
The key assumptions of the BlackScholes model are:
These lead to the following formula for the price of a call on a stock currently trading at price S, where the option has an exercise price of K, i.e. the right to buy a share at price K, at T years in the future. The constant interest rate is r and the constant stock volatility is v:
where
The price of a put option may be computed from this by putcall parity and simplifies to:
The Greeks under the BlackScholes model are also easy to calculate.
The above option pricing formula is used for pricing European put and call options on nondividend paying stocks.
American options are more difficult to value, and a choice of models is available (for example Whaley, binomial options model).
1) The BlackScholes PDE
In this section we derive the partial differential equation (PDE) at the heart of the BlackScholes model via a noarbitrage or deltahedging argument. The presentation given here is informal and we do not worry about the validity of moving between dt meaning an small increment in time and dt as a derivative.
As in the model assumptions above we assume that the underlying (typically the stock) follows a geometric Brownian motion. That is,
where W Brownian. Now let V be some sort of option on S  mathematically V is a function of S and t. By Ito's Lemma for two variables we have
Now consider a portfolio <math>\Pi</math> consisting of one unit of the option V and dV/dS units of the underlying. The composition of this portfolio, called the deltahedge portfolio, will vary from timestep to timestep. Now consider the change in value
of the portfolio by subbing in the equation above:
This equation contains no <math>dW</math> term. That is, it is entirely riskless. Thus, assuming no arbitrage (and also no transaction costs and infinite supply and demand) the rate of return on this portfolio must be equal to the rate of return on any other riskless instrument. Now assuming the riskfree rate of return is <math>r</math> we must have over the time period <math>[t,t+\delta t]</math>
If we now substitute in for <math>\Pi</math> and divide through by <math>dt</math> we obtain the BlackScholes PDE
With the assumptions of the BlackScholes model, this equation holds whenever V has two derivatives with respect to S and one with respect to t.
2) From the general BlackScholes PDE to a specific valuation
We now show how to get from the general BlackScholes PDE to a specific valuation for this option. Consider as an example the BlackScholes price of a call on a stock currently trading at price S. The option has an exercise price of K, i.e. the right to buy a share at price K, at T years in the future. The constant interest rate is r and the constant stock volatility is v(all as at top). Now, for a call option the PDE above has boundary conditions[?]:
In order to solve the PDE we transform thee equation into a standard diffusion equation which may be solved using standard methods. To this end set
Thus our BlackScholes PDE becomes
where <math>c=2r/\sigma^2</math>. The terminal condition <math>V(S,T)=max(SK,0)</math> now becomes an initial condition <math>v(x,0) = max(e^x1,0)</math>. If we now make a further transformation such that
Using the standard method for solving a diffusion equation we have
where u_{0} is the initial condition defined in the line above. This integral may be further transformed until we obtain
Substituting v for u and the V for v, we finally obtain the the value of a call option in terms of the BlackScholes parameters:
where
The formula for the price of a put option, follows from this via putcall parity.
3) Other derivations
Above we used the method of arbitragefree pricing ("deltahedging") to derive a PDE governing option prices given the BlackScholes model. It is also possible to use a risk neutrality argument. This latter method gives the price as the expectation of the option payoff under a particular probability measure, called the riskneutral measure[?], which differs from the real world measure.
The use of the BlackScholes formula is pervasive in the markets. In fact the model has become such an integral part of market conventions that it is common practice for the implied volatility rather than the price of an instrument to be quoted. (All the parameters in the model other than the volatility  that is the time to expiry, the strike, the riskfree rate and current underlying price  are unequivocably observable. This means there is onetoone relationship between the option price and the volatility.) Traders prefer to think in terms of volatility.
However, the BlackScholes model can not be modelling the real world completely accurately. If the BlackScholes model held, then the implied volatility of an option on a particular stock would be constant, even as the strike and maturity varied. In practice, the volatility surface[?] (the twodimensional graph of implied volatility against strike and maturity ) is not flat. In fact, in a typical market, the graph of strike against implied volatility for a fixed maturity is typically smileshaped (see volatility smile[?]). That is, atthemoney (the option for which the underlying price and strike coincide) the implied volatility is lowest; outofthemoney or inthemoney the implied volatility tends to be higher. The reason for this smile is still the subject of much speculation and research. A prominent proposed explanation is that the market in options away from the money is less liquid than atthemoney: traders demand a premium for these options because they know it may be more difficult to reverse an option position in illiquid markets. This view is consistent with the fact the smile was first observed shortly after the stock market crash of 1987[?]. Before this crash, the first and most severe since the introduction of options, the BlackScholes was more widely trusted.
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