The Black–Scholes–Merton Model
Suppose you're looking to value an option contract over a five-year period. For the binomial model, that's a lot of nodes to get back from the expiration to the initiation period. But it's much easier for the BSM model since it can break down that five-year period into a continuous time period.
But there's a catch: it takes many, many calculations. What do you think it uses to perform them?
Exactly!
The BSM model uses any of these tools to calculate the value of the option. Recall that the BSM model assumes a continuously compounded rate, denoted by _r_, with volatility, denoted by σ, also expressed in annualized-percentage terms.
For a non-dividend-paying stock, the BSM model for a call is
$$\displaystyle c=SN(d_1) - e^{-rT}XN(d_2)$$.
And a put is
$$\displaystyle p= e^{-rT}XN(-d_2) - SN(-d_1)$$,
where
$$\displaystyle d_1 = \frac{ln(\frac{S}{X}) + (r + \frac{\sigma ^2}{2})T}{\sigma \sqrt{T}}$$
$$\displaystyle d_2 = d_1 - \sigma \sqrt{T}$$.
_N_(_x_) denotes the standard normal cumulative distribution function.
Upon first glance, it looks like quite a lot. But after breaking down the equations, it's not so bad.
First, note that _N_(_x_) is typically estimated by a computer or spreadsheet, or approximated by a lookup table. The first term is the spot price of the underlying times a cumulative distribution function. And _N_(_d_2) is also a measure of the probability that a call expires in the money. For a put, it's
$$\displaystyle 1 - N(d_2) ~\mbox{or}\ N(-d_2)$$.
And _e_-_rT_ is simply the present value term.
Putting it together, the equations are simply the present value of the expected future cash flows based on a risk-neutral probability.
Or, breaking down the equations another way, the BSM model can be viewed as having two components. The first is clearly a stock, given that it's the underlying. The second would need to capture the continuously compounded interest rate to represent the present value. What might that component be?
That's not it.
A corporate bond couldn't capture the continuous compounding.
Not quite.
The fixed-rate swap can be broken down into other instruments, so that's not it.
Bingo, that's it!
The BSM model can be viewed as a portfolio of dynamically managed zero-coupon bonds and a stock. For the call option BSM model, it's the stock component,
$$\displaystyle SN(d_1)$$,
less the bond component,
$$\displaystyle e^{-rT}XN(d_2)$$.
For the put, it's the bond component,
$$\displaystyle e^{-rT}XN(-d_2)$$,
less the stock component,
$$\displaystyle SN(-d_1)$$.
The goal in breaking down the BSM model into a stock and zero-coupon bonds is to replicate the option payoffs. The initial cost can be thought of as
$$\displaystyle \mbox{Replicating Cost Strategy} = n_SS + n_bB$$
where the equivalent number of shares for calls is
$$\displaystyle n_s = N(d_1) > 0$$
and
$$\displaystyle n_S = -N(-d_1) < 0$$
for puts.
The equivalent number of bonds for calls is
$$\displaystyle n_B = -N(d_2) < 0$$
and
$$\displaystyle n_B = N(-d_2) > 0$$
for puts.
So that leaves the zero-coupon bond _B_ as
$$\displaystyle B = e^{-rT}X$$.
For the calls, note that if _n_ is positive, then you're buying the underlying, and if it's negative, then you're selling the underlying. So it's no surprise that the calls have a positive _n_ for the value of the stock.
Why's that?
Not quite.
You got it!
A call is long the stock's performance. As the stock rises, the value of the call rises too, so the replicating strategy for a call is simply buying shares with financing. For this reason, the replicating strategy for calls requires continually buying shares in a rising market and selling shares in a falling market.
And for puts, you're simply buying bonds with the proceeds from short selling the underlying. So _n_ is negative.
For the BSM model, instead of buying low and selling high, you're really buying higher and selling lower, so the strategy will essentially lead to losses, but that's supposed to happen with the replication strategy.
Why is that the case?
Not quite.
The losses don't capture the change in the underlying because the price is moving higher.
No.
An option won't have a negative present value.
That's it!
Those losses over the life of the option represent the BSM model option premium, so that way there's no arbitrage opportunity.
But in real life, the replicating strategy will clearly incur costs from the constant transactions, along with the fact that prices don't always move continuously, like when a merger's announced, and the price jumps.
Another big assumption is that volatility can be known in advance, but clearly that's not the case. Typically, when using the BSM model, the volatility formula is higher than what's expected, so options are typically more expensive than what the BSM model indicates.
To sum it up:
[[summary]]
Computer
Lookup table
Spreadsheet
Corporate bond
Fixed-rate swap
Zero-coupon bond
A long call is long the stock's performance
A long call is short the stock's performance
The losses represent the option cost
The losses represent the change in the underlying
The losses represent the present value of the option
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