"use strict"; const {Fragment: _Fragment, jsx: _jsx, jsxs: _jsxs} = arguments[0]; const {useMDXComponents: _provideComponents} = arguments[0]; function _createMdxContent(props) { const _components = { annotation: "annotation", blockquote: "blockquote", br: "br", em: "em", h2: "h2", h3: "h3", li: "li", math: "math", mfrac: "mfrac", mi: "mi", mn: "mn", mo: "mo", mrow: "mrow", mspace: "mspace", msqrt: "msqrt", mstyle: "mstyle", msub: "msub", msubsup: "msubsup", msup: "msup", mtext: "mtext", ol: "ol", p: "p", path: "path", semantics: "semantics", span: "span", strong: "strong", svg: "svg", ul: "ul", ..._provideComponents(), ...props.components }; return _jsxs(_Fragment, { children: [_jsx(_components.h2, { children: _jsx(_components.strong, { children: "Learning Outcomes" }) }), "\n", _jsx(_components.p, { children: "This article explains how option values are built from no-arbitrage principles using discrete-time binomial trees and the continuous-time Black–Scholes–Merton (BSM) model. After reading, you will be able to:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Build and value options using a binomial stock-price tree under risk-neutral valuation" }), "\n", _jsx(_components.li, { children: "Identify when early exercise can be optimal and how it changes American option valuation" }), "\n", _jsx(_components.li, { children: "Apply and interpret the Black–Scholes–Merton formulas for European calls and puts (with and without dividends)" }), "\n"] }), "\n", _jsx(_components.h2, { children: _jsx(_components.strong, { children: "GARP FRM Part 1 Syllabus" }) }), "\n", _jsx(_components.p, { children: "For the GARP FRM Part 1 exam, you are expected to understand option valuation with a focus on the following syllabus points:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Distinguish European vs American exercise features and relate them to valuation methods (binomial trees vs BSM)" }), "\n", _jsx(_components.li, { children: "Compute option payoffs, interpret moneyness, and connect payoff asymmetry to risk exposures" }), "\n", _jsx(_components.li, { children: "Use no-arbitrage, replication, and risk-neutral valuation to price options and compare American and European values" }), "\n"] }), "\n", _jsx(_components.h2, { children: _jsx(_components.strong, { children: "Test Your Knowledge" }) }), "\n", _jsx(_components.p, { children: _jsx(_components.em, { children: "Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision." }) }), "\n", _jsxs(_components.ol, { children: ["\n", _jsxs(_components.li, { children: ["\n", _jsx(_components.p, { children: "Which statement is most accurate?" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "a)" }), " American options can be priced only with BSM because they allow early exercise"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "b)" }), " European options are typically priced with BSM, while American options are commonly valued with binomial trees"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "c)" }), " American options must always be worth exactly the same as European options"] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "d)" }), " Binomial trees require real-world probabilities, not risk-neutral probabilities"] }), "\n"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsxs(_components.p, { children: ["One-step binomial: ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mo, { children: "=" }), _jsx(_components.mn, { children: "100" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_0=100" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", 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className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { className: "mord", children: "100" })] })] })] }), ". What is the risk-neutral probability ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "p" }), _jsx(_components.mo, { children: "∗" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "p^*" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8831em", verticalAlign: "-0.1944em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "p" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.6887em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mbin mtight", children: "∗" }) })] }) }) }) }) })] })] }) })] }), "?"] }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsx(_components.p, { children: "In a binomial tree, how do you compute the value of a European option at a nonterminal node from its two successor node values?" }), "\n"] }), "\n", _jsxs(_components.li, { children: ["\n", _jsx(_components.p, { children: "True/False (brief explanation required): For a non-dividend-paying stock, it can be optimal to exercise an American call option early." }), "\n"] }), "\n"] }), "\n", _jsx(_components.h2, { children: _jsx(_components.strong, { children: "What Do You Know Already?" }) }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Can you explain why option valuation uses a “risk-neutral” probability rather than your forecast probability?" }), "\n", _jsxs(_components.li, { children: ["Can you state (from memory) the BSM call and put formulas, including the definitions of ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mn, { children: "1" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "d_1" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8444em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "d" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), " and ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mn, { children: "2" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "d_2" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8444em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "d" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), "?"] }), "\n", _jsx(_components.li, { children: "Can you describe, in words, why American options are at least as valuable as otherwise identical European options?" }), "\n"] }), "\n", _jsx(_components.p, { children: "Use these prompts to gauge your starting point; then focus your revision where your answers were least confident." }), "\n", _jsx(_components.h2, { children: _jsx(_components.strong, { children: "Introduction" }) }), "\n", _jsx(_components.p, { children: "Options are non-linear derivatives: they provide the right, but not the obligation, to buy or sell a given asset at a pre-agreed strike price. This asymmetry makes options central to financial risk management—both for hedging downside risk (long puts) and for transferring risk to an option writer (short options)." }), "\n", _jsx(_components.p, { children: "From a valuation standpoint, the exercise feature matters. European options can be exercised only at expiration; American options can be exercised any time up to and including expiration. As highlighted in FRM coverage, European options are often valued using the Black–Scholes–Merton model, while American options typically require numerical methods such as binomial trees. A key inequality follows directly: an American option must be worth at least as much as an otherwise identical European option, because the holder has weakly more rights." }), "\n", _jsxs(_components.blockquote, { children: ["\n", _jsxs(_components.p, { children: [_jsx(_components.strong, { children: "Key Term: Moneyness" }), _jsx(_components.br, {}), "\n", "Moneyness classifies an option by comparing the current asset price ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "S" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" })] }) })] }), " with strike ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "K" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "K" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" })] }) })] }), ": a call is in-the-money if ", 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}), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" })] })] })] }), " (put: ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mo, { children: "<" }), _jsx(_components.mi, { children: "K" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S" }), _jsx(_components.mi, { children: "K" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: 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marginRight: "0.13889em" }, children: "T" })] }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.10903em" }, children: "N" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "d" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "2" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] }) })] }), " can be read as discounted “risk-adjusted” components of owning the asset and of paying the strike."] }), "\n"] }), "\n", _jsx(_components.h3, { children: "Worked Example 3.1" }), "\n", _jsxs(_components.p, { children: ["Use BSM to price a European call and put with ", _jsx(_components.span, { className: "katex-error", title: "ParseError: KaTeX parse error: Unexpected character: '\\' at position 5: S_0=\\̲", style: { color: "#cc0000" }, children: "S_0=\\" }), "100", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: 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_jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.3em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mpunct", children: "," })] }) })] }), "r=5%", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "o" }), _jsx(_components.mi, { children: "n" }), _jsx(_components.mi, { children: "t" }), _jsx(_components.mi, { mathvariant: "normal", children: "." }), _jsx(_components.mi, { children: "c" }), _jsx(_components.mi, { children: "o" }), _jsx(_components.mi, { 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marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "100" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsx(_components.span, { className: "mord", children: "0.951229" }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6444em" } }), _jsx(_components.span, { 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_jsx(_components.span, {}) }) })] }) })] }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.888em" }, children: _jsxs(_components.span, { style: { top: "-3.063em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mopen mtight", children: "(" }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.02778em" }, children: "r" }), _jsx(_components.span, { className: "mbin mtight", children: "−" }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.03588em" }, children: "q" }), _jsx(_components.span, { className: "mclose mtight", children: ")" }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" })] }) })] }) }) }) }) })] })] })] })] }), ", parity can be written in forward language:"] }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "C" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mo, { children: "−" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "P" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "e" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "r" }), _jsx(_components.mi, { children: "T" })] })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "F" }), _jsx(_components.mn, { children: "0" })] }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "K" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "C_0 - P_0 = e^{-rT}(F_0 - K)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", 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className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "−" }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.02778em" }, children: "r" }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "T" })] }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "F" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.1389em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "0" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.07153em" }, children: "K" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }) }), "\n", _jsx(_components.p, { children: "This is exam-relevant because it lets you:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "Check internal consistency of quoted call and put prices" }), "\n", _jsxs(_components.li, { children: ["Infer a missing option price from the other, given ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "S" }), _jsx(_components.mn, { children: "0" })] }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "S_0" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.05764em" }, children: "S" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "-0.0576em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "0" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] })] }) })] }), ", ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "r" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "r" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.4306em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.02778em" }, children: "r" })] }) })] }), ", ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "q" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "q" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "q" })] }) })] }), ", and ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "T" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "T" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.6833em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.13889em" }, children: "T" })] }) })] })] }), "\n", _jsx(_components.li, { children: "Link option valuation to forward pricing (a common theme across FRM derivatives readings)" }), "\n"] }), "\n", _jsx(_components.h2, { children: _jsx(_components.strong, { children: "Practical Risk Management Uses" }) }), "\n", _jsx(_components.h3, { children: "Model choice in a risk function" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "European vanilla options" }), ": BSM is fast and widely used for quoting, benchmarking, and extracting implied volatility."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "American options" }), ": binomial trees are a standard tool because they explicitly test early exercise at each node."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Path-dependent or embedded-option products" }), ": many instruments (for example, prepayment options in mortgage-backed securities) are path-dependent, which is one reason Monte Carlo simulation is widely used in fixed-income option-adjusted spread (OAS) workflows. A simple binomial equity tree is not naturally built for complex path dependence."] }), "\n"] }), "\n", _jsx(_components.h3, { children: "Hedging and sensitivities" }), "\n", _jsx(_components.p, { children: "Even when an option is priced with BSM, its risk is managed through sensitivities (“Greeks”). For example, the BSM delta of a European call is:" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "Δ" }), _jsx(_components.mtext, { children: "call" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "e" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "q" }), _jsx(_components.mi, { children: "T" })] })] }), _jsx(_components.mi, { children: "N" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\Delta_{\\text{call}} = e^{-qT}N(d_1)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.8333em", verticalAlign: "-0.15em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord", children: "Δ" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3361em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: 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height: "1.1413em", verticalAlign: "-0.25em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsxs(_components.span, { className: "mord mtight", children: [_jsx(_components.span, { className: "mord mtight", children: "−" }), _jsx(_components.span, { className: "mord mathnormal mtight", style: { marginRight: "0.13889em" }, children: "qT" })] }) })] }) }) }) }) })] }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.10903em" }, children: "N" }), _jsx(_components.span, { className: "mopen", children: "(" }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "d" }), _jsx(_components.span, { className: "msupsub", children: _jsxs(_components.span, { className: "vlist-t vlist-t2", children: [_jsxs(_components.span, { className: "vlist-r", children: [_jsx(_components.span, { className: "vlist", style: { height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }) }), "\n", _jsx(_components.p, { children: "For a European put:" }), "\n", _jsx(_components.span, { className: "katex-display", children: _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", display: "block", children: _jsxs(_components.semantics, { children: [_jsxs(_components.mrow, { children: [_jsxs(_components.msub, { children: [_jsx(_components.mi, { mathvariant: "normal", children: "Δ" }), _jsx(_components.mtext, { children: "put" })] }), _jsx(_components.mo, { children: "=" }), _jsxs(_components.msup, { children: [_jsx(_components.mi, { children: "e" }), _jsxs(_components.mrow, { children: [_jsx(_components.mo, { children: "−" }), _jsx(_components.mi, { children: "q" }), _jsx(_components.mi, { children: "T" })] })] }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsx(_components.mi, { children: "N" }), _jsx(_components.mo, { stretchy: "false", children: "(" }), _jsxs(_components.msub, { children: [_jsx(_components.mi, { children: "d" }), _jsx(_components.mn, { children: "1" })] }), _jsx(_components.mo, { stretchy: "false", children: ")" }), _jsx(_components.mo, { children: "−" }), _jsx(_components.mn, { children: "1" }), _jsx(_components.mo, { stretchy: "false", children: ")" })] }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "\\Delta_{\\text{put}} = e^{-qT}(N(d_1)-1)" })] }) }) }), _jsxs(_components.span, { className: "katex-html", "aria-hidden": "true", children: [_jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.9694em", verticalAlign: "-0.2861em" } }), _jsxs(_components.span, { className: 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style: { height: "0.2861em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } }), _jsx(_components.span, { className: "mrel", children: "=" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2778em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1.1413em", verticalAlign: "-0.25em" } }), _jsxs(_components.span, { className: "mord", children: [_jsx(_components.span, { className: "mord mathnormal", children: "e" }), _jsx(_components.span, { className: "msupsub", children: _jsx(_components.span, { className: "vlist-t", children: _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.8913em" }, children: _jsxs(_components.span, { style: { top: "-3.113em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: 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{ height: "0.3011em" }, children: _jsxs(_components.span, { style: { top: "-2.55em", marginLeft: "0em", marginRight: "0.05em" }, children: [_jsx(_components.span, { className: "pstrut", style: { height: "2.7em" } }), _jsx(_components.span, { className: "sizing reset-size6 size3 mtight", children: _jsx(_components.span, { className: "mord mtight", children: "1" }) })] }) }), _jsx(_components.span, { className: "vlist-s", children: "" })] }), _jsx(_components.span, { className: "vlist-r", children: _jsx(_components.span, { className: "vlist", style: { height: "0.15em" }, children: _jsx(_components.span, {}) }) })] }) })] }), _jsx(_components.span, { className: "mclose", children: ")" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } }), _jsx(_components.span, { className: "mbin", children: "−" }), _jsx(_components.span, { className: "mspace", style: { marginRight: "0.2222em" } })] }), _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "1em", verticalAlign: "-0.25em" } }), _jsx(_components.span, { className: "mord", children: "1" }), _jsx(_components.span, { className: "mclose", children: ")" })] })] })] }) }), "\n", _jsx(_components.p, { children: "These are used to form hedges that reduce first-order exposure to small moves in the asset price. In risk measurement (for example, VaR approximations), delta and gamma often appear as inputs to approximations of option P&L over short horizons." }), "\n", _jsx(_components.h3, { children: "Early exercise intuition you should be able to state" }), "\n", _jsx(_components.p, { children: "For common FRM-style statements:" }), "\n", _jsxs(_components.ul, { children: ["\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "American call on a non-dividend-paying stock" }), ": early exercise is typically not optimal. Exercising early gives up remaining time value and forces payment of the strike earlier (you lose the interest benefit of delaying the strike payment)."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "American put" }), ": early exercise can be optimal, especially when the put is deep in-the-money and interest rates are not negligible (receiving the strike earlier can be valuable)."] }), "\n", _jsxs(_components.li, { children: [_jsx(_components.strong, { children: "Dividend-paying stock calls" }), ": early exercise can become optimal around ex-dividend dates because owning the stock (after exercise) is necessary to receive the dividend."] }), "\n"] }), "\n", _jsx(_components.h2, { children: _jsx(_components.strong, { children: "Summary" }) }), "\n", _jsx(_components.p, { children: "Binomial trees and Black–Scholes–Merton are both no-arbitrage valuation frameworks, differing mainly in time structure and exercise features. Binomial trees model price evolution in discrete steps and value options by backward induction using risk-neutral probabilities; they naturally incorporate early exercise and are therefore suited to American options." }), "\n", _jsxs(_components.p, { children: ["Black–Scholes–Merton provides closed-form European call and put prices under strong assumptions about the asset’s return process and market frictions. In practice, BSM is widely used for European pricing and for extracting implied volatility, while binomial trees are used when early exercise must be checked explicitly. For exam purposes, you should be able to compute payoffs, set up risk-neutral valuation, explain why American options are at least as valuable as European options, and apply the BSM formulas correctly (including dividends via ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: _jsxs(_components.semantics, { children: [_jsx(_components.mrow, { children: _jsx(_components.mi, { children: "q" }) }), _jsx(_components.annotation, { encoding: "application/x-tex", children: "q" })] }) }) }), _jsx(_components.span, { className: "katex-html", "aria-hidden": "true", children: _jsxs(_components.span, { className: "base", children: [_jsx(_components.span, { className: "strut", style: { height: "0.625em", verticalAlign: "-0.1944em" } }), _jsx(_components.span, { className: "mord mathnormal", style: { marginRight: "0.03588em" }, children: "q" })] }) })] }), ")."] }), "\n", _jsx(_components.h2, { children: _jsx(_components.strong, { children: "Key Point Checklist" }) }), "\n", _jsx(_components.p, { children: _jsx(_components.em, { children: "This article has covered the following key knowledge points:" }) }), "\n", _jsxs(_components.ul, { children: ["\n", _jsx(_components.li, { children: "American options can be exercised early; European options can only be exercised at maturity" }), "\n", _jsx(_components.li, { children: "American options must be worth at least as much as otherwise identical European options" }), "\n", _jsx(_components.li, { children: "Binomial trees price options by replication or by discounted risk-neutral expected payoffs" }), "\n", _jsxs(_components.li, { children: ["Risk-neutral probability is ", _jsxs(_components.span, { className: "katex", children: [_jsx(_components.span, { className: "katex-mathml", children: _jsx(_components.math, { xmlns: "http://www.w3.org/1998/Math/MathML", children: 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Options - Binomial trees and Black–Scholes–Merton