Black Scholes European Option Pricing Formula

thm. BSM Formula for European Call Options The Fair price of a European call option is $C(t)$ given by

$$\begin{array}{rlr}C(t)& =e^{-\tau q}S(t)N(d_1)-e^{-\tau r}KN(d_{-})& \text{where...}\\ & & \\ d_{+}& =\frac{\ln (S(t)/K)+[(r-q)+\frac{{\sigma }^2}{2}]\tau }{\sigma \sqrt{\tau }}& \\ d_{-}& =d_{+}-\sigma \sqrt{\tau }=\frac{\ln (S(t)/K)+[(r-q)-\frac{{\sigma }^2}{2}](\tau )}{\sigma \sqrt{\tau }}& \\ \tau & =T-t& \text{time to expiration}\end{array}$$

where

• $t$ is entry date, $T$ is execution date
• $r$ is the risk-free rate
• $K$ is the strike price
• $N(d)$ is the CDF of the Standard Normal Distribution
• $S(t)$ are the price of the underlier, modeled as log-normal R.V.s (=geometric brownian motion) as $S(t)=S_0{e}^{\mu t+\sigma \sqrt{t}{\mathfrak{B}}_t}$

There are other derivations of BSM, notably:

• Using the Binomial Option Pricing Model
• Risk-Neutral Derivation of BSM

Derivation Using Replicating Portfolio §

Proof (Derivation) Sketch. We use a similar strategy to when we priced Forwards—by constructing a portfolio whose value is always equal to the derivative we want to price, and then using the Law of One Price to find its current price. We first outline the replicating portfolio as a combination of cash and a certain number of securities. We have $b_t$ units of cash and $n_t$ units of cash. We model them as the following:

• Cash value: $B_t$
  • ! Not ${\mathfrak{B}}_t$. It’s just denoted $B$ because it’s a bond with risk-free
  • $b_t$ units; changes along time. (How? See below)
  • $B_t=B_0{e}^{rt}$ where $r$ is the risk-free rate
    • Thus $d{B}_t=r{B}_{t}dt$
• Stock value: $S_t$
  • $n_t$ units; changes along time. (How? See below)
  • Stochastic process with $d{S}_t=(m-q)S_{t}dt+\sigma S_{t}d{\mathfrak{B}}_t$
    • This is a simple geometric brownian motion process. $m$ is the instantaneous rate of return, and $q$ is the dividend yield
  • Simplify the stock paying the dividend by re-modeling it as $S_t^C=S_t{e}^{qt}$
• Stock value with dividend: $S_t^C$
  • same $n_t$ units
  • Stochastic process with $d{S}_t^C=(m-q)S_{t}dt+\sigma S_{t}d{\mathfrak{B}}_t$ Thus we have the following value of our portfolio $V_t$

$$V_t=b_t{B}_t+n_t{S}_t^C{=}^{\text{must be}}f(S_t,t)$$

Valuing the Self-fiancing Portflio This portfolio must be self-financing; i.e. we must use cash to buy stock, or sell stock to buy cash, in order to exactly track the value of the derivative. This condition translates to:

$$\begin{array}{rlr}\underset{\text{ before dt }}{\underbrace{n_t{S}_{t+dt}^C+b_t{B}_{t+dt}}}& =\underset{\text{ after dt }}{\underbrace{n_{t+dt}{S}_{t+dt}^C+b_{t+dt}{B}_{t+dt}}}& \text{self-financing cond.}\\ \underset{\text{ change in asset value }}{\underbrace{(d{n}_t)S_{t+dt}^C}}& =\underset{\text{ change in cash value }}{\underbrace{-(d{b}_t)B_{t+dt}}}& \text{compact form}\\ d{V}_t& =n_{t}d{S}_t^C+b_{t}d{B}_t& \text{trust the textbook}\end{array}$$

Now, expanding this using the properties of $d{S}_t^C$, $d{B}_t$ , and also that $b_t{=}^{\text{must be}}\frac{f(S_t,t)-n_t{S}_t^C}{B_t}$, we simplify to:

$$d{V}_t=\left(\underset{\text{ A }}{\underbrace{n_t(m-r)S_t^C+rf(S_t,t)}}\right)dt+\underset{\text{ B }}{\underbrace{n_t\sigma S_t^C}}d{\mathfrak{B}}_t$$

Valuing the Security Itself Alternatively, we can use Ito’s lemma to value $f(S_t,t)$ directly. (Recall that we always have $\frac{\partial f}{\partial x}(S_t,t):=\frac{\partial f}{\partial x}(x,t){\mid }_{x=S_t}$

$$df(S_t,t)=\left(\underset{\text{ C }}{\underbrace{\frac{\partial f}{\partial t}+(m-q)S_t\frac{\partial f}{\partial x}(S_t,t)+\frac{1}{2}{\sigma }^2{S}_t^2\frac{{\partial }^{2}f}{\partial x^2}(S_t,t)}}\right)dt+\underset{\text{ D }}{\underbrace{\sigma S_t\frac{\partial f}{\partial x}}}(S_t,t)d{\mathfrak{B}}_t$$

Equating the two Now, due to the fact that an Ito process as a unique representation in the differential form, we know that $A=C$ and $B=D$ in the above formulae. Using the latter:

$$n_t=\frac{S_t}{S_t^C}\underset{:={\Delta }_f}{\underbrace{\frac{\partial f}{\partial x}(S_t,t)}}$$

Using the former:

$$n_t(m-r)S_t^C+rf(S_t,t){=}_{\text{set}}\frac{\partial f}{\partial t}+(m-q)S_t\frac{\partial f}{\partial x}(S_t,t)+\frac{1}{2}{\sigma }^2{S}_t^2\frac{{\partial }^{2}f}{\partial x^2}(S_t,t)$$

And substituting the former $n_t$ into the latter, we have the partial differential equation:

$$\frac{1}{2}{\sigma }^2{S}_t^2\frac{{\partial }^{2}f}{\partial x^2}(S_t,t)+(r-q)S_t\frac{\partial f}{\partial x}(S_t,t)+\frac{\partial f}{\partial t}(S_t,t)-rf(S_t,t)=0$$

Now, this PDE will satisfy any derivative; we can use any boundary conditions on it. But if we solve it for a European Call option, we will get the BSM formula.

• $f(S_t,t):=C(S_t,t)$ starting at $t$ and ending at $T$. Strike price is $K$
• $C(x,T)=\text{max}\{x-K,0\}$
• $C(0,t)=0$ and $C(x,t)\stackrel{x\to \infty }{\to }{e}^{-q(T-t)}$

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