One of the Chapter 3 assignments X3.7 considers a swaption with an underlying swap of pay 4% fixed and receive floating at a future date. Given the information in the question the value of the forward swap rate for this swap is about 4.33%. Is the person holding this option 'in the money'? The question then asks if the vol of the swap rate increased would the swaption increase in value: I understand that the value of the option would always be higher, the solution indicates that as the volatility of the swap rate increases the probability of the option expiring in the money is higher. Would this be the case if the option was out of the money? Secondly the solution indicates that the higher volatility would result in a higher probability of the swap rate falling further which would result in greater profits on exercise. Is this correct? Surely if the swap rate fell below 4% I would prefer not to pay 4%, but the lower rate? Pls help, maybe I am completely misunderstanding the basic principles.
Hi, I'm at work right now and don't have my assignments to hand so I'd rather look at them before answering a couple of your points - I'll get back to you this evening. However, for your volatility question I'm going to have a go at giving you an intuitive way of thinking about it. Imagine a sine curve oscillating about the y axis. If you were just to look at everything above the axis - this could (with a bit of imagination) represent the intrinsic value of a derivative as the underlying fluctuates over time. Where the option is in the money (i.e. the sine curve sits above the axis) you have a positive value, and where the option is out of the money - the intrinsic value is equal to zero and the value just sits on the axis. At expiry, the process will stop. If the option is out of the money you wouldn't exercise it and you would recieve zero, if the option is in the money, you would exercise the option and it would be worth an amount of money. The expected payoff would depend on the height of the sine curve but the ultimate payoff would depend whereabouts it finished on the curve. i.e. how high up the x axis. The thing that determines the height of the sine curve (and therefore the expected payoff) is the volatility. The greater the volatility the bigger the curve and the greater the expected finishing position or payoff. Hopefully you should see that whether the option is in or out of the money is irrelevant - as the volatility increases the expected payoff increases. You have a one-sided payoff function where the probability of a more extreme result increases with the volatility underlying it. That's kind of how I think about it, I hope I haven't confused you further though. Best of luck.
Vol Thank you for your reply. I am not sure if I understood your reply correctly, but I am more concerned about the statement that : 'the probability of making a profit increases'. Am I interpreting it correctly if I think of it as a normal distribution around the current price (or is it around the forward price E(V)) and if the strike price is below the current price (the middle of distribution) then increasing the volatility would decrease the probability of making a profit, but increase the expected value of the profit. If the strike price is above the current price then increasing the volatility would increase the probability of making a profit and increase the expected value of the profit. I am not to worried about this point (maybe because I don't understand it too well), but pls help on the others. Thank you.
I'll have another go. Sorry about my answer earlier - I hadn't read your questions properly. I think that you get the volatility principle fine and it's best not to kill your head thinking about it too much. I just took a look at that question and would agree with your interpretation. I think that the answer goes the wrong way. A lower swap rate would make the swaption less valuable as you are paying fixed and receiving floating. Clearly a lower floating rate is less valuable. The irony is, I did this question over the weekend (correctly), read the answer quickly, thought I might have misread it, frowned and moved on without giving it any further thought. So to answer your qns: "Given the information in the question the value of the forward swap rate for this swap is about 4.33%. Is the person holding this option 'in the money'?" Yes, because the swap rate of 4.33% is above the strike rate of 4%. Although be aware that an option can have positive value whilst being out of the money as an option value consists of two parts, intrinsic value (the degree to which it is in or out of the money) + time value (a measure of value based on the probability of going further into the money) I understand that the value of the option would always be higher, the solution indicates that as the volatility of the swap rate increases the probability of the option expiring in the money is higher. Would this be the case if the option was out of the money? NB It's probably easier to think of the underlying interest rate having the volatility rather than the swap rate, which is a function of the yield curve. Anyway, I don't think that as the volatility increases, it increases the probability of expiring in the money - I think that IF it expires in the money, the expected payoff is higher. Consider 2 simple examples. a) Toss a coin to see whether you win £1 or lose a £1. b) toss for £10 or -£10. Both have E(x) = 0 b) has a higher volatility (represented by standard deviation of returns). In b) the chance of winning is the same but the expected payoff, if you win, is higher. Therefore, going back to your question, whether the option is in or out of the money is irrelvant when you consider the effect of a change in volatility.
Having said the above, maybe a higher volatility does also increase the probability of finishing in the money. Consider an option deep out of the money with very low volatility. It would not be expected to finish in the money, as the underlying would not be expected to fluctuate that much. The same deep out of the money option with a very high volatility could be expected to have a wildly fluctuating underlying, increasing the chances considerably of finishing in the money. Hmm. Think I just confused myself.
1. The person holding the swaption is in-the-money. This is because they have the option to pay 4% fixed and receive floating. The forward swap rate value of 4.33% indicates that (based on today's term structure), the expected floating rate payments over the term of the swap have the same present value as a fixed rate of 4.33% over that period. So, the swap holder has the option to pay 4% fixed in return for floating payments with a value equivalent to 4.33% fixed. The swap holder is therefore looking at a possible profit of 0.33% on each payment (based on the current term structure, which will of course change between now and the strike date of the swaption). 2. There is a typo in the solution to part (iii)(b), for which I apologise. The value of any option always increases with the volatility of the underlying asset. This is because the additional volatility means that there is more likelihood of the underlying asset moving in your favour (which here means the swap rate increasing) between now and the strike date, thereby generating extra profits for you. Although there is also more chance of it moving against you (which here means the swap rate falling), the downside risk is limited by the choice you have not to exercise the option once it goes out-of-the-money. It is the asymmetry resulting from the choice which means the additional value of the extra chance of more profits outweighs the reduction in value due to the possibility of more losses. So, the solution should say: The value of the swaption would be higher ... … because a higher volatility would mean a higher probability that the swap rate increases further between now and the strike date leading to a greater profit on exercise. Although there would also be a higher probability of the swap rate falling by the strike date, which acts to make the swaption less valuable, this downside risk is limited by the choice that the swaption confers. Apologies again for this error and the resulting confusion. I will ensure that it is corrected for next year.
