Reset Strike Options-Type 2 (Gray Whaley) [Loxx]For a reset option type 2, the strike is reset in a similar way as a reset option 1. That is, the strike is reset to the asset price at a predetermined future time, if the asset price is below (above) the initial strike price for a call (put). The payoff for such a reset call is max(S - X, 0), and max(X - S, 0) for a put, where X is equal to the original strike X if not reset, and equal to the reset strike if reset. Gray and Whaley (1999) have derived a closed-form solution for the price of European reset strike options. The price of the call option is then given by (via "The Complete Guide to Option Pricing Formulas")
c = Se^(b-r)T2 * M(a1, y1; p) - Xe^(-rT2) * M(a2, y2; p) - Se^(b-r)T1 * N(-a1) * N(z2) * e^-r(T2-T1) + Se^(b-r)T2 * N(-a1) * N(z1)
p = Se^(b-r)T1 * N(a1) * N(-z2) * e^-r(T2-T1) + Se^(b-r)T2 * N(a1) * N(-z1) + Xe^(-rT2) * M(-a2, -y2; p) - Se^(b-r)T2 * M(-a1, -y1; p)
where b is the cost-of-carry of the underlying asset, a is the volatility of the relative price changes in the asset, and r is the risk-free interest rate. K is the strike price of the option, T1 the time to reset (in years), and T2 is its time to expiration. N(x) and M(a,b; p) are, respectively, the univariate and bivariate cumulative normal distribution functions. Further
a1 = (log(S/X) + (b+v^2/2)T1) / v*T1^0.5 ... a2 = a1 - v*T1^0.5
z1 = ((b+v^2/2)(T2-T1)) / v*(T2-T1)^0.5 ... z2 = z1 - v*(T2-T1)^0.5
y1 = (log(S/X) + (b+v^2/2)T1) / v*T1^0.5 ... y2 = a1 - v*T1^0.5
and p = (T1/T2)^0.5. For reset options with multiple reset rights, see Dai, Kwok, and Wu (2003) and Liao and Wang (2003).
Inputs
Asset price ( S )
Strike price ( K )
Reset time ( T1 )
Time to maturity ( T2 )
Risk-free rate ( r )
Cost of carry ( b )
Volatility ( s )
Numerical Greeks or Greeks by Finite Difference
Analytical Greeks are the standard approach to estimating Delta, Gamma etc... That is what we typically use when we can derive from closed form solutions. Normally, these are well-defined and available in text books. Previously, we relied on closed form solutions for the call or put formulae differentiated with respect to the Black Scholes parameters. When Greeks formulae are difficult to develop or tease out, we can alternatively employ numerical Greeks - sometimes referred to finite difference approximations. A key advantage of numerical Greeks relates to their estimation independent of deriving mathematical Greeks. This could be important when we examine American options where there may not technically exist an exact closed form solution that is straightforward to work with. (via VinegarHill FinanceLabs)
Numerical Greeks Outputs
Delta D
Elasticity L
Gamma G
DGammaDvol
GammaP G
Vega
DvegaDvol
VegaP
Theta Q (1 day)
Rho r
Rho futures option r
Phi/Rho2
Carry
DDeltaDvol
Speed
Strike Delta
Strike gamma
Things to know
Only works on the daily timeframe and for the current source price.
You can adjust the text size to fit the screen
Resetstrikeoption
Reset Strike Options-Type 1 [Loxx]In a reset call (put) option, the strike is reset to the asset price at a predetermined future time, if the asset price is below (above) the initial strike price. This makes the strike path-dependent. The payoff for a call at maturity is equal to max((S-X)/X, 0) where is equal to the original strike X if not reset, and equal to the reset strike if reset. Similarly, for a put, the payoff is max((X-S)/X, 0) Gray and Whaley (1997) x have derived a closed-form solution for such an option. For a call, we have
c = e^(b-r)(T2-T1) * N(-a2) * N(z1) * e^(-rt1) - e^(-rT2) * N(-a2)*N(z2) - e^(-rT2) * M(a2, y2; p) + (S/X) * e^(b-r)T2 * M(a1, y1; p)
and for a put,
p = e^(-rT2) * N(a2) * N(-z2) - e^(b-r)(T2-T1) * N(a2) * N(-z1) * e^(-rT1) + e^(-rT2) * M(-a2, -y2; p) - (S/X) * e^(b-r)T2 * M(-a1, -y1; p)
where b is the cost-of-carry of the underlying asset, a is the volatil- ity of the relative price changes in the asset, and r is the risk-free interest rate. X is the strike price of the option, r the time to reset (in years), and T is its time to expiration. N(x) and M(a, b; p) are, respec- tively, the univariate and bivariate cumulative normal distribution functions. The remaining parameters are p = (T1/T2)^0.5 and
a1 = (log(S/X) + (b+v^2/2)T1) / vT1^0.5 ... a2 = a1 - vT1^0.5
z1 = (b+v^2/2)(T2-T1)/v(T2-T1)^0.5 ... z2 = z1 - v(T2-T1)^0.5
y1 = log(S/X) + (b+v^2)T2 / vT2^0.5 ... y2 = y1 - vT2^0.5
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)
Inputs
Asset price ( S )
Initial strike price ( X1 )
Extended strike price ( X2 )
Initial time to maturity ( t1 )
Extended time to maturity ( T2 )
Risk-free rate ( r )
Cost of carry ( b )
Volatility ( s )
Numerical Greeks or Greeks by Finite Difference
Analytical Greeks are the standard approach to estimating Delta, Gamma etc... That is what we typically use when we can derive from closed form solutions. Normally, these are well-defined and available in text books. Previously, we relied on closed form solutions for the call or put formulae differentiated with respect to the Black Scholes parameters. When Greeks formulae are difficult to develop or tease out, we can alternatively employ numerical Greeks - sometimes referred to finite difference approximations. A key advantage of numerical Greeks relates to their estimation independent of deriving mathematical Greeks. This could be important when we examine American options where there may not technically exist an exact closed form solution that is straightforward to work with. (via VinegarHill FinanceLabs)
Numerical Greeks Ouput
Delta
Elasticity
Gamma
DGammaDvol
GammaP
Vega
DvegaDvol
VegaP
Theta (1 day)
Rho
Rho futures option
Phi/Rho2
Carry
DDeltaDvol
Speed
Things to know
Only works on the daily timeframe and for the current source price.
You can adjust the text size to fit the screen
