# CS代写 STAT0013 – cscodehelp代写

Pricing of American Options.

Department of Statistical Science, University College London
Nov 2, 2021, STAT0013

Copyright By cscodehelp代写 加微信 cscodehelp

ABeskos Meeting5 Nov2,2021,STAT0013 1/32

Binomial Tree
Pricing of European Options Calibrating the Binomial Tree
Pricing of American Options
Value of Forward Contract (Re-Visited)
ABeskos Meeting5 Nov2,2021,STAT0013 2/32

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. ABeskos Meeting5 Nov2,2021,STAT0013 3/32 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 ABeskos Meeting5 Nov2,2021,STAT0013 4/32 1. n-Step Binomial Model 􏰊 Example: We sketch below the Binomial tree for n = 3. ABeskos Meeting5 Nov2,2021,STAT0013 5/32 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. ABeskos Meeting5 Nov2,2021,STAT0013 6/32 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. ABeskos Meeting5 Nov2,2021,STAT0013 7/32 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. ABeskos Meeting5 Nov2,2021,STAT0013 8/32 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). ABeskos Meeting5 Nov2,2021,STAT0013 9/32 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. ABeskos Meeting5 Nov2,2021,STAT0013 9/32 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−rδt􏰋pˆfn +(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. ABeskos Meeting5 Nov2,2021,STAT0013 10/32 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−rδt􏰋pˆfn +(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 0, there is an arbitrage opportunity.
ABeskos Meeting5 Nov2,2021,STAT0013 18/32

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.
ABeskos Meeting5 Nov2,2021,STAT0013 19/32

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 =max􏰓max(K−Sm,0), e−rδt􏰋pˆPm+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.
ABeskos Meeting5 Nov2,2021,STAT0013 20/32

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.
ABeskos Meeting5 Nov2,2021,STAT0013 21/32

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
ABeskos Meeting5 Nov2,2021,STAT0013 21/32

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.
ABeskos Meeting5 Nov2,2021,STAT0013 22/32

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 ABeskos Meeting5 Nov2,2021,STAT0013 23/32 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 Meeting5 Nov2,2021,STAT0013 24/32 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 ABeskos Meeting5 Nov2,2021,STAT0013 25/32 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 . ABeskos Meeting5 Nov2,2021,STAT0013 26/32 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 ] ABeskos Meeting5 Nov2,2021,STAT0013 27/32 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 ABeskos Meeting5 Nov2,2021,STAT0013 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. ABeskos Meeting5 Nov2,2021,STAT0013 29/32 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. ABeskos Meeting5 Nov2,2021,STAT0013 30/32 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. ABeskos Meeting5 Nov2,2021,STAT0013 31/32 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’. ABeskos Meeting5 Nov2,2021,STAT0013 32/32 程序代写 CS代考 加微信: cscodehelp QQ: 2235208643 Email: kyit630461@163.com