Black-Scholes Formula PDF

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Derivation of the Black-Scholes Formula ➾ Consider a Black-Scholes stock whose price, under the risk-neutral measure Q, follows the stochastic differential equation (SDE) dSt = rSt dt + σSt dZt , where r is the annual continuously compounded risk-free rate, σ is the volatility, and Zt is a standard Brownian motion. Then,   St = S0 exp (r − 21 σ 2 )t + σZt . (1)

➾ Consider a (European) call option on the stock with strike price K and expiry at time T . Our goal is to find the initial fair price C0 of the call option. By risk-neutral pricing,    (2) C0 = e−rT EQ CT ] = e−rT EQ max{ST − K, 0} ➾ We can decompose a call option into a long asset-or-nothing option together with a short cash-or-nothing option. An asset-or-nothing option with strike K and expiry T pays ST if ST > K and pays nothing otherwise. A cash-or-nothing option with strike K and expiry T pays 1 if ST > K and pays nothing otherwise. Suppose we hold 1 asset-or-nothing option and have sold K cash-or-nothing options, both with strike K and expiry T . If ST ≤ K then all options expire worthless. If ST > K then the asset-or-nothing option returns ST while the cash-ornothing options leave us with an obligation of K × 1 = K, a net position of ST − K. Thus: Call option = (1 × asset-or-nothing option) − (K × cash-or-nothing option) In what follows, Z will always refer to a standard normal random variable, Φ to its cumulative distribution function (CDF) and φ to its probability density function so that, for t, x ∈ R, Z x 1 −t2 /2 φ(t) = √ e and Φ(x) = φ(t) dt. 2π −∞ 1

➾ Price of a cash-or-nothing option Let V c be the initial fair price of a cash-or-nothing option. Then, again by risk-neutral pricing    c ST K c −rT −rT −rT −rT V =e EQ VT = e . Q{ST > K} = e Q EQ 1{ST >K } = e > S0 S0 By (1), ST /S0 = eX where √   X = (r − 21σ 2 )T + σZT ∼ N a, b2 , where a = (r − 21σ 2 )T, b = σ T .

Lemma 1. If X ∼ N (a, b2 ) under Q and x ∈ R then    X  log(1/x) + a . Q e >x =Φ b Proof. 

X



 X −a log x − a > b b   log x − a =Q Z> b     log(1/x) + a log x − a =Φ . =Q Z x = Q X > log x = Q

If we apply Lemma 1 with the values of a and b given above and with x = K/S0 , we find that V c = e−rT Φ(d2 ) where d2 =

log(S0 /K) + (r − 21 σ 2 )T √ . σ T

(3)

➾ Price of an asset-or-nothing option Let V a be the initial fair price of an asset-or-nothing option. Again by riskneutral pricing     V a = e−rT EQ VTa = e−rT EQ ST 1{ST >K }   = e−rT S0 EQ eX 1{eX >x} (4) where X = log(ST /S0 ) and x = log(K/S0 ). . 2

Lemma 2. If X ∼ N (a, b2 ) under Q and x ∈ R,   b2  X  log(1/x) + a + b2 a+ EQ e 1{eX >x} = e 2 Φ b Proof. We write X = a + bZ where Z ∼ N (0, 1) so that, as above, {eX > x} = {Z > (log x − a)/b}.   We can compute EQ f (Z) for a general function f according to the formula Z ∞   f (t) φ(t) dt. EQ f (Z) = −∞

This gives     EQ eX 1{eX >x} = EQ ea+bZ 1{Z>(log x−a)/b} Z ∞ = ea+bt 1{t>(log x−a)/b} φ(t) dt Z −∞ ∞ = ea+bt φ(t) dt (log x−a)/b

The integrand is 1 2 1 2 2 1 1 1 2 ea+bt φ(t) = ea+bt √ e−t /2 = √ e− 2 (t −2bt)+a = √ e− 2 [(t−b) −b ]+a, 2π 2π 2π

which leads to Z ∞ 2 1  X  2 b +a 1 EQ e 1{eX >x} = √ e− 2 (t−b) + 2 dt 2π (log x−a)/b Z ∞ 1 b2 1 2 a+ 2 √ e−2 (t−b) dt =e 2π (log x−a)/b Z ∞ b2 2 1 a+ 2 e−s /2 ds √ =e 2π ((log x−a)/b)−b    b2 log x − a a+ 2 −b Q Z> =e b    b2 log x − a a+ =e 2 Q Z x}   log(S0 /K) + (r + 21σ 2 )T = e−rT erT S0 Φ(d1 ) = S0 Φ(d1 ) where d1 = √ σ T Combining the formulas for V c and V a gives Black-Scholes Formula for Price of a Call Option The initial fair price C0 of a European call option on a Black-Scholes stock is, in terms of the time to expiry T , the risk-free rate r, the strike price K, the current price of the stock S0 and the volatility of the stock σ , C0 = S0 Φ(d1 ) − Ke−rT Φ(d2 ) where log(S0 /K) + (r + 21 σ 2 )T √ d1 = σ T √ log(S0 /K) + (r − 21 σ 2 )T √ d2 = = d1 − σ T . σ T Exercise Use Put-Call Parity to derive the Black-Scholes formula for the price P0 of the corresponding European put option, namely P0 = Ke−rT Φ(−d2 ) − S0 Φ(−d1 ).

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