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Binomial Pricing Model

Tom Sun

Tom Sun

Joined
2/9/15
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42
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I'm currently taking the Financial Engineering and Risk Management course on Coursera. And for one of the week's assignment, I was asked to build this:

Build a 15-period binomial model whose parameters should be calibrated to a Black-Scholes geometric Brownian motion model with: (time to maturity)T=.25 years, (initial stock price)S0=100, (interest rate) r=2%, (volatility) σ=30%, and a dividend yield of c=1%.

I have the model here. But I can't figure out what the price for the call option and put option is. Can someone take a look and direct me to the right way? Maybe I'm not calibrating the parameters to BS brownian motion, but how would I do that?
 
Calculate the price using the BS model. The price from the Binomial Model should close that.

Using BS model, for a call option, I get 2.560117 and 12.26118 for a put option.

Using Binomial model with 15 time steps, for a call option, I get 2.677662 and 12.22603 for a put option.
 
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Felix Kotey said:
Calculate the price using the BS model. The price from the Binomial Model should close that.

Using BS model, for a call option, I get 2.560117 and 12.26118 for a put option.

Using Binomial model with 15 time steps, for a call option, I get2.677662 and12.22603 for a put option.
Click to expand...

For your binomial model, is the price just the payoff at t=0?
 
Tom Sun said:
I'm currently taking the Financial Engineering and Risk Management course on Coursera. And for one of the week's assignment, I was asked to build this:

Build a 15-period binomial model whose parameters should be calibrated to a Black-Scholes geometric Brownian motion model with: (time to maturity)T=.25 years, (initial stock price)S0=100, (interest rate) r=2%, (volatility) σ=30%, and a dividend yield of c=1%.

I have the model here. But I can't figure out what the price for the call option and put option is. Can someone take a look and direct me to the right way? Maybe I'm not calibrating the parameters to BS brownian motion, but how would I do that?
Click to expand...
Does Coursera give feedback? Just curious.
 
The worksheet is named 'AmericanCall' but what you did is for a European call. However the 2 prices should not differ too much, since the dividend yield is quite small. This is what I got for the call option
- Black Scholes (European): 2.560116588272926
- BTM (European): 2.604077132966570
- BTM (American): 2.604077132966570
 
Le Van said:
The worksheet is named 'AmericanCall' but what you did is for a European call. However the 2 prices should not differ too much, since the dividend yield is quite small. This is what I got for the call option
- Black Scholes (European): 2.560116588272926
- BTM (European): 2.604077132966570
- BTM (American): 2.604077132966570
Click to expand...

The binomial model result for the call option passed. Is it possible for you to send me the worksheet? I just want to see how American Put/Call Options are correctly modeled.
 
Tom Sun said:
The binomial model result for the call option passed. Is it possible for you to send me the worksheet? I just want to see how American Put/Call Options are correctly modeled.
Click to expand...

Sure I have included the file here. The idea is simple and can be extended to other types of options that have American features: at time t you compare the value of holding, which is the risk neutral discounted payoff at time (t+delta t), against the value of early exercise, which is the immediate payoff (at time t) and take the larger.
 

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  • AmericanCall.zip
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Daniel Duffy said:
And the reason is?

For what problems would you use BM?
Click to expand...

I was trying to say that for a method that doesn't require much setup like BM there's only so much it can do. BM only works well for recombining trees, which is why I would only use BM for American vanilla options and barrier options, though in the latter case the number of time steps N must be chosen with care or else it will take very long to converge.

For lookback and Asian options, the tree grows exponentially with time so BM is not a wise choice unless you manage, through similarity reduction or whatever, to reduce it into a single state variable tree. However if such a reduction is not feasible then BM alone cannot handle the extra state variable (this is where the forward shooting grid method comes in, but it's not simple anymore!!!).
 
Le Van said:
I was trying to say that for a method that doesn't require much setup like BM there's only so much it can do. BM only works well for recombining trees, which is why I would only use BM for American vanilla options and barrier options, though in the latter case the number of time steps N must be chosen with care or else it will take very long to converge.

For lookback and Asian options, the tree grows exponentially with time so BM is not a wise choice unless you manage, through similarity reduction or whatever, to reduce it into a single state variable tree. However if such a reduction is not feasible then BM alone cannot handle the extra state variable (this is where the forward shooting grid method comes in, but it's not simple anymore!!!).
Click to expand...

Thanks that worked really well. As a follow up question, how does one build a BM for Options written on futures?
 
Tom Sun said:
I got through the first bunch of chapters, but not too deep into it.
Click to expand...

For options on futures, the underlying asset is now the futures contract. The terminal payoff will be the same as in the case of stock options, but this time the risk neutral probability will change, so you have to start from the hedging argument.
 
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I've computed the payoff of the American Put Option; the only doubt regards when is optimal exercise the option before the maturity? Have you computed this period?
 
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