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Since American style options allow early exercise, put-call parity will not hold for American options (unless they are held to expiration). In practice, there is also a difference between calls and puts for European options as well. The full description is here: What causes the call and put volatility surface to differ?

3

First, we have $P(t)+S(t)=C(t)+B(t,T)\cdot K$, Then, $\frac{\partial P(t)}{\partial S(t)} + \frac{\partial S(t)}{\partial S(t)} = \Delta^{\text{put}}_{t}+1$ and $\frac{\partial C(t)}{\partial S(t)} + \frac{\partial [B(t,T)\cdot K]}{\partial S(t)} = \Delta^{\text{call}}_{t}+0$. Finaly, $\Delta^{\text{call}}_{t}-\Delta^{\text{put}}_{t}=1$. This relationship ...

2

Let's talk about your first equation: If you exercised your option early, you got this payoff. But if you are a rational investor you'd realize that this is less than what you would get if you would just sell your option itself. i.e. the payoff at time t will be more than S(t)-K because the option is worth more than that as it also has some time value. so ...

1

If $PV_{t, T}(\text{Divs}) \ge K\big(1-e^{-r(T-t)}\big)$, since $P_{Eur}(S_t, K, T-t) >0$, the identity \begin{align*} C_{Eur}(S_t, K, T-t) = P_{Eur}(S_t, K, T-t) + (S_t-K) -PV_{t, T}(\text{Divs}) +K\big(1-e^{-r(T-t)}\big), \end{align*} implies that \begin{align*} C_{Eur}(S_t, K, T-t) > (S_t-K). \end{align*} That is, it is not rationale to exercise the ...

1

Let $\{X_t \mid t \ge 0\}$ be the foreign exchange rate rate from $£$ to \. Moreover, let C(X_0, K, T) and P(X_0, K, T) be the prices of the respective call and put options with strike K and maturity T. Then \begin{align*} \frac{1}{X_0}P(X_0,\, K,\, T) = K C\left(\frac{1}{X_0},\, \frac{1}{K},\, T \right). \end{align*} Based on the given condition, ... 1 What a difficult problem. The first line gave 165 e^{-rt} -3 S e^{-dt} = 15 [since 50+55+60 = 165]. In the second line we want to evaluate 110 e^{-rt} -2 S e^{-dt} . We notice that this is exactly two thirds of the left side of the above, because 110 is two thirds of 165 and 2 is two thirds of 3. So we take two thirds of the right hand side of the first ... 1 In this case, call option is deep in the money while put option is deep out of money. As maturity is very near, any change in stock price would have equivalent impact on the call option price. Decline in one dollar in stock price lead to almost one dollar decline in call option price. Whereas for put, its worth would increase but put is still deep out of ... 1 Futures payoff is indeed S_t-F_0, but the t in question is the maturity date of futures. In this derivation t denotes maturity date of the option, which is always before the futures maturity. Therefore, on the day of option maturities, the futures did not expire yet, but the value of the futures position is F_t-F_0 (in mark-to-market sense, you can ... 1 the call version pays I_{S_T > K } S_T $$the put version pays$$ -I_{S_T < K } S_T $$Subtract to get a pay-off$$ S_T.$(ignoring the probability zero event of$S_T=K.$) So the prices subtract to give$S_0.\$

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