| Course | GF561 Derivatives and Hedging |
|---|---|
| Unit | Unit 4 |
| Paper type | Option valuation walkthrough |
| Length | About 850 words, 6 pages double-spaced |
| Format | APA 7 student paper |
| School | Purdue University Global |
| Program | MS in Finance |
| Updated | October 2026 |
Free sample paper for GF561 Unit 4
Every Node Checkable: A Unit 4 Option Valuation Walkthrough From Binomial Tree to Black-Scholes
[Student Name]
Purdue University Global
GF561: Derivatives and Hedging
Unit 4 Option Valuation Walkthrough
[Instructor Name]
[Date]
The share and every input are illustrative; the articles and textbook cited are real.
Inputs
Share price $48. Strike $50. Nine months to expiry, split into three steps of three months. Continuously compounded risk-free rate 4 percent. Volatility 32 percent a year. No dividends.
Step 1: Building the Tree's Factors
Up factor. e raised to volatility times the square root of the step length: e raised to 0.32 times 0.5, or about 1.17351.
Down factor. The reciprocal, about 0.85214.
Risk-neutral probability of an up move. The growth factor for one step, e raised to 0.04 times 0.25, or 1.01005, less the down factor, divided by the up factor less the down factor: 0.15791 divided by 0.32137, about 0.49136.
Discount factor per step. e raised to negative 0.01, about 0.99005.
These are the parameters Cox, Ross and Rubinstein proposed, chosen so that the tree's volatility matches the share's as the steps shrink (Cox et al., 1979).
Step 2: The Share Price Tree
1. Today: $48.00.
2. After three months: $40.90 or $56.33.
3. After six months: $34.86, $48.00 or $66.10.
4. At expiry: $29.70, $40.90, $56.33 or $77.57.
Step 3: The European Call
At expiry. Payoffs are the share price less $50 where positive: $0, $0, $6.33 and $27.57.
Six months. Each node is the discounted probability-weighted average of the two nodes it leads to. At $66.10: 0.99005 times 0.49136 times $27.57 plus 0.50864 times $6.33, about $16.60. At $48.00: about $3.08. At $34.86: $0.
Three months. At $56.33: about $9.63. At $40.90: about $1.50.
Today. About $5.44.
Step 4: The European Put
At expiry. $50 less the share price where positive: $20.30, $9.10, $0 and $0.
Six months. At $34.86: about $14.65. At $48.00: about $4.58. At $66.10: $0.
Three months. At $40.90: about $9.60. At $56.33: about $2.31.
Today. About $5.96.
Check by put-call parity. The call less the put, $5.44 less $5.96, is negative $0.52. The share less the present value of the strike, $48 less $50 discounted nine months at 4 percent, $48.52, is also about negative $0.52. The tree agrees with parity to within rounding.
Step 5: The American Put
An American put can be exercised at any node. At each node, its value is the larger of the discounted continuation value and the immediate exercise value.
At the $34.86 node after six months. Continuing is worth 0.99005 times 0.49136 times $9.10 plus 0.50864 times $20.30, about $14.65. Exercising pays $50 less $34.86, or $15.14. Exercise is worth more, so the node's value becomes $15.14.
Working back. At $40.90 after three months, the value rises from $9.60 to about $9.86. Today the American put is worth about $6.09, 13 cents more than the European put.
Early exercise pays at the lowest node because holding a deep in-the-money put means waiting to receive a fixed $50 that could be earning interest now, and once the share is low enough the interest outweighs the small chance of the share falling further.
Checking the Tree With a Replicating Portfolio
The call's value can be confirmed without probabilities. At the first step, the call is worth $9.63 if the share rises to $56.33 and $1.50 if it falls to $40.90. A portfolio holding about 0.527 shares, the change in call value divided by the change in share price, and borrowing about $19.85 would be worth $9.63 in the up state and $1.50 in the down state. Today that portfolio costs 0.527 times $48, about $25.30, less the $19.85 borrowed, or about $5.44, the same as the tree's value. Because the portfolio copies the call exactly, no other price for the call could persist without an arbitrage.
Sensitivity to Volatility
Raising volatility from 32 to 40 percent widens the tree: the up factor becomes about 1.2214 and the extreme expiry prices move further apart. Both the call and the put become more valuable, since an option holder gains from large moves in one direction and loses at most the premium in the other.
Step 6: Black-Scholes Beside the Tree
The Black-Scholes model values the European options continuously rather than in three steps (Black & Scholes, 1973). With the same inputs it gives a call of about $5.06 and a put of about $5.58.
Why the tree differs. Three steps allow only four possible prices at expiry, so the tree approximates the smooth distribution of outcomes roughly. The tree overstates both values here by about 38 cents. With more steps, the binomial values converge toward Black-Scholes; a tree of fifty or more steps typically lands within a cent or two (Hull, 2022).
Why the American put has no formula. Black-Scholes assumes exercise only at expiry. The tree's ability to compare continuation and exercise at every node is why lattices remain the standard tool for American options.
Conventions
Node values are rounded to cents; intermediate products carry five decimals. Up moves are listed from the lowest node upward at each date.
References
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637-654. https://doi.org/10.1086/260062
Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of Financial Economics, 7(3), 229-263. https://doi.org/10.1016/0304-405X(79)90015-1
Hull, J. C. (2022). Options, futures, and other derivatives (11th ed.). Pearson.
GF561 Unit 4 instructions, in plain terms
Unit 4 of GF561 asks you to value options with a binomial tree and compare the result with Black-Scholes. List the inputs, then derive the up and down factors, the risk-neutral probability and the per-step discount factor, citing the model they come from. Build the share price tree in full. Value a European call and put by working backward from expiry, showing each node, and check them against put-call parity. Value an American option by comparing continuation and exercise at every node, and explain where and why early exercise occurs. Confirm the tree with a replicating portfolio. Then compute Black-Scholes values and explain why they differ from the tree and why the formula cannot value the American option. State rounding conventions. Cite the original model papers and the course text in APA 7.
Inside the GF561 Unit 4 example
The sample starts with an inputs list and derives each tree parameter, citing Cox, Ross and Rubinstein. The share tree lists every node from $48 to the four expiry prices. The call and put sections work backward step by step, and a parity check confirms the put. The American put section compares $14.65 of continuation with $15.14 of exercise at the lowest middle node, and the highlighted sentence explains why interest on the strike makes early exercise pay. A replicating portfolio of about 0.527 shares and $19.85 of borrowing reproduces the call, and a short note shows a wider tree under 40 percent volatility. The Black-Scholes section, citing Black and Scholes and Hull, explains the 38-cent gap and why lattices value American options.
Where the points sit in the GF561 Unit 4 rubric
GF561 graders award most points for a tree that can be checked at every node, with parameters derived correctly from the inputs. European values must follow from backward induction with the risk-neutral probability, and a parity check earns credit for verification. The American option section is the analytical center: comparing continuation with exercise at each node and explaining the economic reason for early exercise shows real understanding. A replicating portfolio confirmation demonstrates that the value rests on no-arbitrage, not only on a formula. Instructors in the MS in Finance program value a clear explanation of the gap between tree and formula. Stated rounding rules, a tidy layout and APA 7 entries for Black and Scholes, Cox and Hull account for the rest.
GF561 Unit 4 help from the desk
The commonest GF561 Unit 4 mistakes are using real-world rather than risk-neutral probabilities, or discounting at the wrong per-step rate. Students also compute the down factor incorrectly or forget that it is the reciprocal of the up factor in this model. American options are valued like European ones, missing the exercise comparison. Parity checks are skipped, so errors in the put go unnoticed. The gap between tree and Black-Scholes is blamed on a mistake rather than on the small number of steps. For any option your prompt supplies, follow the same order: parameters, share tree, European values with a parity check, the American comparison, a replication check and the formula comparison.
Get a GF561 Unit 4 example written to your instructions
Include the share price, strike, rate, volatility, expiry and step count from the GF561 Unit 4 prompt, plus the rubric, and note whether an American option is required. Each node gets printed and the closed form is checked against the tree. The first custom sample costs nothing, with 24-48h the usual wait. The paper above is an original model document written by our desk, not a submitted student paper and not an official Purdue University Global document.
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GF561 Unit 4 questions, answered
Why is the American put worth more than the European one?
Because the holder can exercise before expiry, and in this case doing so beats waiting at one node. Deep in the money, the interest earned on the strike received early outweighs what the option could still gain. The sample shows the exact comparison, 15.14 against 14.65, and the extra 0.13 of value that single decision produces at the root of the tree.
Can Black-Scholes value the American put?
Not directly. The formula assumes exercise only at expiry, so applying it to an American put understates the value whenever early exercise pays. For a call on a share paying no dividends, early exercise is never optimal and the formula still works. The sample states this limit beside the calculation and relies on the lattice for the American figure.
Where can I find a free GF561 Unit 4 sample paper?
This page carries a full GF561 Unit 4 option valuation walkthrough with a three-step binomial tree, European and American puts, a replicating portfolio and a Black-Scholes comparison, with APA 7 references.
Why does the three-step tree differ from Black-Scholes?
With only three steps, the tree allows four possible prices at expiry, a rough stand-in for a smooth distribution. As steps increase, the tree's values converge toward the formula's.
Where should the American put be exercised early?
At the node where the share has fallen to $34.86 after six months. Exercising pays $15.14, more than the $14.65 the put is worth if held, because the strike received now can earn interest.