194 Mathematics for Finance
withd 2 is given by (8.9).
Let us analyse the following example, which will subsequently be expanded
and modified. Suppose that the risk-free rate is 8% and consider a 90-day call
option with strike priceX= 60 dollars written on a stock with current price
S= 60 dollars. Assume that the stock volatility isσ= 30%. The Black–Scholes
formula gives the option priceCE=4.14452 dollars, the delta of the option
being equal to 0.581957.
Suppose that we write and sell 1,000 call options, cashing a premium of
$4, 144 .52. To construct the hedge we buy 581.96 shares for $34, 917. 39 ,borrow-
ing $30, 772 .88. Our portfolio will be (x, y, z) withx= 581.96,y=− 30 , 772 .88,
z=− 1 ,000 and with total value zero. (While it might be more natural math-
ematically to consider a single option withz=− 1 ,in practice options are
traded in batches.)
We shall analyse the value of the portfolio after one day by considering some
possible scenarios. The time to expiry will then be 89 days. Suppose that the
stock volatility and the risk-free rate do not vary, and consider the following
three scenarios of stock price movements:
- The stock price remains unaltered,S( 3651 ) = 60 dollars. A single option is
now worth $4.11833, so our liability due to the short position in options is
reduced. Our debt on the money market is increased by the interest due.
The position in stock is worth the same as initially. The balance on day one
is
stock 34 , 917. 39
money − 30 , 779. 62
options − 4 , 118. 33
TOTAL 19. 45
Without hedging (x=0,y=4, 118. 33 ,z=− 1 ,000) our wealth would have
been $27.10, that is, we would have benefited from the reduced value of the
option and the interest due on the premium invested without risk. - The stock price goes up toS( 3651 ) = 61 dollars. A single option is now worth
$4.72150, which is more than initially. The unhedged (naked) position would
have suffered a loss of $576.07. On the other hand, for a holder of a delta
neutral portfolio the loss on the options is almost completely balanced out
by the increase in the stock value:
stock 35 , 499. 35
money − 30 , 779. 62
options − 4 , 721. 50
TOTAL − 1. 77