Black-Scholes Option Pricing

From ThetaWiki

1 Pricing and Hedging in Complete Markets

[edit] Pricing and Hedging in Complete Markets

[edit] Options and their fair values

Before we present a general framework for option valuation, we focus on the instruments we seek to valuate. All financial options presented here depend on a single underlying. A simple underlying is also called asset, stock, spot price or equity and is denoted as $S$ with specific values at times $t i$ . We denote the value of the underlying at time $t i$ as $S_{t_{i}}$

Now a financial option is a contract between an option issuer and the option holder. An option gives the holder the right, but not the obligation, to receive future cash flows dependent on the value of the underlying $S$ . A financial option is also called a derivative, a security or simply an option.

As a result, options are characterized by cash flows to the option holder. In this section, we introduce a few essential instruments to which we will add others in the subsequent sections if required.

A European call (put) option is the right but not the obligation to receive $S_{t_T}-K~~$ ( $K-S_{t_T}$ ) at maturity time $t T$ . The variable $K$ is called the strike price. European call (put) options are also called vanilla options. (See European Option for details.)

There are many other ways to formulate option contracts. One of the most common option features is an exercise opportunity by the option's holder. This can be specified for a certain date (discrete) or a specific time interval (continuous). We can also make the payoff dependent on the asset history to get a so-called path dependent option.

An American call (put) option is the right to receive $S_{t_i}-K~~$ ( $K-S_{t_i}$ ) at any time $t i$ from initial time $t 0$ until maturity time $t T$ . The variable $K$ is called the strike price. This option is said to be early exercisable because the option holder can receive the exercise value $S_{t_i}-K~~$ ( $K-S_{t_i}$ ) prior to the option's maturity date $t T$ . (See American Option for details.)

In order to determine the so-called fair values for these and other options, we proceed with a general pricing framework.

[edit] General Framework

The valuation of a derivative security is a common task in mathematical finance. Here we provide a brief summary of the model derivation which can also be found in the literature, e.g., Wilmott or Hull.

For the Black-Scholes model, we need to make some assumptions and simplifications. The basic assumptions are

A frictionless market
No transaction costs
Risk-less assets earn the risk-free rate $r$ and
Short selling is allowed without restrictions.

nother more conceptual assumption is that the price of the underlying $S$ behaves with geometric Brownian motion. This means that future stock prices cannot be predicted and that their assets evolves as

d S t = μ S t d t + σ S t d W t

where $μ$ is the drift rate, $σ$ the volatility of $S$ and $d W t$ the increment of a Wiener process. This is the equation of Geometric Brownian Motion

We can now establish a portfolio $Π$ consisting of the security of interest and a short position of $φ$ shares. The price of the security clearly depends on the underlying stock price $S t$ and time $t$ and will be denoted as $V (S t, t)$ or just $V t$ , i.e.,

Π t = V (S t, t) - φ t S t .

Using Itô's Lemma, the portfolio changes can be described by

$d \Pi_t=d V_t-\phi_t d S_t =\frac{\partial V_t}{\partial t}d t + \frac{\partial V_t}{\partial S_t}d S_t + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V_t}{\partial S_t^2} d t -\phi_t d S_t.$

If we chose

$\phi_t=\frac{\partial V_t}{\partial S_t}$

we can make the portfolio independent of the movements of the stock price $d S t$ . This means that the portfolio is completely deterministic and inhabits no risk. Using the no-arbitrage principle, this risk-free portfolio earns the risk-free rate $r$ since no-arbitrage means, in this case, that risk-less investments earn the risk-free rate and investments that earn more than the risk-free rate must by risky. Consequently, the changes in $Π t$ are

d Π t = r Π t d t .

Using \eqref{eqn:dPi1_basic}, \eqref{eqn:Delta_basic} and \eqref{eqn:dPi2_basic}, we obtain the relationship

$\frac{\partial V_t}{\partial t} + \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 V_t}{\partial S_t^2}+rS_t\frac{\partial V_t}{\partial S_t} -rV_t=0.$

Knowing that at maturity date $t T$ the option value equals the payoff $P (S T, t T)$ ,

$V(S_{t_T},t_T) = P(S_{t_T},t_T),$

the partial differential equation can be solved by numerical methods as an initial value problem in backwards time. This general framework was first published in 1973 by Fischer Black and Myron Scholes.

[edit] Exercisable Options

We now look at a possible early exercise by the holder of the security. By the no-arbitrage assumption, an exercise where the holder obtains the payoff $P (S t, t)$ leads to

$V(S_t,t)\geq P(S_t,t),$

i.e., the early exercisable option will always have a value greater or equal to the immediate exercise value. Were this not true, an investor would buy the option, exercise it immediately and make a risk-less profit.

In order to find a representation of an exercisable option value similar to Equation ?, a little thought leads to the observation that the portfolio $Π$ can at most earn the risk-free rate under the no-arbitrage assumption. Consequently Equation ? is restated as

$d\Pi_t \leq r \Pi_t d t \Leftrightarrow \frac{\partial V_t}{\partial t} + \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 V_t}{\partial S_t^2}+rS_t\frac{\partial V_t}{\partial S_t} -rV_t \leq 0.$

In the case of an American option, the value $V$ is given by the solution to Equations ? and ? where at least one of the inequalities holds with equality on the complete solution. This is an initial value problem or Cauchy problem in backwards time $τ = t T - t$ with a free boundary and starting with a boundary condition, i.e. for an American put option with strike $K$ we have that

$V(S_{t_T},t_T) = \max(K-S_{t_T}, 0) ~.$

Numerical schemes for this kind of PDE solution have been presented by several authors, but an efficient one was presented by Forsyth and Vetzal which also forms the foundation for the PDE reference methods used in this thesis. The next section proceeds with an overview of the numerical methods currently used in the field of derivatives pricing.

[edit] Example of a European Option

The price of a European put (or call) option is given by

function [V] = myBlsprice(S, K, sigma, r, T, call)
%function [V] = myBlsprice(S, K, sigma, r, T, call)
% Computes the price of a put (call) with 
% S : asset price
% K : strike
% sigma : volatility
% r : risk-free rate
% T : maturity
% call: true or "1" for call, otherwise put
 
if T<=0
  if call
      V = max(S-K,0);
  else
      V = max(K-S,0);
  end
  return
end
 
  d1 = ( log(S./K) + (r + (sigma.^2)/2)*T ) ./ (sigma * sqrt(T));
  d2 = d1 - sigma*sqrt(T);
 
if call
  V = S.*normcdf(d1,0,1) - K .* exp(-r.*T).*normcdf(d2);
else
  V = -S.*normcdf(-d1,0,1) + K .* exp(-r.*T).*normcdf(-d2);
end

Retrieved from "http://wiki.thetaris.com/index.php/Black-Scholes_Option_Pricing"

Black-Scholes Option Pricing

From ThetaWiki

Contents

[edit] Pricing and Hedging in Complete Markets

[edit] Options and their fair values

[edit] General Framework

[edit] Exercisable Options

[edit] Example of a European Option

Views

Personal tools

Navigation

Search

Toolbox