dstefan
03-22-2008, 01:19 PM
Chapter 1, Problem 15:
A stock with spot price 40 pays dividends continuously at a rate of 3%. The four months at-the-money put and call options on this asset are trading at $2 and $4, respectively. The risk-free rate is constant and equal to 5% for all times.
Show that the Put-Call parity is not satisfied and explain how would you take advantage of this arbitrage opportunity.
Solution:
Data: S=40; K=40; T=1/3; r=0.05; q=0.03; P=2; C=4.
The Put-Call parity is not satisfied:
P + S e^{-qT} - C = 39.5821 ~>~ 39.3389 = K e^{-rT}
``Buy low, sell high": short the portfolio on the left hand side and go long the right hand side portfolio (which is cash only); the riskless profit at maturity will be the future value of the mispricing from the put call parity, i.e.,
(39.5821-39.3389) e^{rT} = 0.2473
To show this, start with no money and sell one put option, short e^{-qT} shares, and buy one call option. This will generate the following cash amount:
P + S e^{-qT} - C = 39.5821
since shorting the shares means that e^{-qT} shares are borrowed and sold on the market for cash. (The short will be closed at maturity T by buying shares on the market and returning them to the borrower; see below for more details.)
Overall, the portfolio is
\bullet short one put option with strike K and maturity T
\bullet short e^{-qT} shares
\bullet long one call option with strike K and maturity T
\bullet cash position: USD +39.5821
The initial value of the portfolio is zero, since no money were invested:
V(0) ~=~ -P(0) - S(0) e^{-qT} + C(0) + 39.5821 ~=~ 0
Note that by shorting the shares you are responsible for paying the accrued dividends. We assume that the owner of the stock will have reinvested the dividends in buying more shares. As dividends are paid continuously at rate q, this means that e^{-qT} shares at time 0 will become 1 share at time T, through continuous purchases of (fractions of) shares using the dividend payments.
The value of the portfolio at maturity is
V(T) ~=~ -P(T) - S(T) + C(T) + 39.5821 e^{r T} ~=~ -\max(K-S(T),0) - S(T) + \max(S(T)-K,0) + 39.5821 e^{r T}
As shown when proving the Put-Call parity,
P(T) + S(T) - C(T) = \max(K-S(T),0) + S(T) - \max(S(T)-K,0) = K,
regardless of the value S(T) of the underlying asset at maturity.
Therefore,
V(T) ~=~ -K + 39.5821 e^{r T} ~=~ -40 + 40.2473 = 0.2473
A stock with spot price 40 pays dividends continuously at a rate of 3%. The four months at-the-money put and call options on this asset are trading at $2 and $4, respectively. The risk-free rate is constant and equal to 5% for all times.
Show that the Put-Call parity is not satisfied and explain how would you take advantage of this arbitrage opportunity.
Solution:
Data: S=40; K=40; T=1/3; r=0.05; q=0.03; P=2; C=4.
The Put-Call parity is not satisfied:
P + S e^{-qT} - C = 39.5821 ~>~ 39.3389 = K e^{-rT}
``Buy low, sell high": short the portfolio on the left hand side and go long the right hand side portfolio (which is cash only); the riskless profit at maturity will be the future value of the mispricing from the put call parity, i.e.,
(39.5821-39.3389) e^{rT} = 0.2473
To show this, start with no money and sell one put option, short e^{-qT} shares, and buy one call option. This will generate the following cash amount:
P + S e^{-qT} - C = 39.5821
since shorting the shares means that e^{-qT} shares are borrowed and sold on the market for cash. (The short will be closed at maturity T by buying shares on the market and returning them to the borrower; see below for more details.)
Overall, the portfolio is
\bullet short one put option with strike K and maturity T
\bullet short e^{-qT} shares
\bullet long one call option with strike K and maturity T
\bullet cash position: USD +39.5821
The initial value of the portfolio is zero, since no money were invested:
V(0) ~=~ -P(0) - S(0) e^{-qT} + C(0) + 39.5821 ~=~ 0
Note that by shorting the shares you are responsible for paying the accrued dividends. We assume that the owner of the stock will have reinvested the dividends in buying more shares. As dividends are paid continuously at rate q, this means that e^{-qT} shares at time 0 will become 1 share at time T, through continuous purchases of (fractions of) shares using the dividend payments.
The value of the portfolio at maturity is
V(T) ~=~ -P(T) - S(T) + C(T) + 39.5821 e^{r T} ~=~ -\max(K-S(T),0) - S(T) + \max(S(T)-K,0) + 39.5821 e^{r T}
As shown when proving the Put-Call parity,
P(T) + S(T) - C(T) = \max(K-S(T),0) + S(T) - \max(S(T)-K,0) = K,
regardless of the value S(T) of the underlying asset at maturity.
Therefore,
V(T) ~=~ -K + 39.5821 e^{r T} ~=~ -40 + 40.2473 = 0.2473