Stochastic Calculus for Quants | Risk Neutral Pricing
Mathematical insights behind Risk Neutral Expectation Pricing Formula
Risk neutral pricing
Stochastic Calculus may have many applications, I wrote general overview week ago but today what I would like to introduce as more in-depth example is the risk neutral pricing, have you ever wondered why models like Monte Carlo Simulation or Black and Scholes model actually work, risk neutral pricing theory lie at very core of those as well as many more latest asset pricing models. Despite most Quantitative professionals do not create new financial products, I believe that risk neutral pricing lies in the foundation of understanding the mathematics behind stochastic calculus.
NOTE: This article like last week’s is quite math heavy, if you don’t enjoy that much dive in, stay in the loop, next week we will focus on top 10 Machine Learning models used in quantitative finance and the we’ll have a break from heavy math for a while
Risk neutral pricing is a method for pricing financial derivatives that assumes that investors are indifferent to risk. This means that the price of a derivative is based on the expected value of the derivative's payoff, discounted at a risk-free rate.
In this article we will try to examine math insights of Risk Neutral Expectation formula, you don’t really need it when you are tackling European option because there is already very well defined Black and Scholes formula however, when it comes to more complex financial products such as American option or different variations of exotic ones, knowing in and our risk neutral pricing theory and Risk Neutral Expectation Pricing Formula may really come in handy!
Broad definition
Let’s start from the basis, the risk neutral part of it is that in stochastic calculus for pricing derivatives change in underling actually doesn’t matter (check Binominal Model explanation!). Key property of this model is that:
There is perfect correlation between value of an option and underlying
And that assumption or property we are going to exploit.
Let’s take an example, having portfolio of:
Long Option Contract
Shorting number of units in the underlying
And by that simple trick we essentially eliminate risk from the equation, at least in theory. If hedging is done perfectly then payoff would look similarly to a bond.
In comparison to Partial Differential Equations (PDEs) such as Black and Scholes described in previous article, risk neutral pricing does not require such an amount of computational power, also PDEs realistically due to their multidimensionality with complex contract features don’t have closed-form solution. Finally, by taking the expectation we can approximate value of an option based on famous Monte Carlo simulation (which I will describe in the next article).
Math behind the risk neutral pricing
1 period binominal model
Starting from portfolio binominal model we will attempt to get a mathematical formula for fair price of an option based on probability. Having closer look on probabilities, from start time, t=0, at the maturity (t), Spot Price S and option price C can go in 2 directions, upwards or downwards. Probability of option price going up discounted with respect to risk-free rate, rate of change should be equal to the option price at time t=0.
Having portfolio of n units and k stock and assumption that there are no arbitrage opportunities, P=1, this can be represented by formula:
Where:
C0 = Option price at the beginning
qu/d = Risk-free probability of potential upside/downside
Cu/d = Value of an option at potential upsite/downside
And can be wrapped up as:
Where:
EQ = Expectation probability
Ct = Future payoff
The fundamental theorem of asset pricing
Having our formula for 1 period binominal model leads us to the fundamental theorem of entire asset pricing that is:
While there is no arbitrage opportunities (assumption), there exists a probability measure Q, equivalent to P (physical probability, that is probability resulting from individual properties of the underlying), such that the discounted prices of all tradable assets are martingales with respect to Q.
Before we will go any further let’s clarify what martingales are.
Martingales:
In simple terms, a martingale is a stochastic process that, on average, neither gains nor loses value over time.
Now, imagine a gambler playing a fair game where they have a 50-50 chance of winning or losing on each bet. A betting strategy that is a martingale would be one where the amount of money the gambler bets on each round depends on their current wealth, but such that their expected wealth after the next round is the same as their current wealth. This means that, on average, the gambler neither gains nor loses money over time.
Mathematically, a martingale is a stochastic process that satisfies a property called the "martingale property." This property states that, given the information available up to some point in time, the expected value of the process at any future time is equal to its current value. In other words, a martingale is a process that is "fair" in the sense that its expected value does not change over time, given what has already been observed
Q and P
Having considered the properties above we can come to conclusion that there is such a risk free probability (Q) that is equal to real-life probability (P): Q=P, and that is what models like Black and Sholes call fair price!
Radon-Nikodym
Having the kitchen behind fair price set, it is apparent that in order to apply risk-free model we need certain kind of shift between P and Q, here Radon-Nikodym derivate theorem comes in handy.
Firstly, let’s strip down our assumption a bit more, in our set of all probabilities (Ω) we have probability of upsite and downside (ὠu/d), therefore Ω = (ὠu, ὠd). And as there are no arbitrage opportunities: P(ὠu) = pu >0 and P(ὠd) = pd = 1 - pu
Having those symbols in mind, we can perform following computations:
Q(ὠu) = qu = pu(qu/pu) = (qu/pu)*P(ὠu)
Respectfully, modifying downside we will come to the following equation: (1- qu)/(1-pu) *P(ὠu)
Having that on place, let’s perform our probability shift using Radon-Nikodym derivative which states that
Z – dQ/dP
Where Z can be read as the difference between new and old probability measure
Above equation is the bridge between real-world probability and our risk neutral formulas, with Radon-Nikodym we can translate our Co formula to:
Having above in mind, objective for neutral pricing will be to find a measure Zt so that discounted stock price process is a martingale.
Looking for the martingale
Revising the Geometric Brownian Motion (GBM) discussed in last article, as it is under the Physical probability P, {St} is not a martingale because, straight from the assumptions μS is not equal zero. Therefore, we need to identify change of measure in that is a Brownian motion under probability Q.
Stock market dynamics, under the GBM discounted stock process is a martingale so {St/Bt} under GB dynamics is a Q martingale (According to First Fundamental Theorem)
Therefore we need to find d(St/Bt) and we could use Ito Calculus for that:
As little reminder, our Brownin Motion orginal formula goes:
For help comes the main part of the Risk Pricing part of this article, the Girsanov’s Theorem which would enable us to change the measure from the real world to risk neutral world. The process of shifting from the physical probability measure to the risk-neutral measure involves applying the Girsanov kernel to the drift component of the underlying stochastic process.
Under the measure Q, the process X(t) = B(t) - ∫[0,t] θ(u) du is a Brownian motion, where θ(u) is a predictable process defined by:
θ(u) = σ(u) * (dZ(u) / Z(u))
Here, σ(u) is a matrix or vector of the instantaneous volatilities of the underlying assets. The process X(t) is known as the Girsanov process or the exponential martingale.
The formula for pricing a derivative under the risk-neutral measure Q is:
V(t) = e^(-r(T-t)) * E_Q [h(S(T)) | F(t)]
where V(t) is the price of the derivative at time t, r is the risk-free interest rate, T is the time to maturity, S(T) is the price of the underlying asset at maturity, h(.) is the payoff function of the derivative, and F(t) is the filtration of the underlying asset up to time t under the physical probability measure P.
To use the formula, we first calculate the Girsanov kernel or hat, which is the process Z(t) that allows us to transform from the physical probability measure P to the equivalent risk-neutral measure Q. We then use the Girsanov process X(t) to modify the drift of the underlying asset's stochastic process, and calculate the expected value of the derivative under the risk-neutral measure Q
Risk Neutral Expectation Pricing Formula
Now let’s wrap the theory back into universal Risk Neutral Expectation Pricing Formula. We just displayed that there exist a measurable shift between physical and risk neutral probability, option fair price can be set as the martingale of discounted expected probability. All of that leads us to the Risk Neutral Expectation Pricing Formula, which remains relevant not only measuring European options (Like Black and Scholes) but helps you to examine more sophisticated variations of payoff structure. You are most likely to not go through reasoning above on daily basis but it all concludes in formula I am about to present.
We can conclude that it is possible to construct a replicating portfolio (S,B) with value {Vt} with t within [0-T] as discounted stock process {St/Bt} is a Q martingale. Therefore any discounted portfolio (S,B) is also a Q martingale
Under Q, discounted value process exists that is a martingale {e-rTVt}
e-rTCT = EQ[e-rTCT|Ft] = EQ[e-rtVT|Ft] = e-rTVt
Therefore any options or Derivate price can be constructed with the above replication approach which means that we may assign fair value to ANY option under this formula according to the prove above.
Above formula is more popularly stated as follows and that’s how you probably would use it in most cases:
Ct/Bt = EQ[Ct/Bt|Ft]
And with the very handy help of Risk Neutral Pricing you can go ahead and enrich your measures regarding non-European option.
Next articles
We will see more practical use of the maths in this article with Monte Carlo Simulation model which we will construct in following weeks, but before that happens next 2 articles will give us a little breath from math as we will dive more into technological aspect of quantitative finance, that is I will list Top 10 most widely used Machine Learning Algorithms in Quantitative Finance something that every quant should have at some level in their skill set.
Thank you for your time
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Tomasz