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Black-Scholes Option Pricing

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Pricing and Hedging in Complete Markets

Options and their fair values

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

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 $.

As a result, options are characterized by contingent cash flows to the option holder. In this section, we introduce a couple of basic instruments upon which we will add others in the subsequent sections if desired.

A European call (put) option is the right but not the obligation to receive $ max(0, K-S_{T}) $) at maturity time $ T $. The variable $ K $ is called the strike price. European call (put) options are vanilla options. (See European Option for details.)

There are many other ways to design option contracts. One of the most common option features is flexible exercise opportunity to the option's holder. The exercise dates can be a certain discrete dates or any time during option lifetime. We can also make the payoff dependent on the asset history to get a path dependent option.

An American call (put) option is the right to receive $ max(0, K-S_{t}) $ at any time $ t $ from option initiation $ t=0 $ until maturity time $ t=T $. The variable $ K $ is called the strike price. The early exercisable feature enable the option holder to receive the exercise value $ S_{t}-K $ ($ K-S_t $) prior to the option's maturity date $ 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.

General hedge pricing 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(2000)[1] or Hull(2006)[2].

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.

Other more conceptual assumption is that the price of the underlying $ S $ follows a Geometric Brownian Motion process. The stochastic differential equation for the underlying is

$ d S_t = \mu S_t d t + \sigma S_t d W_t, \qquad (BS.1) $

where $ \mu $ is the drift rate, $ \sigma $ the volatility of $ S $ and $ d W_t $ the increment of a standard Wiener process under the physical measure $ P $.

We can now establish a portfolio $ \Pi $ consisting of a bank account and a short position of $ \phi $ 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.,

$ \Pi_t = V(S_t,t)-\phi_t S_t. \qquad (BS.2) $

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.\qquad (BS.3) $

If we chose

$ \phi_t=\frac{\partial V_t}{\partial S_t}. \qquad (BS.4) $

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 exhibits no risk. Using the no-arbitrage principle, this risk-free portfolio earns the risk-free rate $ r $. Consequently, the changes in $ \Pi_t $ are

$ d\Pi_t = r \Pi_t d t.\qquad (BS.5) $

Using equations (BS.3), (BS.4) and (BS.5), 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.\qquad (BS.6) $

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

$ V(S_{T},T) = P(S_{T},T),\qquad (BS.7) $

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

The case of exercisable options

We now look at an early exercisable option. By the no-arbitrage assumption, upon exercise the holder obtains the payoff $ P(S_t,t) $ leads to

$ V(S_t,t)\geq P(S_t,t),\qquad (BS.8) $

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 for the valuation of an early exercisable option similar to Equation (BS.3), a little thought leads to the observation that the portfolio $ \Pi $ can at most earn the risk-free rate under the no-arbitrage assumption. Consequently Equation (BS.8) 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. \qquad (BS.9) $

In the case of an American option, the value $ V $ is given by the solution to Equations (BS.9) where at least one of the inequalities holds with equality for the complete solution. This is an initial value problem or Cauchy problem in backwards time $ \tau = T\,-\,t $ with a free boundary. The time $ T $ boundary condition, i.e. for an American put option with strike $ K $ is

$ V(S_T,T) = \max(K-S_{T}, 0). \qquad (BS.10) $

Numerical schemes for solving this kind of PDE problem have been proposed by several authors, but an efficient one was presented by Forsyth and Vetzal which is also the reference for the PDE approach used for solving equation (BS.10).

Pricing options using probabilistic expectations

We now price options using an alternative approach, i.e. as an expectation of discounted future payoffs. Specifically, in the Black-Scholes framework, with all the assumptions above remain valid, the price of a European call under the risk-neutral measure $ Q $ is

$ V(t) = \mathrm{E}^Q \left [ e^{-r\,(T\,-\,t)}V(T)\mid \mathcal{F}_t \right ] $
$ = \mathrm{E}^Q \left [ e^{-r\,(T\,-\,t)}max(S_T\,-\,K,\,0) \mid \mathcal{F}_t \right ] $
$ = e^{-r\,(T\,-\,t)} \mathrm{E}^Q \left [ max(S_T)\,-\,K,\,0) \mid \mathcal{F}_t \right ]. \qquad (BS.11) $

To evaluate the expectation term, we need to know the distribution of $ S(T) $ under the risk-neutral measure $ Q $. The stock price process in equation (BS.1) under the risk-neutral measure is

$ d S_t = r\, S_t d t\, +\, \sigma \, S_t d {W_t}^Q, \qquad (BS.12) $

where $ r $ is the risk-free interest rate, and $ {W_t}^Q $ is a standard wiener process under the risk-neutral measure $ Q $. Equation (BS.12) is derived as in Geometric Brownian Motion.

The stock price at time $ T $ is the Ito integral of equation (BS.12) from time $ t $ to time $ T $:

$ S_T = S_t e^{(r\,-\, \frac{1}{2} \sigma^2)(T\,-\,t) \, +\, \sigma \, (W_T \,-\,W_t)}, \qquad (BS.13) $

where $ S_t $ is the current stock price.

Substituting equation (BS.13) into equation (BS.11), we have

$ V(t) = e^{-r\,(T\,-\,t)} \mathrm{E}^Q \left [ max(S_t e^{(r\,-\, \frac{1}{2} \sigma^2)(T\,-\,t) \, +\, \sigma \, (W_T \,-\,W_t)}\,-\,K,\,0) \right ] $
$ = e^{-r\,(T\,-\,t)} \int_{-\infty}^{+\infty}\frac {1}{\sqrt {2 \pi }}e^{-\frac {y^2}{2}} max(S_t e^{(r\,-\, \frac{1}{2} \sigma^2)(T\,-\,t) \, +\, \sigma \, \sqrt{T\,-\,t} \, y} \,-\,K,\,0)\,dy. \qquad (BS.14) $

To eliminate the $ max(\, , 0) $ operator in equation (BS.14), we transform the integration limit as follows. First, we find the value of $ d^* $ that makes the term $ S_t e^{(r\,-\, \frac{1}{2} \sigma^2)(T\,-\,t) \, +\, \sigma \, \sqrt{T\,-\,t}\, d^*}\,-\,K = 0 $, i.e.

$ S_t e^{(r\,-\, \frac{1}{2} \sigma^2)(T\,-\,t) \, +\, \sigma \, \sqrt{T\,-\,t}\, d^*} = K $
$ \Rightarrow \ln S_t \,+\,(r\,-\, \frac{1}{2} \sigma^2)(T\,-\,t) \, +\, \sigma \, \sqrt{T\,-\,t} \,d^* = \ln K $
$ d^* = \frac {- \ln \frac{S_t}{K} \,-\,(r\,-\, \frac{1}{2} \sigma^2)(T\,-\,t) }{\sigma \, \sqrt{T\,-\,t}} $

Define

$ d2 = -d^* = \frac { \ln \frac{S_t}{K} \,+\,(r\,-\, \frac{1}{2} \sigma^2)(T\,-\,t) }{\sigma \, \sqrt{T\,-\,t}}, $

equation (BS.14) then becomes

$ V(t) = e^{-r\,(T\,-\,t)} \int_{d^*}^{+\infty} \frac {1}{\sqrt {2 \pi }}e^{-\frac {y^2}{2}} (S_t e^{(r\,-\, \frac{1}{2} \sigma^2)(T\,-\,t) \, +\, \sigma \, \sqrt{T\,-\,t} \, y}\,-\,K)\,dy. \qquad (BS.15) $

Next we evaluate the integral in equation (BS.15)

$ V(t) = e^{-r\,(T\,-\,t)} \int_{-\infty}^{d2} \frac {1}{\sqrt {2 \pi }}e^{-\frac {y^2}{2}} S_t e^{(r\,-\, \frac{1}{2} \sigma^2)(T\,-\,t) \, +\, \sigma \, \sqrt{T\,-\,t} \, y} dy \,-\,e^{-r\,(T\,-\,t)}\int_{-\infty}^{d2} K \frac {1}{\sqrt {2 \pi }}e^{-\frac {y^2}{2}} \,dy $
$ = e^{-r\,(T\,-\,t)} S_t e^{(r\,-\, \frac{1}{2} \sigma^2)(T\,-\,t) }\int_{-\infty}^{d2} \frac {1}{\sqrt {2 \pi }}e^{-\frac { (y\,-\,\sigma \sqrt{T\,-\,t})^2}{2}} e^{ \frac {\sigma^2 \,(T\,-\,t)}{2} } dy \,-\, e^{-r\,(T\,-\,t)} K \, \Phi(d2) $
$ = S_t \int_{-\infty}^{d2} \frac {1}{\sqrt {2 \pi }}e^{-\frac { (y\,-\,\sigma \sqrt{T\,-\,t})^2}{2}} dy \,-\, e^{-r\,(T\,-\,t)} K \, \Phi(d2) $
$ = S_t \int_{-\infty}^{d2\,+\, \sigma \sqrt{T\,-\,t} } \frac {1}{\sqrt {2 \pi }}e^{-\frac { u^2}{2}} du \,-\, e^{-r\,(T\,-\,t)} K \, \Phi(d2) $
$ = S_t \Phi(d2\,+\, \sigma \sqrt{T\,-\,t} ) \,-\, e^{-r\,(T\,-\,t)} K \, \Phi(d2) $
$ = S_t \Phi(d1) \,-\, e^{-r\,(T\,-\,t)} K \, \Phi(d2), \qquad (BS.16) $

where $ d1 = d2\,+\, \sigma \sqrt{T\,-\,t} = \frac { \ln \frac{S_t}{K} \,+\,(r\,+\, \frac{1}{2} \sigma^2)(T\,-\,t) }{\sigma \, \sqrt{T\,-\,t}} $ in the last equality.

Having derived the classical Black-Scholes equation for European call option, we now proceed to implement the above closed-form solution in Matlab.

Matlab implementation for the closed-form solution 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)
% This function computes the price of a put (call) with
% S : underlying asset price
% K : option strike price
% sigma : volatility of stock price
% r : risk-free rate
% T : option time-to-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
  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

References

  1. Wilmott, Paul, 2000, Paul Wilmott On Quantitative Finance. John Wiley & Sons, 2000.
  2. Hull, John C, 2006, Options, Futures, and Other Derivatives (6th ed. ed.). Upper Saddle River, N.J: Prentice Hall. p657–658. ISBN 0131499084.