Pricing a Call That Is Certain to Finish In the Money
Summary
This note explains how to value a call when the stock is guaranteed to finish above its strike. Since exercise is then certain, the payoff at maturity is the stock price minus the strike. The call can therefore be treated as a forward-like claim and replicated with one share of stock plus a short position in a risk-free zero-coupon bond whose maturity value equals the strike. Discounting the strike at the stated T-bill yield gives the present value used in the example.
The argument does not require a volatility estimate because uncertainty about whether the call will be exercised has been removed by the stated lower bound on the terminal stock price. It relies on the guarantee that the stock finishes above the strike and on borrowing or lending at the quoted risk-free rate. If that guarantee does not hold, the payoff is nonlinear and volatility can matter; the note also cautions that a standard Black–Scholes model may not suit the stated stock-price setting.
Key ideas
- When the stock is certain to finish above the strike, the call payoff is linear in the terminal stock price.
- The payoff can be replicated by holding one share and financing the strike with a zero-coupon bond.
- The strike amount is discounted at the risk-free rate to obtain the call's value in the example.
- The argument depends on guaranteed exercise and does not establish a price for a call that may expire out of the money.
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# Pricing a call when minimum stock price above strike with certainty # Pricing a call when minimum stock price above strike with certainty I am editing this question because it was originally unclear, and I didn't get the answers I was hoping for. In my finance book I have the following question T-bills currently yield 5.5 percent. Stock in Maria's Manufacturing is currently selling for $70 per share. There is no possibility that the stock will be worth less than $65/share in one year. What is the value of a call option with a $60 exercise price? The answer book is: C0=70–[60/1.055] = 13.13. I don't understand how the option price was found like this. We don't know the volatility for the stock, so how is it possible to calculate the option price. Does it have something to do with the fact that the option will definitely be exercised? Thanks a lot for your help. ## Answer by SmallChess (score 1) https://quant.stackexchange.com/a/17250 The intrinsic value is $70 - $60. However, we don't know exactly what the stock price will end up in a one-year time. But we know that it it is the best estimate for the future price in one-year. Profit in one-year = ($70 * 1/D - $60) where D is the discount factor. This profit needs to be discounted: $70 - $60 * D. You should be able to relate the discount factor to your example yourself. Note: I don't think Black-Scholes is an appropriate model for this case. The stock is not a GBM. ## Answer by Mark Joshi (score 1) https://quant.stackexchange.com/a/17694 If you know the stock will finish above the strike, then the call option becomes a forward contract since it will always be exercised. We therefore price it as a forward. Its value at maturity is $$S_T - 60.$$ We can synthesize $S_T$ with one unit of stock costing $70.$ We can synthesize $60$ with $60$ ZC bonds which costs $60/1.055$ since the yield is $5.5\%.$ So the value is $$ 70 - 60/1.055. $$
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.