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Derifree

Derivative pricing for the lactose side-effect intolerant.

Derifree is an experimental library exploring the use of free monads to implement a contract definition language for (equity) derivatives.

This is work in progress!

Intro

A contract is encoded as a value of type dsl.RV[A] where dsl is an instance of derifree.Dsl[T] for some time-like T (e.g,. java.time.Instant). The type A is typically Unit and we provide a convenient alias:

dsl.ContingentClaim = dsl.RV[Unit]

Returning Unit makes sense if you think of a contract as having the "side-effect" of exchanging a sequence of cash flows between the long and the short party over some time horizon. There isn't a meaningful value to return, in general. A generic return type A can still be useful. In that case RV[A] should be interpreted as a random variable taking value in A. This is used internally, e.g., to compute factors in the Longstaff-Schwartz regression.

The type dsl.RV is a free monad. In particular, values are sequenced using flatMap, allowing us to write contracts using for-comprehensions (see the examples below).

Rules

When writing contracts using the derifree DSL, some rules need to be followed to ensure pricing doesn't fail or, worse, return incorrect results.

  1. The occurrences of any of the keywords (cashflow, spot, etc.) should be unconditional. Note that this does not preclude things like conditional cash flows. Indeed, to model a conditional cash flow a at time t, we write cashflow(if p then a else 0.0, ccy, t) instead of Monad[RV].whenA(p)(cashflow(a, ccy, t)). To give another example, if we need a certain spot observation spot("ACME", t) only conditionally, we simply ignore its result when it isn't needed.

    The reason for this rule is that we want to be able to collect all meta information about the contract (spot/discount observations, barriers, callability, etc) by evaluating the contract once using an arbitrary realization of the underlying assets.

    For the read-like keywords (spot, hitProb, survivalProb) this means simply ignoring their result when it isn't needed. For write-like keywords (cashflow, callable, puttable) this means using a neutral value (0.0 or None).

    (Side note: another approach would be to "discover" this information at pricing time and restart the pricing until we arrive at a "fix point". This would come at a loss in efficiency, however.)

  2. Causality: the DSL doesn't prevent you from having a cash flow or call/early exercise at time t1 depend on an observation at time t2 > t1. It's the user's responsibility to ensure all relationships are causal.

Examples

import cats.*
import cats.syntax.all.*
import derifree.*
import derifree.literals.*
import derifree.syntax.*

// use any time type with a `TimeLike` instance
val dsl = Dsl[java.time.Instant]
import dsl.*

val refTime = i"2023-12-27T18:00:00Z"
val expiry = refTime.plusDays(365)
val settle = expiry.plusDays(2)

val europeanCall = for
  s0 <- spot("AAPL", refTime)
  s <- spot("AAPL", expiry)
  _ <- cashflow(max(s / s0 - 1, 0.0), Ccy.USD, settle)
yield ()

val europeanUpAndOutCall = for
  s0 <- spot("AAPL", refTime)
  s <- spot("AAPL", expiry)
  p <- survivalProb(
    Barrier.Discrete(
      Barrier.Direction.Up,
      Map("AAPL" -> List((expiry, 1.5 * s0)))
    )
  )
  _ <- cashflow(p * max(s / s0 - 1, 0.0), Ccy.USD, settle)
yield ()

val europeanPut = for
  s0 <- spot("AAPL", refTime)
  s <- spot("AAPL", expiry)
  _ <- cashflow(max(1 - s / s0, 0.0), Ccy.USD, settle)
yield ()

val bermudanPut = for
  s0 <- spot("AAPL", refTime)
  _ <- List(90, 180, 270, 360)
    .map(refTime.plusDays)
    .traverse_(d =>
      spot("AAPL", d).flatMap(s =>
        exercisable(max(1 - s / s0, 0.0).some.filter(_ > 0.0), Ccy.USD, d)
      )
    )
  s <- spot("AAPL", expiry)
  _ <- cashflow(max(1 - s / s0, 0.0), Ccy.USD, settle)
yield ()

val quantoEuropeanCall = for
  s0 <- spot("AAPL", refTime)
  s <- spot("AAPL", expiry)
  _ <- cashflow(max(s / s0 - 1, 0.0), Ccy.EUR, settle)
yield ()

val worstOfContinuousDownAndInPut = for
  s1_0 <- spot("AAPL", refTime)
  s2_0 <- spot("MSFT", refTime)
  s3_0 <- spot("GOOG", refTime)
  s1 <- spot("AAPL", expiry)
  s2 <- spot("MSFT", expiry)
  s3 <- spot("GOOG", expiry)
  p <- hitProb(
    Barrier.Continuous(
      Barrier.Direction.Down,
      Map("AAPL" -> 0.8 * s1_0, "MSFT" -> 0.8 * s2_0, "GOOG" -> 0.8 * s3_0),
      from = refTime,
      to = expiry
    )
  )
  _ <- cashflow(p * max(0.0, 1 - min(s1 / s1_0, s2 / s2_0, s3 / s3_0)), Ccy.USD, settle)
yield ()

val worstOfEuropeanDownAndInPut = for
  s1_0 <- spot("AAPL", refTime)
  s2_0 <- spot("MSFT", refTime)
  s3_0 <- spot("GOOG", refTime)
  s1 <- spot("AAPL", expiry)
  s2 <- spot("MSFT", expiry)
  s3 <- spot("GOOG", expiry)
  p <- hitProb(
    Barrier.Discrete(
      Barrier.Direction.Down,
      Map(
        "AAPL" -> List((expiry, 0.8 * s1_0)),
        "MSFT" -> List((expiry, 0.8 * s2_0)),
        "GOOG" -> List((expiry, 0.8 * s3_0))
      )
    )
  )
  _ <- cashflow(p * max(0.0, 1 - min(s1 / s1_0, s2 / s2_0, s3 / s3_0)), Ccy.USD, settle)
yield ()

val barrierReverseConvertible =
  val relBarrier = 0.7
  val couponTimes = List(90, 180, 270, 360).map(refTime.plusDays)
  for
    s1_0 <- spot("AAPL", refTime)
    s2_0 <- spot("MSFT", refTime)
    s3_0 <- spot("GOOG", refTime)
    s1 <- spot("AAPL", expiry)
    s2 <- spot("MSFT", expiry)
    s3 <- spot("GOOG", expiry)
    p <- hitProb(
      Barrier.Continuous(
        Barrier.Direction.Down,
        Map(
          "AAPL" -> relBarrier * s1_0,
          "MSFT" -> relBarrier * s2_0,
          "GOOG" -> relBarrier * s3_0
        ),
        from = refTime,
        to = expiry
      )
    )
    _ <- couponTimes.traverse_(t => cashflow(5.0, Ccy.USD, t))
    _ <- cashflow(
      100 * (1 - p * max(0.0, 1 - min(s1 / s1_0, s2 / s2_0, s3 / s3_0))),
      Ccy.USD,
      settle
    )
  yield ()

val callableBarrierReverseConvertible =
  val relBarrier = 0.7
  val callTimes = List(90, 180, 270, 360).map(refTime.plusDays)
  val couponTimes = List(90, 180, 270, 360).map(refTime.plusDays)
  for
    s1_0 <- spot("AAPL", refTime)
    s2_0 <- spot("MSFT", refTime)
    s3_0 <- spot("GOOG", refTime)
    s1 <- spot("AAPL", expiry)
    s2 <- spot("MSFT", expiry)
    s3 <- spot("GOOG", expiry)
    p <- hitProb(
      Barrier.Continuous(
        Barrier.Direction.Down,
        Map(
          "AAPL" -> relBarrier * s1_0,
          "MSFT" -> relBarrier * s2_0,
          "GOOG" -> relBarrier * s3_0
        ),
        from = refTime,
        to = expiry
      )
    )
    _ <- callTimes.traverse_(t => callable(100.0.some, Ccy.USD, t))
    _ <- couponTimes.traverse_(t => cashflow(5.0, Ccy.USD, t))
    _ <- cashflow(
      100 * (1 - p * max(0.0, 1 - min(s1 / s1_0, s2 / s2_0, s3 / s3_0))),
      Ccy.USD,
      settle
    )
  yield ()

val couponBarrier =
  val relBarrier = 0.95
  val couponAmount = 5.0
  val couponTimes = List(90, 180, 270, 360).map(refTime.plusDays)
  for
    s1_0 <- spot("AAPL", refTime)
    s2_0 <- spot("MSFT", refTime)
    s3_0 <- spot("GOOG", refTime)
    s1 <- spot("AAPL", expiry)
    s2 <- spot("MSFT", expiry)
    s3 <- spot("GOOG", expiry)
    _ <- couponTimes.traverse: t =>
      (spot("AAPL", t), spot("MSFT", t), spot("GOOG", t))
        .mapN((s1_t, s2_t, s3_t) => min(s1_t / s1_0, s2_t / s2_0, s3_t / s3_0) > relBarrier)
        .flatMap: isAbove =>
          cashflow(if isAbove then couponAmount else 0.0, Ccy.USD, t)
  yield ()

val couponBarrierWithMemoryEffect =
  val relBarrier = 0.95
  val couponAmount = 5.0
  val couponTimes = List(90, 180, 270, 360).map(refTime.plusDays)
  for
    s1_0 <- spot("AAPL", refTime)
    s2_0 <- spot("MSFT", refTime)
    s3_0 <- spot("GOOG", refTime)
    s1 <- spot("AAPL", expiry)
    s2 <- spot("MSFT", expiry)
    s3 <- spot("GOOG", expiry)
    _ <- couponTimes.foldLeftM(0.0): (acc, t) =>
      (spot("AAPL", t), spot("MSFT", t), spot("GOOG", t))
        .mapN((s1_t, s2_t, s3_t) => min(s1_t / s1_0, s2_t / s2_0, s3_t / s3_0) > relBarrier)
        .flatMap: isAbove =>
          cashflow(if isAbove then acc + couponAmount else 0.0, Ccy.USD, t)
            .as(if isAbove then 0 else acc + couponAmount)
  yield ()

// let's price using a simple Black-Scholes model

val discount = YieldCurve.fromContinuouslyCompoundedRate(0.05.rate, refTime)

val aapl = models.blackscholes.Asset(
  "AAPL",
  Ccy.USD,
  Forward(195.0, divs = Nil, discount = discount, borrow = YieldCurve.zero),
  0.23.vol
)

val msft = models.blackscholes.Asset(
  "MSFT",
  Ccy.USD,
  Forward(370.0, divs = Nil, discount = discount, borrow = YieldCurve.zero),
  0.25.vol
)

val goog = models.blackscholes.Asset(
  "GOOG",
  Ccy.USD,
  Forward(135.0, divs = Nil, discount = discount, borrow = YieldCurve.zero),
  0.29.vol
)

val correlations =
  Map(
    ("AAPL", "MSFT") -> 0.7,
    ("AAPL", "GOOG") -> 0.6,
    ("MSFT", "GOOG") -> 0.65,
    ("EURUSD", "AAPL") -> -0.2
  )

// for quantos
val fxVols = Map(CcyPair(Ccy.EUR, Ccy.USD) -> 0.07.vol)

val dirNums = Sobol.directionNumbers(10000).toTry.get

val sim = models.blackscholes.simulator(
  TimeGrid.Factory.almostEquidistant(YearFraction.oneDay),
  NormalGen.Factory.sobol(dirNums),
  refTime,
  List(aapl, msft, goog),
  correlations,
  discount,
  Map.empty
)

// the number of Monte Carlo simulations
val nSims = 32767
europeanCall.fairValue(sim, nSims)
// res0: Either[Error, PV] = Right(value = 0.11573966972403645)

// should be cheaper than plain European call
europeanUpAndOutCall.fairValue(sim, nSims)
// res1: Either[Error, PV] = Right(value = 0.08550169593261212)

europeanPut.fairValue(sim, nSims)
// res2: Either[Error, PV] = Right(value = 0.06699973963703337)

// should be more expensive than European put
bermudanPut.fairValue(sim, nSims)
// res3: Either[Error, PV] = Right(value = 0.07095113729727867)

// what are the probabilities of early exercise?
bermudanPut.putProbabilities(sim, nSims)
// res4: Either[Error, Map[Instant, Double]] = Right(
//   value = Map(
//     2024-09-22T18:00:00Z -> 0.130283516953032,
//     2024-03-26T18:00:00Z -> 0.05578783532212287,
//     2024-12-21T18:00:00Z -> 0.07791375469222084,
//     2024-06-24T18:00:00Z -> 0.11328470717490158
//   )
// )

quantoEuropeanCall.fairValue(sim, nSims)
// res5: Either[Error, PV] = Left(
//   value = Error(message = "missing vol for USDEUR")
// )

worstOfContinuousDownAndInPut.fairValue(sim, nSims)
// res6: Either[Error, PV] = Right(value = 0.11952072431608426)

// should be cheaper than continuous barrier
worstOfEuropeanDownAndInPut.fairValue(sim, nSims)
// res7: Either[Error, PV] = Right(value = 0.09498951398431003)

barrierReverseConvertible.fairValue(sim, nSims)
// res8: Either[Error, PV] = Right(value = 106.09064030012115)

// should be cheaper than non-callable BRC
callableBarrierReverseConvertible.fairValue(sim, nSims)
// res9: Either[Error, PV] = Right(value = 105.51905901142916)

// what are the probabilities of being called?
callableBarrierReverseConvertible.callProbabilities(sim, nSims)
// res10: Either[Error, Map[Instant, Double]] = Right(
//   value = Map(
//     2024-09-22T18:00:00Z -> 0.15588854640339367,
//     2024-03-26T18:00:00Z -> 0.0011597033600878933,
//     2024-12-21T18:00:00Z -> 0.16977446821497238,
//     2024-06-24T18:00:00Z -> 0.003143406476027711
//   )
// )

// should be cheaper than sum of discounted coupons
couponBarrier.fairValue(sim, nSims)
// res11: Either[Error, PV] = Right(value = 8.314781740288916)

// should be more expensive than w/o memory, still cheaper than discounted sum of coupons
couponBarrierWithMemoryEffect.fairValue(sim, nSims)
// res12: Either[Error, PV] = Right(value = 10.370530640687656)

Building

README.md in the root directory is generated from ./docs/readme.md by running sbt docs/mdoc. The example code in ./docs/readme.md is a copy of the code in ./examples/src/main/scala/examples/readme.scala and should be kept in sync.

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Derivative pricing ❤️ free monads

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scala functional-programming derivatives quantitative-finance

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