13

It is indeed. The price of an American option is the Bermuda option in the limit that the exercising interval approaches zero. The Bermuda option at any exercising time can be evaluated inductively via the dynamic programming principle as the maximum of the payoff and the risk-neutral expected value of the Bermuda option price at the next exercise time. The ...


8

Here is a much more straightforward proof of the convexity of the American option with respect to a parameter, if it is independent of time and sample, than my previous one, though I am happy to have made the connection amongst the dynamics programming principle, the discrete time process and the continuous time process there. Let $g(t,\omega,x)$ be the ...


6

European Contracts It's a really important question and as @noob2 commented, the FTAP is normally applied to European-style derivatives, even if they are (strongly) path-dependent, including barrier options and Asian options. The idea is always the same, $V_t=B_t\mathbb{E}^\mathbb{Q}\left[\frac{\xi_T}{B_T}\Big|\mathcal{F}_t\right]$, that is the derivative's ...


6

Let $\mathscr{T}$ be the set of stopping times with values in $[0, T]$. Note that, for any $\tau \in \mathscr{T}$, $\lambda_1\ge 0$, $\lambda_2 \ge 0$, and $\lambda_1+\lambda_2 =1$, \begin{align*} &\ \max(\lambda_1 K_1+\lambda_2 K_2 -S_{\tau}, 0) \\ =&\ \max\big(\lambda_1 (K_1-S_{\tau})+\lambda_2 (K_2 -S_{\tau}), 0\big)\\ =&\ \lambda_1\max(K_1-...


4

That’s because in the case of a non dividend paying asset (the usual studied case), an American call is worth the same as a European call. Conversely for a non dividend paying asset the American put is different from the European put, so the American put needs special methods. Nonetheless, once you study a pricing algorithm on an American put such as a tree,...


4

He circled the volume of the option in question. It was roughly 50,000 and one option contract is usually for 100 shares. Price was 0.13. $50k*100*0.13 = 650k$


4

You can use the standard black-scholes formula to price an european option. The only parameter you do not know to use the formula is the volatility. If you have the price of an american option then you can use the Cox-Ross-Rubinstein (CRR) model to backout the implied volatility. Then just use black scholes. The CRR model: In the framework of the CRR model, ...


3

With $V$ American option value, $H$ holding (aka continuation) value, and $B$ bank account value, we have: $$V_N(S_N) = (K-S_N)^+$$ and for $i$ backwards from $N-1$ down to 0, we have: $$ H_i(S_i) = \mathbf{E}\left[B_{i}B_{i+1}^{-1}V_{i+1}(S_{i+1})|S_i\right]$$ $$ V_i(S_i) = \max (K-S_i, H_i(S_i)) $$ The algorithm result is $V_0(S_0)$. The different ...


3

1) Do we need to deal with infinite dimensional spaces? Yes, I think you need an infinite dimensional pay-off space. Your remark that a finite sample spans a finite dimensional space of pay-offs is true. But you would like to prove convergence of the method for any pay-off, i.e. for all possible samples of all sizes. 2) In the case that we want to insist ...


3

This is a good question. See my answer to a question here The point is that under Black-Scholes (and also many SV models) not only European prices but also American options prices are homogeneous of degree 1 in strike and spot as the optimal exercise time does not affect the homogeneity property in strike and spot price. Hence also for American options ...


3

Theoretically, this is a more difficult problem than it looks like at first glance. Unfortunately, existing literature taking into account a proper dividend consideration is rare (at least from a practical viewpoint). There are several options: 1) Use what is called "De-Americanization": In this case, based on your input dividends (maybe based on other ...


3

The advantage is that you get to keep the option premium. The obvious drawback is that your option can be exercised. You’re effectively capping your maximum gains on stock price increase.


2

From put-call parity we have $C_t =P_t +S_t - K e^{-r(T-t)},$ so $$C_t \geq S_t - K e^{-r(T-t)} > S_t - K.$$ This means that the price of the call $C_t$ at any time $0 < t<T$ is always greater than the value of exercising the call which is $S_t - K.$ Therefore, the optionality of exercising an American call option (with no dividends) before $T$ has ...


1

In both cases, you should argue along the lines of options arbitrage. If you think an asset (or a portfolio) is relatively cheap (as in: arbitrageable) then you simply buy low, sell high. In your first case, it seems that the call is too cheap, as in: $$ C^E<P^E+S-Ke^{-rT} $$ So let's buy the call, sell a put, short a unit of stock, and borrow some money: ...


1

Usually, you would use the volatility from a fitted volatility curve or surface. Those are based on implied volatilities. You can use historical volatility, but then your valuation is likely to be off because the volatility curve/surface is not constant and at historical vol. You should use a yield curve to present value nodes. This is unlikely to make a big ...


1

Following the notation in Hull, let $H$ be the barrier level. I list the prices of European-style down-and-out barrier options with continuously observed barrier. If $H\leq K$, then $$c_{di}=S_0e^{-qT}(H/S_0)^{2\lambda}N(y)-Ke^{-rT}(H/S_0)^{2\lambda-2}N(y-\sigma\sqrt{T})$$ and $$c_{do}=c-c_{di}.$$ If $H>K$, then $$c_{do}=S_0N(x_1)e^{-qT}-Ke^{-rT}N(x_1-\...


1

The Bermudan (American) callable/swaption is convex with respect to the strike. The payoff function of the Bermudan (American) callable/swaption is of the form, with implicit dependence on sample $\omega$, $$g(t,K)=\big(a(t)-b(t)K\big)_+$$ where $t$ is the time the swap (interest) rate is set and $K$ is the strike. $g(t,K)$ is obviously convex with respect ...


1

If the underlying is driftless (think futures) and the value of the option is not discounted (think future style options with daily bilateral variation margin or CSA's with zero collateral interest rates) then the value of an american put and a european put would be the same by Jensen's inequality.


1

In practice, you will not be able to find assets on which both types of options are written save for rare exceptions where there might have been a transition between one type to the other. For equity, options on individual titles tend to be American while options on indexes tend to be European... so, you can't really run a test of this idea. In theory, if ...


1

If there is no interest rate, the european and american put prices are the same for every strike. More details can be found in my answer for the question below: Longstaff Schwartz Algrorithm in R


1

Working in discrete time or continuous time is mostly a matter of convenience. What most people do in some field of finance or economics is suggestive of what tends to be easier, though it's a kind of rule of thumb. Off the top of my head, CT has the convenience of easily handling uneven time steps and allowing easy aggregation. It also handles changing ...


1

It depends of the convexity of the function f. I guess you already heard about the fact that american call price is the same as european call price when there is no dividends. It is still valid for bermudan call as its price is between american call price and european call price. Please have a look on this document for more details: http://www.stat....


1

From the same source (Introduction to Quantitative Finance by Stephen Blyth), before proving that both American and European call options on non-dividend paying stock have the same value, the author proves the following bound for non-dividend European call option value at page $57.$ Result: The European call price on a non-dividend paying stock satisfies ...


1

It is due the number of timestamps in your case. Actually, as the ZC rate is zero, the price of European and american options should be the same. EDIT PROOF: You know that for american options (see proof in pages 4,5 HERE): $S_T-K\leqslant C-P \leqslant S_T-Ke^{-rT}$ When the risk free rate is zero, you get that the call put parity remains valid $S_T-K= ...


1

Your understanding is correct. It makes no sense to exercise a call (as well as a put) when there is time premium remaining because you are throwing away that time premium by doing so. Sell the call and buy the stock if you want to own it in order to capture the dividend. Buying the stock to capture the dividend makes no sense to me if it's a non sheltered ...


1

To add to @DM63's answer, as a secondary characterisitc, vol may also matter in deciding the european approximation impact. As vol goes to 0, you want to exercise as soon as possible, because the underlying future rate becomes a constant (a martingale with no vol). As you'll receive the same payoff at at date, better to get it earlier (if rates are.positive)....


1

For this, let us consider two portfolios: A:One American Call (c) and Kexp(-r(T-t)) in money mkt and B: The stock S. Now, let us say we decide to exercise the Call early, Value of portfolio A (by exercising at t1): (S(t1) - K) + Kexp(-r(T-t1)) < S = B Now if we consider the portfolio A at expiry (time T): Value of portfolio A (by exercising at T): max(S-...


Only top voted, non community-wiki answers of a minimum length are eligible