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Pricing of American Options.
Department of Statistical Science, University College London
Nov 2, 2021, STAT0013
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Binomial Tree
Pricing of European Options Calibrating the Binomial Tree
Pricing of American Options
Value of Forward Contract (Re-Visited)
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1. n-Step Binomial Model
The n-step Binomial model for a stock price is a discrete-time model with the following properties:
The asset (henceforth referred to as a stock) has value S at time 0.
Thestockpricechangesonlyatdiscretetimesδt,2δt,3δt,…,nδt.
At each time step mδt, with m = 1,2,…,n, the stock can either move up, to a price:
u × (stock value at previous time step) or it can move down to a price:
d × (stock value at previous time step)
d < 1 < erδt < u, where r is the risk-free interest rate.
At each step, the probability of an up movement is p and of a down movement is 1 − p.
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1. n-Step Binomial Model
We can build up a ‘tree’ of possible stock prices.
The tree is called a Binomial Tree, because the stock price will either
move up or down at the end of each time period.
Each node represents a possible future stock price. Note that u and d
are the same at every node of the tree.
We divide the time to expiration, T , into n steps, each of same duration
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1. n-Step Binomial Model
Example: We sketch below the Binomial tree for n = 3.
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1. n-Step Binomial Model
The Binomial model becomes more realistic as we divide the periods to maturity into a larger number of sub-periods.
As the number of time steps increases:
The length of each time period becomes smaller.
The number of possible stock values at maturity increases, thereby adding realism to the model.
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1. n-Step Binomial Model
We introduce the following notation:
Skm is the k-th possible value of the stock price at time mδt, with
m ≤ n. Notice that we have m + 1 possible prices:
Skm = ukdm−kS, k = 0,1,2,...,m.
Thus, k represents the number of upward steps among the m steps taken up to the instance mδt.
For example, at the 3rd time-step, at time 3δt, there are 4 possible stock prices: S03 = d3S, S13 = ud2S, S23 = u2dS, S3 = u3S.
At the final time-step T = nδt, there are n + 1 possible values of the stock price.
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1.1 Pricing of European Options
The following general strategy can be used for the valuation of European call options, under a n-step Binomial tree model, for any choice of steps n ≥ 1:
Computetherisk-neutralprobabilitypˆforevery1-stepBinomial ‘sub-tree’ (part of the complete large tree).
Compute the option values at the terminal nodes – these are just the payoff function values.
Work backwards and compute the option values at each internal node using risk-neutral valuations.
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1.1 Pricing of European Options
We will price an European call option that gives the right to buy a stock at strike price K, at time T. Recall that the call option payoff is max(ST −K,0).
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1.1 Pricing of European Options
We will price an European call option that gives the right to buy a stock at strike price K, at time T. Recall that the call option payoff is max(ST −K,0).
Let fkm be the kth possible value (equivalently, payoff) of the call option at time-step mδt, where m ≤ n and k = 0,1,2,...,m.
At the final step we have a payoff fkn = max(Skn − K, 0) for the call option, with possible values:
Skn = ukdn−kS, k = 0,1,2,...,n.
Let f be the price of the option at time 0.
Theaimistofindf.
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1.1 Pricing of European Options
At each step the risk-neutral probability pˆ – same at all steps – is: pˆ=erδt−d, δt=T/n
Via risk-neutral valuation (recursion backward in time), we have: Atstepn−1:
fn−1 =e−rδtpˆfn +(1−pˆ)fn, 0≤k≤n−1 k k+1 k
where, we recall that:
fkn=max(Skn−K,0), ST=Skn=ukdn−kS for 0 ≤ k ≤ n.
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1.1 Pricing of European Options
At each step the risk-neutral probability pˆ – same at all steps – is: pˆ=erδt−d, δt=T/n
Via risk-neutral valuation (recursion backward in time), we have: Atstepn−1:
fn−1 =e−rδtpˆfn +(1−pˆ)fn, 0≤k≤n−1 k k+1 k
where, we recall that:
fkn=max(Skn−K,0), ST=Skn=ukdn−kS for 0 ≤ k ≤ n.
Atstepm(m
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2. Pricing of American Options
The general pricing method for an American option is as follows.
Compute the risk-neutral probability for every 1-step Binomial
sub-tree (part of the complete large tree).
Compute the option values at the terminal nodes using the payoff function.
Work backwards and compute the option values at each internal node using risk-neutral valuation. Test if early exercise at each node is optimal. If it is, replace the value from the risk-neutral valuation with the payoff from early exercise.
Continue with the nodes one step earlier.
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2. Pricing of American Options
We denote by Pkm the kth possible value of a put option at time mδt. In the case of a European put option:
Pm = e−rδt pˆPm+1 + (1 − pˆ)Pm+1 k k+1 k
for0≤k≤mandpˆ= erδt−d. u−d
In the case of an American put option:
Pm =maxmax(K−Sm,0), e−rδtpˆPm+1 +(1−pˆ)Pm+1
k k k+1 k where Skm is the k-th possible value of the stock price at time-step
Finalcondition:Pkn =max(K−Skn,0), 0≤k≤n.
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2. Pricing of American Options
Example: We assume that over each of the next 2 years a stock price either moves up by 20% or moves down by 20%. The risk-free interest rate is 5%.
Find the price of a 2-step, 2-year American put option with a strike price of $52 on a stock with current price $50.
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2. Pricing of American Options
Example: We assume that over each of the next 2 years a stock price either moves up by 20% or moves down by 20%. The risk-free interest rate is 5%.
Find the price of a 2-step, 2-year American put option with a strike price of $52 on a stock with current price $50.
Inthiscaseu=1.2,d=0.8,r=0.05,K=52,Su=60,Sd=40, Su2 =72,Sud=48,Sd2 =32.
Risk-neutral probability: pˆ = e0.05 −0.8 = 0.6282 1.2−0.8
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2. Pricing of American Options
Example (Continued): At the terminal node, we compute:
Pk2 =max(52−Sk2,0),fork=0,1,2.
Thatis:P02 =max(52−S02,0)=max(52−Sd2,0)=20, P12 =max(52−S12,0)=max(52−Sud,0)=4and
P2 =max(52−S2,0)=max(52−Su2,0)=0.
e−0.05×1 (0.6282 × 0 + 0.3718 × 4) = 1.4147 e−0.05×1 (0.6282 × 4 + 0.3718 × 20) = 9.4636.
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2. Pricing of American Options
Example (Continued):
Payoff: K − S01 = 52 − 40 = 12 > 9.4636. Early exercise is optimal!
So P01 = max(12, 9.4636) = 12.
Payoff: K − S1 = max(52 − 60, 0) = 0 < 1.4147. Early exercise is not
optimal! So P1 = max(0, 1.4147) = 1.4147.
e−0.05×1 (0.6282 × 1.4147 + 0.3718 × 12) = 5.0894.
Payoff: K − S0 = 52 − 50 = 2 < 5.0894. Early exercise is not optimal at the initial node. Therefore:
P0 = max(2, 5.0894) = 5.0894
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3. Value of Forward Contract
A forward contract with delivery price K is pictured in the diagram below. Its payoff fT is given by:
fu =Su−K fT = fd =Sd−K
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3. Value of Forward Contract
We can now determine the no-arbitrage price for the forward in the Binomial model by replicating the contract using an amount of stock and an amount of riskless investment (risk-free bonds).
Consider the portfolio that consists of being long at 1 unit of stock, andbeingshortatKe−rT unitsoftherisk-freeinvestment.
LetST isthepriceofthestockattimeT.AttimeT thevalueofthe portfolio will be:
ST −Ke−rT ×erT =ST −K
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3. Value of Forward Contract
This is exactly the payoff of the forward derivative at time T , so we have constructed a replicating portfolio for the forward.
Thus, the price of the derivative at time 0 is the value of this portfolio at time 0, which is S − Ke−rT . Note that this is the usual value for a forward contract we saw in a previous Meeting.
For the contract to be fair at inception, this price must be 0, that is: S−Ke−rT =0
which implies:
This finding is consistent with the result from a previous Meeting.
K = SerT .
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3. Value of Forward Contract
We can now use the risk-neutral approach in the Binomial model and verify that the formula for the value f at time 0 gives the correct deliverypriceSerT (forwardprice).
Under the usual risk-neutral equation, applied now for a different payoff function, the value of the contract at time 0 is given by:
f = e−rT [ pˆ fu + (1 − pˆ)fd ]
This gives the following expression, once we plug in the appropriate
values for fu and fd:
f =e−rT[pˆ(Su−K)+(1−pˆ)(Sd−K)]
which simplifies to:
f = e−rT [ pˆ Su + (1 − pˆ)Sd − K ]
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3. Value of Forward Contract
Use now the formula for the risk-neutral probability pˆ: pˆ = e r T − d
We obtain:
−rT erT −d u−erT u−d ×Su+ u−d ×Sd−K
−rT erTSu−Sud+Sud−erTSd −rT erTS(u−d)
=e u−d −K = S − Ke−rT
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3. Value of Forward Contract
This is the same value for a forward contract as obtained with the replicating portfolio approach.
As with the replicating portfolio approach, the delivery price which gives zero initial value to the forward contract is:
Therefore, we see that the Binomial model and the related risk-neutral equations give consistent results also for other derivatives (not only options).
It is only the expression for the payoff function that changes.
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Further Reading for the Binomial Model
For further examples and explanations of the Binomial model and its applications, see the suggested reading below.
. Hull (2003, 5th Edition). Options Futures and Other Derivative Securities. Section 9.1.
& (1996). Financial Calculus. Chapter 2.
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Short Call Options
There are two counter-parties in every option contract.
The one party is taking a long position (i.e., buys the option).
The other party is taking a short position (i.e, sells or ‘writes’ the option).
The seller/writer of the option receives a premium up-front in exchange for potential liabilities later on.
The profit or loss for the seller/writer of the option is the opposite of that for the buyer of the option.
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Short Call Options
The seller of a call (short call) might be obliged (depending on the movement of the stock) to deliver stock at the strike price.
When the seller of the call already owns the underlying stock, the setting is referred to as ‘writing a covered call’. If the stock movement favors the buyer, the seller is obliged to hand over the stock. If it favors the seller, the seller retains the shares.
When the seller of the call does not own the underlying stock, the setting is referred to as ‘writing a naked call’.
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