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Arbitrage Relationship for American Options

分类： 其他平台

2017-12-05 16:27:52




9.1 Arbitrage Relationship for American Options

It is complex to price American options since they can be exercised at any time point up to expiry date. The time the holder chooses to exercise the options depends on the spot price of the underlying asset $ S_t$. In this sense the exercising time is a random variable itself. It is obvious that the Black-Scholes differential equations still hold as long as the options are not exercised. However the boundary conditions are so complicated that an analytical solution is not possible. In this section we study American options in more detail. The numerical procedures of pricing will also be discussed in the next section.

As shown in Section , the right to early exercise implies that the value of an American option can never drop below its intrinsic value. For example the value of an American put should not go below $ .max(K-S_t,0)$ with the exercise price $ K$. In contrast this condition does not hold for European options. Thus American puts would be exercised before expiry date if the value of the option would drop below the intrinsic value.

Let's consider an American put on a stock with expiry date $ T$. If the stock price $ S_{t^*}$ at time $ {t^*}$ is zero, then $ S_t=0$ holds for $ t .geq t^*$ since the price process follows a geometric Brownian motion. It is then not worth waiting for a later exercise any more. If the put holder waits, he will loose the interests on the value $ K$ that can be received from a bond investment for example. If $ S_{t^*}=0$, the value of the put at $ t^*$ is $ K$ which is the same as the intrinsic value. Since the respective European put cannot be exercised early, e.g. at time $ t^*$, we can only get $ K$ on the expiry date. If we discount it to time $ t^*$ with $ .tau^* = T-t^*$, we only get $ K e^{-r.tau^*}$ that is the value of the European put at time $ t^*$. Obviously this value is smaller than the value of an American put and its intrinsic value. Figure  shows the put value with a continuous cost of carry $ b$.



Fig.: European put and early exercise of an American put with costs of carry $ b_1$ and $ b_2$.
.includegraphics[width=1.defpicwidth]{euput.ps}


As we can see an early exercise of the put is maybe necessary even before $ S_t=0$. For a certain critical stock price $ S^{**}$, the lost of interests on the intrinsic value, which the holder can receive by exercising it immediately, is higher than the possible increase of the option value due to the eventual underlying fluctuations. That is one of the reasons why the critical underlying price is dependent on time: $ S^{**} = S^{**}(t)$.

From the derivation of the Black-Scholes differential equations it follows that they are valid as long as the option is not exercised. Given that there are no transaction costs in perfect markets, a revenue can be realized from an early exercise, which equals to the intrinsic value of the option. One says in this case that the option falls back to its intrinsic value by early exercising. Thus the pricing of American options is an open boundary problem. The Black-Scholes differential equations are valid where the underlying $ S$ is either higher than the critical put-price $ S^{**} = S^{**}(t)$ or lower than the critical call-price $ S^* = S^*(t)$. The boundaries defined through $ S^{**}(t)$ and $ S^*(t)$ are unknown.

Figure  shows the regions where the option price $ C = C(S,t)$ for an American call satisfies the Black-Scholes differential equations.


  • In the interior $ \{ (S,t) \vert 0 \le S < S^*(t), t the Black-Scholes differential equations hold. </li> <li> At the boundaries <img width= the call falls back to the intrinsic value $ .max(S-K,0)$.
  • $ C(S,t)$ and $ .displaystyle.frac{.partial C(S,t)}{.partial S}$ are continuous in the whole region including the boundaries.



Fig.: The exercise boundary $ S^{*}(t)$ for an American call.
.includegraphics[width=1.defpicwidth]{euput2.ps}


The numerical solution for such boundary problems is described in the next section. Based on the assumptions of perfect markets and arbitrage free argument in Section  we derive some properties of American options without considering any specific mathematical models for the price process $ S_t$.



Theorem 9.1   


  1. An American call on an asset that does not bring any positive incomes is not early exercised and has the same value as an equivalent European call.
  2. For an American call on an asset that brings positive incomes at discrete time points $ t_1,.cdots ,t_n$ the optimal exercise time lays just before these points. Consequently in the case of continuous positive payments, any time point can be an optimal exercise time.


Proof: 
Let $ K$ denote the exercise price, $ T$ the expiry date, $ .tau = T - t$ the time to maturity of a call and $ S_t$ the price of the underlying asset. $ C_{am}(S,.tau)$ and $ C_{eur}(S,.tau)$ denote the value of the respective American and European calls at time $ t$ with time to maturity $ .tau = T - t$ and the spot price $ S_t=S$. Using the put-call parity for European options we find


  1. in the case of discrete dividends $ D$ with the discounted value $ D_t .le 0$ at time $ t$, i.e. they are costs. Furthermore it holds based on Theorem :


    $.displaystyle C(S_t,.tau) = P(S_t,.tau) + S_t - D_t - K e^{-r.tau} .ge S_t - K e^{-r.tau} > S_t - K$ (9.1)


    where $ C = C_{eur}$ and $ P$ is the respective put price. 
    In the case of continuous dividends with rate $ d .le 0$, it follows from $ b - r = -d .ge 0$ that: 
    $.displaystyle C(S_t,.tau)$ $.displaystyle =$ $.displaystyle P(S_t,.tau) + S_te^{(b-r).tau} - K e^{-r.tau}$  
      $.displaystyle .ge$ $.displaystyle S_te^{(b-r).tau} - K e^{-r.tau} > S_t - K$ (9.2)


    In both cases we verify that $ C(S_t,.tau) > S_t - K$ for European calls. Since $ C_{am} .ge C$, we conclude that


    $.displaystyle C_{am}(S_t,.tau) > S_t - K,$


    i.e. the value of an American call is higher than the intrinsic value during the whole maturity. Therefore early exercise is avoided.


  2. without any restriction we consider the case where $ t_1$ is the next payment time. $ \tilde{t}  represents any early time. <img width= with $ .tilde{.tau}= .tilde{t} -t$ is the value of a European call with the same exercise price $ K$ but with a different maturity date at $ .tilde{t}$. Since there are no payments before $ .tilde{t}$ at all, it follows from part 1 that $ .tilde{C}(S_t,.tilde{.tau}) > S_t - K$ for $ t . Due to the longer time to maturity and the possibility of early exercise of American calls, it follows <p> <br /> </p> <div align=
    $.displaystyle C_{am}(S_t,.tau) .ge .tilde{C}(S_t,.tilde{.tau}) > S_t -K$ (9.3)


As in part 1, the value of an American call at any time $ t  lays strictly over the intrinsic value, which excludes an early exercise. Since <img width= falls to the intrinsic value just at time $ t_1$ (or in a respectively later time point). $ {.Box}$ 

Figure  shows a graphical representation of the first part of the theorem.


  • If $ b.ge r$ which is equivalent to $ d .le 0$, then $ C_{am}=C_{eur}$.
  • If $ b  which is equivalent to <img width=, then $ C_{am}>C_{eur}$.


Fig.: A European call and an early exercised American Call.
.includegraphics[width=1.defpicwidth]{euput3.ps}



It is also possible to derive a formula similar to the put call parity for American options. Given that the model is unknown for the critical price $ S^*(t)$, $ S^{**}(t)$ and consequently the time point for early exercise, the formula is just an inequality.



Theorem 9.2 (Put-Call Parity for American options)   
We consider an American call and an American put with the same maturity date $ T$ and the same exercise price $ K$ on the same underlying asset. Let $ C_{am}(S,.tau)$ and $ P_{am}(S,.tau)$ denote the option prices at time $ t$ when the spot price is $ S_t=S$ and the time to maturity is $ .tau = T - t$. It holds


  1. if there are incomes or costs during the time to maturity $ .tau = T - t$ with the discounted value $ D_t$ at time $ t$, then


    $.displaystyle P_{am}(S_t,.tau) + S_t - Ke^{-r.tau} .ge C_{am}(S_t,.tau) .ge P_{am}(S_t,.tau)+ S_t - D_t - K$ (9.4)


  2. if there are continuous costs of carry with rate $ b$ on the underlying asset, then 
    $.displaystyle P_{am}(S_t,.tau) + S_t - K e^{-r.tau}$ $.displaystyle .ge$ $.displaystyle C_{am}(S_t,.tau)$  
      $.displaystyle .ge$ $.displaystyle P_{am}(S_t,.tau) + S_te^{(b-r).tau}- K$  
           if $\displaystyle b </td> <td width= (9.5)
    $.displaystyle P_{am}(S_t,.tau) + S_te^{(b-r).tau}- Ke^{-r.tau}$ $.displaystyle .ge$ $.displaystyle C_{am}(S_t,.tau)$  
      $.displaystyle .ge$ $.displaystyle P_{am}(S_t) + S_t - K$   if $.displaystyle b .ge r$  



Proof: 
Supposing without any restriction that the underlying asset is a stock paying dividends $ D_1$ at time $ t_1$. 


Table: Portfolio value at time $ .tilde{t}$, $ t_1$ and $ T$.
$ .tilde{t}= t_1 -.Delta t$ $ t_1$ $ T$
early Call is Call is not
exercise exercised early exercised early
Position of the call $ S_{T} .le K$ $ S_{T}>K$ $ S_{T} .le K$ $ S_{T}>K$
1. $ .ge 0$ - $ K-S_{T} $ 0 $ K-S_{T} $ 0
2. $ S_{.tilde{t}}$ $ D_1$ - - $ S_{T}$ $ S_{T}$
3. $ -Ke^{-r(T-.tilde{t})}$ - $ -K$ $ -K$ $ -K$ $ -K$
4. $ -(S_{.tilde{t}}-K)$ - $ Ke^{r(T-.tilde{t})}$ $ Ke^{r(T-.tilde{t})}$ 0 $ -(S_{T}-K)$
Sum $ .ge 0$ $ .ge 0$ $ .ge 0$ $ .ge 0$ 0 0
11683 



1. We show first the left inequality. We consider a portfolio consisting of the following four positions:

  1. buy an American put
  2. buy a stock
  3. sell bonds (i.e. borrow money) with the nominal value $ K$ and the maturity date $ T$
  4. sell an American call
In this portfolio, the position 1 is held until time $ T$ despite the underoptimal conditions, i.e. the put is not exercised early even when the call holder exercises it early. Note from Theorem  that an early exercise of the call is only possible at time $ .tilde{t} := t_1 - .Delta t$ where $ .Delta t .approx 0$, i.e. directly before the payment at time $ t_1$. In this case we deliver the stock of the portfolio. The value of the portfolio at time $ T$ is given in the Table .

Therefore it holds for every time $ t$ as mentioned:


$.displaystyle P_{am}(S_t,.tau) + S_t - Ke^{-r.tau} - C_{am}(S_t,.tau) .ge 0$ (9.6)


The proof of the second inequality is analogous but with opposite positions. Here we consider that the put can be exercised early, see Table .


  1. buy an American call
  2. sell a stock
  3. buy a bond (i.e. lend money) at present value $ K+D_t$ with maturity date $ T$
  4. sell an American put


Table: Portfolio value at time $ .tilde{t}$, $ t_1$ and $ T$.
$ .tilde{t}= t_1 -.Delta t$ $ T$
early Put is Put is not
exercise exercised early exercised early
Pos. of a put $ S_{T} .le K$ $ S_{T}>K$ $ S_{T} .le K$ $ S_{T}>K$
1. $ .ge 0$ 0 $ S_{T}-K $ 0 $ S_{T}-K $
2. $ -S_{.tilde{t}}-D_t e^{r(.tilde{t}-t)}$ $ -S_{T}$ $ -S_{T}$ $ -S_{T}$ $ -S_{T}$
$ -D_t e^{r.tau}$ $ -D_t e^{r.tau}$ $ -D_t e^{r.tau}$ $ -D_t e^{r.tau}$
3. $ (D_t +K)e^{r(.tilde{t}-t)}$ $ (D_t +K)e^{r.tau}$ $ (D_t +K)e^{r.tau}$ $ (D_t +K)e^{r.tau}$ $ (D_t +K)e^{r.tau}$
4. $ -(K-S_{.tilde{t}})$ 0 0 $ -(K-S_T)$ 0
Sum $ .ge 0$ $ .ge 0$ $ .ge 0$ $ .ge 0$ $ .ge 0$



Therefore we have for every time $ t$ as mentioned:


$.displaystyle C_{am}(S_t,.tau)-P_{am}(S_t,.tau) - S_t + K + D_t .ge 0$ (9.7)


2. For continuous cost of carry we first consider the case where $ b .ge r .iff d.le 0$. We prove the left inequality at first


$.displaystyle P_{am}(S_t,.tau) + S_te^{(b-r).tau}- Ke^{-r.tau} .ge C_{am}(S_t,.tau)$ (9.8)


Consider the following portfolio at time $ t$:
  1. buy an American put
  2. buy $ e^{(b-r).tau}$ stocks
  3. sell bonds at nominal value $ K$ with expiring date $ T$
  4. sell an American call
As in part 1 it follows that the value of the portfolio at time $ T$ is zero if the call is not exercised early. The continuous costs of carry ($ d .le 0$) are financed through the sell of the stocks so that only one stock is left in the portfolio at time $ T$.

If, on the other hand, the call is exercised early at time $ .tilde{t}$, the whole portfolio is then liquidated and we get:


.begin{displaymath}.begin{array}{ccc} P_{am}(S_{.tilde{t}},.tau) - (S_{.tilde{t}... ... S_{.tilde{t}} (e^{(b-r)(T-.tilde{t})}-1) & .ge & 0 .end{array}.end{displaymath} (9.9)


The value of the portfolio at time $ t$ is:


$.displaystyle P_{am}(S_t,.tau) + S_te^{(b-r).tau}- Ke^{-r.tau} - C_{am}(S_t,.tau) .ge 0$ (9.10)


If $ b<span style= the left inequality is similarly proved,


$.displaystyle P_{am}(S_t,.tau) + S_t - K e^{-r.tau}.ge C_{am}(S_t,.tau)$ (9.11)


it is enough to hold one stock in the portfolio because of $ d > 0$.

We show now the right inequality for the case $ b.ge r$


$.displaystyle C_{am}(S_t,.tau) .ge P_{am}(S_t,.tau) + S_t - K$ (9.12)


We consider the following portfolio at time $ t$:
  1. buy an American call
  2. sell an American put
  3. sell a stock (short sales)
  4. buy a bond with nominal value $ Ke^{r.tau}$ and expiring at time $ T$
If the put is not exercised early, it holds at time $ T$:


.begin{displaymath}.begin{array}{cccrclcc} 0& -&(K-S_{T})&-S_{T}e^{-(b-r).tau}+K... ....tau}+Ke^{r.tau} &.ge& 0 &.text{.rm if}& S_{T}.ge K .end{array}.end{displaymath} (9.13)


If the put is exercised early at time $ .tilde{t}$, the whole portfolio is liquidated and we get:


$.displaystyle C_{am}(S_{.tilde{t}},.tau) - (K-S_{.tilde{t}})-S_{.tilde{t}} e^{-(b-r)(.tilde{t}-t)}+Ke^{r(.tilde{t}-t)} .ge 0$ (9.14)


Thus the value of the portfolio at time $ t$ is:


$.displaystyle C_{am}(S_t,.tau) - P_{am}(S_t,.tau) - S_t + K .ge 0$ (9.15)


Analogously one gets for the right inequality when $ b <span style=


$.displaystyle C_{am}(S_t,.tau) .ge P_{am}(S_t,.tau) + S_te^{(b-r).tau}- K$ (9.16)


where the position of the stock is reduced to $ e^{(b-r).tau}$. $ {.Box}$ 

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