FX Forwards

FXForwards.html

<h1 id="fx-forwards">FX Forwards</h1>

<p>The standard no-arbitrage price <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-1-Frame" role="textbox" aria-readonly="true"><nobr><span class="math" id="MathJax-Span-1" style="width: 2.069em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.715em; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(1.243em 1000.003em 2.6em -0.469em); top: -2.122em; left: 0.003em;"><span class="mrow" id="MathJax-Span-2"><span class="msubsup" id="MathJax-Span-3"><span style="display: inline-block; position: relative; width: 1.656em; height: 0px;"><span style="position: absolute; clip: rect(1.243em 1000.003em 2.305em -0.469em); top: -2.122em; left: 0.003em;"><span class="mi" id="MathJax-Span-4" style="font-family: MathJax_Math; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.121em;"></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span><span style="position: absolute; top: -1.945em; left: 0.711em;"><span class="texatom" id="MathJax-Span-5"><span class="mrow" id="MathJax-Span-6"><span class="mi" id="MathJax-Span-7" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-8" style="font-size: 70.7%; font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-9" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.062em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.289em; vertical-align: -0.425em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-1">F_{t,T}</script> for currency forwards when the spot price is <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-2-Frame" role="textbox" aria-readonly="true"><nobr><span class="math" id="MathJax-Span-10" style="width: 1.302em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.066em; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(1.243em 1000.003em 2.482em -0.469em); top: -2.122em; left: 0.003em;"><span class="mrow" id="MathJax-Span-11"><span class="msubsup" id="MathJax-Span-12"><span style="display: inline-block; position: relative; width: 1.006em; height: 0px;"><span style="position: absolute; clip: rect(1.243em 1000.003em 2.305em -0.469em); top: -2.122em; left: 0.003em;"><span class="mi" id="MathJax-Span-13" style="font-family: MathJax_Math; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.062em;"></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span><span style="position: absolute; top: -1.945em; left: 0.652em;"><span class="mi" id="MathJax-Span-14" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">t</span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.218em; vertical-align: -0.282em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-2">S_t</script>, the domestic rate is <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-3-Frame" role="textbox" aria-readonly="true"><nobr><span class="math" id="MathJax-Span-15" style="width: 1.243em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.006em; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(1.538em 1000.003em 2.482em -0.528em); top: -2.122em; left: 0.003em;"><span class="mrow" id="MathJax-Span-16"><span class="msubsup" id="MathJax-Span-17"><span style="display: inline-block; position: relative; width: 0.947em; height: 0px;"><span style="position: absolute; clip: rect(1.538em 1000.003em 2.305em -0.528em); top: -2.122em; left: 0.003em;"><span class="mi" id="MathJax-Span-18" style="font-family: MathJax_Math; font-style: italic;">r</span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span><span style="position: absolute; top: -1.945em; left: 0.475em;"><span class="mi" id="MathJax-Span-19" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 0.861em; vertical-align: -0.282em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-3">r_d</script> and the foreign rate is <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-4-Frame" role="textbox" aria-readonly="true"><nobr><span class="math" id="MathJax-Span-20" style="width: 1.243em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.006em; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(1.538em 1000.003em 2.6em -0.528em); top: -2.122em; left: 0.003em;"><span class="mrow" id="MathJax-Span-21"><span class="msubsup" id="MathJax-Span-22"><span style="display: inline-block; position: relative; width: 0.947em; height: 0px;"><span style="position: absolute; clip: rect(1.538em 1000.003em 2.305em -0.528em); top: -2.122em; left: 0.003em;"><span class="mi" id="MathJax-Span-23" style="font-family: MathJax_Math; font-style: italic;">r</span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span><span style="position: absolute; top: -1.945em; left: 0.475em;"><span class="mi" id="MathJax-Span-24" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.062em;"></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.075em; vertical-align: -0.425em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-4">r_f</script> is</p>

<p><span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-5-Frame"><nobr><span class="math" id="MathJax-Span-25" style="width: 16.65em; display: inline-block;"><span style="display: inline-block; position: relative; width: 13.757em; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(1.243em 1000.003em 2.659em -0.469em); top: -2.181em; left: 0.003em;"><span class="mrow" id="MathJax-Span-26"><span class="msubsup" id="MathJax-Span-27"><span style="display: inline-block; position: relative; width: 1.656em; height: 0px;"><span style="position: absolute; clip: rect(1.243em 1000.003em 2.305em -0.469em); top: -2.122em; left: 0.003em;"><span class="mi" id="MathJax-Span-28" style="font-family: MathJax_Math; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.121em;"></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span><span style="position: absolute; top: -1.945em; left: 0.711em;"><span class="texatom" id="MathJax-Span-29"><span class="mrow" id="MathJax-Span-30"><span class="mi" id="MathJax-Span-31" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-32" style="font-size: 70.7%; font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-33" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.062em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span></span></span><span class="mo" id="MathJax-Span-34" style="font-family: MathJax_Main; padding-left: 0.298em;">=</span><span class="msubsup" id="MathJax-Span-35" style="padding-left: 0.298em;"><span style="display: inline-block; position: relative; width: 1.006em; height: 0px;"><span style="position: absolute; clip: rect(1.243em 1000.003em 2.305em -0.469em); top: -2.122em; left: 0.003em;"><span class="mi" id="MathJax-Span-36" style="font-family: MathJax_Math; font-style: italic;">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.062em;"></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span><span style="position: absolute; top: -1.945em; left: 0.652em;"><span class="mi" id="MathJax-Span-37" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">t</span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span></span></span><span class="mo" id="MathJax-Span-38" style="font-family: MathJax_Main;">(</span><span class="mn" id="MathJax-Span-39" style="font-family: MathJax_Main;">1</span><span class="mo" id="MathJax-Span-40" style="font-family: MathJax_Main; padding-left: 0.239em;">+</span><span class="mo" id="MathJax-Span-41" style="font-family: MathJax_Main; padding-left: 0.239em;">(</span><span class="msubsup" id="MathJax-Span-42"><span style="display: inline-block; position: relative; width: 0.947em; height: 0px;"><span style="position: absolute; clip: rect(1.538em 1000.003em 2.305em -0.528em); top: -2.122em; left: 0.003em;"><span class="mi" id="MathJax-Span-43" style="font-family: MathJax_Math; font-style: italic;">r</span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span><span style="position: absolute; top: -1.945em; left: 0.475em;"><span class="mi" id="MathJax-Span-44" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span></span></span><span class="mo" id="MathJax-Span-45" style="font-family: MathJax_Main; padding-left: 0.239em;">−</span><span class="msubsup" id="MathJax-Span-46" style="padding-left: 0.239em;"><span style="display: inline-block; position: relative; width: 0.947em; height: 0px;"><span style="position: absolute; clip: rect(1.538em 1000.003em 2.305em -0.528em); top: -2.122em; left: 0.003em;"><span class="mi" id="MathJax-Span-47" style="font-family: MathJax_Math; font-style: italic;">r</span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span><span style="position: absolute; top: -1.945em; left: 0.475em;"><span class="mi" id="MathJax-Span-48" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.062em;"></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span></span></span><span class="mo" id="MathJax-Span-49" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-50" style="font-family: MathJax_Main;">(</span><span class="mi" id="MathJax-Span-51" style="font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.121em;"></span></span><span class="mo" id="MathJax-Span-52" style="font-family: MathJax_Main; padding-left: 0.239em;">−</span><span class="mi" id="MathJax-Span-53" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.239em;">t</span><span class="mo" id="MathJax-Span-54" style="font-family: MathJax_Main;">)</span><span class="mo" id="MathJax-Span-55" style="font-family: MathJax_Main;">)</span></span><span style="display: inline-block; width: 0px; height: 2.187em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.432em; vertical-align: -0.425em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-5">F_{t,T} = S_t (1 + (r_d-r_f)(T-t) )</script></p>

<p>and the price time <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-6-Frame" role="textbox" aria-readonly="true"><nobr><span class="math" id="MathJax-Span-56" style="width: 1.066em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.888em; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(1.243em 1000.003em 2.305em -0.469em); top: -2.122em; left: 0.003em;"><span class="mrow" id="MathJax-Span-57"><span class="mi" id="MathJax-Span-58" style="font-family: MathJax_Math; font-style: italic;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-59" style="font-family: MathJax_Math; font-style: italic;">t</span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.004em; vertical-align: -0.068em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-6">\delta t</script> later is</p>

<p><span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-7-Frame"><nobr><span class="math" id="MathJax-Span-60" style="width: 3.368em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.777em; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(1.243em 1000.003em 2.6em -0.469em); top: -2.122em; left: 0.003em;"><span class="mrow" id="MathJax-Span-61"><span class="msubsup" id="MathJax-Span-62"><span style="display: inline-block; position: relative; width: 2.718em; height: 0px;"><span style="position: absolute; clip: rect(1.243em 1000.003em 2.305em -0.469em); top: -2.122em; left: 0.003em;"><span class="mi" id="MathJax-Span-63" style="font-family: MathJax_Math; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.121em;"></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span><span style="position: absolute; top: -1.945em; left: 0.711em;"><span class="texatom" id="MathJax-Span-64"><span class="mrow" id="MathJax-Span-65"><span class="mi" id="MathJax-Span-66" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-67" style="font-size: 70.7%; font-family: MathJax_Main;">+</span><span class="mi" id="MathJax-Span-68" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">δ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-69" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">t</span><span class="mo" id="MathJax-Span-70" style="font-size: 70.7%; font-family: MathJax_Main;">,</span><span class="mi" id="MathJax-Span-71" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.062em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.128em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.361em; vertical-align: -0.425em;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-7">F_{t+\delta t, T}</script></p>