Why Option Pricing Uses Risk-Neutral Expected Payoffs

# Why is option value different from discounted CF


# Why is option value different from discounted CF












as stated: why other assets' value can be determined by taking into consideration their expected cash flow (CF)? I read an argument which refers to arbitrage, but I wonder is there an additional simple argument from a theoretical point of view. thanks,

## Answer by Olórin (score 1, accepted)

https://quant.stackexchange.com/a/33486

Discouting a cash-flow to get its present value only works for non random cash-flows. In the option case, the cash-flow (the option's pay-off) is unknown as it depends on the value of the underlying at maturity, value that you don't know. Therefore you can't simply discount the option pay-off/cash-flow to get the option price/present value of the cash-flow.

You have to discount first indeed and then take an expectation (with respect to the risk neutral measure) of the discounted payoff to get the option's price. This is general, and for non-random future cash-flow (fix payments) you recover the discounting method. That is what essentialy states the fundamental theorem of asset pricing which is connected to the notion of arbitrage (to the non existence of it, precisely).

To be more theoretical, under the hypothesis that arbitrage do not exist, there exists a numéraire $N$ and a probability measure $\mathbf{Q}^N$ associated to it such that each tradable asset (of expiry $T$) price $X$ is a (local) martingale under the numéraire $N$, that is $X/N$ is a (local) martingale : $$\forall t\in [0,T], X_t = N_t \mathbf{E}^{\mathbf{Q}^N}\left[ \left.\frac{X_T}{N_T}\right| \mathscr{F}_t\right].$$ (Please note I don't give full precision here, just rough ideas allowing to get a formula for the price.) In particular, today's price at $t=0$ is the expectation $$X_0 = N_0 \mathbf{E}^{\mathbf{Q}^N}\left[ \frac{X_T}{N_T}\right].$$

If your asset pays a known cash-flow $c$ at $T$ then $$X_0 = N_0 c \mathbf{E}^{\mathbf{Q}^N}\left[ \frac{1}{N_T}\right].$$

Often the measure $N$ is the bank-account numéraire measure (also called risk neutral measure), the numéraire being $N_t = e^{\int_0^t r_s ds}$ where $r_s$ is the time $s$ instantaneous interest rate, and then $$X_0 = c\mathbf{E}^{\mathbf{Q}^N}\left[ e^{-\int_0^T r_s ds}\right].$$ In this case $\mathbf{E}^{\mathbf{Q}^N}\left[ e^{-\int_0^T r_s ds}\right]$ is the price today of the zero-coupon of maturity $T$ (the product that pays you $1$ at $T$), this price is noted $P_{0,T}$ and called the discount factor of maturity $T$, so that simply $$X_0 = c P_{0,T}.$$

You see : to get the price today of a known (that is, non random) cash-flow $c$ payed at $T$, you simply multiply $c$ by the discount factor $P_{0,T}$.

Of course, it can be more complex, think of call options for instance, for which for a strike $K$ you'd have $$\textrm{Call price}_{t=0} = k\mathbf{E}^{\mathbf{Q}^N}\left[ e^{-\int_0^T r_s ds}\left( S_T - K \right)_{+}\right]$$ would $S$ be the underlying.

For a neat introduction to all of this and to quantitative finance, you have the following references :

- chapter 2 of Pierre Henry-Labordère's "Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing"

- chapter 1 of volume 1 ("Foundations and Vanilla Models") of Andersen's and Piterbarg's "Interest Rate Modeling"

They can be a bit "rough", but there are really worth the pain.

## Answer by Daneel Olivaw (score 0)

https://quant.stackexchange.com/a/33490

I guess the existence of a market is the gist of the question. It is easier to understand if you compare the insurance industry with the investment banking industry:

- Broadly speaking, when an insurer underwrites an insurance policy, it is unable to hedge its risk by subscribing the same insurance policy it just underwrote $-$ there exists reinsurance treaties that allow insurers to offload part of their risk, but they tend to be aggregate (i.e. on a portfolio of policies basis) and non-linear (see non-proportional reinsurance article in Wikipedia if you are interested). Hence, a risk management approach based on expected cash flows and the law of large numbers is a sensible approach.

- On the contrary, investment banks can hedge their derivative exposure by entering a trade which corresponds to a contrary, offsetting position to the one they just underwrote.

The main difference between these 2 situations is the existence of a (reasonably) complete, (reasonably) liquid and (reasonably) well-functioning market, which leads to the following claims:

- When there is a functioning market in the risk you trade, your price must be determined by the absence of arbitrage.

- When there is no functioning market for this risk, your price must rely on your expected cash flows and the law of large numbers.

So, getting back to the purely financial realm and forgetting about insurance, I guess you can apply this type of reasoning to:

- Other assets for which there is no sufficiently liquid market for them, such as Commercial Real Estate;

- In general, corporate finance projects, companies investment plans, etc. which have traditionally been evaluated through the discounted cash flows method.

To finish off, I will include some excerpts from the introduction (Chapter 1) of Baxter's and Rennie's excellent book Financial Calculus':

> With markets where the stock can be bought and sold freely and arbitrarily positive and negative amounts of stock can be maintained without cost, trying to trade forward using the strong law would lead to disaster […]. […] But the existence of an arbitrage price, however surprising, overrides the strong law. To put it simply, if there is an arbitrage price, any other price is too dangerous to quote. […] The strong law and expectation give the wrong price for forwards. But in a certain sense, the forward is a special case. The construction strategy $-$ buying the stock and holding it $-$ certainly wouldn’t work for more complex claims. The standard call option which offers the buyer the right but not the obligation to receive the stock for some strike price agreed in advance certainly couldn’t be constructed this way. If the stock price ends up above the strike, then the buyer would exercise the option and ask to receive the stock – having it salted away in a drawer would then be useful to the seller. But if the stock price ends up below the strike, the buyer will abandon the option and any stock owned by the seller would have incurred a pointless loss. Thus maybe a strong-law price would be appropriate for a call option, and until 1973, many people would have agreed. Almost everything appeared safe to price via expectation and the strong law, and only forwards and close relations seemed to have an arbitrage price. Since 1973, however, and the infamous Black-Scholes paper, just how wrong this is has slowly come out. Nowhere in this book will we use the strong law again. […] All derivatives can be built from the underlying $-$ arbitrage lurks everywhere.