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A Derivation of the Black–Scholes Pricing Equations for Vanilla Options

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A Derivation of the Black–Scholes Pricing Equations for Vanilla Options A Derivation of the Black–Scholes Pricing Equations for Vanilla Options Junior quant: ‘Should I be surprised that μ drops out?’ Senior quant: ‘Not if you want to keep your job.’ This appendix describes the procedure for deriving closed-form expressions for the prices of vanilla call and put options, by analytically performing integrals derived in Chapter 2. In that chapter, we derive two integral expressions, either of which may be used to calculate the Black–Scholes value of a European option. One of the integral expressions yields the value in terms of the transformed variable X (Equation 2.64): ∞ ( − )2 1 − X X ¯ v(X,τ)= √ e 2σ2τ P(X )dX −∞ 2πσ2τ The other integral expression yields the value in terms of the original financial variable S (spot price) (Equation 2.65): ∞ 2 − ( − ) 1 1 (lnF − lnS ) V(S,t) = e rd Ts t exp − P(S )dS 2 σ 2( − ) 0 2πσ (Te − t) S 2 Te t We will here perform the integral in Equation 2.64 in order to obtain the pricing formulae. The transformed payoff function P¯ (X) takes the following form for a vanilla option: ¯ X P(X) = max 0,φ F0e − K (A.1) where φ is the option trait (+1foracall;−1foraput)andF0 is the arbitrary quantity that was introduced in the transformation process for the purpose of dimensional etiquette. The zero floor in the payoff function has the result that the integrand vanishes for a semi-infinite range of X values. For φ =+1, the integrand vanishes for X < ln(K/F0), whereas for φ =−1, the integrand vanishes for X > ln(K/F0). In general, we may write the 215 216 FX Barrier Options integral in terms of lower limit a and upper limit b: ( − )2 b − X X 1 2 X v(X,τ) = √ e 2σ τ max 0,φ F0e − K dX (A.2) a 2πσ2τ where the limits a and b depend on the option trait φ in the following way: ( ln K φ =+1 a = F0 (A.3) −∞ φ =−1 ( ∞ φ =+1 b = (A.4) ln K φ =−1 F0 Let us write the two terms in Equation A.2 explicitly as such: v(X,τ) = I1 − I2 (A.5) where ( − )2 b − X X . 1 2 X I1 = φF0 √ e 2σ τ e dX (A.6) a 2πσ2τ and ( − )2 b − X X . 1 2 I2 = φK √ e 2σ τ dX (A.7) a 2πσ2τ The integrand of Expression I2 is the probability density function (PDF) of a normal distribution with mean X and variance σ 2τ. This can be written in terms of the special function N(·), which is the cumulative distribution function (CDF) of a standard normal distribution: − − = φ b √X − a √X I2 K N σ τ N σ τ (A.8) In Expression I1, completion of the square in the exponent gives: 2 X − X+σ2τ 1 2 b − X+ σ τ 1 2 I1 = φF0e 2 √ e 2σ τ dX (A.9) a 2πσ2τ As with I2, the integrand is again the probability density function (PDF) of a normal distribution, and the variance is again σ 2τ, but this time the mean is (X + σ 2τ). Again, this can be written in terms of N(·): 2 2 + 1 σ 2τ b − X − σ τ a − X − σ τ = φ X 2 √ − √ I1 F0e N σ τ N σ τ (A.10) Derivation of the Black–Scholes Pricing Equations for Vanilla Options 217 The closed-form expressions for I1 and I2 given by Equations A.10 and A.8 respectively can now be inserted into Equation A.5 to give a closed-form expression for v. Since the quantities a and b depend on the option trait φ (see Equations A.3 and A.4), we will separate the call and put cases. For calls (φ =+1), I1, I2 and v are given as follows: ⎛ ⎛ ⎞⎞ ln K − X − σ 2τ + 1 σ 2τ F0 = X 2 ⎝ (∞) − ⎝ √ ⎠⎠ I1 (call) F0e N N σ τ ⎛ ⎛ ⎞⎞ ln K − X − σ 2τ + 1 σ 2τ F0 = X 2 ⎝ − ⎝ √ ⎠⎠ F0e 1 N σ τ ⎛ ⎞ −ln K + X + σ 2τ + 1 σ 2τ F0 = X 2 ⎝ √ ⎠ F0e N σ τ (A.11) ⎛ ⎛ ⎞⎞ ln K − X = ⎝ (∞) − ⎝ F0√ ⎠⎠ I2 (call) K N N σ τ ⎛ ⎛ ⎞⎞ ln K − X = ⎝ − ⎝ F0√ ⎠⎠ K 1 N σ τ ⎛ ⎞ −ln K + X = ⎝ F√0 ⎠ KN σ τ (A.12) ⎛ ⎞ ⎛ ⎞ −ln K + X + σ 2τ −ln K + X + 1 σ 2τ F0 F0 ⇒ = X 2 ⎝ √ ⎠ − ⎝ √ ⎠ vcall F0e N σ τ KN σ τ (A.13) Forputs(φ =−1), I1, I2 and v are given as follows: ⎛ ⎛ ⎞ ⎞ ln K − X − σ 2τ + 1 σ 2τ F0 =− X 2 ⎝ ⎝ √ ⎠ − (−∞)⎠ I1 (put) F0e N σ τ N ⎛ ⎞ ln K − X − σ 2τ + 1 σ 2τ F0 =− X 2 ⎝ √ ⎠ F0e N σ τ (A.14) 218 FX Barrier Options ⎛ ⎛ ⎞ ⎞ ln K − X =− ⎝ ⎝ F0√ ⎠ − (−∞)⎠ I2 (put) K N σ τ N ⎛ ⎞ ln K − X =− ⎝ F0√ ⎠ KN σ τ (A.15) ⎛ ⎞ ⎛ ⎞ ln K − X − σ 2τ ln K − X + 1 σ 2τ F0 F0 ⇒ =− X 2 ⎝ √ ⎠ + ⎝ √ ⎠ vput F0e N σ τ KN σ τ (A.16) The similarities between Equations A.13 and A.16 allow us to recombine the call and put results into a single vanilla result, like so: ⎡ ⎛ ⎞ ⎛ ⎞⎤ X − ln K + σ 2τ X − ln K + 1 σ 2τ F0 F0 = φ⎣ X 2 ⎝φ √ ⎠ − ⎝φ √ ⎠⎦ vvanilla F0e N σ τ KN σ τ (A.17) We now have a closed-form expression for the transformed value variable v(X,τ).To obtain an expression for the original value variable V(S,t), it only remains for us to undo the four transformations of Section 2.7.1 one by one. Undoing Transformation 4 gives us an expression for undiscounted vanilla prices in terms of the forward: ⎡ ⎛ ⎞ ⎛ ⎞⎤ ln F + 1 σ 2τ ln F − 1 σ 2τ ˜ ( τ)= φ⎣ ⎝φ K √ 2 ⎠ − ⎝φ K √ 2 ⎠⎦ U F, FN σ τ KN σ τ (A.18) Undoing Transformation 3 gives us an expression for undiscounted prices in terms of spot: ⎡ ⎛ ⎞ S + ( − + 1 σ 2)τ ( − )τ ln K rd rf 2 ( τ)= φ⎣ rd rf ⎝φ √ ⎠ U S, Se N σ τ ⎛ ⎞⎤ S 1 2 ln + (rd − rf − σ )τ − ⎝φ K √ 2 ⎠⎦ KN σ τ (A.19) Derivation of the Black–Scholes Pricing Equations for Vanilla Options 219 Undoing Transformation 2 gives us an expression for discounted prices in terms of spot: ⎡ ⎛ ⎞ S + ( − + 1 σ 2)τ − τ ln K rd rf 2 ˜ ( τ)= φ⎣ rf ⎝φ √ ⎠ V S, Se N σ τ ⎛ ⎞⎤ S + ( − − 1 σ 2)τ − τ ln K rd rf 2 − rd ⎝φ √ ⎠⎦ Ke N σ τ (A.20) Lastly, undoing Transformation 1 gives us an expression for discounted prices in terms of spot, with explicit reference to the time variable t: ⎡ ⎛ ⎞ S + ( − + 1 σ 2)( − ) − ( − ) ln K rd rf 2 T t V(S,t) = φ⎣Se rf T t N⎝φ √ ⎠ σ T − t ⎛ ⎞⎤ S + ( − − 1 σ 2)( − ) − ( − ) ln K rd rf 2 T t −Ke rd T t N⎝φ √ ⎠⎦ (A.21) σ T − t If it seems that we have laboured the working, it is for a reason: each of the forms of expression we have presented can be useful in its own right. All of the forms of expression shown above may be found in the literature. The long expressions that form the arguments of the normal CDF are commonly given their own symbols. For example, following the conventions in Hull [2], we define: S + ( − + 1 σ 2)( − ) ln K rd rf 2 T t d1 = √ (A.22) σ T − t S + ( − − 1 σ 2)( − ) ln K rd rf 2 T t d2 = √ (A.23) σ T − t whereupon our formula for the discounted prices in terms of spot becomes: −r (T−t) −r (T−t) V(S,t) = φ Se f N(φd1) − Ke d N(φd2) (A.24) The value V here is for an option with unit Foreign principal (Af = 1); to get the value for non-unit-principal options, we simply need to multiply V by Af. B Normal and Lognormal Probability Distributions B.1 Normal distribution N In the case where Z follows a normal distribution, its density function fZ has the form: ( − μ )2 N ( ) = 1 − z Z fZ z exp (B.1) πσ2 2σ 2 2 Z Z where z may take any real value. μ σ 2 The mean of the distribution equals Z , and its variance equals Z .Thestandard normal distribution is a normal distribution which has mean equal to zero and variance equal to one. We denote the PDF and CDF of the standard normal distribution by special functions n(·) and N(·) respectively: 1 z2 n(z) = √ exp − (B.2) 2π 2 z 1 x2 N(z) = √ exp − dx (B.3) −∞ 2π 2 B.2 Lognormal distribution LN In the case where Z follows a lognormal distribution, its density function fZ has the form: ( − μ )2 LN ( ) = 1 1 − lnz ln Z fZ z exp (B.4) πσ2 z 2σ 2 2 Z Z where z > 0. 220 C Derivation of the Local Volatility Function C.1 Derivation in terms of call prices Our aim here is to derive an expression for the local volatility (lv) function σ(S,t) that appears in the local volatility model of Equation 4.21: dS = (rd − rf)S dt + σ(S,t)S dWt Central to the derivation is the probability density function (PDF) of spot. This quantity provides the crucial link between the dynamics of spot and the values of options. We introduced the PDF in the special case of the Black–Scholes model, in Section 2.7.2.
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