Risk neutral

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In economics, risk neutral behavior is in between risk aversion and risk seeking. If offered either 50 EUR or a 50% chance of 100 EUR, a risk averse person will take the 50 EUR, a risk seeking person will take the 50% chance of 100 EUR, and a risk neutral person would have no preference between the two options.

In finance, when pricing an asset, a common technique is to figure out the probability of a future cash flow, then to discount that cash flow at the risk free rate. For example, if the probability of receiving 1 USD one instant from now is 50%, the value is .50 USD. This is called 'expected value', using real world probabilities. Risk neutral demonstrates that when pricing some assets, the real world probabilities assigned to future cash flows are irrelevant.

The fundamental assumption behind risk-neutral valuation is to use a replicating portfolio of assets with known prices to remove any risk. The amounts of assets needed to hedge determine the risk-neutral probabilities.

[edit] Example

Assume that a coin will be flipped, and if you play the game, heads will win 100 EUR 50% of the time, or tails requires you to pay 50 EUR 50% of the time. The expected value of playing this game is 50% * 100 EUR - 50% * 50 EUR = 25 EUR, the game has a positive expected outcome, but there is a risk of losing money.

Next suppose products are created which allow you to buy the winnings or loss of the coin toss. If you buy the winnings (name this product a "call option"): a win will give you 100 EUR, and loss will give you nothing. If you buy the loss (call this product a "put option"): a loss will give you 50 EUR, a win will give you nothing.

If these two options are priced using real-world probabilities, the call option should be 50 EUR (50% expected win x 100 EUR) and the put option should be 25 EUR (50% expected loss x 50 EUR).

In order to price these options using risk-neutral valuation, we will construct a portfolio that will have a constant payout, regardless of the outcome of the game. Suppose you play the game, you buy the put option, and you sell the call option. If the prices were of the options were the same as those established above with real-world probability, you would enter the game with 25 EUR (from selling the call for 50 EUR minus buying the put for 25 EUR). If you win the game, you receive 100 EUR, which you deliver to the person who bought the call, giving you an overall profit of 25 EUR. If you lose the game, you must pay 50 EUR, but since you bought the put, you also receive this amount, giving you an overall profit of 25 EUR. This strategy therefore guarantees that you will make a profit of 25 EUR regardless of the outcome of the coin toss. This is pure arbitrage profit.

In order to figure out the arbitrage-free prices of the options, there are two steps. The first is to recognize that a call and a put must be worth the same thing. If you play the game (which takes zero to enter) and buy a put and sell a call, you have completely hedged your exposure. If the call and put were not equivalently priced, you would be able to get an instant arbitrage profit. The second step is to replicate the value of the call using known instruments. If we play the game, we know the payouts. If we borrow money and instantly pay it back, there is no interest. So we can use a combination of these two instruments to replicate the call.

If we denote the variable s as playing the game, and b as borrowing money, the equations we need to solve are: s-b=1 and -.5s-b=0 since the value of the call should be 1 when s=1 and should be 0 when s=-.5. Solving these equations tells us that s=2/3 and b=-1/3. Therefore playing 2/3 of the game will be the equivalent of buying a call. This means that buying a call and 'going short' 2/3 of the game (being the payer of the game rather than the player), should be a neutral strategy, meaning both payouts should be the same as the price of entering the game. Therefore, the call and the put options should cost 1/3 EUR each.

To illustrate that this is correct, suppose you play the game, and you sell 1.5 call options. This should generate income of 1.5 x 1/3 EUR = .5 EUR. If you win the game, you will win 1 EUR, but you will owe 1.5 for the call options, and will end up with -.5. Likewise if you lose the game, you will have -.5, but the call options will expire worthless. In both cases, the results of the game and option payouts exactly equal the income you generated from selling the options. The risk-neutral valuations generated arbitrage-free prices. The same will also hold if you sell 1.5 put options.