Derivatives Pricing I: Pricing under the Black-Scholes model

In this new article series QuantStart returns to the discussion of pricing derivative securities, a topic which was covered a few years ago on the site through an introduction to stochastic calculus.

Imanol Pérez, a PhD researcher in Mathematics at Oxford University, and an expert guest contributor to QuantStart will mathematically describe the Black-Scholes model for options pricing in this article and then subsequently outline its limitations in future posts.

Derivatives are financial instruments whose price depends on the performance of some underlying asset or assets. The world of financial derivatives is very complex, and derivatives can be very different from each other. In this article we will focus on a particular type of derivatives, options, and we will show how stochastic analysis can be used to find the fair price, a notion that will be made clear later, of a class of options.

The class of options we will be interested in are European options. These options give the owner the right, but not the obligation, to make a predetermined transaction at an agreed maturity time at a specific price. One can distinguish two types of European options:

  • European call options with underlying asset $S$, maturity time $T$ and exercise price $K$ give the contract owner the right (but not the obligation) to buy one share of $S$ at the price $K$ at time $T$.
  • Similarly, European put options give the contract owner the right (but not the obligation) to sell one share of $S$ at the price $K$ at time $T$.

Naturally, if the price of the asset at time $T$ is greater than the exercise price, that is,

\begin{equation} S_T>K, \end{equation}

then it is in the interest of the owner of a European call option to buy the asset at the price $K$. If the price at time $T$ is lower than the exercise price, buying the asset at the price $K$ is against the interest of the option owner, since the asset can be bought at a cheaper price directly in the market. Therefore, the option owner will earn $\Phi_c(S_T)$, with $\Phi_c(x):=\max(x-K, 0)$. Similarly, one can check that the owner of a European put option will receive $\Phi_p(S_T)$, with $\Phi_p(x):=\max(K-x, 0)$, from the option contract.

After defining these contracts, one would like to answer the following question: If $\Pi_t$ represents the price of a European option (either a put or call option), how can one find a fair value for $\Pi_t$?

To answer the question, we will first introduce our market, which will consist of three assets: a risk free asset $B$ (which can be interpreted to be the money in the bank), a stock $S$, and the European option $\Pi$ whose underlying asset is $S$. Moreover, we will assume that the price of $B_t$ and $S_t$ follow the dynamics

$$\begin{cases} dB_t=rB_tdt\\ dS_t=\alpha S_tdt+\sigma S_tdW_t \end{cases}$$

where $r$ is the short rate of interest, $\alpha$ is the local mean rate of return of the stock, $\sigma$ is its volatility and $W$ is a Brownian motion. We will assume that $r$ is constant and $\alpha$ and $\sigma$ are functions of $t$ and $S_t$. As we see, we will assume that $S$ follows a geometric Brownian motion, which was mentioned in this article.

An investor can hold a portfolio $h_t=(h_t^1, h_t^2, h_t^3)$. The value $h_t^1$ will correspond to the money invested in the risk free asset $B$, $h_t^2$ the money invested in the stock $S$ and $h_t^3$ the money invested in the option contract. A negative value indicates short selling. We will also denote by $V^h_t$ the wealth of the portfolio $h$ at time $t$.

In order to give a fair price to $\Pi_t$, we need to define what we mean by fair. We would like to price $\Pi$ in such a way that our market is arbitrage free, in the sense that there exists no self-financed portfolio $h$ (see [1] for the definition) such that

\begin{align*} &V^h(0)=0,\\ &\mathbb{P}(V^h(T)\geq 0)=1,\\ &\mathbb{P}(V^h(T)>0)>0. \end{align*}

The conditions above avoid the possibility of being able to make a positive amount of money, without investing any money, and without taking any risk.

It turns out that this condition is enough to be able to price our option $\Pi$:

Theorem 0.1 (Black–Scholes Equation). Assume that our market is arbitrage free. Then, if we assume that the price of the option $\Pi$ is given by $\Pi_t=F(t, S_t)$ for some smooth function $F$, then $F(t,s)$ is the solution of the following PDE:

$$\begin{cases} \partial_t F+rs\partial_s F+\dfrac{1}{2}s^2\sigma^2 \partial_s^2 F-rF=0 \\ F(T, s)=\Phi(s) \end{cases}\quad \mbox{in }[0,T]\times [0,\infty)$$

where $\Phi$ can be either $\Phi_c$ or $\Phi_p$, although more general contingent claims $\Phi$ can be used as well.

Limitations of the Black–Scholes model

The Black–Scholes model is a simple model that can be very useful, but it has many limitations and it should therefore be treated with care. In subsequent articles we shall see some of the limitations of the model, and how one could solve them in order to obtain a model that adjusts better to reality.

References

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