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Log Contract Ln(S/X) [Loxx]A log contract, first introduced by Neuberger (1994) and Neuberger (1996), is not strictly an option. It is, however, an important building block in volatility derivatives (see Chapter 6 as well as Demeterfi, Derman, Kamal, and Zou, 1999). The payoff from a log contract at maturity T is simply the natural logarithm of the underlying asset divided by the strike price, ln(S/ X). The payoff is thus nonlinear and has many similarities with options. The value of this contract is (via "The Complete Gu

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Log Option [Loxx]A log option introduced by Wilmott (2000) has a payoff at maturity equal to max(log(S/X), 0), which is basically an option on the rate of return on the underlying asset with strike log(X). The value of a log option is given by: (via "The Complete Guide to Option Pricing Formulas") e^−rT * n(d2)σ√(T − t) + e^−rT*(log(S/K) + (b −σ^2/2)T) * N(d2) where N(*) is the cumulative normal distribution function, n(*) is the normal density function, and d = ((log(S/X) + (b - v^2/2)*T) / (v*T^0.5) b=

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Log Contract Ln(S) [Loxx]A log contract, first introduced by Neuberger (1994) and Neuberger (1996), is not strictly an option. It is, however, an important building block in volatility derivatives (see Chapter 6 as well as Demeterfi, Derman, Kamal, and Zou, 1999). The payoff from a log contract at maturity T is simply the natural logarithm of the underlying asset divided by the strike price, ln(S/ X). The payoff is thus nonlinear and has many similarities with options. The value of this contract is (via "The Complete Gu

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Powered Option [Loxx]At maturity, a powered call option pays off max(S - X, 0)^i and a put pays off max(X - S, 0)^i . Esser (2003 describes how to value these options (see also Jarrow and Turnbull, 1996, Brockhaus, Ferraris, Gallus, Long, Martin, and Overhaus, 1999). (via "The Complete Guide to Option Pricing Formulas") b=r options on non-dividend paying stock b=r-q options on stock or index paying a dividend yield of q b=0 options on futures b=r-rf currency options (where rf is the rate in the second currency)

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Capped Standard Power Option [Loxx]Power options can lead to very high leverage and thus entail potentially very large losses for short positions in these options. It is therefore common to cap the payoff. The maximum payoff is set to some predefined level C. The payoff at maturity for a capped power call is min . Esser (2003) gives the closed-form solution: (via "The Complete Guide to Option Pricing Formulas") c = S^i * (e^((i - 1) * (r + i*v^2 / 2) - i * (r - b))*T) * (N(e1) - N(e3)) - e^(-r*T) * (X*N(e2) - (C + X) * N(e4))

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Standard Power Option [Loxx]Standard power options (aka asymmetric power options) have nonlinear payoff at maturity. For a call, the payoff is max(S^i - X, 0), and for a put, it is max(X - S^i , 0), where i is some power (i > 0). The value of this power call is given by (see Heynen and Kat, 1996c; Zhang, 1998; and Esser, 2003). (via "The Complete Guide to Option Pricing Formulas") c = S^i * (e^((i - 1) * (r + i*v^2 / 2) - i * (r - b))*T) * N(d1) - X*e^(-r*T) * N(d2) while the value of a put is p = X*e^(-r*T) * N(-d2

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Power Contract [Loxx]There are two main categories of power options. Standard power options' payoff depends on the price of the underlying asset raised to some power. For powered options, the "standard" payoff (stock price in excess of the exercise price) is raised to some power. A power contract is a simple derivative instrument paying (S/ X)^i at maturity, where i is some fixed power. The value of such a power contract is given by Shaw (1998) as: (via "The Complete Guide to Option Pricing Formulas") VPower = (

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Moneyness Options [Loxx]A moneyness option is basically a plain vanilla option where the strike is set to a percentage of the future/forward price. For example, a 120% moneyness call would have a strike equal to 120% of the forward price. A 120% moneyness put would have a spot equal to 120% of the strike. The value of this option is given in percent of the forward. The value of a moneyness call or put is thus given by: (via "The Complete Guide to Option Pricing Formulas") c = p = c^-rT * (N(d1) - LN(d2)) where L =

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Forward Start Options [Loxx]A forward start option with time to maturity T starts at-the-money or proportionally in- or out-of-the-money after a known elapsed time t in the future. The strike is set equal to a positive constant a times the asset price S after the known time t. If a is less than unity, the call (put) will start 1 - a percent in-the-money (out-of-the- money); if a is unity, the option will start at-the-money; and if a is larger than unity, the call (put) will start a - 1 percentage out-of-the- money (in-the-

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Executive Stock Options [Loxx]The Jennergren and Naslund (1993) formula takes into account that an employee or executive often loses her options if she has to leave the company before the option's expiration: (via "The Complete Guide to Option Pricing Formulas") c = e^(-lambda*T) * (Se^((b-r)T) * N(d1) - Xe^-rT * N(d2)) p = e^(-lambda*T) * (Xe^(-rT) * N(-d2) - Se^(b-r)T * N(-d1)) where d1 = (log(S/X) + (b + v^2/2)T) / vT^0.5 d2 = d1 - vT^0.5 lambda is the jump rate per year. The value of the executive option equals

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Variable Purchase Options [Loxx]Handley (2001) describes how to value variable purchase options (VPO). A VPO is basically a call option, but where the number of underlying shares is stochastic rather than fixed, or more precisely, a deterministic function of the asset price. The strike price of a VPO is typically a fixed discount to the underlying share price at maturity. The payoff at maturity is equal to max , where N is the number of shares. VPOs may be an interesting tool for firms that need to raise capital relatively far

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Perpetual American Options [Loxx]Perpetual American Options is Perpetual American Options pricing model. This indicator also includes numerical greeks. American Perpetual Options While there in general is no closed-form solution for American options (except for non-dividend-paying stock call options) it is possible to find a closed-form solution for options with an infinite time to expiration. The reason is that the time to expiration will always be the same: infinite. The time to maturity, therefore, does not depend on a

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American Approximation Bjerksund & Stensland 2002 [Loxx]American Approximation Bjerksund & Stensland 2002 is an American Options pricing model. This indicator also includes numerical greeks. You can compare the output of the American Approximation to the Black-Scholes-Merton value on the output of the options panel. The Bjerksund & Stensland (2002) Approximation The Bjerksund and Stensland (2002) approximation divides the time to maturity into two parts, each with a separate flat exercise boundary. It is thus a straightforward generalization of

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American Approximation Bjerksund & Stensland 1993 [Loxx]American Approximation Bjerksund & Stensland 1993 is an American Options pricing model. This indicator also includes numerical greeks. You can compare the output of the American Approximation to the Black-Scholes-Merton value on the output of the options panel. The Bjerksund and Stensland (1993) approximation can be used to price American options on stocks, futures, and currencies. The method is analytical and extremely computer-efficient. Bjerksund and Stensland's approximation is based on an

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American Approximation: Barone-Adesi and Whaley [Loxx]American Approximation: Barone-Adesi and Whaley is an American Options pricing model. This indicator also includes numerical greeks. You can compare the output of the American Approximation to the Black-Scholes-Merton value on the output of the options panel. An American option can be exercised at any time up to its expiration date. This added freedom complicates the valuation of American options relative to their European counterparts. With a few exceptions, it is not possible to find an ex

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Samuelson 1965 Option Pricing Formula [Loxx]Samuelson 1965 Option Pricing Formula is an options pricing formula that pre-dates Black-Scholes-Merton. This version includes Analytical Greeks. Samuelson (1965; see also Smith, 1976) assumed the asset price follows a geometric Brownian motion with positive drift, p. In this way he allowed for positive interest rates and a risk premium. c = SN(d1) * e^((rho - omega) * T) - Xe^(-omega * T)N(d2) d1 = (log(S / X) + (rho + v^2 / 2) * T) / (v * T^0.5) d2 = d1 - (v * T^0.5) where rho is the

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Boness 1964 Option Pricing Formula [Loxx]Boness 1964 Option Pricing Formula is an options pricing model that pre-dates Black-Scholes-Merton. This model includes Analytical Greeks. Boness (1964) assumed a lognormal asset price. Boness derives the following value for a call option: c = SN(d1) - Xe^(rho * T)N(d2) d1 = (log(S / X) + (rho + v^2 / 2) * T) / (v * T^0.5) d2 = d1 - (v * T^0.5) where rho is the expected rate of return to the asset. Analytical Greeks Delta Greeks: Delta, DDeltaDvol, Elasticity Gamma Greeks: Gamma,

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Generalized Black-Scholes-Merton on Variance Form [Loxx]Generalized Black-Scholes-Merton on Variance Form is an adaptation of the Black-Scholes-Merton Option Pricing Model including Numerical Greeks. The following information is an excerpt from Espen Gaarder Haug's book "Option Pricing Formulas". This version is to price Options using variance instead of volatility. Black- Scholes- Merton on Variance Form In some circumstances, it is useful to rewrite the BSM formula using variance as input instead of volatility, V = v^2: c = S * e^((b - r) *

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Asay (1982) Margined Futures Option Pricing Model [Loxx]Asay (1982) Margined Futures Option Pricing Model is an adaptation of the Black-Scholes-Merton Option Pricing Model including Analytical Greeks and implied volatility calculations. The following information is an excerpt from Espen Gaarder Haug's book "Option Pricing Formulas". This version is to price Options on Futures where premium is fully margined. This means the Risk-free Rate, dividend, and cost to carry are all zero. The options sensitivities (Greeks) are the partial derivatives of th

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Black-76 Options on Futures [Loxx]Black-76 Options on Futures is an adaptation of the Black-Scholes-Merton Option Pricing Model including Analytical Greeks and implied volatility calculations. The following information is an excerpt from Espen Gaarder Haug's book "Option Pricing Formulas". This version is to price Options on Futures. The options sensitivities (Greeks) are the partial derivatives of the Black-Scholes-Merton ( BSM ) formula. Analytical Greeks for our purposes here are broken down into various categories: Del

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Garman and Kohlhagen (1983) for Currency Options [Loxx]Garman and Kohlhagen (1983) for Currency Options is an adaptation of the Black-Scholes-Merton Option Pricing Model including Analytical Greeks and implied volatility calculations. The following information is an excerpt from Espen Gaarder Haug's book "Option Pricing Formulas". This version of BSMOPM is to price Currency Options. The options sensitivities (Greeks) are the partial derivatives of the Black-Scholes-Merton ( BSM ) formula. Analytical Greeks for our purposes here are broken down i

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BSM OPM 1973 w/ Continuous Dividend Yield [Loxx]Generalized Black-Scholes-Merton w/ Analytical Greeks is an adaptation of the Black-Scholes-Merton Option Pricing Model including Analytical Greeks and implied volatility calculations. The following information is an excerpt from Espen Gaarder Haug's book "Option Pricing Formulas". The options sensitivities (Greeks) are the partial derivatives of the Black-Scholes-Merton ( BSM ) formula. Analytical Greeks for our purposes here are broken down into various categories: Delta Greeks: Delta, DDelt

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Black-Scholes 1973 OPM on Non-Dividend Paying Stocks [Loxx]Black-Scholes 1973 OPM on Non-Dividend Paying Stocks is an adaptation of the Black-Scholes-Merton Option Pricing Model including Analytical Greeks and implied volatility calculations. Making b equal to r yields the BSM model where dividends are not considered. The following information is an excerpt from Espen Gaarder Haug's book "Option Pricing Formulas". The options sensitivities (Greeks) are the partial derivatives of the Black-Scholes-Merton ( BSM ) formula. For our purposes here are, Anal

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Generalized Black-Scholes-Merton w/ Analytical Greeks [Loxx]Generalized Black-Scholes-Merton w/ Analytical Greeks is an adaptation of the Black-Scholes-Merton Option Pricing Model including Analytical Greeks and implied volatility calculations. The following information is an excerpt from Espen Gaarder Haug's book "Option Pricing Formulas". The options sensitivities (Greeks) are the partial derivatives of the Black-Scholes-Merton (BSM) formula. Analytical Greeks for our purposes here are broken down into various categories: Delta Greeks: Delta, DD

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