🧮Calculators

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Amihud Illiquidity (ILLIQ)

Computes the Amihud (2002) illiquidity measure: the average of the ratio of the absolute daily return to the dollar trading volume, scaled by one million as is convention. The intuition is that, in illiquid assets, a small volume already moves the price a lot — so the higher the ratio, the more illiquid the asset. It's one of the most used liquidity measures in research, since it needs only daily price and volume data. Enter the lists of returns in percent and dollar volumes.

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Up Capture Ratio

Computes a portfolio's up capture ratio: how much it kept pace with the benchmark during periods when the index rose, using the geometric compounded method (Morningstar standard). A value above one hundred percent means the portfolio captured more than the market's rise in good months; below, that it lagged on the way up. It's one half of the pair with down capture, which measures behavior on the way down. Enter the lists of portfolio and benchmark returns, in the same order.

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Down Capture Ratio

Computes a portfolio's down capture ratio: how much it fell alongside the benchmark during periods when the index dropped, by the geometric compounded method. Here, less is better — a value below one hundred percent means the portfolio lost less than the market in declines, a sign of good protection. A negative value means the portfolio rose while the market fell. Together with up capture, it describes the manager's asymmetric profile. Enter the lists of portfolio and benchmark returns.

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R-Multiple

Computes the R-multiple of a long trade: the result expressed in multiples of the initial risk, (exit − entry) divided by (entry − stop). It's system traders' favorite unit of measure, popularized by Van Tharp, because it normalizes any trade by the risk it took — a 3R gain means profiting three times what was risked. Thinking in R, rather than in currency, keeps the focus on risk management. Enter the entry, stop and exit prices.

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Sharpe Ratio from Series

Computes the Sharpe ratio directly from a series of returns: the mean return minus the risk-free rate, divided by the sample standard deviation. It's the most practical way to get the Sharpe when you have the history at hand, without computing the mean and volatility separately. Remember the result comes in the frequency of the data entered — to annualize monthly returns, multiply by the square root of twelve. Enter the list of returns and the risk-free rate for the same period.

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Volatility Drag (Variance Drag)

Computes volatility drag: the loss of the compound (geometric) return relative to the average (arithmetic) return, approximately volatility squared divided by two. It's the reason a portfolio that does +50% and then −50% doesn't return to the start: volatility erodes compound growth. The more volatile the asset, the larger the drag, even with the same average return. Enter the arithmetic mean return and the volatility.

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Loan Constant (Mortgage Constant)

Computes the loan constant, also called the mortgage constant: the annual debt service divided by the original loan amount, as a percentage. It shows directly what fraction of the principal you pay per year, combining interest and amortization. In real estate, comparing the loan constant with a property's cap rate immediately reveals whether the financing produces positive or negative leverage. Enter the annual debt service and the loan amount.

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Combined Ratio (Insurance)

Computes an insurer's combined ratio: the sum of the loss ratio and the expense ratio. It's the central measure of an insurance operation's technical profitability — below one hundred percent, the insurer had an underwriting profit, making money on operations alone, before investments. Above one hundred percent, it had a technical loss and relied on investment income to finish in the black. Enter the loss ratio and the expense ratio.

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TVPI (Total Value to Paid-In)

Computes the TVPI of a private equity or venture capital fund: the total value created, adding what has already been distributed to investors to the residual value still in the portfolio (NAV), divided by the paid-in capital. It's a fund's most complete multiple, summing realized and unrealized — a TVPI of 1.5x means each dollar invested became one and a half in total value. It decomposes into DPI plus RVPI. Enter the distributions, the NAV and the paid-in capital.

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DPI (Distributed to Paid-In)

Computes the DPI of a private equity fund: the cumulative distributions to investors divided by the paid-in capital. It's the realized multiple, the money that has actually returned to the investor's pocket, not counting what's still locked in unsold holdings. A DPI of 1.0x marks the point where the fund has returned all contributed capital; above that, it's realized profit. It complements RVPI, which measures the unrealized part. Enter the distributions and the paid-in capital.

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RVPI (Residual Value to Paid-In)

Computes the RVPI of a private equity fund: the residual value in the portfolio (NAV) divided by the paid-in capital. It's the unrealized multiple, how much is still alive in the holdings the fund hasn't sold, waiting to turn into cash. In a fund's early years, RVPI dominates; as it divests, RVPI falls and DPI rises. The sum of the two is the TVPI. Enter the NAV and the paid-in capital.

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Operating Expense Ratio (Real Estate)

Computes a property's operating expense ratio: operating expenses divided by gross operating income, as a percentage. It measures how much of the rental income is consumed by the costs of operating the property, such as maintenance, management, taxes and insurance, not counting financing. The lower it is, the more efficient the property; a very low value may signal deferred maintenance. It's a key indicator in income-property analysis. Enter the operating expenses and the gross operating income.

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Black-Scholes-Merton Call (Dividends)

Computes the price of a European call option with the Black-Scholes-Merton model, the extension that incorporates a continuous dividend yield q. The premium is S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2), and the dividend lowers the call's value, because part of the asset's return leaks to whoever holds the stock. It's the standard model for options on dividend-paying stocks and on indices. Enter the spot price, the strike, the interest rate, the dividend yield, the term and the volatility.

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Bachelier Model Call

Computes the price of a call option with the Bachelier model, which assumes the price follows normal (arithmetic) motion instead of lognormal. Because it allows negative prices, it came back into fashion for pricing options on assets that can go negative, as happened with oil in 2020 and with some spreads. The volatility here is absolute, in price units, not a percentage. The premium is e^(−rT)·[(F−K)·N(d) + σ√T·φ(d)]. Enter the forward price, the strike, the interest rate, the normal volatility and the term.

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Digital Call (Cash-or-Nothing)

Computes the price of a cash-or-nothing digital call option: it pays a fixed amount if the asset finishes above the strike, and nothing otherwise. The price is the discounted payout multiplied by the risk-neutral probability of finishing in the money, Q·e^(−rT)·N(d2). It's the purest form of a binary bet and the building block of many exotic structures. Enter the spot price, the strike, the interest rate, the term, the volatility and the payout.

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Nelson-Siegel Yield Curve

Computes the spot rate for any maturity using the Nelson-Siegel model, the most widely used way to fit the yield curve with few parameters. The three betas control the long-run level, the slope and the curvature, while lambda sets where the curve takes shape. Central banks and fixed-income desks use this model to smooth and interpolate curves from traded bonds. Enter the three betas in percent, the lambda and the desired maturity.

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Geometric Asian Call (Kemna-Vorst)

Computes the price of a geometric-average Asian call option with the Kemna-Vorst closed form. Asian options pay based on the average price over the period, which reduces the impact of expiry manipulation and makes the premium cheaper. The geometric-average version has an exact solution: it's a Black-Scholes with volatility adjusted to σ/√3 and an adapted cost of carry. Enter the spot price, the strike, the interest rate, the term and the volatility.

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Vasicek Bond Price

Computes the price of a zero-coupon bond with the Vasicek model, the first short-rate interest-rate model with mean reversion. It describes the short rate oscillating around a long-run mean and yields a closed form for the bond price from four parameters: reversion speed, mean, volatility and current rate. Despite allowing negative rates, it's the foundation of the whole family of term-structure models. Enter the parameters and the maturity, and see the price and implied yield.

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Implied Volatility (Black-Scholes)

Computes the implied volatility of a European call option by inverting the Black-Scholes formula via bisection: given the market price, it finds the volatility the model would need to reach it. It's the volatility the market is actually pricing in, the number behind the volatility smile and the VIX. Unlike the other Greeks, it has no closed form and requires a numerical solution. Enter the spot price, the strike, the interest rate, the term and the market price of the call.

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Svensson Yield Curve

Computes the spot rate with the Svensson curve, the extension of the Nelson-Siegel model that adds a second hump to fit more complex yield curves. With six parameters (four betas and two lambdas), it captures shapes Nelson-Siegel can't, which is why it's the choice of central banks like the ECB and the Bundesbank to publish their curves. Enter the four betas in percent, the two lambdas and the desired maturity.

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CIR Bond Price (Cox-Ingersoll-Ross)

Computes the price of a zero-coupon bond with the Cox-Ingersoll-Ross model, the successor to Vasicek that fixes its biggest flaw: CIR prevents negative interest rates, because the volatility shrinks as the rate approaches zero. It also has mean reversion and yields an affine closed form for the bond price. It's one of the most used short-rate models in practice. Enter the reversion speed, the long-run mean, the volatility, the current rate and the maturity.

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Down-and-Out Barrier Call

Computes the price of a down-and-out barrier call: an option that ceases to exist if the asset touches a barrier below the current price before expiry. Because of that knockout risk, it costs less than a plain call, and the difference is exactly the value of the down-and-in version. The Reiner-Rubinstein closed form holds for a barrier at or below the strike. Enter the spot price, the strike, the barrier, the interest rate, the term and the volatility.

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Par Swap Rate

Computes the par swap rate from discount factors: the fixed rate that makes the interest rate swap's value zero at inception, equating the fixed and floating legs. The formula is (1 − last discount factor) divided by the sum of discount factors weighted by the period. It's a swap's market quote and the basis for marking existing positions to market. Enter the list of discount factors by payment date and the year-fraction of each period.

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Futures/Forward Convexity Adjustment

Computes the convexity adjustment between a futures rate and the equivalent forward rate, using the Ho-Lee approximation: forward = futures − ½·σ²·T1·T2. Interest rate futures and forwards don't carry the same rate because of the daily marking to market of futures, and the convexity adjustment corrects that difference, always pushing the forward rate below the futures rate. Enter the absolute short-rate volatility, the time to the futures maturity and the end of the rate period.

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Margrabe Exchange Option

Computes the price of an exchange option with the Margrabe formula: the right to exchange one asset for another at expiry. It's the generalization of Black-Scholes to two risky assets, where the strike stops being fixed and becomes the price of a second asset. The relevant volatility is that of the ratio between the two, combining the individual volatilities and the correlation. It shows up in mergers, spread options and executive compensation. Enter the two prices, the two volatilities, the correlation and the term.

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Payer Swaption (Black Model)

Computes the premium of a payer swaption with the Black model: the right to enter an interest rate swap paying a pre-agreed fixed rate. The price is the swap's annuity multiplied by a Black formula on the forward swap rate. Swaptions are the central instrument for those managing long-term interest rate risk, like banks and insurers. Enter the forward swap rate, the strike, the volatility, the expiry, the annuity (PV01) and the notional.

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Interest Rate Caplet (Black Model)

Computes the premium of a caplet with the Black model: an option that pays when a period's interest rate exceeds a cap. A full interest rate cap is a sum of caplets, one for each payment period. It's the classic protection for someone who took a floating-rate loan and wants to limit how much they can pay. The price discounts the expected payoff to the payment date. Enter the forward rate, the cap rate, the volatility, the fixing time, the accrual fraction, the discount factor and the notional.

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Floating-Strike Lookback Call

Computes the price of a floating-strike lookback call with the Goldman-Sosin-Gatto formula: an option that pays the difference between the final price and the lowest price observed during the contract's life. In practice, it's like buying at the best possible price in hindsight, which makes it expensive but eliminates the risk of mistiming the purchase. It requires a positive interest rate. Enter the spot price, the observed minimum, the interest rate, the volatility and the term.

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Asset-or-Nothing Call

Computes the price of an asset-or-nothing call: it delivers the asset itself if the price finishes above the strike, and nothing otherwise. It's the sibling of the cash-or-nothing option, and together they decompose the plain Black-Scholes call — a call equals exactly an asset-or-nothing minus a strike's worth of cash-or-nothing. The price is simply S·N(d1). Enter the spot price, the strike, the interest rate, the volatility and the term.

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Chooser Option

Computes the price of a simple chooser option with the Rubinstein formula: an option that lets the holder decide, on a future date, whether it will be a call or a put, both with the same strike and expiry. It's the ideal bet for someone expecting a big move but not yet knowing the direction, costing more than a plain option and less than buying a call and a put separately. Enter the spot price, the strike, the interest rate, the volatility, the expiry and the choice date.

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Spread Option (Kirk Approximation)

Computes the price of a spread option with Kirk's approximation: a call on the difference between two assets, S1 minus S2, with a strike. Spreads are everywhere in commodities (oil crack spread, power spark spread) and no exact formula exists, so Kirk proposed a clever approximation that reduces the problem to a Black-Scholes with an effective volatility combining the two vols and the correlation. Enter the two forward prices, the strike, the volatilities, the correlation, the rate and the term.

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Power Option

Computes the price of a power call option, whose payoff is the asset price raised to a power, minus the strike: max(S^n − K, 0). Raising the price to a power hugely amplifies the moves, so these options have explosive payoffs and high premiums. They're used for leveraged bets on volatility and in structured products. The growth prefactor already incorporates the discounting, with no double counting. Enter the spot price, the strike, the power, the rate, the volatility and the term.

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Gap Option

Computes the price of a gap call, where the strike that triggers exercise differs from the strike that sets the payoff. The option pays (S − K1) when the price exceeds K2, and that separation creates a jump (gap) in the payoff exactly at K2: the option can start paying with a positive or negative value. It's the theoretical basis of many discontinuous-payoff options. Enter the spot price, the payment strike, the trigger strike, the rate, the volatility and the term.

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Forward-Start Option

Computes the price of a forward-start call: an option granted now, but whose strike is only set on a future date, usually as a proportion of the price at that moment. It's the structure behind employee option plans and cliquets, where new at-the-money options are issued periodically. Since the strike tracks the future price, the value doesn't depend on the current level in a trivial way. Enter the spot price, the moneyness, the rate, the volatility, the grant date and the expiry.

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Quanto Option

Computes the price of a quanto call: an option on a foreign asset, but whose payoff is paid in domestic currency at a fixed exchange rate. This removes the FX risk for the investor but introduces a drift adjustment that depends on the correlation between the asset and the exchange rate. It's widely used by investors who want exposure to a foreign index without the currency risk. Enter the asset price, the strike, the two rates, the asset and FX volatilities, the correlation, the term and the fixed exchange rate.

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Cash-or-Nothing Put

Computes the price of a cash-or-nothing put: it pays a fixed amount if the asset finishes below the strike, and nothing otherwise. It's the downside version of the digital option, the complement of the cash-or-nothing call. The price is the payout discounted and multiplied by the risk-neutral probability of the asset finishing below the strike, Q·e^(−rT)·N(−d2). Enter the spot price, the strike, the interest rate, the term, the volatility and the payout.

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Compound Option (Call-on-Call, Geske)

Computes the price of a compound call-on-call option with the Geske (1979) model: a call option whose underlying is, itself, another call option. It's the structure behind many real-world contracts — an option to extend a project, for example, is an option on an option. The calculation requires finding the critical price at which exercising the first option is worthwhile and uses the bivariate normal. Enter the spot price, the two strikes, the two expiries, the rate and the volatility.

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Rainbow Option on the Maximum (Stulz)

Computes the price of a call on the maximum of two assets with the Stulz (1982) formula: the option pays based on the better performer of two correlated assets, minus the strike. It's a bet on the winner of a race between two assets, and its price depends heavily on the correlation between them — the less correlated, the more valuable, because there's a greater chance at least one takes off. It uses the bivariate normal. Enter the two prices, the strike, the two volatilities, the correlation, the rate and the term.

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Rainbow Option on the Minimum (Stulz)

Computes the price of a call on the minimum of two assets with the Stulz (1982) formula: the option pays based on the worse performer of two assets, minus the strike. It's the more conservative option of the rainbow pair, valuable when you want both assets to perform. An elegant identity holds: the sum of the call on the maximum and the call on the minimum equals the sum of two plain calls. It uses the bivariate normal. Enter the two prices, the strike, the two volatilities, the correlation, the rate and the term.

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American Call with Dividend (Roll-Geske-Whaley)

Computes the price of an American call option on a stock paying a discrete dividend, with the Roll-Geske-Whaley model. Unlike the European call, the American one can be exercised early, and that's only optimal precisely an instant before the dividend, when the price will drop. The model finds the critical exercise price and combines probabilities via the bivariate normal. Enter the spot price, the strike, the rate, the volatility, the term, the dividend and the date it's paid.

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Two-Asset Correlation Binary

Computes the price of a two-asset binary option: it pays a fixed amount if and only if the first asset finishes above its strike AND the second asset finishes above its own. It's a conditional double bet whose price depends critically on the correlation between the assets — the more correlated, the more likely both conditions happen together. It uses the bivariate normal. Enter the two prices, the two strikes, the two volatilities, the correlation, the rate, the term and the payout.

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Complex Chooser Option (Rubinstein)

Computes the price of a complex chooser option with the Rubinstein (1991) formula: on the choice date, the holder decides between a call and a put that may have different strikes and expiries. It's the general version of the simple chooser, and because it allows distinct parameters for each side it requires the bivariate normal and a search for a critical price. When the call and put share the same strike and expiry, it collapses to the simple chooser. Enter the spot price, the call and put strikes, the choice date, the two expiries, the rate and the volatility.

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Asset-or-Nothing Put

Computes the price of an asset-or-nothing put: it delivers the asset itself if the price finishes below the strike, and nothing otherwise. It's the downside counterpart of the asset-or-nothing call, and together they always sum to the asset's value, because one or the other always pays. The price is simply S·N(−d1). Enter the spot price, the strike, the interest rate, the volatility and the term.

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Gap Put Option

Computes the price of a gap put, where the strike that triggers exercise differs from the strike that sets the payoff. The option pays (K1 − S) when the price falls below K2, creating a jump in the payoff exactly at K2. It's the downside version of the gap option, the theoretical basis of many discontinuous-payoff contracts. Enter the spot price, the payment strike, the trigger strike, the rate, the volatility and the term.

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Down-and-In Barrier Call

Computes the price of a down-and-in barrier call: an option that only comes into existence if the asset touches a barrier below the current price before expiry. It's the counterpart of the down-and-out, and the sum of the two is exactly a plain call: either the barrier is touched (activating the in) or it isn't (keeping the out). Because it depends on a trigger, it costs less than a plain call. Enter the spot price, the strike, the barrier, the rate, the term and the volatility.

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Up-and-Out Barrier Call

Computes the price of an up-and-out barrier call: an option that ceases to exist if the asset touches a barrier above the current price before expiry. It's a curious case — the option is a bet on the upside, but it dies if the price rises too far, which makes it cheap when the barrier is close. The Reiner-Rubinstein formula holds for a barrier above the strike. Enter the spot price, the strike, the barrier, the rate, the term and the volatility.

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Supershare Option

Computes the price of a supershare option (Hakansson): it pays a fraction of the asset if the price finishes within a band between a lower and an upper bound, and nothing outside it. It was proposed as the building block of a state-contingent mutual fund system, and is an elegant example of a range-dependent option. The price is the asset fraction multiplied by the probability of landing in the band. Enter the spot price, the lower and upper bounds, the rate, the volatility and the term.

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Option Dual Gamma

Computes the dual gamma of a call option: the second derivative of the price with respect to the strike, ∂²C/∂K². While gamma measures the price's curvature relative to the asset, dual gamma measures the curvature relative to the strike, and it's tied to the risk-neutral probability density of the future price — in fact, dual gamma is exactly that density discounted. It's used to extract implied price distributions from option prices. Enter the spot price, the strike, the rate, the volatility and the term.

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American Put (Barone-Adesi-Whaley)

Computes the price of an American put option with the Barone-Adesi-Whaley quadratic approximation. Unlike the European put, the American one can be exercised at any time, and that right has value — the so-called early-exercise premium. The method iteratively finds the critical price below which exercising already pays off, and adds that premium to the European put value. It's fast and accurate, with no need for a binomial tree. Enter the spot price, the strike, the rate, the cost of carry, the volatility and the term.

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American Call (Bjerksund-Stensland)

Computes the price of an American call option with the Bjerksund-Stensland (1993) approximation, valid when the asset pays dividends (cost of carry below the interest rate). It defines an exercise trigger price and, below it, combines exponential terms to approximate the value with early exercise. It's faster than a binomial tree and widely used in practice for American calls on dividend-paying stocks. Enter the spot price, the strike, the rate, the cost of carry, the volatility and the term.

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Arithmetic Asian Option (Turnbull-Wakeman)

Computes the price of an arithmetic-average Asian call with the Turnbull-Wakeman approximation. Unlike the geometric average, the arithmetic average has no exact closed form, so Turnbull and Wakeman match the first two moments of the average's distribution and apply a Black-Scholes with adjusted volatility and carry. It's the market-standard approximation for arithmetic Asians, common in commodities and FX. Enter the spot price, the strike, the rate, the cost of carry, the volatility and the term.

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One-Touch Option

Computes the price of a one-touch option: it pays a fixed amount if the asset touches a barrier above the current price at any time before expiry, and nothing if it never touches. It's one of the most traded American binary options in the FX market, and its price is the risk-neutral probability of the price reaching the barrier, brought to present value. Enter the spot price, the barrier, the rate, the volatility, the term and the payout.

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No-Touch Option

Computes the price of a no-touch option: it pays a fixed amount if the asset does NOT touch a barrier before expiry, and nothing if it touches. It's the opposite bet to the one-touch — you win as long as the price behaves and stays away from the barrier. The two are complementary: the sum of a one-touch and a no-touch with the same barrier is always the discounted payout. Enter the spot price, the barrier, the rate, the volatility, the term and the payout.

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Floating-Strike Lookback Put

Computes the price of a floating-strike lookback put with the Goldman-Sosin-Gatto formula: an option that pays the difference between the highest price observed during the contract's life and the final price. It's like always selling at the top, in hindsight, eliminating the risk of mistiming the sale. It's the counterpart of the lookback call, and the high price reflects that power to pick the best moment. Enter the spot price, the observed maximum, the rate, the cost of carry, the volatility and the term.

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Basket Option (Levy Approximation)

Computes the price of a call on a basket of two assets with the Levy approximation. The weighted sum of two lognormal assets isn't lognormal, so there's no exact formula; Levy matches the basket's mean and variance to an equivalent lognormal and applies a Black-Scholes. It's the practical way to price options on indices and portfolios, where the correlation between assets is decisive. Enter the two prices, the weights, the strike, the volatilities, the correlation, the rate and the term.

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Variance Swap (Settlement)

Computes the settlement of a variance swap: the variance notional multiplied by the difference between the realized variance and the variance strike (the square of volatility). It's the pure volatility derivative — unlike an option, it has no directional exposure to the asset, only to the variance that actually occurred. The buyer profits if the market swings more than expected. Enter the realized volatility, the strike volatility and the variance notional.

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Volatility Swap (Settlement)

Computes the settlement of a volatility swap: the volatility notional multiplied by the difference between the realized volatility and the strike, in volatility points. It's a cousin of the variance swap, but pays linearly in volatility, not its square, which makes it more intuitive yet harder to replicate. The difference between the two fair strikes is the convexity adjustment. Enter the realized volatility, the strike volatility and the volatility notional.

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Binomial Tree (Cox-Ross-Rubinstein)

Prices an option with the Cox-Ross-Rubinstein binomial tree, the most teachable numerical method for options. At each step, the price moves up or down by factors calibrated to the volatility, and the option value is computed backward, from expiry to today. Unlike Black-Scholes, the tree prices American options, checking early exercise at each node. Choose call or put, European or American, and enter the price, strike, rate, volatility, term and number of steps.

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Trinomial Tree (Boyle)

Prices an option with the Boyle trinomial tree, an evolution of the binomial where the price, at each step, can move up, down or stay flat. That third path gives the tree more flexibility and faster, more stable convergence than the binomial for the same number of steps. It's widely used for options whose features demand numerical precision. Choose call or put and enter the price, strike, rate, volatility, term and number of steps.

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Forward Volatility

Computes the implied forward volatility between two maturities from the corresponding spot volatilities. Just as there's a forward interest rate embedded in two spot rates, there's a forward volatility embedded in two volatilities of different terms, given by √((σ2²·T2 − σ1²·T1)/(T2 − T1)). It's what the market prices for the volatility of the period between the two dates. Enter the two spot volatilities and their terms.