Systems and methods for callable options values determination using deep machine learning
Systems, apparatuses, methods, and computer program products are disclosed for pricing a callable instrument. A plurality of corresponding pairs of Brownian motion paths and index value paths are determined corresponding to a set of dates. A deep neural network (DNN) of a backward DNN solver is trained until a convergence requirement is satisfied by for each pair of corresponding Brownian motion path and index value path, using the backward DNN solver to determine by iterating in reverse time order from a final discounted option payoff to an initial value. A statistical measure of spread of a set of initial values is determined and parameters of the DNN are modified based on the statistical measures of spread. Pricing information is determined by the backward DNN solver and provided such that a representation thereof is provided via an interactive user interface of a user computing device.
1. A method for pricing a callable instrument, the method comprising:
defining a set of dates comprising a plurality of time-ordered dates, one or more dates of the set of dates being an exercise date of the callable instrument;
determining a plurality of corresponding pairs of Brownian motion paths and index value paths, each path corresponding to the set of dates;
training a deep neural network (DNN) of a backward DNN solver until a convergence requirement is satisfied by:
for each pair of corresponding Brownian motion path and index value path, using the backward DNN solver, determining a discounted option payoff for each of the plurality of dates of the set of dates, wherein for each pair of corresponding Brownian motion path and index value path, (a) a final discounted option payoff, corresponding to a final date t N of the set of dates, is determined, (b) intermediate discounted payoff options, corresponding to the dates of the set of dates between an initial date and the final date of the set of dates, are determined in reverse time order, and (c) an initial value, corresponding to the initial date t 0 of the set of dates, is determined, wherein an i th intermediate discounted payoff option corresponding to a date t i of the set of dates is determined based on an (i+1) th intermediate discounted payoff option corresponding to a date t i+l , wherein the set of dates is time ordered such that t 0 <t i+1 <t N ,
defining a set of initial values comprising the initial value determined for each pair of corresponding Brownian motion path and index value path and determining one or more statistical measures of spread based on the set of initial values, and
modifying one or more parameters of the DNN based on the one or more statistical measures of spread;
after the convergence requirement is satisfied, determining pricing information for the callable instrument; and
providing at least a portion of the pricing information such that a user computing device receives the at least a portion of the pricing information and provides a representation of the at least a portion of the pricing information via an interactive user interface provided via a display of the user computing device.
2. The method of claim 1 , wherein the set of dates comprises a plurality of exercise dates for the callable instrument.
3. The method of claim 1 , wherein the convergence requirement is at least one of (a) a set number of iterations performed or (b) at least one of the one or more statistical measures of spread satisfies a spread threshold requirement.
4. The method of claim 1 , wherein the index value path is a Libor rate path.
5. The method of claim 1 , wherein the pricing information comprises at least one of an initial discounted option price or a discounted option payoff for one or more exercise dates of the callable instrument.
6. The method of claim 1 , wherein the DNN of the backward DNN solver determines a gradient of a discount option payoff with respect to an index value path.
7. The method of claim 1 , wherein the one or more parameters of the DNN are modified using a stochastic gradient descent training model.
8. The method of claim 1 , wherein the DNN comprises a plurality of sub-networks, each sub-network corresponding to a date of the set of dates.
9. The method of claim 1 , wherein the DNN is a feedforward DNN.
10. The method of claim 1 , wherein the intermediate discounted payoff options and the initial value for a pair of corresponding Brownian motion paths and index value paths are determined by performing an iteration of the discounted payoff option backward in time based on the index value path.
11. An apparatus for pricing a callable instrument, the apparatus comprising:
at least one processor configured to:
define a set of dates comprising a plurality of time-ordered dates, one or more dates of the set of dates being an exercise date of the callable instrument;
determine a plurality of corresponding pairs of Brownian motion paths and index value paths, each path corresponding to the set of dates;
train a deep neural network (DNN) of a backward DNN solver until a convergence requirement is satisfied by:
for each pair of corresponding Brownian motion path and index value path, using the backward DNN solver, determining a discounted option payoff for each of the plurality of dates of the set of dates, wherein for each pair of corresponding Brownian motion path and index value path, (a) a final discounted option payoff, corresponding to a final date of the set of dates, is determined, (b) intermediate discounted payoff options, corresponding to the dates of the set of dates between an initial date and the final date of the set of dates, are determined in reverse time order, and (c) an initial value, corresponding to the initial date of the set of dates, is determined, wherein an i th intermediate discounted payoff option corresponding to a date t i of the set of dates is determined based on an (i+1) th intermediate discounted payoff option corresponding to a date t i+1 , wherein the set of dates is time ordered such that t 0 <t i+1 <t N ,
defining a set of initial values comprising the initial value determined for each pair of corresponding Brownian motion path and index value path and determining one or more statistical measures of spread based on the set of initial values, and
modifying one or more parameters of the DNN based on the one or more statistical measures of spread;
after the convergence requirement is satisfied, determine pricing information for the callable instrument; and
communication interface circuitry configured to:
provide at least a portion of the pricing information such that a user computing device receives the at least a portion of the pricing information and provides a representation of the at least a portion of the pricing information via an interactive user interface provided via a display of the user computing device.
12. The apparatus of claim 11 , wherein the set of dates comprises a plurality of exercise dates for the callable instrument.
13. The apparatus of claim 11 , wherein the convergence requirement is at least one of (a) a set number of iterations performed or (b) at least one of the one or more statistical measures of spread satisfies a spread threshold requirement.
14. The apparatus of claim 11 , wherein the index value path is a Libor rate path.
15. The apparatus of claim 11 , wherein the pricing information comprises at least one of an initial discounted option price or a discounted option payoff for one or more exercise dates of the callable instrument.
16. The apparatus of claim 11 , wherein the DNN of the backward DNN solver determines a gradient of a discount option payoff with respect to an index value path.
17. The apparatus of claim 11 , wherein the one or more parameters of the DNN are modified using a stochastic gradient descent training model.
18. The apparatus of claim 11 , wherein the DNN comprises a plurality of sub-networks, each sub-network corresponding to a date of the set of dates.
19. The apparatus of claim 11 , wherein the DNN is a feedforward DNN.
20. The apparatus of claim 11 , wherein the intermediate discounted payoff options and the initial value for a pair of corresponding Brownian motion paths and index value paths are determined by performing an iteration of the discounted payoff option backward in time based on the index value path.