Chapter 8 Option pricing
Introduction
Seminal articles: Scholes and Black (1973) and Merton (1973).
First, Black and Scholes put forward assumptions related to so-called perfect markets, i.e. markets in which participants have unrestricted access to credit and where securities are perfectly divisible and can be traded without transaction costs. For simplicity’s sake, interest rates are assumed to be constant. To these assumptions, which are extremely commonplace in economics, the two authors added a truly innovative dimension: A) continuous time and B) brownian motion.
While securities trading is sometimes referred to as”continuous quotation”, observations of prices and trading opportunities are basically discrete. One of Black and Scholes’ main ideas was to work within a continuous timeframe. As a result, they obtained differential equations, which are easier to work with than discrete equations.
The second key assumption was that relative price movements should be represented by Brownian motion. In fact, this was a default option because Brownian motion is the most “natural” process within a continuous time framework. In particular, it allows us to give meaning to the infinitesimal variations of a function that depends on this process. An option price is one example of such a function. Black and Scholes’ choice of Brownian motion is one way of encapsulating an efficient market guided by a random walk.
Outline of the section…
8.1 Model and assumptions
We assume an asset which is not paying any dividend.
- \(S\) be the price process with mean \(\mu\) and volatility \(\sigma\)
- \(K\) be the strike price of the option
- \(r\) the constant positive interest rate
- \(T\) the time to maturity
- \(W\) a brownian motion
\(C\) or \(C(t,S_t)\) will be used for the Call price while \(P\) or \(C(t,S_t)\) for the Put price. When necessary we might use \(C^a\) or\(C^e\) to differentiate the American or European type of exercise.
One has: \(C(T,x)=(x-K)^+\) and \(P(T,x)=(K-x)^+\)
The spot is solution of the following Stochastic Differential Equation: \[dSt=S_t(\mu dt+\sigma dW_t) \] with initial value \(S_0\).
At date \(T\),
Some words on the model. Solution of the SDE.
8.2 Arbitrage bounds and payoff generation
Beware of negative interest rates, they change the below bounds.
Note that \(K+(x-K)^+=x-(K-x)^+\) In plain English: Zero-Coupon + call = underlying + put. By arbitrage: \[B_T+C=P+S_0\]
- \((S_t-K)^+\leq (S_t-Ke^{-rT})^+\leq C^e_t=C^a_t\leq S_t\) (call/Put parity)
- \((Ke^{-rT}-S_t)^+\leq P^e_t\)
- \((K-S_t)^+\leq P^a_t\)
Early exercise of the american put. Various kind of payoffs: straddle, call-spread. More generally, any payoff can be generated by a portfolio of bond, stock, puts and calls (see P. Carr and Madan (2001)):
\[ \begin{align}f(S) & =f(S_{0})-f'(S_{0})S_{0}\\ & +f'(S_{0})\times S\\ & +\int_{0}^{S_{0}}f''\left(K\right)\left(K-S\right)^{+}dK\\ & +\int_{S_{0}}^{+\infty}f''\left(K\right)\left(S-K\right)^{+}dK \end{align} \]
8.3 The Black-Scholes-Merton analysis
8.3.1 The Black-Scholes formula
Locally, option price variations are described by the Ito formula: \[ dC=\left[\frac{\partial C}{\partial t}+\mu S\frac{\partial C}{\partial S}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}C}{\partial S^{2}}\right]dt+\frac{\partial C}{\partial S}\sigma SdW_{t} \]
Now consider a portfolio comprising one option and \(\delta\) stocks. Its value \(V\) varies as follows:
\[ dV=\left[\frac{\partial C}{\partial t}+\mu S\frac{\partial C}{\partial S}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}C}{\partial S^{2}}+\delta \mu S \right]dt+\sigma S(\frac{\partial C}{\partial S}+\delta)dW_{t} \]
At each moment \(t\), if \(\delta = - \frac{\partial C}{\partial S}\) then brownian terms disappear and the portfolio bears no risk between \(t\) and \(t+dt\). Accordingly, its instantaneous return must be equal to \(rdt\) because any other value would imply an arbitrage opportunity. So the price of the option satisfies a partial differential equation, which no longer depends on \(\mu\):
\[\begin{align} \frac{\partial C}{\partial t}+rS\frac{\partial C}{\partial S}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}C}{\partial S^{2}} & =rdt\\ C(T,x) & =(x-K)^{+} \tag{8.1} \end{align}\]
Equation (8.2) is known as the Black and Scholes equation. How can this equation be solved ?
Two possibilities:
- Feynman-Kac / link between PDE and expectation
- Use fundamental theorem of asset pricing
In both case, one has to compute \(B(0,T)\mathbb E((S_T-K)^)\) which yields
\[\begin{align} C\left(T,S_{0}\right) & =e^{-rT}\left(\mathbb{E}\left(S_{T}\mathbb{I}_{S_{T}>K}\right)-KPr\left(S_{T}>K\right)\right)\\ & =e^{-rT}\left(FN(d_{1})-KN(d_{2})\right) \tag{8.2} \end{align}\]
where
\[\begin{align} F & =S_{0}e^{rT}\\ d_{1} & =\frac{ln\left(\frac{F}{K}\right)+\frac{\sigma^{2}T}{2}}{\sigma\sqrt{T}}\\ d_{2} & =\frac{ln\left(\frac{F}{K}\right)-\frac{\sigma^{2}T}{2}}{\sigma\sqrt{T}} \end{align}\]
Comments
8.3.2 The greeks
Formula: See: Wikipedia B&S
Graphical analysis:
rate sensitivity: lending/borrowing Delta: graphical representation vega
8.4 Time value and local time: the difference between a stop-loss and a put strategy
This section borrows from P. P. Carr and Jarrow (1990).
8.4.1 The paradox
What is the difference between a put option and a stop-loss strategy ? It seems that the downside risk of an equity position might be offset by a stop-order almost as efficiently as with a put option. This would be puzzling since the put option has a cost whereas the stop-order can be placed for free.
A natural reaction would be to think that “jumps” might happen which make a difference between the “best effort” stop order and the contractual put payoff. But, where does this difference come from in a framework where trajectories are continuous and trading can be done continuously ?
In the sequel, we will imagine a simplified “local” Gaussian framework. For short time horizons, there are hardly any differences with the Black-Scholes framework: \(S_t=S_0+\sigma W_t\) where \(\sigma\) in this framework is homogeneous to \(\sigma S\) in a BS framework.
8.4.2 Resolution
The delta of the stop-loss strategy of strike \(K\) is \(-1_{S_t<K}\). Hence the dynamic hedging strategy value, if it exists, should be: \[V_t=\int_0^t-1_{S_t<K}dS_t\] This quantity might be approached by applying Ito formally to \((K-S_t)^+\): \[(K-S_t)^+-(K-S_0)^+=V_t-\frac{1}{2}\int_0^t\delta_{S_t=K}d<S>_t\]
The last term exists and is called the **local time* of S spent in K. It is absolutely continuous w.r.t. to dt. The above equation provides an other interpretation of the time value of an option
8.5 Some implications in Corporate Finance
Merton (1974)
In the graphical representation of the fundamental accounting equation, the limited liability aspect of the equity is not taken into account. If \(L\) is the liability level the liability are equal to \(L+(A-L)^+\). Who is paying the put ? This has to be the lenders and the risky debt has the following payoff w.r.t. the assets of the firm: \(L-(L-A)^+\)
It is clear that:
- shareholders have incentive to increase risk (long call)
- lenders have incentive to lower risk (short put)
Same for fund managers performance fees