}$(the lease payment decreases). Then this is economically equivalent to maintaining the unmodified lease and entering into another lease with notional$\Delta N_{0,\tau}:=N_{0}-N_{\tau}$as a lessor at the time of modification$\tau$. FASB would then require the lessor to put the resulting embedded derivative on their balance sheet that time. The value of this derivative (since it is equivalent to a long position in CUR1/CUR2 or short CUR2/CUR1) is $$\tilde{Z_{t}}(\Delta N_{0,\tau},K_{\tau},T)=\Delta N_{0,\tau}\cdot(F_{t}(T)-K_{\tau})\cdot D_{t}(T).$$ Now, from an operational lease accounting point of view, the P/L at the cash flow date is just$N_{0}-\Delta N_{0,\tau}=N_{\tau}.$Therefore, the net embedded derivative of this overall lease contract is$Z+\tilde{Z}$(the +'' is actually a -'' since we modeled$\tilde{Z}$as a long position). Thus, the derivative's value at all times$\tau<t<Tis \begin{align*} Z^{\tau}_{t}(N_{\tau},K_{\tau},T) &=Z_{t}(N_{0},K_{0},T)+\tilde{Z_{t}}(\Delta N_{0,\tau},K_{\tau},T)\\ &=N_{0}\cdot(K_{0}-F_{t}(T))\cdot D_{t}(T)+\Delta N_{0,\tau}\cdot(F_{t}(T)-K_{\tau})\cdot D_{t}(T)\\ &=D_{t}(T)\Big[N_{0}K_{0}-N_{0}F_{t}(T)+N_{0}F_{t}(T)-N_{0}K_{\tau}-N_{\tau}F_{t}(T)+N_{\tau}K_{\tau}\Big]\\ &=N_{0}\cdot(K_{0}-K_{\tau})\cdot D_{t}(T)+N_{\tau}\cdot(K_{\tau}-F_{t}(T))\cdot D_{t}(T)\\ &=N_{0}\Delta K_{0,\tau}D_{t}(T)+N_{\tau}\cdot(K_{\tau}-F_{t}(T))\cdot D_{t}(T)\\ &=N_{\tau}(K_{\tau}-F_{t}(T))\cdot D_{t}(T)+C \end{align*} where the constantC$is the settlement price of the original embedded derivative established at time$\tau$, but settled at time$T$and discounted back to time$t$. The value of the new derivative is then the sum of this settlement and newly entered into forward with the revised notional. There is more intuitive way to rewrite this result. Let us introduce the notation$\Delta N_{0,\tau}$as with$\Delta K_{0,\tau}.Then \begin{align*}Z^{\tau}_{t}(N_{\tau},K_{\tau},T) &=Z_{t}(N_{0},K_{0},T)+\tilde{Z_{t}}(\Delta N_{0,\tau},K_{\tau},T)\\ &=N_{0}\cdot(K_{0}-F_{t}(T))\cdot D_{t}(T)+\Delta N_{0,\tau}\cdot(F_{t}(T)-K_{\tau})\cdot D_{t}(T)\\ &=N_{\tau}\cdot(K_{0}-F_{t}(T))\cdot D_{t}(T)+\Delta N_{0,\tau}\Delta K_{0,\tau}\\ &=N_{\tau}\cdot(K_{0}-F_{t}(T))\cdot D_{t}(T)+C, \end{align*} where nowC$is the settlment price of the cancelled notional and the value of the new embedded derivative is the sum of this settlement cost and the original forward contract with reduced notional. In other words, it is the value of the same forward, but with a revised notional that represents the decreased exposure, and a constant settlement that carries with the valuation. This constant is the cost of reducing the exposure at time$\tau$from$N_{0}$to$N_{\tau}$. The entity can account for this in two ways. Either P/L the settlement cost$C$on the modification date, or value expense it over time by carrying the settlement cost valuation period to valuation period up-to expiration of the lease. The two are equivalent from an accounting point of view, but obviously from a valuation point of view one will lead to a jump in the value of the embedded derivative and the other will maintain continuity of the valuation. A similar approach can be taken to deal with lease increases and extensions - these valuations and related accounting issues will be taken up in a subsequent post. Click here to read full post » ## 17 March, 2015 ### Can a Derivative's Value Exceed the Underlying Notional Value? On a recent project I valued some derivatives, the results of which the client balked at because the values exceeded the notional on which they were written. So is it ever possible for a derivative's valuation to exceed its underlying notional? The answer, of course, depends, and first we need to clarify what we mean by a derivative's value exceeding its notional. A typical derivative (like an option or future) is written on some underlying asset with price$S_{t}$and some quantity or notional$N$. The term quantity is frequently used for assets like stocks and the term notional for assets like currencies - so in the latter case, if I have USD/EUR call option, then I view the asset as the US dollar (that I want to buy a call option on) priced in the European Euro (that is what the USD/EUR exchange rate is - the cost of a US dollar in Euros), with a notional (i.e. quantity) equal to (say)$\$100,000,000$ USD.

Notice that the quantity/notional has units in the underlying asset and that the spot price $S_{t}$ has units of value in the numeraire/settlement currency per 1 asset.  Hence, when we ask if a derivative's value can exceed its underlying notional, we are really asking whether the value at time $t$, denoted $V_{t}$, can exceed the quantity $NS_{t}$, which has units in the settlement currency (EUR in the above example).  In other words, we ask whether
$$V_{t}>NS_{t}$$
can hold without introducing an arbitrage.

Essentially, the purpose of the above discussion was to express precisely what we mean for the valuation to exceed the underlying notional and moreover, to emphasize that derivative's valuation cannot be directly compared to the notional in order to answer the question, since since the units are not the same - the underlying notional $N$ needs to be multiplied by the spot price $S_{t}$ so that each has units in the valuation currency.

The classic counter-example to answering this question affirmatively in all instances comes from considering a call on option written on $N$ of some asset with price $S_{t}$.  If you value this option at time $t$, then it is clear that
$$V_{t}<NS_{t},$$
for otherwise one could short a covered option at no cost and an arbitrage would exist.  But this argument no longer holds for instruments with payoffs that are not artificially bounded by some optionality mechanism.  Indeed, the example from my experience involved an FX forward on CUR1/CUR2 (to be generic).  For such a forward, let $K$ be the strike, $N$ the notional (denominated in CUR1), $D$ the discount factor and $\alpha$ the CUR2/CUR1 exchange rate ($1/S_{t}$).  Then if the entity is in the short position we have
$$-\alpha N\cdot(F-K)\cdot D>N$$
$$(F-K)<-\frac{1}{\alpha D}$$
$$F<K-\frac{1}{\alpha D}.$$
Hence, if the forward rate is sufficiently small (i.e. price of CUR1 declined) with respect to the inception strike, then the value of the forward will exceed the notional as an asset. Conversely,
$$-\alpha N\cdot(F-K)\cdot D<-N$$
$$(F-K)>\frac{1}{\alpha D}$$
$$F>K+\frac{1}{\alpha D}$$
Hence, if the forward rate is sufficiently large (i.e. the price of CUR1 increased) with respect to the inception strike, then the value of the forward will exceed the notional as a liability.

It is not difficult to come up with similar bounds for other basic instruments such as swaps either.