Yield to Maturity
(TODO) YTM for a cash flow with cfl::Root
Input:
- : the cash flow. Some payments can be negative, which may lead to counter-intuitive results
- : the payment times,
- : the initial value of the cash flow
- : the initial time
- : the initial lower bound for the YTM
- : the initial upper bound for the YTM
cfl::Root: the root-finding routing of bracketing type
Output:
- : the yield to maturity for the cash flow
Remark: the YTM solves the equation:
(TODO) Forward YTM curve for an annuity with cfl::RootD
Input:
- : the coupon rate
- : the time interval between coupon payments
- : time to maturity,
- : the discount curve
- : the initial time
cfl::RootD: the root-finding routing of polishing type. It requires the computation of the function and its derivative. We can use the coupon rate as the initial guess for the YTM
Output:
- : the forward yields to maturity for the annuity
The annuity pays coupons at times such that
For , the forward yield to maturity solves the equation:
where is the “dirty” forward price computed at for delivery at . In the forward contract, the buyer pays forward price at delivery time and then receives coupon at payments times .
Prices of options
Price of a call in the BS model
Input:
- : the strike
- : the time to maturity
- : the discount factor
- : the forward price
- : the volatility
Output:
- : the price of the call option in the Black model
Recall that
where is the standard Gaussian random variable with mean 0 and variance 1,
is the standard Gaussian cumulative distribution function (cdf), and
(TODO) Price of the clique option in the BS model
Input:
- : the strike
- : the averaging times,
- : the maturity
- : the initial discount curve
- : the initial forward curve
- : the mean-reversion rate
- : the volatility of the spot
- : the initial time
Output:
The price of the clique option in the BS model at .
Remark: the payoff of the clique option at the maturity is given by
where is the spot price at .
We use the stationary form of the general BS model, where the interest rates are deterministic. The forward price computed at time for the contract with delivery time evolves as
where is a Brownian motion. Solving this linear SDE, we obtain that
where the average variance of the forward price on is given by
Greeks of options
Vega for a call option in the BS model
Input:
- : the strike
- : the time to maturity
- : the discount factor
- : the forward price
- : the volatility
Output:
- : the vega of the call option in the Black model, where is the price of the call option
Algorithm:
We can deduce that
where is the standard Gaussian random variable with mean 0 and variance 1,
are the standard Gaussian density and distribution functions.
and
(TODO) Vega for the clique option in the BS model
Input:
- : the strike
- : the averaging times,
- : the maturity
- : the initial discount curve
- : the initial forward curve
- : the mean-reversion rate
- : the volatility of the spot
- : the initial time
Output:
- the derivative of the price of the clique option in the BS model at w.r.t .
Implied volatility of options
Implied volatility for a call option in the BS model with Newton method
Input:
- : the strike
- : the time to maturity
- : the discount factor
- : the forward price
- : the price of the call
- : the initial guess for the volatility
- : the absolute error for the price
Output:
the implied volatility such that , where is the value of the call option in the Black-Scholes model with volatility :
where is the standard Gaussian random variable with mean 0 and variance 1.
Algorithm:
To find root using the Newton method (gradient descent):
We stop as soon as .
Remark: The idea behind the Newton (gradient descent) method comes from the first order expansion of function:
in a neighborhood of a root where :
so that
Implied volatility for a call option in the BS model with clf::RootD
Input:
- : the strike
- : the time to maturity
- : the discount factor
- : the forward price
- : the price of the call
- : the initial guess for the volatility
cfl::RootD: the root-finding routing of polishing type. It requires the computation of the function and its derivative.
Output:
- : the implied volatility of the call option with the given parameters computed in the BS model
where is the standard Gaussian random variable with mean 0 and variance 1.
Algorithm:
Run the root-finding algorithm with the function
its derivative (i.e. the vega), and the initial guess .
Implied volatility for a portfolio of call options in the BS model with clf::Root
Input:
- Portfolio of call options: , where
- : the number of options (can be negative, but better to be positive to avoid counter-intuitive results)
- : the strike
- : the maturity according to the absolute calendar
- : the market price of the portfolio
- : the discount curve
- : the forward curve
- : the initial time
- : the initial lower bound for the volatility
- : the initial upper bound for the volatility,
cfl:Root: the root-finding routing of bracketing type. It relies on the computation of the function only
Output:
- : the implied volatility of the portfolio of call options in the BS model:
Algorithm:
Run the root-finding algorithm of bracketing type with the initial interval for the function
where is the BS price for the call option with strike and time to maturity under discount factor , forward price , and volatility .
(TODO) Implied volatility for the clique option in the BS model
Input:
- : the strike
- : the averaging times,
- : the maturity
- : the initial discount curve
- : the initial forward curve
- : the mean-reversion rate
- : the initial guess for the volatility
- : the initial time
cfl::RootD: the root-finding routing of polishing type. It requires the computation of the function and its derivative
Output:
- : the implied volatility of the clique option with the given parameters computed in the BS model
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