Black-Scholes model shares common intuitions with risk-neutral option pricing mo

Question

Black-Scholes model shares common intuitions with risk-neutral option pricing model (also known as the binomial option pricing model). One of the biggest underlying assumptions of risk-neutral (binomial) model is that we live in a risk-neutral world. In a risk-neutral world, all investors only demand a risk-free return on all assets. Although the risk-neutral assumption is counterfactual, it is brilliant and desirable because the risk-neutral assumption doesn’t affect the prices of an option. In another word, prices of an option are exactly the same with or without the risk-neutral assumption. Explain why that is the case, and how risk-neutral assumption greatly simplifies the calculations of risk-neutral option pricing approach.

Explanation / Answer

Probability can be called physical and risk neutral.

Physical probability is like probability of every outcome. But these probabilities are less intuitive in Finance and pricing of financial assets.Lets say if a stock have 50% chance of having worth $20 and a 50% chance of being worth $10 at some future time t1.So weightage avg of future outcome is (.5*20 +.5*10 =15) $15 but in markets this stock should not worth $15 because there is a risk involved with a considerable chance of being it only 10.So to attract investor this stock should be worth less than $15 and this premium actually compensate the risk neutral investor.

So when we deal with physical probabilities then to compute value you have to take the probability-weighted average of all the prices @t1 and then add value that compensate for risk. But this value is hard to compute.

Risk-Neutral Probability

Instead of computing like this, we can embed risk-compensation into our probabilities. That is we can adjust the probability of good market outcome downward and bad market outcome to upward. These probabilities can be used then to price financial asset like Option and bond. A price can be arrived by multiplying every outcome by its risk neutral probabilities and then discount the sum by risk free rate.

Risk-Neutral Pricing

Now let’s assume that a stock had two possible value tomorrow U and D.Risk neutral probability for u is Q.

(U:up and D:down)

Price = [ Uq + D(1-q) ] / e^(rt)

The exponential there is just continuous discounting by the risk-free rate.

Now in above equation if one has prespecified that what U and D is going to be(Assume 20 n 10 given at top ex) then we can simply compute Q by looking at risk free rate and spot price in current market.

Now let assume an option. In the case of an option we do know that what will be worth of option if prices go up or down. We have fixed time(expiry) and risk free rate along with spot price of underlying that is the reason that pricing of option will be not be affected much even if we don’t adjust our probabilities. Another reason to this is that one side movement is also capped as if you bought a call option and your downward payoff will always be zero and weightage avg will not be affected.

Computed option pricing thorough this method is easy as we need to do the three step process.

1-Identify the model’s risk-neutral probability.(Does not affect much)

2-Take expectations of the option’s payoffs under the risk-neutral probability.

3-Discount these expectations back to the present at the risk-free rate

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