A hedge funds wants to use a binomial tree to replicate a put option on a stock. The time step is 0.25 years and the interest rate for continuous compounding is 1.5%. At time t, the stock price is $94 per share, the put's delta is -0.4378 per share, and B = 47.3813 per share. Over the first time step, suppose the stock price moves up. In the upper node at t+0.25, the stock price is $97.499 per share, the put's delta is -0.3604 per share, and B = 40.0129 per share.

The option contract size is 400 shares.

What trades should be done at time t+0.25 to rebalance the replicating portfolio?

Question

A hedge funds wants to use a binomial tree to replicate a put option on a stock. The time step is 0.25 years and the interest rate for continuous compounding is 1.5%. At time t, the stock price is $94 per share, the put's delta is -0.4378 per share, and B = 47.3813 per share. Over the first time step, suppose the stock price moves up. In the upper node at t+0.25, the stock price is $97.499 per share, the put's delta is -0.3604 per share, and B = 40.0129 per share.

The option contract size is 400 shares.

What trades should be done at time t+0.25 to rebalance the replicating portfolio?

BlackTom AI · Accepted Answer

Delta per option went from -0.4378 to -0.3604, so the replicating stock position must be adjusted immediately.

1) Relevant concepts
- Delta gives the number of shares of stock to hold per option (negative for a long put implies a short stock position).
- For N = 400 option units, total stock holding = delta_per_option * N.
- To change the stock position by Δshares at the new node, buy (if Δshares > 0) or sell (if Δshares < 0) that many shares at the node price. The cash needed (or freed) equals Δshares × current stock price, and is financed by adjusting the cash/lending account.

2) Step-by-step calculation
- Initial total stock holding at time t: -0.4378 × 400 = -175.12 shares (short 175.12).
- New total stock holding at t+0.25 (upper node): -0.3604 × 400 = -144.16 shares (short 144.16).
- Change in stock holding = -144.16 − (−175.12) = +30.96 shares. So you must buy 30.96 shares to reduce the short position.

3) Cash/ lending required
- Stock price at the upper node: $97.499 per share.
- Cash required to buy 30.96 shares = 30.96 × 97.499 = 3,018.56904 ≈ $3,018.57.
- That cash is obtained by reducing the lending (bank) position by $3,018.57.

4) Why this answer is correct
- The buy of 30.96 shares exactly moves the replicating portfolio from the old delta to the new delta for 400 option units.
- The precise cash change equals the purchase cost at the node price and therefore reduces the bank/lending amount by $3,018.57.

5) Why the other options are incorrect
- Options that say “short” 30.96 are wrong sign: the portfolio must buy shares (reduce the short), not increase it.
- “Short 36.04” or other share counts do not equal the computed change (30.96) from the delta difference.
- Answers involving lending/borrowing amounts like $4,001.29 or $16,005.14 correspond to different (incorrect) share quantities or misinterpretations of per-share B totals.

Conclusion: The correct rebalancing is to buy 30.96 shares of stock and reduce lending by $3,018.57, which matches the provided answer.

类似问题

A single branch on a Binomial tree is represented by the following diagram. Calculate how many shares of the underlying asset (delta) are required to delta hedge.Enter your answer to two decimal places. If your answer is negative, then enter the negative sign.

A project has a required return of 12.6 percent, an initial cash outflow of $42,100, and cash inflows of $16,500 in Year 1, $11,700 in Year 2, and $10,400 in Year 4. What is the net present value?

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