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FRM- Binomial (one & two step) | Option Pricing Strategies | Python implementation - YouTube
Channel: Quantra
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Welcome to the first section
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of this course!
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Here,
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we will be equipping ourselves with
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essential mathematical knowledge required to
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understand how Options pricing
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models are derived.
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In this video lecture,
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we will discuss a
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popular technique for pricing an
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option using Binomial Trees.
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A Binomial tree is a diagram
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that flows from one starting
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node into two nodes and
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continues the same for n-layers.
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In this case,
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the binomial tree represents different
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possible paths that the underlying
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might follow in the life
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span of the option.
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An assumption here is that the
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underlying follows a random walk,
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i.e. future price movements are
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independent of past price movements.
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At each step in the
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binomial tree there is a
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defined probability of the underlying
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either moving up or moving
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down by a certain percentage amount.
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There could be many
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steps in the tree depending
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on how complex a model
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we want to create.
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Let us start by looking at
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a simple one step model
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using an example of a
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European call option.
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We are interested in finding the price
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of a European call option
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with a strike price of
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INR 110 expiring in 3 months.
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Assume that the underlying
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stock price is valued at
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INR 100 at present.
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At the end of the three
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months, there will be two
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scenarios, either the stock price
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moves to INR 120, which
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makes the option worth INR
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10, or the stock price
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moves to INR 80, in
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which case the option expires worthless.
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We assume two conditions,
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the first being that arbitrage
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opportunities do not exist.
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We will set up a portfolio
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in such a way that
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there is no uncertainty about
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the value of the portfolio
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at the end of 3 months.
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The next condition is
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that because the portfolio has
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zero risk, the return it
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earns must be equal to
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the risk-free interest rate.
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This means, this portfolio will
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give us the same returns
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as a risk-free bank deposit.
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Let us understand this better with an example.
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First let us understand the
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“no arbitrage” assumption,
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which implies that quantr
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there should not exist any
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opportunity to make risk free profit.
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Assuming this happens, irrespective
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of the market movement, that
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is, stock price moving up
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or down, the payoff from
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the risk-free portfolio containing
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the underlying and the option,
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should remain the same.
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Consider a position where we buy
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delta number of shares of
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the underlying stock and sell
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one call option.
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We will compute the value of delta,
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which makes this a risk-free portfolio.
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We will equate the values of the two
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outcomes after 3 months.
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In the first case when the
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stock price is at
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INR 120 and the value of
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the option is INR 10,
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the value of our portfolio
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is 120 times delta minus 10.
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In the second case, when the
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stock price is at INR
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80, the value of
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our long position is 80 delta
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and the option expires worthless,
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hence the portfolio has a
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value of 80 delta.
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On equating these two cases,
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120 delta minus 10 is equal to 80 delta,
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which implies 40 delta is equal to 10.
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We see that when delta is equal to
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0.25, we can create a
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risk-free portfolio.
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You can substitute this value of delta
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in both the cases and
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see that the value of
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our portfolio remains to be
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INR 20 after three months
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in either outcomes.
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Let us use the assumption that
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“a risk-free portfolio must
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have its returns equal to
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the risk-free interest rate”.
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The present value of the
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portfolio will be calculated by
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discounting the value of the
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portfolio three months in the
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future, which we have seen to be INR 20.
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Let us assume the risk-free
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interest rate to be 10%
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and the time period to
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be discounted by is 3 months or 1/4 years.
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The present value of the portfolio
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would be 20 multiplied by
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exponential raised to the power
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of negative 0.10 multiplied by 1/4.
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Therefore, the present value
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of the portfolio is equal to INR 19.50.
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Next, substituting
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this value in the equation
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that we had constructed, we
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will get 19.50, which is
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equal to 100 multiplied by
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0.25 minus the price of the option.
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By rearranging,
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we can compute the price of
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the option as 25 minus
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19.50, i.e.
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INR 5.50.
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We can extend this analysis to
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a two step binomial tree
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as shown in the diagram.
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At each step, we make
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an assumption, that the price
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can either go up or down by 20%.
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The time scale for each step is
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3 months and the option
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has 6 months to expiration.
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At the end of the
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first step, the underlying might
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either be priced at INR
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120 or INR 80, this
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is the same as the
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example we have seen previously.
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At the next step, i.e.
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at the time of expiration,
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the underlying might be at
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either INR 144, INR 96
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or INR 64, this is
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computed using the assumption that
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the prices can either go
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up or down by 20%.
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Now, for computing the option
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price, we will sequentially compute
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the price of the option,
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first at three months in
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the future at points B
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and C following which we
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will then compute the price
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of the option today at point A.
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The value or price of the option at
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points C, E and F will be 0,
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as the option will expire out of
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the money if it takes the bottom path.
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The price of the option at point
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B will be computed by
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considering the branch B-D-E to
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be a one step binary tree.
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Once we compute the price at point B,
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we will consider the A-B-C branch
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and compute the option price
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at point A. We will
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apply our learnings, from the
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one step binary tree example,
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twice, to arrive at the
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option price at point A.
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Applying this methodology, the fair
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value of the option is
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computed to be INR 10.26.
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You can try this as
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an exercise by computing the values yourself.
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In the upcoming units,
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there will be a couple of multiple choice
questions
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to test your understanding of
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binomial trees.
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Following which, you
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will be provided with some
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reading material on the derivation
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of BSM using Binomial Tree
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and another reading unit on
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Wiener Process and Ito’s Lemma.
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