Question

You have been asked to create a synthetic short position in a forward contract that permits you to sell 10 units of the underlying one year from now at a price of $50 per unit. (1) Describe the positions you need to take in call and put options to achieve the synthetic short forward position. (2) If the underlying is selling for $48 today (i.e. So = 48), what is the cost of your synthetic short position?

Answer 1

Solution :

$\text{Short forward = buy a put + short a call on the same stock}$ with the same exercise price.

X = exercise price = 50

1). Position to be taken :

-- buy 10 numbers of Put options with strike price of $ 50 per unit.

--- short (sell) 10 numbers of Call option with strike price of $ 50 per unit.

2). Cost of synthetic short position = $10 \times (P-C)$,

where, P = price of 1 put ption

           C = price of 1 call option

The Call - Put parity equation :

$\frac{C+X}{(1+r)^t}=S_0+P$

Here, C = Call premium

          X = strike price of call and Put

          r = annual rate of interest

           t = time in years

          $S_0$ = initial price of underlying

          P = Put premium

Therefore,

$P-C=PV(X)-S_0=\frac{X}{(1+r)^t}-S_0$

Here, t = 1, $S_0$ = 48, X = 50

So the cost of the position is given as : $\frac{50}{(1+r)} -48$