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### Black-Scholes

The Black-Scholes model, often simply Black-Scholes, is a model of the
varying price over time of financial instruments, and in particular stocks.
The Black-Scholes formula is a mathematical formula for the theoretical
value of European put and call stock options that may be derived from the
assumptions of the model. The equation was derived by Fisher Black and Myron
Scholes; the paper that contains the result was published in 1973. They
built on earlier research by Paul Samuelson and Robert Merton. The
fundamental insight of Black and Scholes was that the call option is
implicitly priced if the stock is traded. The use of the Black-Scholes model
and formula is pervasive in financial markets.

The model

The key assumptions of the Black-Scholes model are:

* The price of the underlying instrument is a geometric Brownian motion,
in particular with constant drift and volatility.
* It is possible to short sell the underlying stock
* There are no riskless arbitrage opportunities.
* Trading in the stock is continuous.
* There are no transaction costs.
* All securities are perfect divisible (e.g. it is possible to buy
1/100th of a share).
* The risk free interest rate is constant, and the same for all maturity dates.

The formula

The above 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:

V(S,t) = SN(d1) - Ke - rTN(d2)

where

[d_1=\frac{\log \frac{S}{K}+\left( r+v^{2}/2\right) T}{v\sqrt{T}}]
[d_2=d_1-v\sqrt{T}].

N is the cumulative Normal distribution function.

The price of a put option may be computed from this by put-call parity and
simplifies to:

P(S,t) = Ke - rTN( - d2) - SN( - d1)

The Greeks under the Black-Scholes model are also easy to calculate.

Extensions of the formula

The above option pricing formula is used for pricing European put and call
options on non-dividend paying stocks. The Black-Scholes model may be easily
extended to options on instruments paying dividends. For options on indexes
(such as the FTSE) where each of 100 constituent companies may pay a
dividend twice a year and so there is a payment nearly every business day,
it is reasonable to assume that the dividends are paid continuously. The
dividend payment paid over the time period [t,t + δt] is then modelled as

qStdt

for some constant q. Under this formulation the arbitrage-free price under
the Black-Scholes model can be shown to be

C(S,T) = e - qTS0N(d1) - e - rTKN(d2)

where now

F = e(r - q)TS0

is the modified forward price that occurs in the d. terms.

Exactly the same formula is used to price options on foreign exchange rates,
except now q plays the role of the foreign risk-free interest rate and S is
the spot exchange rate. This is the Garman-Kohlhagen model (1983).

It is also possible to extend the Black-Scholes framework to options on
instruments paying discrete dividends. This is useful when the option is
struck on a single stock. A typical model is to assume that a proportion
δ of the stock price is paid out at pre-determined times T1,T2,....
The price of a stock is then modelled as

[ S_t = S_0(1-\delta)^{n(t)}e^{\sigma W_t + \mu t}]

where n(t) is the number of dividends that have been paid at time t. The
price of a call option on a such stock is again

C(S,T) = FN(d1) - Ke - rTN(d2)

where now

F = S0(1 - δ)n(T)erT

is the forward price for the dividend paying stock.

American options are more difficult to value, and a choice of models is
available (for example Whaley, binomial options model).

Formula derivation

1) The Black-Scholes PDE

In this section we derive the partial differential equation (PDE) at the
heart of the Black-Scholes model via a no-arbitrage or delta-hedging
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,

dSt = μSdt + σSdWt

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

dV = \sigma S \frac{\partial V}{\partial S}dW + ( \mu S \frac{\partial V}{\partial S}+ \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + \frac{\partial V}{\partial

Now consider a portfolio Π consisting of one unit of the option V and
-dV/dS units of the underlying. The composition of this portfolio, called
the delta-hedge portfolio, will vary from time-step to time-step. Now
consider the change in value

[d\Pi = dV - \frac{\partial V}{\partial S} dS]

of the portfolio by subbing in the equation above:

[ d\Pi = ( \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2})dt ]

This equation contains no dW 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 risk-free rate of
return is r we must have over the time period [t,t + δt]

[ r\Pi dt = ( \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2})dt ]

If we now substitute in for Π and divide through by dt we obtain the
Black-Scholes PDE

[ \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - r V = 0]

With the assumptions of the Black-Scholes model, this equation holds
whenever V has two derivatives with respect to S and one with respect to t.

2) From the general Black-Scholes PDE to a specific valuation

We now show how to get from the general Black-Scholes PDE to a specific
valuation for this option. Consider as an example the Black-Scholes 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:

V(0,t) = 0 for all t
[ V(S,t) \rightarrow S ] as [S\rightarrow\infty]
V(S,T) = max(S - K,0)

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

[ x\ s.t.\ S = Ke^{x}]
[ \tau\ s.t. \ t=T-\frac{\tau}{\frac{1}{2}\sigma^2}]
[ v(x,\tau)\ s.t.\ V = K.v(x,\tau)]

Thus our Black-Scholes PDE becomes

[ \frac{\partial v}{\partial \tau}=\frac{\partial^2 v}{\partial x^2} + (c-1)\frac{\partial v}{\partial x} - cv = 0]

where c = 2r / σ2. The terminal condition V(S,T) = max(S - K,0) now
becomes an initial condition v(x,0) = max(ex - 1,0). If we now make a
further transformation such that

[ v(x,\tau)=e^{-\frac{1}{2}(c-1)x -\frac{1}{4}(c+1)^2\tau}u(x,\tau)]

then

[ \frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x^2}]

a standard diffusion equation as desired. Our initial condition has
translated to

[ u(x,0) = max(e^{\frac{1}{2}(c+1)x}-e^{\frac{1}{2}(c-1)x},0)]

Using the standard method for solving a diffusion equation we have

[u(x,\tau) = \frac{1}{2\sqrt{\pi\tau}}\int_{-\infty}^{\infty} u_0(y) e^{-\frac{(x-y)^2}{4\tau}}dy]

where u0 is the initial condition defined in the line above. This integral
may be further transformed until we obtain

u(x,τ) = I1 - I2

where

[ I_1 = e^{\frac{1}{2}(c+1)x+\frac{1}{4}(c+1)^2\tau}N(d_1)]
[ d_1 = \frac{x}{\sqrt{2\tau}}+\frac{1}{2}(c+1)\sqrt{2\tau}]

and I2 is identical to I1 except that (c+1) is replaced by (c-1) everywhere.

Substituting v for u and the V for v, we finally obtain the the value of a
call option in terms of the Black-Scholes parameters:

V(S,t) = SN(d1) - Ke - rTN(d2)

where

[d_1=\frac{\log \frac{S}{K}+\left( r+v^{2}/2\right) T}{v\sqrt{T}}]
[d_2=d_1-v\sqrt{T}].

N is the cumulative Normal distribution function.

The formula for the price of a put option, follows from this via put-call
parity.

3) Other derivations

Above we used the method of arbitrage-free pricing ("delta-hedging") to
derive a PDE governing option prices given the Black-Scholes 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 risk-neutral measure, which differs from the
real world measure.

* The risk neutrality argument:
o Underlying logic
o Pricing derivation
* Arbitrage-free pricing:
o Underlying logic
o Pricing derivation or an alternative treatment
* Detailed solutions of the Black-Scholes equation or a further
treatment.

Black-Scholes in practice

The use of the Black-Scholes 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 risk-free rate and
current underlying price - are unequivocably observable. This means there is
one-to-one relationship between the option price and the volatility.)
Traders prefer to think in terms of volatility.

However, the Black-Scholes model can not be modelling the real world
completely accurately. If the Black-Scholes 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
two-dimensional 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 smile-shaped (see volatility
smile). That is, at-the-money (the option for which the underlying price and
strike co-incide) the implied volatility is lowest; out-of-the-money or
in-the-money 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 at-the-money: 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
Black-Scholes was more widely trusted.

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