VisuApps
Options, swaps and forwards — payoffs, valuation and risks interactively
Apps
Monte-Carlo pricing with a quantifiable error: the running price estimate with its ±1.96·SE band cinches in at 1/√M; the European estimate from the same paths validates against the analytic Black-Scholes reference. “+1000 paths” animated, “+10000” fast. Debunks “simulation is guessing”.
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Path dependence made tangible: a random price path and its sluggish running average Ā(t) in course color — Asian pays on Ā, European on the final price. Roll paths, compare payoff bars, and watch mean Asian < mean European across many paths. Debunks “only where the price ends up matters”.
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The American put’s exercise boundary S*(t) from a 200-step tree: tinted exercise region, boundary curve rising to K at expiry. A sample path exercises exactly on first entry into the region. Debunks “there is one fixed trigger price” — and for a call without dividends no boundary exists at all.
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European, American, Bermudan — the same tree, three exercise windows (tinted columns). Warm rings show where early exercise happens; the price comparison shows European ≤ Bermudan ≤ American including the early-exercise premium. Debunks “in the money ⇒ exercise immediately” — and for a call without dividends early exercise never pays.
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The binomial price V(N) for N = 1…150 as a dot sequence: it sawtooths around the Black-Scholes line and cinches onto it — “converge” animates N; u/d/p tick along. Debunks “binomial and Black-Scholes are different theories”: BS is the limit of the tree.
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Option valuation by backward induction: the tree spans price paths (×u/×d), “value it” lets the value travel column by column from the payoff to the root. Clicking a node shows the one-step formula with real numbers. Debunks “you need a price forecast” — the risk-neutral p follows from u, d and r alone.
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A rate cap is not a single instrument but a series of caplets — one bar per period. A floor bounds from below, a collar = buy cap + sell floor (free via the zero-cost snap). The forward staircase in the corridor shows which caplets bite. Debunks “a cap is one instrument with one price”.
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The hazard curve is bootstrapped from market spreads across maturities: on top the term-spread curve (input) with exactly-hit par control points, below the distilled default probability 1−Q(t). A steeper spread curve steepens the hazard curve. Debunks “one number λ / one spread is enough”.
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Credit insurance as two cash-flow streams: the buyer pays the spread (premium leg, shrinking with survival probability), receives (1−R)·N on default (protection leg). “Set to fair” balances the present values. Debunks “the spread is the default probability” — credit triangle: ≈ λ·(1−R).
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The hedge-frequency trade-off over 300 paths: the histogram of hedged P/L tightens with N (gamma residual risk ~ 1/√N), but transaction costs push the mean linearly negative — the curves over N reveal the RMSE optimum N★. Debunks “hedging more often is always better”.
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The writer neutralizes price risk: price path with rebalance points on top, below the P/L scissors — unhedged swings wildly, hedged hugs zero, tighter with every increase in rebalancing frequency. Debunks “option sellers bet against the client” and “hedging once is enough”.
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Why digitals are almost unhedgeable: the delta is not a hump but a spike at the strike that grows ever taller and narrower as T → 0 (pin risk). The call-spread replication (two vanillas) has a bounded delta instead — the contrast is the aha. Debunks “you hedge a digital with its delta”.
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Cash-or-nothing: a fixed amount if the price lands above the strike — otherwise nothing. The payoff is a step function, the fair value today the smooth S-curve above it; “time decay” stiffens it into the step. Debunks “the payoff grows with the distance to the strike” — it is a cliff.
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The volatility smile: market prices (OTM quote board) visibly gape above the flat Black-Scholes line at the wings — below, the smile σ_impl(K) distilled point by point. Presets “BS world”, “equity skew” (crash fear), “FX smile”; skew/curvature/ATM controls. Debunks “one σ per stock is enough”.
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From the market price back to σ: the price-over-σ curve rises strictly monotonically (vega > 0) — the movable market-price line intersects it in exactly one point, the implied volatility. “Solve” shows bisection as a visible search; no-arbitrage zones bound admissible prices. Debunks “σ is measurable like the price”.
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The barrier has a price: the MC price curve of the up-and-out call over B (4000 paths, common random numbers) climbs to the European value; below, the knock probability mirrors the discount. At B = 125 the barrier costs over 90% of the value. Debunks “the barrier is fine print”.
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One excursion above the threshold wipes the payoff: up-and-out call with barrier B, KO marker at the touch point, €0 despite ending in the money — next to it the lookback that buys at the path low (running-min line). Debunks “only the final price matters”: the path extreme decides.
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The four basic positions (long/short × call/put) as a payoff diagram: kink at the strike, profit and loss zones, draggable spot marker with live P/L. Debunks “past the strike I’m in profit” — the break-even is shifted by the premium, and the buyer’s loss is capped at the premium.
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The fair option value before expiry (Black-Scholes) as a smooth curve above the payoff — the gap is the time value. Sliders for K, σ, T, r; “time decay” runs T → 0 and the curve collapses onto the hockey stick. Delta as a tangent, gamma/vega/theta live. Debunks “out of the money = worthless”.
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A strategy’s risk profile today: value curve above the payoff plus the aggregated greek curve over S (Δ/Γ/vega/Θ toggle) in two cards. Debunks “a straddle is market-neutral, hence riskless” — Δ ≈ 0, but Γ/vega are large and Θ bleeds daily; the short straddle mirrors it.
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Straddle, strangle, bull spread, butterfly, short straddle — combine basic positions and shape the payoff landscape: legs as dashed lines, combined curve with break-evens, net premium and max P/L. Debunks “options are directional bets” — the straddle wins in both directions.
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If the market call price deviates from parity, conversion or reversal locks in a riskless profit: four legs whose sum is a horizontal line above zero — wherever the price ends up. With a deviation slider, automatic strategy choice and a step recipe. Debunks “surely some price risk remains”.
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Long call + short put = forward: two kinked payoffs add up to the perfect straight line S − K — the addition stack at the spot marker shows it pointwise. In the values panel C − P = S₀ − K·e^(−rT): turning σ changes C and P, never the difference. Debunks “calls and puts are independent products”.
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Black-76, the market convention: the price curve over the strike sits above intrinsic value — time value as an area, numerical vega, and “time decay” collapses the curve onto the concept app’s hockey stick. Markets quote swaptions in vol. Debunks “the price is today’s swap advantage”.
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The right to enter a swap: the hockey stick lives over the par rate instead of a price — the payer swaption wins when rates rise. The tenor slider changes the slope (annuity as leverage), the forward marker shows where the market stands. Debunks “a swaption is a stock option on a price”.
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A fair swap does not stay fair: the NPV curve over the parallel shift runs through the inception point, its slope IS the DV01 (tangent), and maturity steepens it. Below, today’s curve vs. the scenario with both par rates. Debunks “rising rates are always bad” — the payer wins.
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Fixed versus floating on a notional that never changes hands: cash-flow bars show the constant fixed leg against the rising forwards; par rate and fixed rate K sit as lines in the same picture (“€ per 100 notional = % rate”). “Set to par” makes the exchange exactly fair (NPV 0). Debunks “the fair fixed rate is today’s rate”.
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