Early exercise on a dice game

(post from old site)

Rules:
you get to throw a fair die up to 3 times
the number of dollars you're gonna get is determined by the number you get on you last throw
How much would you pay to play this game?

=====================================================
First of all, if the game is free, what would your strategy be?

1st throw -----> 2nd throw -------------------> 3rd throw
              |-> early exercise     |-> early exercise

The above will be the decisions you'd have to make
after first throw, do you 
go for a 2nd throw, or
early exercise (ie taking the $ shown as on your first throw)?
Similarly, do you make the 3rd throw if you have done the 2nd one?


Let's determine the expected value (EV) for only one toss.  easy enough => 3.5.

part A
If my 2nd throw gets me 1, 2, or 3, I'll go for the 3rd throw.  Take the $ (exercise) otherwise.
  
After my 1st throw, it's a bit trickier.  Obviously, I'll go for the 2nd throw if my first throw is 1, 2, or 3, applying the same logic above.  If my first throw is 6, I will not go for the 2nd throw as I would have already achieved the highest payoff possible.

How about if you get 5 on your first toss? intuitively, I'd see my chance of getting a 6 on either 2nd throw or 3rd throw => 1 - p(1 to 5)*p(1 to 5), which essentially gives the chance of NOT (having first throw resulting in 1 TO 5 AND 2nd throw resulting in 1 TO 5)
you only have 11/36 to beat getting 5.  So, if your first toss gives 5, you'd early exercise

How about if you get 4 on your first toss?
similar calculation, you'd have 20/36 to beat getting 4.  In this case, you'd NOT exercise and go for the 2nd throw.

part b
So, that's the strategy.  Now, how much would you pay to play this game?

Again, working backward:
right before 2nd throw, 
50% chance to go for 3rd throw, which will result in EV of 3.5
50% of getting 4, 5, or 6, giving an EV of 5
=> before 2nd throw, the game is worth 4.25 

right before first throw,
66% chance to go for 2nd throw, which gives an EV of 4.25, as shown above
33% chance of getting 5 or 6, giving an EV of 5.5 
=> the game is worth $4.66666666...

So, if the game is offered to you below that price, go for it.  Or you can offer that game above that price (casino!).

One more note, we could have gotten the strategy by calculating the EV alone.  ie, part b alone will be sufficient.  since we get the EV of 2nd throw = $4.25.  If our first throw is above that number, we exercise.  If not, we play on.

How does interest rate affect option price?

(post from old sites)

not as simple as many web sites suggest.

interest rate affects option price in 2 folds:

1. forward price of the spot stock
• as interest rate increases, forward price of the stock underlying, aka no arbitrage price of the stock underlying at option expiry, increases, and in turns, increases call value and decreases put value
2. cost of carrying of the option
• as interest rate increases, cost of carrying the option increases and in turns, decreases the option value
• a small amount, as option price is a fraction of the underlying price

FAQ:

how about future underlying?

stock option prices on the spot contract as the underlying.  Spot underlying and the interest rate determines the forward price, aka no arbitrage price of the stock underlying at option expiry.

Future options prices uses the FUTURE contract as the underlying.  Future price is effectively the equivalent of the forward price of the spot underlying and that's why when interest rate changes, the forward of the future doesn't change.  In fact and again, we use future as the underlying, not the forward.

Increase in interest does not drive up the forward price of future, but it should drive up future price because of no arbitrage, right?

Yes, but the change in future price is in fact a change in UNDERLYING price.   Perhaps another way to think of it is that the change in interest rate directly affects the underlying price of future option, but not affect the the future option itself.

As opposed to change in interest rate does not directly affect the underlying of a stock option, aka spot price of stock (economist may beg to differ), but directly affects the option itself as it changes the ATM forward.

How about far month future option?

Future option uses simple offset mode instead of forward price.  Far month future options still use front-month future as the base contract, which is most liquid.  To account for the different in expiry, traders put in offset for different months, which embeds the interest and dividends?

Reference:

Option volatility & pricing by Natenberg

Note:
confirmed using Orc.  push up interest rate, then

• both index call and put TV go down
• stock call TV goes up and put TV goes down

CAPM & Efficient frontier "results in error-maximizing investment-irrelevant portfolios"?

(post from old site)

How?  parameter estimation error in expected return and covariance.

Example, with a 2 asset portfolio, let's say you overestimates A1 return by e and underestimates A2 by 2. Your average error is=0, which is pretty good.

With modern portfolio theory, you would invest a lot more in A1 and less in A2 and thus maximizing your error...

Quick review on CAPM, efficient frontier, etc.

In a nutshell, modern portfolio theory with CAPM picks a portfolio of risky assets (efficient frontier) and risk-free asset and maximizes the expected return for any given level of risk (volatility) / minimizes the risk for any given level of expected return.

How to maximize expected return? by investing more in assets with higher expected return.

How to minimize risk? by diversifying among assets have low correlation.

example, if you invest in oil company and airline company, you can minimize your risk due to oil price fluctuation.  While both companies expect to make money, if oil price increases, earning of the oil company is expected to increase and that of the airline company is expected decrease due to higher cost and vice versa.

How to improve the accuracy of prediction based on past performance?

(post from old site)

One way is, instead of using the expected value, to calculate the confidence interval of the expected value and use the lower end of the value.

Another way is to find a period (period A)in the past when the conditions and the value similar to present, then use the period A+1 data to estimate the next period.

As usual, back testing with out of sample data.

Value at Risk vs. Conditional value at risk

(post from old site)

VaR
95% VaR is the 95th quantile of loss
Say it's = $1 mil. then it means 95% chance the loss is under $1 mil.

CVaR
Aka expected shortfall or tail conditional expectation
With the above 95% VaR, there is a 5% probability that the loss will be >= 1 mil.
Let's say if we fall into that 5%, what's the loss that we should expect?
Now, that's what CVaR is.

Think about it. 95% VaR is essentially the MINIMUM loss of the portfolio for the 5% worst case scenario.
CVaR will actually tell u the mean (expected) loss for the 5% worst case scenario.

Monty Hall, goats and 3 prisoners

(posts from old site)

Welcome to the game show!  I'm Monty Hall.  As you can see, there are 3 doors in front of you, A, B and C.

You are in for a prize!  Behind one of the doors is $1 mil!  For the other 2 doors, there is a goat behind each.  :D

Now, pick a door!

... Say you pick door A...

Ok, with door B and C, let me open a door with a goat!  (Monty opened door C.)

Now, contestant, let me give you 2 choices.  You may stick with door A or switch door B for $1 mil.

What do you do?

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There are 3 condemned prisoners, with 2 to be executed and 1 to be pardoned.

Prisoner A begs the warden to tell him which of the other prisoners, B & C, will be executed, arguing that this reveals no information to his own fate, but secretly thinking that it'll increase his odds of being pardoned from 1/3 to 1/2.

what do u think?

PHILOSOPHY in pricing derivatives

(posts from old site)

1. Specify a model under the Q(theta)-dynamics
• theta is a vector of parameters, e.g. volatility, drift, etc.
• Q() is the risk neutral framework
2. Price all securities at time t by discounting the next period (t+1) risk neutral prices
3. Calibrate the model by choosing theta so that market prices of appropriate liquid securities agree with model prices of those securities
• this calibration procedure to market prices will incorporate the factors not specified in the model, e.g. policy risk, market expectation, economic outlook, etc.