Last edited on Oct. 13, 2020.
콜옵션과 풋옵션 사이에 관계가 있을 것이라는 생각은 쉽게 생각해 볼 수 있겠지만, 실제로 어떤 관계가 있는지를 배우지 않고 알기는 매우 힘들다.
Options have been traded for centuries without explicitly stating the hidden relationship between puts and calls, known as the put-call parity in modern terms. However, there is an indication that the relationship has been well exploited by a few financiers. The official first appearance of the put-call parity in the academic literature was in 1969.
To explore the relationship, let’s take the difference between the payoffs of the call and the put options. Without difficulty, we can derive the RHS. This is just a mathematical operation.
LHS of the equation is just the difference between the payoff of the call option and the payoff of the put option. But RHS of the equation looks like the payoff of the forward contract.
Note that this is the payoff at time T. Now we want to translate this formula in the current time 0.
This formula is just an equation. We can exploit this formula by treating LHS as the target position and RHS as its implementation. For instance, suppose we want to have a short position of the underlying asset. Under some regulations, it is not allowed. However, we can accomplish this by rearranging the equation.
The equation says if we want a short position of the underlying asset, then we just need to make the portfolio of one short call, one long put, and borrowing money.
Let’s analyze the equation from the RHS. We borrow the present value of K from our bank and then buy the put option and sell the call option. How much does it cost to build the RHS portfolio? We will see it costs nothing.
def call_option(S,K,u,d,r):
p = (r-d)/(u-d)
return 1/r * (p * max(u*S - K, 0) + (1-p) * max(d*S - K, 0))def put_option(S,K,u,d,r):
p = (r-d)/(u-d)
return 1/r * (p * max(K - u*S, 0) + (1-p) * max(K - d*S, 0))S = 100
K = 100
u = 1.1
d = 0.9
r = 1.05call_option(S, K, u, d, r)
# 7.142857142857152put_option(S, K, u, d, r)
# 2.380952380952382call_option(S, K, u, d, r) - put_option(S, K, u, d, r)
# 4.76190476190477S - K * r ** -1
# 4.761904761904773- K * r ** -1
# -95.23809523809523
The fact that it costs nothing does not mean we are not in a risk-taking position. In fact, we just make a risky speculative position. We are betting on the downside movement of the underlying asset.
You may review this video.
