feat(bs-approx): uses cdf approximation for bs formula

src/BlackScholes.ts

@@ -1,3 +1,4 @@
+import { getCDFSolidity } from '.'
 import { std_n_cdf } from './CumulativeNormalDistribution'
 
 /**
@@ -10,7 +11,7 @@ export function moneyness(strike: number, spot: number): number {
 }
 
 /**
- * @notice Calculates a common expression
+ * @notice Calculates a common equation
  * @param sigma Volatility as a float
  * @param tau Time until expiry, in years, as a float
  * @returns volatilty * sqrt(tau)
@@ -60,6 +61,23 @@ export function callDelta(strike: number, sigma: number, tau: number, spot: numb
   return delta
 }
 
+/**
+ * @notice D1 = (Log(spot / strike) + sigma^2 / 2 * tau) / (sigma * sqrt(tau))
+ * D2 = D1 - sigma * sqrt(tau)
+ * @returns D1 and D2, auxiliary variables of the black-scholes formula
+ */
+export function getD1AndD2(
+  strike: number,
+  sigma: number,
+  tau: number,
+  spot: number,
+  rate: number = 0
+): { d1: number; d2: number } {
+  const d1 = calculateD1(strike, sigma, tau, spot, rate)
+  const d2 = d1 - getProportionalVol(sigma, tau)
+  return { d1, d2 }
+}
+
 /**
  * @param strike Strike price of option, as a float
  * @param sigma Implied volatility of option, as a float
@@ -69,7 +87,24 @@ export function callDelta(strike: number, sigma: number, tau: number, spot: numb
  * @returns Black-Scholes price of call option with parameters
  */
 export function callPremium(strike: number, sigma: number, tau: number, spot: number, rate: number = 0): number {
-  const d1 = calculateD1(strike, sigma, tau, spot, rate)
-  const d2 = d1 - getProportionalVol(sigma, tau)
+  const { d1, d2 } = getD1AndD2(strike, sigma, tau, spot, rate ?? rate)
   return Math.max(0, std_n_cdf(d1) * spot - std_n_cdf(d2) * strike * Math.exp(-tau * rate))
 }
+
+/**
+ * @notice Uses solidity CDF approximation
+ * @dev Use to determine the theoretical theta which can accrue to a position
+ * @returns Premium of a call option using the solidity CDF approximation formula
+ */
+export function callPremiumApproximation(
+  strike: number,
+  sigma: number,
+  tau: number,
+  spot: number,
+  rate: number = 0
+): number {
+  const { d1, d2 } = getD1AndD2(strike, sigma, tau, spot, rate ?? rate)
+  const cdfD1 = getCDFSolidity(d1)
+  const cdfD2 = getCDFSolidity(d2)
+  return Math.max(0, cdfD1 * spot - cdfD2 * strike * Math.exp(-tau * rate))
+}

test/blackScholes.test.ts

@@ -25,23 +25,23 @@ describe('Black Scholes', () => {
   })
 
   describe('calculateD1', () => {
-    it('return 0 if tau is 0', () => {
-      const tau = 0
-      expect(math.calculateD1(1, 1, tau, 1)).toEqual(0)
+    it('return 0.5 if tau is 1', () => {
+      const tau = 1
+      expect(math.calculateD1(1, 1, tau, 1)).toEqual(0.5)
     })
   })
 
   describe('calculateD2', () => {
-    it('return 0 if tau is 0', () => {
-      const tau = 0
-      expect(math.calculateD2(1, 1, tau, 1)).toEqual(0)
+    it('return -0.5 if tau is 1', () => {
+      const tau = 1
+      expect(math.calculateD2(1, 1, tau, 1)).toEqual(-0.5)
     })
   })
 
   describe('callDelta', () => {
-    it('return 0.5 if tau is 0', () => {
-      const tau = 0
-      expect(math.callDelta(1, 1, tau, 1)).toBeCloseTo(0.5)
+    it('return 0.6914624612740131036377 if tau is 1', () => {
+      const tau = 1
+      expect(math.callDelta(1, 1, tau, 1)).toBeCloseTo(0.6914624612740131036377)
     })
   })
 
@@ -50,5 +50,36 @@ describe('Black Scholes', () => {
       const tau = 0
       expect(math.callPremium(1, 1, tau, 1)).toEqual(0)
     })
+
+    it('return 0.3829249225480262072754 if tau is 1', () => {
+      const tau = 1
+      expect(math.callPremium(1, 1, tau, 1)).toBeCloseTo(0.3829249225480262072754)
+    })
+  })
+
+  describe('approximations', () => {
+    it('call premium using solidity CDF approximation for 0 tau', () => {
+      const tau = 0
+      expect(math.callPremiumApproximation(1, 1, tau, 1)).toBeCloseTo(math.callPremium(1, 1, tau, 1), 6)
+    })
+
+    it('call premium using solidity CDF approximation for 1 year tau', () => {
+      const tau = 1
+      expect(math.callPremiumApproximation(1, 1, tau, 1)).toBeCloseTo(math.callPremium(1, 1, tau, 1), 6)
+    })
+
+    it('premium for ITM', () => {
+      const tau = 1
+      const spot = 2
+      const strike = 1
+      expect(math.callPremiumApproximation(strike, 1, tau, spot)).toBeCloseTo(math.callPremium(strike, 1, tau, spot), 6)
+    })
+
+    it('premium for OTM', () => {
+      const tau = 1
+      const spot = 1
+      const strike = 2
+      expect(math.callPremiumApproximation(strike, 1, tau, spot)).toBeCloseTo(math.callPremium(strike, 1, tau, spot), 6)
+    })
   })
 })